A quick introduction to quantum states and unitary operations

qc_intro.py

# the state of N qubits is a complex vector of dimension 2^N
n_qubits = 2
dimension = 2 ** n_qubits
state = [1j, 0, 0, 0]

def print_probs(state, n_qubits):
    # the elements of this state are squared to calculate outcome probabilities
    for bitstring in range(n_qubits ** 2):
        probability = np.abs(state[bitstring]) ** 2

        print("Bitstring", format(bitstring, "0" + str(n_qubits) + "b"), " has probability ", probability)
    print()

print_probs(state, n_qubits)
# an example with a "superposition" over outcomes
print_probs([0, -1j / np.sqrt(2), 0, 1 / np.sqrt(2)], n_qubits)

# evolution is then given by a unitary matrix
identity = np.array([[1, 0], [0, 1]]) # identity on one qubit
flip = np.array([[0, 1], [1, 0]]) # a flip or X-gate on one qubits
flip_first = np.kron(flip, identity) # tensor products make this a two qubit operation

new_state = flip_first@state
print_probs(new_state, n_qubits)

flip_second = np.kron(identity, flip)
print_probs(new_state, n_qubits)

# if we start in the state with all qubits in zero
# then we can take a shortcut to get the probabilities of any particular bitstring
all_zeros = [1] + [0]*(dimension-1)
bs = np.random.choice(range(dimension))
assert (flip_second@all_zeros)[bs] == flip_second[bs, 0]