Source code for sympy.physics.quantum.shor

"""Shor's algorithm and helper functions.

Todo:

* Get the CMod gate working again using the new Gate API.
* Fix everything.
* Update docstrings and reformat.
* Remove print statements. We may want to think about a better API for this.
"""
from __future__ import print_function, division

import math
import random

from sympy import Mul, S
from sympy import log, sqrt
from sympy.core.numbers import igcd
from sympy.core.compatibility import range
from sympy.ntheory import continued_fraction_periodic as continued_fraction
from sympy.utilities.iterables import variations

from sympy.physics.quantum.gate import Gate
from sympy.physics.quantum.qubit import Qubit, measure_partial_oneshot
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.qft import QFT
from sympy.physics.quantum.qexpr import QuantumError


class OrderFindingException(QuantumError):
    pass


class CMod(Gate):
    """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
    """

    @classmethod
    def _eval_args(cls, args):
        # t = args[0]
        # a = args[1]
        # N = args[2]
        raise NotImplementedError('The CMod gate has not been completed.')

    @property
    def t(self):
        """Size of 1/2 input register.  First 1/2 holds output."""
        return self.label[0]

    @property
    def a(self):
        """Base of the controlled mod function."""
        return self.label[1]

    @property
    def N(self):
        """N is the type of modular arithmetic we are doing."""
        return self.label[2]

    def _apply_operator_Qubit(self, qubits, **options):
        """
            This directly calculates the controlled mod of the second half of
            the register and puts it in the second
            This will look pretty when we get Tensor Symbolically working
        """
        n = 1
        k = 0
        # Determine the value stored in high memory.
        for i in range(self.t):
            k = k + n*qubits[self.t + i]
            n = n*2

        # The value to go in low memory will be out.
        out = int(self.a**k % self.N)

        # Create array for new qbit-ket which will have high memory unaffected
        outarray = list(qubits.args[0][:self.t])

        # Place out in low memory
        for i in reversed(range(self.t)):
            outarray.append((out >> i) & 1)

        return Qubit(*outarray)


def shor(N):
    """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.
    """
    a = random.randrange(N - 2) + 2
    if igcd(N, a) != 1:
        print("got lucky with rand")
        return igcd(N, a)
    print("a= ", a)
    print("N= ", N)
    r = period_find(a, N)
    print("r= ", r)
    if r % 2 == 1:
        print("r is not even, begin again")
        shor(N)
    answer = (igcd(a**(r/2) - 1, N), igcd(a**(r/2) + 1, N))
    return answer


def getr(x, y, N):
    fraction = continued_fraction(x, y)
    # Now convert into r
    total = ratioize(fraction, N)
    return total


def ratioize(list, N):
    if list[0] > N:
        return S.Zero
    if len(list) == 1:
        return list[0]
    return list[0] + ratioize(list[1:], N)


def period_find(a, N):
    """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.
    """
    epsilon = .5
    #picks out t's such that maintains accuracy within epsilon
    t = int(2*math.ceil(log(N, 2)))
    # make the first half of register be 0's |000...000>
    start = [0 for x in range(t)]
    #Put second half into superposition of states so we have |1>x|0> + |2>x|0> + ... |k>x>|0> + ... + |2**n-1>x|0>
    factor = 1/sqrt(2**t)
    qubits = 0
    for arr in variations(range(2), t, repetition=True):
        qbitArray = arr + start
        qubits = qubits + Qubit(*qbitArray)
    circuit = (factor*qubits).expand()
    #Controlled second half of register so that we have:
    # |1>x|a**1 %N> + |2>x|a**2 %N> + ... + |k>x|a**k %N >+ ... + |2**n-1=k>x|a**k % n>
    circuit = CMod(t, a, N)*circuit
    #will measure first half of register giving one of the a**k%N's
    circuit = qapply(circuit)
    print("controlled Mod'd")
    for i in range(t):
        circuit = measure_partial_oneshot(circuit, i)
    print("measured 1")
    #Now apply Inverse Quantum Fourier Transform on the second half of the register
    circuit = qapply(QFT(t, t*2).decompose()*circuit, floatingPoint=True)
    print("QFT'd")
    for i in range(t):
        circuit = measure_partial_oneshot(circuit, i + t)
    print(circuit)
    if isinstance(circuit, Qubit):
        register = circuit
    elif isinstance(circuit, Mul):
        register = circuit.args[-1]
    else:
        register = circuit.args[-1].args[-1]

    print(register)
    n = 1
    answer = 0
    for i in range(len(register)/2):
        answer += n*register[i + t]
        n = n << 1
    if answer == 0:
        raise OrderFindingException(
            "Order finder returned 0. Happens with chance %f" % epsilon)
    #turn answer into r using continued fractions
    g = getr(answer, 2**t, N)
    print(g)
    return g