Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with HTTPS or Subversion.

Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

219 lines (178 sloc) 6.479 kb
"""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.
"""
import math
import random
from sympy import Mul
from sympy import log, sqrt
from sympy.core.numbers import igcd
from sympy.physics.quantum.gate import Gate
from sympy.physics.quantum.qubit import Qubit, measure_partial_oneshot
from sympy.physics.quantum.applyops import apply_operators
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][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 arr(num, t):
"""This function returns num as an array in binary
It does this with the 0th digit being on the right
>>> from sympy.physics.quantum.shor import arr
>>> arr(5, 4)
[0, 1, 0, 1]
"""
binary_array = []
for i in reversed(range(t)):
binary_array.append((num>>i)&1)
return binary_array
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 0
if len(list) == 1:
return list[0]
return list[0] + ratioize(list[1:], N)
def continued_fraction(x, y):
"""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]
"""
x = int(x)
y = int(y)
temp = x/y
if temp*y == x:
return [temp,]
list = continued_fraction(y, x-temp*y)
list.insert(0, temp)
return list
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 i in range(2**t):
qbitArray = arr(i, t) + 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 = apply_operators(circuit)
print "controlled Mod'd"
for i in range(t):
circuit = measure_partial_oneshot(circuit, i)
# circuit = measure(i)*circuit
# circuit = apply_operators(circuit)
print "measured 1"
#Now apply Inverse Quantum Fourier Transform on the second half of the register
circuit = apply_operators(QFT(t, t*2).decompose()*circuit, floatingPoint = True)
print "QFT'd"
for i in range(t):
circuit = measure_partial_oneshot(circuit, i+t)
# circuit = measure(i+t)*circuit
# circuit = apply_operators(circuit)
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
Jump to Line
Something went wrong with that request. Please try again.