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January 10, 2021 06:06
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Julia notebook for "Quantum computing for Computer scientist"
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| <main><preamble><button class="runallchanged" title="Save and run all changed cells"><span></span></button></preamble><pluto-notebook id="7963b360-4325-11eb-1e5f-e3c4f3a0f7a4"><pluto-cell class="code_folded " id="8423cb12-4273-11eb-2b32-5568d93043dd"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h1>Quantum Computing</h1> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">40.8 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="72d474ec-4327-11eb-26d0-072a3be1e84f"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y rich_output " mime="text/plain"><assignee></assignee><div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">1.6 ms</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="63dae6be-4364-11eb-056a-8785db062735"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><iframe width="560" height="315" src="https://www.youtube.com/embed/F_Riqjdh2oM" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe> | |
| </div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.4 ms</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="3cc2bb2e-43fa-11eb-3945-ff510d51fe72"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p><a href="https://www.microsoft.com/en-us/research/uploads/prod/2018/05/40655.compressed.pdf">Slides</a></p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">8.1 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="6ac48d68-4364-11eb-1012-337857ac86a3"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>In this notebook we will be going over the lecture and creating some necessary functions and math to play around with qbits and operations for it. Feel free to play around and change variable or apply operations to see what actually happens.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">5.7 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="8a5b0632-4325-11eb-018a-f7a3ef9e043d"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h2>Definitions of qubits and other concepts</h2> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">10.2 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="13358096-432f-11eb-3c6d-17b9a562fb39"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h3>Defining classical bits</h3> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.8 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="da0ee23c-4273-11eb-0fbe-87b9d9df7612"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee>cb_0</assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>1.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_0</span> <span class="cm-operator">=</span> [<span class="cm-number">1.0</span> ; <span class="cm-number">0.0</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">1.2 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="a07efc0c-4325-11eb-2cc4-a92d98495bc2"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee>cb_1</assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>1.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_1</span> <span class="cm-operator">=</span> [<span class="cm-number">0.0</span> ; <span class="cm-number">1.0</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">1.5 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="2b598b6c-4326-11eb-1801-3d5765d51bca"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h3>Basic operation on classical bits</h3> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.4 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="a8a5749c-4325-11eb-3e86-cb66466a4617"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y " mime="text/plain"><assignee></assignee><div><pre><code>const_one (generic function with 1 method)</code></pre></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>7</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation" style=""><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">begin</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-def">identity</span>(<span class="cm-variable">qb</span>) <span class="cm-operator">=</span> [<span class="cm-number">1</span> <span class="cm-number">0</span>; <span class="cm-number">0</span> <span class="cm-number">1</span>] <span class="cm-operator">*</span> <span class="cm-variable">qb</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-def">negation</span>(<span class="cm-variable">qb</span>) <span class="cm-operator">=</span> [<span class="cm-number">0</span> <span class="cm-number">1</span>; <span class="cm-number">1</span> <span class="cm-number">0</span>] <span class="cm-operator">*</span> <span class="cm-variable">qb</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" class="cm-tab-wrap-hack" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-def">const_zero</span>(<span class="cm-variable">qb</span>) <span class="cm-operator">=</span> [<span class="cm-number">1</span> <span class="cm-number">1</span>; <span class="cm-number">0</span> <span class="cm-number">0</span>] <span class="cm-operator">*</span> <span class="cm-variable">qb</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-def">const_one</span>(<span class="cm-variable">qb</span>) <span class="cm-operator">=</span> [<span class="cm-number">0</span> <span class="cm-number">0</span>; <span class="cm-number">1</span> <span class="cm-number">1</span>] <span class="cm-operator">*</span> <span class="cm-variable">qb</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 113px;"></div><div class="CodeMirror-gutters" style="height: 148px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">42.5 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="15bce1d2-4326-11eb-150a-9d72bde51c62"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed"><jldict class="NamedTuple"><r><k>before</k><v><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>1.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r></jlarray></jltree></v></r><r><k>after</k><v><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>1.0</pre></v></r></jlarray></jltree></v></r></jldict></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class=" CodeMirror-matchingbracket">(</span><span class="cm-variable">before</span><span class="cm-operator">=</span><span class="cm-variable">cb_0</span>, <span class="cm-variable">after</span><span class="cm-operator">=</span><span class="cm-builtin">negation</span>(<span class="cm-variable">cb_0</span>)<span class=" CodeMirror-matchingbracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.0 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="916735c4-4326-11eb-0fcb-673a126ce8d5"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h3>Reversible computing</h3> | |
| <p>If you know the result and the operation then you can reverse it to get the original input. For example in above <code>negation</code> is a reversible function as if you negate twice then you get the original input. On the other hand things that remove state/info are not reversible e.g. <code>const_one</code> always sets to one so you can't know what the original input can be.</p> | |
| <p>Quantum computing only uses reversible operations.</p> | |
| <p>Fun fact: All quantum operations are their own inverses.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">9.0 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="d3ed69aa-4359-11eb-25dd-4ff19aa31f4d"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y rich_output " mime="text/plain"><assignee></assignee><div></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-meta">@assert</span> (<span class="cm-variable">cb_0</span> <span class="cm-operator">|></span> <span class="cm-variable">negation</span> <span class="cm-operator">|></span> <span class="cm-variable">negation</span>) <span class="cm-operator">==</span> <span class="cm-variable">cb_0</span> <span class="cm-string">"Negation should be reversible"</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">36.4 ms</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="2441b812-4327-11eb-3f75-c10037cd1f04"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h3>Tensor product</h3> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">7.5 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="e921c808-4326-11eb-2ca2-d1eb00bb02ac"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>Currently I'm using the library <code>Tensors</code> which has the <code>\otimes</code> <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="0" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.76ex" height="1.507ex" role="img" focusable="false" viewBox="0 -583 778 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.188ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2297"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⊗</mo></math></mjx-assistive-mml></mjx-container></span> operation which indicates a tensor product. It changes the dimensions of the vector and the left matrix multiply with all on the right.