Simulations of Shor's Algorithm using Matrix Product States

TL;DR

This paper demonstrates that using matrix product states significantly reduces the computational resources needed to simulate Shor's quantum algorithm, enabling larger simulations on classical hardware.

Contribution

It introduces a matrix product state approach to simulate Shor's algorithm efficiently, reducing space complexity and enabling larger-scale simulations than traditional methods.

Findings

Simulated up to 42 qubits on a single processor.

Successfully simulated a 45-qubit case using distributed memory.

Achieved substantial space savings over amplitude formalism.

Abstract

We show that under the matrix product state formalism the states produced in Shor's algorithm can be represented using O(max($4lr^2$, $2^{2l}$)) 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 32GB RAM. This approach is readily adapted to a distributed memory environment, and we have simulated a 45 qubit case using 8 cores with 16GB RAM in approximately one hour.