def shor_factor(n):
"""Shor's factorization algorithm.
Args:
n: Integer >=2 to factor.
Returns:
If n is composite, a non-trivial factor of n. If n is prime, returns 1.
Raises:
ValueError if n is <= 1.
"""
if n <= 1:
raise ValueError('n must be at least 2')
if is_prime(n):
return 1
if n % 2 == 0:
return 2 # even numbers > 2 are trivial
# Need to check that n is not a power of an integer for algorithm to work
root = int_root(n)
if root != -1:
return root
# choose m s.t. n^2 <= 2^m < 2*n^2
# log2(n^2) <= m <= log2(2 * n^2) = 1 + log2(n^2)
m = ceil(log(n**2, 2))
ny = ceil(log(n - 1, 2)) # number of qubits in output of function f
I = QuantumGate(eye(2**ny))
H = kron(hadamard_gate(m), I)
MAX_ITER = 10
niter = 0
while True:
a = n - 1 # arbitrary integer coprime to n
# Initialize a system of qubits long enough to represent the integers 0
# to 2^m - 1 alongside an integer up to n - 1, then apply Hadamard gate
# to the first m qubits to create a uniform superposition.
q = QubitSystem(m + ny)
H * q
# Apply the function f(x) = a^x (mod n) to the system
f = lambda x: pow(a, x, n)
Uf = function_gate(f, m, ny)
Uf * q
# Find the period of f via quantum Fourier transform
qft(q)
r = 1. / q.measure() # period = 1 / frequency
niter += 1
if niter >= MAX_ITER or (r % 2 == 0 and (a**(r / 2)) % n != -1):
break
return gcd(a**(r / 2) + 1, n)
def test_function_gate(self):
f = lambda x: x >> 2
m, k = 4, 2
Uf = function_gate(f, m, k)
# Initial state - M-bit binary string for 6 plus random K-bit register
q = QubitSystem(m + k, 0b011010)
# Expected output state: first M bits unchanged, followed by initial
# string added to f(x) mod 2 (where x = first M bits)
Uf * q
self.assertEqual(q.measure(), 0b011011)