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.8 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="431158a6-4329-11eb-1327-85947b30268c"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y " mime="text/plain"><assignee>⊗</assignee><div><pre><code>kron (generic function with 32 methods)</code></pre></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">⊗</span> <span class="cm-operator">=</span> <span class="cm-variable">kron</span> <span class="cm-comment">#Defining the symbol to be used as tensor product in from std lib</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">42.0 ns</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="ecd34d7c-4327-11eb-171d-2990322bb3f0"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Int64<jlarray class="Array"><r><k>1</k><v><pre>3</pre></v></r><r><k>2</k><v><pre>4</pre></v></r><r><k>3</k><v><pre>6</pre></v></r><r><k>4</k><v><pre>8</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class=" CodeMirror-matchingbracket">[</span><span class="cm-number">1</span>,<span class="cm-number">2</span><span class=" CodeMirror-matchingbracket">]</span> <span class="cm-variable">⊗</span> [<span class="cm-number">3</span>,<span class="cm-number">4</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">1.4 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="e7470d58-4327-11eb-2bd8-7b257c3ad6b1"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Int64<jlarray class="Array"><r><k>1</k><v><pre>0</pre></v></r><r><k>2</k><v><pre>0</pre></v></r><r><k>3</k><v><pre>0</pre></v></r><r><k>4</k><v><pre>0</pre></v></r><r><k>5</k><v><pre>0</pre></v></r><r><k>6</k><v><pre>0</pre></v></r><r><k>7</k><v><pre>1</pre></v></r><r><k>8</k><v><pre>0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class=" CodeMirror-matchingbracket">[</span><span class="cm-number">0</span>,<span class="cm-number">1</span><span class=" CodeMirror-matchingbracket">]</span> <span class="cm-variable">⊗</span> [<span class="cm-number">0</span>,<span class="cm-number">1</span>] <span class="cm-variable">⊗</span> [<span class="cm-number">1</span>,<span class="cm-number">0</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">1.9 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="d28b4466-4326-11eb-3c5e-e1565a299333"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">String<jlarray class="Array"><r><k>1</k><v><pre>"x1y1"</pre></v></r><r><k>2</k><v><pre>"x1y2"</pre></v></r><r><k>3</k><v><pre>"x2y1"</pre></v></r><r><k>4</k><v><pre>"x2y2"</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class=" CodeMirror-matchingbracket">[</span><span class="cm-string">"x1"</span>, <span class="cm-string">"x2"</span><span class=" CodeMirror-matchingbracket">]</span> <span class="cm-variable">⊗</span> [<span class="cm-string">"y1"</span>, <span class="cm-string">"y2"</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">1.7 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="0670935a-432a-11eb-2181-557f26eb11e7"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>Multiple classic bits can be represented using tensor product e.g.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">5.4 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="6eb38ba2-432a-11eb-31d3-2fe98e6a151e"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee>cb_00</assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>1.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r><r><k>3</k><v><pre>0.0</pre></v></r><r><k>4</k><v><pre>0.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>2</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># |00⟩</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_00</span> <span class="cm-operator">=</span> [<span class="cm-number">1.</span> , <span class="cm-number">0.</span>] <span class="cm-variable">⊗</span> [<span class="cm-number">1.</span> , <span class="cm-number">0</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 38px;"></div><div class="CodeMirror-gutters" style="height: 73px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">27.7 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="80974156-432a-11eb-2b15-6982ef60444f"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee>cb_01</assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>1.0</pre></v></r><r><k>3</k><v><pre>0.0</pre></v></r><r><k>4</k><v><pre>0.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>2</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># |01⟩</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_01</span> <span class="cm-operator">=</span> [<span class="cm-number">1.</span> , <span class="cm-number">0.</span>] <span class="cm-variable">⊗</span> [<span class="cm-number">0.</span> , <span class="cm-number">1.</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 38px;"></div><div class="CodeMirror-gutters" style="height: 73px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.2 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="885ad9ac-432a-11eb-3b15-f3c2f64783a3"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee>cb_10</assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r><r><k>3</k><v><pre>1.0</pre></v></r><r><k>4</k><v><pre>0.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>2</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># |10⟩</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_10</span> <span class="cm-operator">=</span> [<span class="cm-number">0.</span> , <span class="cm-number">1.</span>] <span class="cm-variable">⊗</span> [<span class="cm-number">1.</span> , <span class="cm-number">0.</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 38px;"></div><div class="CodeMirror-gutters" style="height: 73px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.0 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="8d9276e6-432a-11eb-1f90-ad762761499e"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee>cb_11</assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r><r><k>3</k><v><pre>0.0</pre></v></r><r><k>4</k><v><pre>1.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>2</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># |11⟩</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_11</span> <span class="cm-operator">=</span> [<span class="cm-number">0.</span> , <span class="cm-number">1.</span>] <span class="cm-variable">⊗</span> [<span class="cm-number">0.</span> , <span class="cm-number">1.</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 38px;"></div><div class="CodeMirror-gutters" style="height: 73px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.3 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="385dde4a-432f-11eb-1b66-977dc7fa634f"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>Hence to represent any classic bit as the vector, we can take tensor product e.g.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="49af253c-432f-11eb-184e-936240d9723c"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r><r><k>3</k><v><pre>0.0</pre></v></r><r><k>4</k><v><pre>0.0</pre></v></r><r><k>5</k><v><pre>0.0</pre></v></r><r><k>6</k><v><pre>1.0</pre></v></r><r><k>7</k><v><pre>0.0</pre></v></r><r><k>8</k><v><pre>0.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>2</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># To represent |101⟩</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_1</span> <span class="cm-variable">⊗</span> <span class="cm-variable">cb_0</span> <span class="cm-variable">⊗</span> <span class="cm-variable">cb_1</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 38px;"></div><div class="CodeMirror-gutters" style="height: 73px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.2 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="7d3269dc-432f-11eb-3126-e1e3388c2327"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r><r><k>3</k><v><pre>0.0</pre></v></r><r><k>4</k><v><pre>0.0</pre></v></r><r><k>5</k><v><pre>0.0</pre></v></r><r><k>6</k><v><pre>0.0</pre></v></r><r><k>7</k><v><pre>0.0</pre></v></r><r><k>8</k><v><pre>1.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>2</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># To represent |111⟩</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_1</span> <span class="cm-variable">⊗</span> <span class="cm-variable">cb_1</span> <span class="cm-variable">⊗</span> <span class="cm-variable">cb_1</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 38px;"></div><div class="CodeMirror-gutters" style="height: 73px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="b92e0540-432a-11eb-2e05-379067c4ab90"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h3>Defining the control not operation</h3> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.1 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="1fb20a1e-432b-11eb-19c2-fd75fee5ccd2"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>Similar to how in classic computing NAND gate is like the universal gate which you can use to construct all other gates, for reversible operations CNOT does something similar. </p> | |
| <p>Though it's not totally universal as there are some gates which need the Toffoli gate to build.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">8.0 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="c0a78e18-432a-11eb-1223-3bad74db13ef"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y " mime="text/plain"><assignee></assignee><div><pre><code>control_not (generic function with 1 method)</code></pre></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-def">control_not</span>(<span class="cm-variable">qb</span>) <span class="cm-operator">=</span> [<span class="cm-number">1</span> <span class="cm-number">0</span> <span class="cm-number">0</span> <span class="cm-number">0</span>; <span class="cm-number">0</span> <span class="cm-number">1</span> <span class="cm-number">0</span> <span class="cm-number">0</span>; <span class="cm-number">0</span> <span class="cm-number">0</span> <span class="cm-number">0</span> <span class="cm-number">1</span>; <span class="cm-number">0</span> <span class="cm-number">0</span> <span class="cm-number">1</span> <span class="cm-number">0</span>] <span class="cm-operator">*</span> <span class="cm-variable">qb</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">20.4 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="e47b7b74-432c-11eb-2912-4f4bac4924f1"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r><r><k>3</k><v><pre>0.0</pre></v></r><r><k>4</k><v><pre>1.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">control_not</span>(<span class="cm-variable">cb_10</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.1 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="33b85acc-432b-11eb-3d69-d31d47266874"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y rich_output " mime="text/plain"><assignee></assignee><div></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>2</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># Passing in |10⟩ and result is |11⟩</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-meta">@assert</span> <span class="cm-builtin">control_not</span>(<span class="cm-variable">cb_10</span>) <span class="cm-operator">==</span> <span class="cm-variable">cb_11</span> <span class="cm-string">"Value is not |11⟩"</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 38px;"></div><div class="CodeMirror-gutters" style="height: 73px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.6 ms</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="3d80c592-432d-11eb-0999-bfc399890c49"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h3>Defining QBits and superposition</h3> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="3e308c98-432d-11eb-1281-e3c7ce349a1c"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>All along we have been using qbits. The cbit vectors we have been using are just special cases of qbits.</p> | |
