Simulations of Shor’s algorithm using matrix product states

Abstract

We show that under the matrix product state formalism the states produced in Shor's algorithm can be represented using $$O(\max (4lr^2, 2^{2l}))$$O(max(4lr2,22l)) space, where l is the number of bits in the number to factorise and r is the order and the solution to the related order-finding problem. The reduction in space compared to an amplitude formalism approach is significant, allowing simulations as large as 42 qubits to be run on a single processor with 32 GB RAM. This approach is readily adapted to a distributed memory environment, and we have simulated a 45-qubit case using 8 cores with 16 GB RAM in approximately 1 h.