Shor’s algorithm and helper functions.
Todo:
A controlled mod gate.
This is black box controlled Mod function for use by shor’s algorithm. TODO implement a decompose property that returns how to do this in terms of elementary gates
This applies the continued fraction expansion to two numbers x/y
x is the numerator and y is the denominator
>>> from sympy.physics.quantum.shor import continued_fraction
>>> continued_fraction(3, 8)
[0, 2, 1, 2]
Finds the period of a in modulo N arithmetic
This is quantum part of Shor’s algorithm.It takes two registers, puts first in superposition of states with Hadamards so: |k>|0> with k being all possible choices. It then does a controlled mod and a QFT to determine the order of a.
This function implements Shor’s factoring algorithm on the Integer N
The algorithm starts by picking a random number (a) and seeing if it is coprime with N. If it isn’t, then the gcd of the two numbers is a factor and we are done. Otherwise, it begins the period_finding subroutine which finds the period of a in modulo N arithmetic. This period, if even, can be used to calculate factors by taking a**(r/2)-1 and a**(r/2)+1. These values are returned.