| <p>A qbit is represented by <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="1" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.527ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 2001 2400" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.149ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd" transform="translate(50, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1265, 0)"><use xlink:href="#MJX-TEX-S3-29"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></span> where <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="2" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.197ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 529 451" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container></span> and <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="3" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.971ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 429 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></mjx-assistive-mml></mjx-container></span> are complext number and it holds this property </p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="4" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="15.432ex" height="2.71ex" role="img" focusable="false" viewBox="0 -948 6821.1 1198" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mi" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(1029, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(1529, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(2154.8, 0)"><use xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(3155, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mi" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(929, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(1429, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(5265.3, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(6321.1, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mi>a</mi><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></p> | |
| <p>The cbits also satisfy this property along with a lot of other varients e.g.</p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="5" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="32.523ex" height="7.486ex" role="img" focusable="false" viewBox="0 -1904.5 14375.4 3308.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.178ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 750.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -760.5)"></use></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 1039.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -814.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(2271.7, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 750.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -760.5)"></use></g></g><g data-mml-node="mo" transform="translate(3424.5, 0)"><use xlink:href="#MJX-TEX-N-21D2"></use></g><g data-mml-node="msup" transform="translate(4702.3, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><svg width="500" height="2540" y="-1020" x="28" viewBox="0 -317.5 500 2540"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.81)"></use></svg></g><g data-mml-node="mfrac" transform="translate(556, 0)"><g data-mml-node="mn" transform="translate(646.5, 676)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -920.5)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 100.5)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="840.5"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2349, 0)"><svg width="500" height="2540" y="-1020" x="28" viewBox="0 -317.5 500 2540"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.81)"></use></svg></g></g><g data-mml-node="mn" transform="translate(2905, 1247.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(8233, 0)"><use xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(9233.3, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><svg width="500" height="2540" y="-1020" x="28" viewBox="0 -317.5 500 2540"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.81)"></use></svg></g><g data-mml-node="mfrac" transform="translate(556, 0)"><g data-mml-node="mn" transform="translate(646.5, 676)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -920.5)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 100.5)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="840.5"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2349, 0)"><svg width="500" height="2540" y="-1020" x="28" viewBox="0 -317.5 500 2540"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.81)"></use></svg></g></g><g data-mml-node="mn" transform="translate(2905, 1247.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(12819.6, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(13875.4, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">⇒</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="6" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="30.218ex" height="7.003ex" role="img" focusable="false" viewBox="0 -1823 13356.4 3095.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.878ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-28"></use></g><g data-mml-node="mtable" transform="translate(792, 0)"><g data-mml-node="mtr" transform="translate(0, 907.3)"><g data-mml-node="mtd" transform="translate(301.6, 0)"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220, -345) scale(0.707)"><use 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data-mml-node="mrow"><g data-mml-node="mo"><svg width="500" height="2184" y="-842" x="28" viewBox="0 -273 500 2184"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.276)"></use></svg></g><g data-mml-node="mfrac" transform="translate(556, 0)"><g data-mml-node="mn" transform="translate(220, 676)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220, -686)"><use xlink:href="#MJX-TEX-N-32"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1496, 0)"><svg width="500" height="2184" y="-842" x="28" viewBox="0 -273 500 2184"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.276)"></use></svg></g></g><g data-mml-node="mn" transform="translate(2052, 1069.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(7214, 0)"><use xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(8214.3, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><svg width="500" height="2750" y="-1125" x="28" viewBox="0 -343.8 500 2750"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 4.125)"></use></svg></g><g data-mml-node="mfrac" transform="translate(556, 0)"><g data-mml-node="msqrt" transform="translate(220, 676)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-33"></use></g></g><g data-mml-node="mo" transform="translate(0, 89)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="829"></rect></g><g data-mml-node="mn" transform="translate(646.5, -686)"><use xlink:href="#MJX-TEX-N-32"></use></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2349, 0)"><svg width="500" height="2750" y="-1125" x="28" viewBox="0 -343.8 500 2750"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 4.125)"></use></svg></g></g><g data-mml-node="mn" transform="translate(2905, 1352.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(11800.6, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(12856.4, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">⇒</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container> | |
| <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="7" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="27.029ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 11946.7 2400" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.149ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd" transform="translate(389, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g></g><g data-mml-node="mo" transform="translate(2014, 0)"><use xlink:href="#MJX-TEX-S3-29"></use></g></g><g data-mml-node="mo" transform="translate(3027.8, 0)"><use xlink:href="#MJX-TEX-N-21D2"></use></g><g data-mml-node="msup" transform="translate(4305.6, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mo" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1278, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(1778, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(2278, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(7209.3, 0)"><use xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(8209.6, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mn" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(1000, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(1500, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(10390.9, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(11446.7, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">⇒</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mn>0</mn><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="8" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="32.523ex" height="7.495ex" role="img" focusable="false" viewBox="0 -1906.5 14375.4 3312.