The Simulation of Shor's Algorithm

The implementation of Shor's algorithm found in shor.C follows the steps outlined in the description of Shor's algorithm found above. There are some significant differences in the behavior of the simulator and the behavior of a actual quantum computer.

The simulator uses 2ⁿ⁺¹ double precision floating point numbers to represent the state of the quantum register. It uses one Complex object to represent the probability amplitude of each of the 2ⁿ eigenstates of a n bit quantum memory register, and each Complex object uses two double precision floating point numbers. On a machine in which a double precision floating point number is represented with 64 bits the simulator uses approximately 2ⁿ⁺⁷ classical bits used to represent the state of the quantum memory register, where n is the number of bits required to represent the number the simulator is trying to factor. A real quantum computer would use exactly n qubits as its memory register. This exponential space usage is apparently unavoidable, as a classical computer cannot simply cannot simulate a quantum computer without suffering from exponential time or space overhead if the Complexity-Theoretic Church-Turing Thesis holds.

A second large difference is that during the modular exponentiation step of Shor's algorithm (Step 6 above) a quantum computer would perform one operation of x^a mod n, where a is the superposition of states in register 1. The simulation must calculate the superposition of values caused by calculating x^a mod n for a = 0 through q - 1 iteratively. The simulation also stores the result of each modular exponentiation, and uses that information to collapse register 1 in step 7 in Shor's algorithm. A quantum computer would not be capable of performing this book keeping, as examining any particular result would collapse the existing superposition to be placed in register 2 at the end of step 6 of Shor's algorithm. A quantum computer would have no need whatsoever to do such bookkeeping, as when when register 2 is measured in step 7, the collapse of register 1 is an automatic and unavoidable consequence of the measurement. A analogous argument follows for the use of the get probability function in step 8 of Shor's algorithm for calculating the discrete Fourier transform.

Aside from these differences, which are necessitated by the inability of a classical computer to accurately depict the behavior of a quantum mechanical system, the operations are performed by the shor.C program are identical to those called for in the description of Shor's algorithm.

As the algorithm runs the state of the quantum memory register changes in the manner laid out in the description of Shor's algorithm. After the final measurement of register 1 in step 9 we obtain some integer m, which has a high probability of being an integer multiple $\lambda$ of q/r, where $\lambda$ is some integer, and r is the period of x^a mod n that we are trying to find, so that we may calculate x^r/2 -1 mod n and x^r/2 +1 mod n in an effort to find numbers which share factors with n.

In the post processing step shor.C takes the integer m and divides it by q, thus yielding some number c which is approximately equal to $\lambda$/r, where $\lambda$ is an integer, and r is the desired period. Then using a helper function it calculates the best rational approximation to c which has a denominator that is less than or equal to q (recall that q is the power of two such that n² $\leq$ q < 2n², where n is the number to be factored by Shor's algorithm).

We take this denominator to be our period. Shor's algorithm can only use even periods in determining factors of n, and so we check to see if r is even, if not we check to see if doubling the period would still yield a period less than q, if so we double our guessed period.

Taking the period, which is now guaranteed to be even, we proceed to calculate x^r/2 -1 mod n and x^r/2 +1 mod n, calling these values a and b, we compute the greatest common denominator of a and n, and b and n, to see if we have attained a nontrivial factor of n.