Alex Gough > Quantum-Entanglement-0.32 > shor

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NAME ^

  shor - A short demonstration of Quantum::Entanglement

SYNOPSIS ^

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

DESCRIPTION ^

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

The important maths

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

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

There will also be some other number, 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 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 n. So long as x**(r/2) != +-1, at least one of the two brackets on the left must share a factor with 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 n.

The steps of the algorithm ^

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.

2. Pick an integer

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

3. Select at random a number coprime to n

Think of some number less than n so that n and x do not share a common factor (if they do, we already know the answer...).

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.

5. Calculate F, store in a second register

We now calculate 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.

6. Look at register2

We now look at the value of register two and get some value 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.

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 q/r.

8. Observe register1

We now observe register1, and use the result to calculate a likely value for 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 n.

Things to remember ^

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).

COPYRIGHT ^

This code is copyright (c) Alex Gough (alex@rcon.org )2001. This is free software, you may use, modify and redistribute it under the same terms as Perl itself.

BUGS ^

This is slow, being run on classical computers, ah well.

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