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.182ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -762.5)"></use></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 1041.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -816.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(246.5, 398) scale(0.707)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(2271.7, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -762.5)"></use></g></g><g data-mml-node="mo" transform="translate(3424.5, 0)"><use xlink:href="#MJX-TEX-N-21D2"></use></g><g data-mml-node="msup" transform="translate(4702.3, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><svg width="500" height="2540" y="-1020" x="28" viewBox="0 -317.5 500 2540"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.81)"></use></svg></g><g data-mml-node="mfrac" transform="translate(556, 0)"><g data-mml-node="mn" transform="translate(646.5, 676)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -920.5)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 100.5)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="840.5"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2349, 0)"><svg width="500" height="2540" y="-1020" x="28" viewBox="0 -317.5 500 2540"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.81)"></use></svg></g></g><g data-mml-node="mn" transform="translate(2905, 1247.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(8233, 0)"><use xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(9233.3, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><svg width="500" height="2540" y="-1020" x="28" viewBox="0 -317.5 500 2540"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.81)"></use></svg></g><g data-mml-node="mfrac" transform="translate(556, 0)"><g data-mml-node="mrow" transform="translate(257.5, 676)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g><g data-mml-node="msqrt" transform="translate(220, -920.5)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 100.5)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="840.5"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2349, 0)"><svg width="500" height="2540" y="-1020" x="28" viewBox="0 -317.5 500 2540"><use xlink:href="#MJX-TEX-S4-2225" transform="scale(1, 3.81)"></use></svg></g></g><g data-mml-node="mn" transform="translate(2905, 1247.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(12819.6, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(13875.4, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">⇒</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msqrt><mn>2</mn></msqrt></mfrac><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">9.5 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="3eb070ca-432d-11eb-0b24-27a2dcfc6795"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>As both <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="9" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.197ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 529 451" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container></span> and <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="10" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.971ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 429 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></mjx-assistive-mml></mjx-container></span> can be non zero e.g. <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="11" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.119ex" height="7.486ex" role="img" focusable="false" viewBox="0 -1904.5 3146.7 3308.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.178ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 750.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -760.5)"></use></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 1039.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -814.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(2271.7, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 750.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -760.5)"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></span> so that is an example of <strong>superposition</strong>. When we measure they collapse to a known value we generally see in cbits. <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="12" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.527ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 2001 2400" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.149ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd" transform="translate(50, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1265, 0)"><use xlink:href="#MJX-TEX-S3-29"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></span> collapses to <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="13" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></mjx-assistive-mml></mjx-container></span> with probability <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="14" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.372ex" height="2.71ex" role="img" focusable="false" viewBox="0 -948 1932.6 1198" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mi" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(1029, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(1529, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mi>a</mi><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container></span> and collapses to <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="15" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></mjx-assistive-mml></mjx-container></span> with probability <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="16" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.146ex" height="2.71ex" role="img" focusable="false" viewBox="0 -948 1832.6 1198" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mi" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(929, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(1429, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container></span></p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">11.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="6e49ccc0-435d-11eb-1742-e9261e144769"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><hr> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">4.3 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="0a9d8200-435d-11eb-01fd-4f72e67df2dc"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>Similar to cbits, multiple qbits are also represented by tensor product <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="17" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.76ex" height="1.507ex" role="img" focusable="false" viewBox="0 -583 778 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.188ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2297"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⊗</mo></math></mjx-assistive-mml></mjx-container></span></p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">7.0 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="ea52eeb4-435d-11eb-2b7e-09c0b6bf70ee"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.5</pre></v></r><r><k>2</k><v><pre>0.5</pre></v></r><r><k>3</k><v><pre>0.5</pre></v></r><r><k>4</k><v><pre>0.5</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class=" CodeMirror-matchingbracket">[</span><span class="cm-number">1</span><span class="cm-operator">/√</span><span class="cm-number">2</span> ; <span class="cm-number">1</span><span class="cm-operator">/√</span><span class="cm-number">2</span><span class=" CodeMirror-matchingbracket">]</span> <span class="cm-variable">⊗</span> [<span class="cm-number">1</span><span class="cm-operator">/√</span><span class="cm-number">2</span> ; <span class="cm-number">1</span><span class="cm-operator">/√</span><span class="cm-number">2</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">3.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="08f1b8c8-435e-11eb-0a5b-d3fd80f7de3b"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>The above also shows that there is <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="18" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.795ex" height="2.737ex" role="img" focusable="false" viewBox="0 -864.9 793.6 1209.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.781ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220, -345) scale(0.707)"><use xlink:href="#MJX-TEX-N-34"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math></mjx-assistive-mml></mjx-container></span> probability of collapsing to <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="19" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="18.104ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 8002 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-30"></use><use xlink:href="#MJX-TEX-N-30" transform="translate(500, 0)"></use></g><g data-mml-node="mo" transform="translate(1278, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g><g data-mml-node="mo" transform="translate(1667, 0)"><use xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(2111.7, 0)"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(2389.7, 0)"><use xlink:href="#MJX-TEX-N-30"></use><use xlink:href="#MJX-TEX-N-31" transform="translate(500, 0)"></use></g><g data-mml-node="mo" transform="translate(3389.7, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g><g data-mml-node="mo" transform="translate(3778.7, 0)"><use xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(4223.3, 0)"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(4501.3, 0)"><use xlink:href="#MJX-TEX-N-31"></use><use xlink:href="#MJX-TEX-N-30" transform="translate(500, 0)"></use></g><g data-mml-node="mo" transform="translate(5501.3, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g><g data-mml-node="mo" transform="translate(5890.3, 0)"><use xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(6335, 0)"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(6613, 0)"><use xlink:href="#MJX-TEX-N-31"></use><use xlink:href="#MJX-TEX-N-31" transform="translate(500, 0)"></use></g><g data-mml-node="mo" transform="translate(7613, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>00</mn><mo fence="false" stretchy="false">⟩</mo><mo>,</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>01</mn><mo fence="false" stretchy="false">⟩</mo><mo>,</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>10</mn><mo fence="false" stretchy="false">⟩</mo><mo>,</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>11</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span></p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.9 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="3440af16-435e-11eb-25dd-af037fc22701"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>The rule we had can hold for any probability state representing qbit e.g.