An example of Shor's algorithm to show that a lot of feasible states cannot be measured.

Shor.py

#!/usr/bin/python3
import cmath, math

a = 7; print('a =', a)
N = 15; print('N =', N)
for ord in range(1, N):
    if (a ** ord) % N == 1:
        break
print('ord =', ord)
m = 2 ** math.ceil(math.log2(N*N))
print('m =', m, '>= N^2')

amp = [[0 for _ in range(m)] for _ in range(m)]
for k in range(m):
    for j in range(m):
        amp[j][(a ** k) % N] += cmath.exp(2j * math.pi * j * k / m) / m
for k in range(1, m): # need to exclude k := 0 to avoid "x.append(p // 0)"
    nonzero = False
    for j in range(m):
        if abs(amp[k][j]) > 1e-10:
            nonzero = True
            # print(k, j, amp[k][j])
            break
    if True: #not nonzero:
        p = k
        q = m
        x = []
        d = math.gcd(p, q)
        p //= d
        q //= d
        while q != 1:
            # print(p, q)
            x.append(p // q)
            p, q = q, p % q
        x.append(p)
        r = [0, 1]
        for i in range(1, len(x)):
            r.append(x[i] * r[-1] + r[-2])
            if r[-1] == ord:
                if nonzero:
                    print('nonzero-prob. success k =', k)
                else:
                    print('zero-prob. success k =', k)
                break

a = 7
N = 15
ord = 4
m = 256 >= N^2
zero-prob. success k = 52
zero-prob. success k = 53
zero-prob. success k = 54
zero-prob. success k = 55
zero-prob. success k = 56
zero-prob. success k = 57
zero-prob. success k = 58
zero-prob. success k = 59
zero-prob. success k = 60
zero-prob. success k = 61
zero-prob. success k = 62
zero-prob. success k = 63
nonzero-prob. success k = 64
zero-prob. success k = 65
zero-prob. success k = 66
zero-prob. success k = 67
zero-prob. success k = 68
zero-prob. success k = 69
zero-prob. success k = 70
zero-prob. success k = 71
zero-prob. success k = 72
zero-prob. success k = 73
zero-prob. success k = 183
zero-prob. success k = 184
zero-prob. success k = 185
zero-prob. success k = 186
zero-prob. success k = 187
zero-prob. success k = 188
zero-prob. success k = 189
zero-prob. success k = 190
zero-prob. success k = 191
nonzero-prob. success k = 192
zero-prob. success k = 193
zero-prob. success k = 194
zero-prob. success k = 195
zero-prob. success k = 196
zero-prob. success k = 197
zero-prob. success k = 198
zero-prob. success k = 199
zero-prob. success k = 200
zero-prob. success k = 201
zero-prob. success k = 202
zero-prob. success k = 203
zero-prob. success k = 204