Simulations of Shor's algorithm with implications to scaling and quantum error correction
Simon J. Devitt, Austin G. Fowler, Lloyd C. L. Hollenberg · arXiv (Cornell University) · 2004
The publication in 1994 of Shor's algorithm, which allows factorisation of composite number N in a time polynomial in its binary length L has been the primary catalyst for the race to construct a functional quantum computer. However, it seems clear that any practical system that may be developed will not be able to perform completely error free quantum gate operations or shield even idle qubits from inevitable error effects. Hence, the practicality of quantum algorithms needs to be investigated to not only determine limitations on such algorithms in a noisy quantum computer, but also to estimate what demands must be made of quantum error correction (QEC). Shor's algorithm is a combination of both classical pre and post-processing, and also a quantum period finding subroutine (QPF) which allows for the exponential speed up of this algorithm on a quantum device. This paper will look at the stability of this quantum subroutine under the effects of several error models. Direct simulation of the entire QPF subroutine required to factorise a given composite number N in the presence of errors shows that the circuit required to implement Shor's algorithm is very sensitive to a small number of errors within the entire calculation. Well designed and efficient error correction codes, quick gate times and very low gate error rates will be essential for any physical realisation of Shor's factoring algorithm.