</p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="20" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="32.207ex" height="13.009ex" role="img" focusable="false" viewBox="0 -3125 14235.5 5750" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -5.939ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 1971)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -1981)"></use><svg width="875" height="2332" y="-916" x="0" viewBox="0 526.6 875 2332"><use xlink:href="#MJX-TEX-S4-239C" transform="scale(1, 5.642)"></use></svg></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 2375)"><g data-mml-node="mtd" transform="translate(35.5, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, 975)"><g data-mml-node="mtd" transform="translate(85.5, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -975)"><g data-mml-node="mtd" transform="translate(161, 0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-22EE"></use></g></g></g></g><g data-mml-node="mtr" transform="translate(0, -2375)"><g data-mml-node="mtd"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D45B"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1475, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 1971)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -1981)"></use><svg width="875" height="2332" y="-916" x="0" viewBox="0 526.6 875 2332"><use xlink:href="#MJX-TEX-S4-239F" transform="scale(1, 5.642)"></use></svg></g></g><g data-mml-node="mo" transform="translate(2627.8, 0)"><use xlink:href="#MJX-TEX-N-21D2"></use></g><g data-mml-node="msup" transform="translate(3905.6, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mi" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(1029, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(1529, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(6060.3, 0)"><use xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(7060.6, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mi" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(929, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(1429, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(9059.8, 0)"><use xlink:href="#MJX-TEX-N-22EF"></use></g><g data-mml-node="msup" transform="translate(10398.4, 0)"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2225"></use></g><g data-mml-node="mi" transform="translate(500, 0)"><use xlink:href="#MJX-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(1100, 0)"><use xlink:href="#MJX-TEX-N-2225"></use></g></g><g data-mml-node="mn" transform="translate(1600, 477.1) scale(0.707)"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(12679.8, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(13735.5, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mrow><mo>⋮</mo></mrow></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">⇒</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mi>a</mi><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>⋯</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">∥</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">∥</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">7.9 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="0c8c0e5a-4360-11eb-042b-db3d0b2e485b"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y " mime="text/plain"><assignee></assignee><div><pre><code>sum_quantum_state (generic function with 1 method)</code></pre></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-def">sum_quantum_state</span>(<span class="cm-variable">qbs</span>) <span class="cm-operator">=</span> <span class="cm-builtin">round</span>(<span class="cm-builtin">sum</span>((<span class="cm-variable">abs</span><span class="cm-operator">.</span>(<span class="cm-variable">qbs</span>))<span class="cm-operator">.^</span><span class="cm-number">2</span>), <span class="cm-variable">digits</span><span class="cm-operator">=</span><span class="cm-number">5</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">27.8 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="2686ae4c-435f-11eb-1441-4bf55d0cc55f"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h3>Hadamard gate <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="21" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.649ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2055 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D43B"></use></g><g data-mml-node="mo" transform="translate(888, 0)"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(1166, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(1666, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>0</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="22" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.649ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2055 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D43B"></use></g><g data-mml-node="mo" transform="translate(888, 0)"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(1166, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(1666, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>1</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span></h3> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.3 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="1704e6da-4361-11eb-1df2-9905c4fef512"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>Hadamard is a special gate which takes in the classic bit i.e. qbit in a state <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="74" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.64ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1167 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>0</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> or <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="75" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.64ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1167 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>1</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> and then puts it into superposition. The mathematical representation of it is</p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="76" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="17.069ex" height="7.495ex" role="img" focusable="false" viewBox="0 -1906.5 7544.4 3312.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.182ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -762.5)"></use></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 1041.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mtd" transform="translate(2396.7, 0)"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -816.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mtd" transform="translate(2396.7, 0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(246.5, 398) scale(0.707)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(4668.4, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -762.5)"></use></g></g><g data-mml-node="mrow" transform="translate(5543.4, 0)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd" transform="translate(50, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1265, 0)"><use xlink:href="#MJX-TEX-S3-29"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd><mtd><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></p> | |
| <p>For <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="77" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.64ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1167 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>0</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> this leads to</p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="78" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="27.14ex" height="7.495ex" role="img" focusable="false" viewBox="0 -1906.5 11995.7 3312.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.182ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -762.5)"></use></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 1041.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mtd" transform="translate(2396.7, 0)"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -816.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mtd" transform="translate(2396.7, 0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(246.5, 398) scale(0.707)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(4668.4, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -762.5)"></use></g></g><g data-mml-node="mrow" transform="translate(5543.4, 0)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1236, 0)"><use xlink:href="#MJX-TEX-S3-29"></use></g></g><g data-mml-node="mo" transform="translate(7793.2, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(8849, 0)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 750.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -760.5)"></use></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 1039.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -814.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(2271.7, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 750.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -760.5)"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd><mtd><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></p> | |
