``````#!/usr/bin/perl -w

die 'usage: ./shor.pl [number to factor]' unless @ARGV;

use strict;
use warnings;
use Quantum::Entanglement qw(:DEFAULT :complex :QFT);
\$Quantum::Entanglement::destroy = 0;

my \$num = \$ARGV;

# do some early die'ing
die "\$num is a multiple of two, here I am, brain the size..." unless \$num %2;
die "\$num is a non-integer, I only have whole numbers of fingers"
unless \$num == int(\$num);
die "\$num is less than 15" unless \$num >= 15;

print "Performing initial classical steps:\n";
# work out q value
my \$q_power = int(2* log(\$num) / log(2)) +1;
my \$q = 2 ** \$q_power;

# pick some x so that x is coprime to n.
my \$x;
do {
\$x = int(rand \$num) + 1;
} until (\$num % \$x != 0 and \$x > 2); #ok.. so this misses the point slightly

print "Using q:\$q, x:\$x\nStarting quantum steps\n";

# fill up a register with integers from 1..q
my \$prob = 1/sqrt(\$q);
my \$register1 = entangle(map {\$prob, \$_} (0..\$q-1));

# apply transformation F = x**|a> mod n, store in register 2
# (need to do a p_func to avoid overflow while **)

sub power_mod {
my (\$state, \$x1, \$num1) = @_;
my \$rt = 1;
return 1 if \$state == 0;
return 1 if \$state == 1;
for (1..\$state) {
\$rt = (\$rt * \$x1) % \$num1;
}
return \$rt;
}
print "Performing F = x**|a> mod n\n";
my \$register2 = p_func(\&power_mod, \$register1, \$x, \$num);

# We now observe \$register2, thus partially collapsing reg1
my \$k = "\$register2";

print "\\$register2 collapsed to \$k\n";
print "Finding period of F (this is where you wish for a QCD)\n";

# take a ft of the amplitudes of reg1, placing result in reg3
my \$register3 = QFT(\$register1);

my \$lqonr = "\$register3"; # observe, this must be multiple of q/r
if (\$lqonr == 0) {
print "Got period of '0', halting\n"; exit(0);
}
my \$period = int(\$q / \$lqonr + 0.5); # rounding

print "Period of F = x**|a> mod n is \$period\n";

# now given the period, we need to work out the factor of n
# work out the two thingies:

if (\$period % 2 != 0) {
print "\$period is not an even number, doubling to";
\$period *=2;
print " \$period\n";
}

my \$one = \$x**(\$period/2) -1;
my \$two = \$x**(\$period/2) +1;

# one and two must have a gcd in common with n, which we now find...
print "\$one * \$two and \$num might share a gcd (classical step)\n";
my (\$max1, \$max2) = (1,1);
for (2..\$num) {
last if \$_ > \$num;
unless ((\$num % \$_) || (\$one % \$_)) {
\$max1 = \$_;
}
unless ((\$num % \$_) || (\$two % \$_)) {
\$max2 = \$_;
}
}
print "\$max1, \$max2 could be factors of \$num\n";

__END__;

shor - A short demonstration of Quantum::Entanglement

./shor.pl [number to factor (>14)]

This program implements Shor's famous algorithm for factoring numbers.  A
brief overview of the algorithm is given below.

Given a number B<n> which we are trying to factor, and some other number
which we have guessed, B<x>, we can say that:

x**0 % n == 1 (as x**0 = 1, 1 % n =1)

There will also be some other number, B<r> such that

x**r % n == 1

or, more specifically,

x**(kr) % n ==1

in other words, the function

F(a) = x**a % n

is periodic with period B<r>.

Now, starting from

x**r = 1 % n

x**(2*r/2) = 1 % n

(x**(r/2))**2 - 1 = 0 % n

and, if r is an even number,

(x**(r/2) - 1)*(x**(r/2) + 1) = 0 mod n

or in nice short words, the term on the left is an integer multiple of B<n>.
So long as x**(r/2) != +-1, at least one of the two brackets on the left
must share a factor with B<n>.

Shor's alorithm provides a way to find the periodicity of the function F
and thus a way to calculate two numbers which share a factor with n, it
is then easy to use a classical computer to find the GCD and thus a
factor of B<n>.

=head1 The steps of the algorithm

=head2 1. Remove early trivial cases

We have efficient classical methods for finding that 2 is a factor of 26,
so we do not need to use this method for this.

Chose a number B<q> so that C<n**2 <= q <= 2n**2>, this is done on a
classical computer. (This is the size we will use for our quantum register.)

=head2 3. Select at random a number coprime to n

Think of some number less than B<n> so that B<n> and B<x> do not share

=head2 4. Fill a quantum register with integers from 0..q-1

This is where we create our first entangled variable, and is the first
non-classical step in this algorithm.

=head2 5. Calculate F, store in a second register

We now calculate C< F(a) = x**a % n> where a represents the superposition
of states in our first register, we store the result of this in our
second register.

We now look at the value of register two and get some value B<k>, this forces
register1 into
a state which can only collapse into values satisfying the equation

x**a % n = k

The probability amplitudes for the remaining states are now all equal to zero,
note that we have not yet looked directly at register1.

=head2 7. Find period of register1

We now apply a fourier transform to the amplitudes of the states in
register1, storing the result as the probability amplitudes for a new
state with the values of register1.  This causes there to be a high
probability that the register will collapse to a value which is some
multiple of C<q/r>.

We now observe register1, and use the result to calculate a likely value
for B<r>.  From this we can easily calculate two numbers, one of which
will have a factor in common with n, by applying an efficient classical
algoirthm for finding the greatest common denominator, we will be able
to find a value which could be a factor of B<n>.

This algorithm does not claim to produce a factor of our number the first
time that it is run, there are various conditions which will cause it
to halt mid-way, for instance, the FT step can give a result of 0 which
is clearly useless.  The algorithm is better than any known classical one
because the expectation value of the time required to get a correct answer
is still O(n).

This also cannot factor a number which is prime (it being, as it were, prime)
and also cannot factor something which is a prime power (25, say).