Classical simulability and the significance of modular exponentiation in Shor's algorithm

TL;DR

This paper demonstrates that classical algorithms can efficiently simulate key parts of Shor's algorithm under certain conditions, challenging assumptions about its quantum advantage.

Contribution

It introduces a classical simulation approach for the modular exponentiation circuit in Shor's algorithm using semi-classical Fourier transform techniques.

Findings

Classical simulation of modular exponentiation is feasible for specific input states.

The semi-classical Fourier transform enables efficient classical simulation.

Implications for the quantum advantage in factoring algorithms.

Abstract

We show that a classical algorithm efficiently simulating the modular exponentiation circuit, for certain product state input and with measurements in a general product state basis at the output, can efficiently simulate Shor's factoring algorithm. This is done by using the notion of the semi-classical Fourier transform due to Griffith and Niu, and further discussed in the context of Shor's algorithm by Browne.