| <p>and for <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="79" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.64ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1167 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>1</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> this leads to</p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="80" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="27.14ex" height="7.495ex" role="img" focusable="false" viewBox="0 -1906.5 11995.7 3312.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.182ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -762.5)"></use></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 1041.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mtd" transform="translate(2396.7, 0)"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -816.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mtd" transform="translate(2396.7, 0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(246.5, 398) scale(0.707)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(4668.4, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -762.5)"></use></g></g><g data-mml-node="mrow" transform="translate(5543.4, 0)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1236, 0)"><use xlink:href="#MJX-TEX-S3-29"></use></g></g><g data-mml-node="mo" transform="translate(7793.2, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(8849, 0)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -762.5)"></use></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 1041.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -816.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(246.5, 398) scale(0.707)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(2271.7, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 752.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -762.5)"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd><mtd><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></p> | |
| <p>The reason we have <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="81" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 1278 748" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></span> in the last element is so that we can differentiate when running the hadamard gate from <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="82" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.64ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1167 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>0</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> or <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="83" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.64ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1167 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>1</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> and make it a <strong>reversible operation</strong>. This is also representative of what happens in real world as the state of when running the gate through <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="84" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.64ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1167 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>0</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> or <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="85" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.64ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1167 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(778, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>1</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> is different and that info is carried on to further operations.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">12.3 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="bc6a103e-435f-11eb-3731-bba3e197f859"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y " mime="text/plain"><assignee></assignee><div><pre><code>hadamard (generic function with 1 method)</code></pre></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.2 ms</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="d9ff7f80-435f-11eb-3e25-9be1324cea5c"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed"><jlarray class="Tuple"><r><k>1</k><v><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.707107</pre></v></r><r><k>2</k><v><pre>0.707107</pre></v></r></jlarray></jltree></v></r><r><k>2</k><v><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.707107</pre></v></r><r><k>2</k><v><pre>-0.707107</pre></v></r></jlarray></jltree></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">hadamard</span>(<span class="cm-variable">cb_0</span>), <span class="cm-builtin">hadamard</span>(<span class="cm-variable">cb_1</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">2.7 ms</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="b63f9f84-4360-11eb-19eb-d7333c8370a4"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y rich_output " mime="text/plain"><assignee></assignee><div></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-meta">@assert</span> <span class="cm-builtin">sum_quantum_state</span>(<span class="cm-builtin">hadamard</span>(<span class="cm-variable">cb_1</span>)) <span class="cm-operator">==</span> <span class="cm-number">1</span> <span class="cm-string">"The quantum state is incorrect"</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">22.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="da6e73a8-4360-11eb-2a00-155b1fac8e98"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>As it's reversible so you can go from superposition back to out of superposition and without measurement.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.8 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="8ea44972-4361-11eb-167f-490694a636d6"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>1.0</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_0</span> <span class="cm-operator">|></span> <span class="cm-variable">hadamard</span> <span class="cm-operator">|></span> <span class="cm-variable">hadamard</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">2.0 ms</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="88eab6b2-4361-11eb-21d6-51deb59a3ffd"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.0</pre></v></r><r><k>2</k><v><pre>1.0</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cb_1</span> <span class="cm-operator">|></span> <span class="cm-variable">hadamard</span> <span class="cm-operator">|></span> <span class="cm-variable">hadamard</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">4.1 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="edcffdc2-4374-11eb-0bb6-b13e3e2a8a4f"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>You can also compose Hadamard operation to process bigger states e.g. for <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="113" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></mjx-assistive-mml></mjx-container></span> and <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="114" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.554ex" role="img" focusable="false" viewBox="0 -665 500 687" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-33"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></mjx-assistive-mml></mjx-container></span> qubits</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.7 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="037b38ee-4375-11eb-3ed4-819e2e7bfbef"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.5</pre></v></r><r><k>2</k><v><pre>-0.5</pre></v></r><r><k>3</k><v><pre>-0.5</pre></v></r><r><k>4</k><v><pre>0.5</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 197.796875px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors" style="visibility: hidden;"><div class="CodeMirror-cursor" style="left: 167.796875px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">(<span class="cm-variable">H_op</span> <span class="cm-variable">⊗</span> <span class="cm-variable">H_op</span>) <span class="cm-operator">*</span> <span class="cm-variable">cb_11</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">2.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="1e68e11a-4375-11eb-1b3f-51f7aa9d84c9"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.353553</pre></v></r><r><k>2</k><v><pre>-0.353553</pre></v></r><r><k>3</k><v><pre>-0.353553</pre></v></r><r><k>4</k><v><pre>0.353553</pre></v></r><r><k>5</k><v><pre>0.353553</pre></v></r><r><k>6</k><v><pre>-0.353553</pre></v></r><r><k>7</k><v><pre>-0.353553</pre></v></r><r><k>8</k><v><pre>0.353553</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 369px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors" style=""><div class="CodeMirror-cursor" style="left: 339px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">(<span class="cm-variable">H_op</span> <span class="cm-variable">⊗</span> <span class="cm-variable">H_op</span> <span class="cm-variable">⊗</span> <span class="cm-variable">H_op</span>) <span class="cm-operator">*</span> (<span class="cm-variable">cb_0</span> <span class="cm-variable">⊗</span> <span class="cm-variable">cb_1</span> <span class="cm-variable">⊗</span> <span class="cm-variable">cb_1</span>) <span class="cm-comment"># H|011⟩</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">4.4 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="43b60a7e-4375-11eb-365b-fdf42513d80f"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y rich_output " mime="text/plain"><assignee></assignee><div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">20.2 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="da788d42-4366-11eb-0398-45966d74f080"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed"><jlarray class="Tuple"><r><k>1</k><v><pre>negation</pre></v></r><r><k>2</k><v><pre>hadamard</pre></v></r></jlarray></jltree></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">749 ns</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="b46103f0-4361-11eb-345b-11d548d51ef7"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h2>The Deutsch oracle</h2> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">5.1 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="b65306ec-4369-11eb-098d-b5d80739dabd"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p><a href="https://youtu.be/F_Riqjdh2oM?t=1994">Video mark</a></p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">8.4 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="99e1431c-4368-11eb-02bc-b7300a33b4de"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>For the basic operations that we descrived earlier i.e. <code>identity</code>, <code>negation</code>, <code>constant-0</code> and <code>constant-1</code> if we had a blackbox which contained either of those and we can input something and see only its output then how would we know what function is in there?</p> | |
| <p>In classical computing we would have to pass in first 0, see its output and then pass in 1, see its output to figure out. In quantum its the same as the state space required 2 bits to differentiate between 4 different functions.</p> | |
| <p>If the question becomes that see if it has a constant function or variable (i.e. identity or negation), now the state we need to distinguish are 2 requiring 1 bit. In this case classical computing still needs two operations but quantum computing needs <strong>only one operation</strong>. This is what the power of quantum computing is.</p> | |
| <p>Though the problem is that as in quantum would we can only work with operations that are reversible so how would we solve that for constant functions. The way is to pass in two wires one marked as output which the blackbox will write the output to and the other as input which the blackbox can use but it won't change. In this way you always know what the input was and the operation so you have a reversible operations.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">10.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="b95d8650-4364-11eb-1058-250c08209de3"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee>blackbox_cnot</assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.5</pre></v></r><r><k>2</k><v><pre>-0.5</pre></v></r><r><k>3</k><v><pre>0.5</pre></v></r><r><k>4</k><v><pre>-0.5</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 307px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors" style="visibility: hidden;"><div class="CodeMirror-cursor" style="left: 277px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">blackbox_cnot</span> <span class="cm-operator">=</span> <span class="cm-builtin">control_not</span>((<span class="cm-variable">cb_0</span> <span class="cm-operator">|></span> <span class="cm-variable">X</span> <span class="cm-operator">|></span> <span class="cm-variable">H</span>) <span class="cm-variable">⊗</span> (<span class="cm-variable">cb_0</span> <span class="cm-operator">|></span> <span class="cm-variable">X</span> <span class="cm-operator">|></span> <span class="cm-variable">H</span>))</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">8.9 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="2a4c4f98-4367-11eb-23f4-d1f73b7bfd35"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y rich_output " mime="text/plain"><assignee></assignee><div></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 20px; left: 174.390625px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true" style="width: 18px; pointer-events: none;"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true" style="height: 18px; pointer-events: none;"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors" style="visibility: hidden;"><div class="CodeMirror-cursor" style="left: 144.390625px; top: 15px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-meta">@assert</span> <span class="cm-variable">blackbox_cnot</span> <span class="cm-operator">==</span> ((<span class="cm-variable">cb_0</span> <span class="cm-operator">|></span> <span class="cm-variable">H</span>) <span class="cm-variable">⊗</span> (<span class="cm-variable">cb_1</span> <span class="cm-operator">|></span> <span class="cm-variable">H</span>)) <span class="cm-string">"The black box CNOT value is not expected |01⟩"</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 38px;"></div><div class="CodeMirror-gutters" style="height: 73px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">17.1 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="51a4fd90-4369-11eb-1f8a-c71736565e8e"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>This example might be contrived but somebody found the generallization of it where for <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="23" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D45B"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container></span> bits in classical computer we would need to perform <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="24" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.204ex" height="1.528ex" role="img" focusable="false" viewBox="0 -675.5 974.3 675.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(500, 363) scale(0.707)"><use xlink:href="#MJX-TEX-I-1D45B"></use></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mi>n</mi></msup></math></mjx-assistive-mml></mjx-container></span> operations but in quantum computing we can limit it to only few operations to figure it out. This leads to more foundation building and eventually also help build the <a href="https://en.wikipedia.org/wiki/Shor%27s_algorithm">Shor's algorithm</a>.</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">10.1 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="84a14ed4-43f4-11eb-1ba2-b9fa511ff9ae"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h2>Quantum entaglement</h2> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">5.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="8ee215d4-43f4-11eb-17de-4ffb8b648570"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>If the product state of <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="155" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></mjx-assistive-mml></mjx-container></span> qbits can't be factored then they are considred to be entangled.</p> | |
| <p>Following is an example where it can't be factored as you can take the <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="156" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.522ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 2882.6 776" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(529, 0)"><use xlink:href="#MJX-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(1326.8, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2382.6, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>d</mi><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container></span> and for that to be right either <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="157" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.345ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2362.6 748" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(806.8, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1862.6, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container></span> or <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="158" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.325ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 2353.6 776" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(797.8, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1853.6, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container></span> but that will invalidate the other values hence it can't be factored.</p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="159" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="34.274ex" height="17.249ex" role="img" focusable="false" viewBox="0 -4062 15149 7624" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -8.059ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 2150.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -2160.5)"></use><svg width="875" height="2690.9" y="-1095.5" x="0" viewBox="0 607.6 875 2690.9"><use xlink:href="#MJX-TEX-S4-239C" transform="scale(1, 6.51)"></use></svg></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 2439.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd" transform="translate(448.4, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd" transform="translate(448.4, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -2214.9)"><g data-mml-node="mtd"><g 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xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(4480.3, 0)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd" transform="translate(50, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1265, 0)"><use xlink:href="#MJX-TEX-S3-29"></use></g></g><g data-mml-node="mo" transform="translate(6703.5, 0)"><use xlink:href="#MJX-TEX-N-2297"></use></g><g data-mml-node="mrow" transform="translate(7703.7, 0)"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd" transform="translate(43.5, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D450"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D451"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1256, 0)"><use xlink:href="#MJX-TEX-S3-29"></use></g></g><g data-mml-node="mo" transform="translate(9973.5, 0)"><use xlink:href="#MJX-TEX-N-21D2"></use></g><g data-mml-node="mtable" transform="translate(10973.5, 0)"><g data-mml-node="mtr" transform="translate(0, 2720)"><g data-mml-node="mtd" transform="translate(87, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(529, 0)"><use xlink:href="#MJX-TEX-I-1D450"></use></g></g><g data-mml-node="mtd" transform="translate(1049, 0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g 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transform="translate(1333.6, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -650)"><g data-mml-node="mtd" transform="translate(187, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(429, 0)"><use xlink:href="#MJX-TEX-I-1D450"></use></g></g><g data-mml-node="mtd" transform="translate(1049, 0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6, 0)"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -2542)"><g data-mml-node="mtd" transform="translate(100, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(429, 0)"><use xlink:href="#MJX-TEX-I-1D451"></use></g></g><g data-mml-node="mtd" transform="translate(1049, 0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8, 0)"><use xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1333.6, 0)"><g data-mml-node="mn" transform="translate(646.5, 676)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -920.5)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 100.5)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="840.5"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g></g></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>⊗</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>d</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">⇒</mo><mtable displaystyle="true" columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" rowspacing="3pt"><mtr><mtd><mi>a</mi><mi>c</mi></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mi>a</mi><mi>d</mi></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>b</mi><mi>c</mi></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>b</mi><mi>d</mi></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">7.8 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="847bf074-43f6-11eb-202f-e14cdc351d31"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>As in the following product state</p> | |
| <p class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" display="true" role="presentation" tabindex="0" ctxtmenu_counter="210" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.119ex" height="13.821ex" role="img" focusable="false" viewBox="0 -3304.5 3146.7 6108.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -6.345ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-S4-239B" transform="translate(0, 2150.5)"></use><use xlink:href="#MJX-TEX-S4-239D" transform="translate(0, -2160.5)"></use><svg width="875" height="2690.9" y="-1095.5" x="0" viewBox="0 607.6 875 2690.9"><use xlink:href="#MJX-TEX-S4-239C" transform="scale(1, 6.51)"></use></svg></g><g data-mml-node="mtable" transform="translate(875, 0)"><g data-mml-node="mtr" transform="translate(0, 2439.5)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd" transform="translate(448.4, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd" transform="translate(448.4, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -2214.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(2271.7, 0)"><use xlink:href="#MJX-TEX-S4-239E" transform="translate(0, 2150.5)"></use><use xlink:href="#MJX-TEX-S4-23A0" transform="translate(0, -2160.5)"></use><svg width="875" height="2690.9" y="-1095.5" x="0" viewBox="0 607.6 875 2690.9"><use xlink:href="#MJX-TEX-S4-239F" transform="scale(1, 6.51)"></use></svg></g></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></p> | |
| <p>As for the above there is <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="211" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.147ex" height="1.824ex" role="img" focusable="false" viewBox="0 -750 1833 806" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.127ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-35"></use><use xlink:href="#MJX-TEX-N-30" transform="translate(500, 0)"></use></g><g data-mml-node="mi" transform="translate(1000, 0)"><use xlink:href="#MJX-TEX-N-25"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mi mathvariant="normal">%</mi></math></mjx-assistive-mml></mjx-container></span> probability for it to be in state <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="212" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.771ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1667 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-30"></use><use xlink:href="#MJX-TEX-N-30" transform="translate(500, 0)"></use></g><g data-mml-node="mo" transform="translate(1278, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>00</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> or <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="213" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.771ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1667 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278, 0)"><use xlink:href="#MJX-TEX-N-31"></use><use xlink:href="#MJX-TEX-N-31" transform="translate(500, 0)"></use></g><g data-mml-node="mo" transform="translate(1278, 0)"><use xlink:href="#MJX-TEX-N-27E9"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>11</mn><mo fence="false" stretchy="false">⟩</mo></math></mjx-assistive-mml></mjx-container></span> which means when measured both the qbits will have the exact same value i.e. <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="214" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></mjx-assistive-mml></mjx-container></span> or <span class="tex"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" role="presentation" tabindex="0" ctxtmenu_counter="215" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mn"><use xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml role="presentation" unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></mjx-assistive-mml></mjx-container></span>.</p> | |
| <p>The following is how to put two qbits into entangled state</p> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">9.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="cc46107c-43f5-11eb-21a6-3f3da3ef827f"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="scroll_y " mime="text/plain"><assignee></assignee><div><pre><code>entangle (generic function with 1 method)</code></pre></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 338px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors" style=""><div class="CodeMirror-cursor" style="left: 308px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-def">entangle</span>(<span class="cm-variable">qb1</span>, <span class="cm-variable">qb2</span>) <span class="cm-operator">=</span> <span class="cm-builtin">control_not</span>(<span class="cm-builtin">H</span>(<span class="cm-variable">qb1</span>) <span class="cm-variable">⊗</span> <span class="cm-variable">qb2</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">20.5 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="" id="f6ea1a2e-43f5-11eb-3dbb-6708cba6153a"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="" mime="application/vnd.pluto.tree+object"><assignee></assignee><div><jltree class="collapsed">Float64<jlarray class="Array"><r><k>1</k><v><pre>0.707107</pre></v></r><r><k>2</k><v><pre>0.0</pre></v></r><r><k>3</k><v><pre>0.0</pre></v></r><r><k>4</k><v><pre>0.707107</pre></v></r></jlarray></jltree></div></pluto-output><pluto-input><button class="delete_cell" title="Delete cell"><span></span></button><div class="CodeMirror cm-s-default CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 5px; left: 34px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-vscrollbar" tabindex="-1" cm-not-content="true"><div style="min-width: 1px; height: 0px;"></div></div><div class="CodeMirror-hscrollbar" tabindex="-1" cm-not-content="true"><div style="height: 100%; min-height: 1px; width: 0px;"></div></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 30px; margin-bottom: -15px; border-right-width: 35px; min-height: 23px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre class="CodeMirror-line-like"><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-cursors" style="visibility: hidden;"><div class="CodeMirror-cursor" style="left: 4px; top: 0px; height: 15px;"> </div></div><div class="CodeMirror-code" role="presentation"><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -30px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 21px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">entangle</span>(<span class="cm-variable">cb_0</span>, <span class="cm-variable">cb_0</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 35px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 58px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 29px;"></div></div></div></div></pluto-input><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">10.8 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="5510dc94-43f8-11eb-15c9-11924e0c8cde"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><h2>Quantum teleportation</h2> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">7.4 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell><pluto-cell class="code_folded " id="5b9ffe32-43f8-11eb-3e3b-55c55ac69d63"><pluto-shoulder draggable="true" title="Drag to move cell"><button class="foldcode" title="Show/hide code"><span></span></button></pluto-shoulder><pluto-trafficlight></pluto-trafficlight><button class="add_cell before" title="Add cell"><span></span></button><pluto-output class="rich_output " mime="text/html"><assignee></assignee><div><div class="markdown"><p>Quantum teleportation allows you to send the state of an arbitrary qbit from one location to another by way of two other entangled qbits. You can transfer qbit states (cut and paste) but not clone them and it is called no-cloning theorem.</p> | |
| <p>The teleportation is not faster-than-light because some classical information must be sent.</p> | |
| <blockquote> | |
| <p>Question: If classical information must be sent then what's the use?</p> | |
| </blockquote> | |
| </div></div></pluto-output><pluto-runarea><button class="runcell" title="Run"><span></span></button><span class="runtime">6.6 μs</span></pluto-runarea><button class="add_cell after" title="Add cell"><span></span></button></pluto-cell></pluto-notebook><dropruler></dropruler></main> | |
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