Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

TL;DR

This paper proves an unconditional correctness for a modern, more efficient quantum algorithm for factoring and discrete logarithms, based on a number-theoretic conjecture, using tools from analytic number theory.

Contribution

It provides the first unconditional proof of correctness for Regev's multi-dimensional quantum algorithm, validating its efficiency without relying on unproven conjectures.

Findings

Unconditional proof of Regev's quantum algorithm correctness

Validation of the algorithm's efficiency in factoring and discrete logs

Application of analytic number theory to quantum algorithm analysis

Abstract

In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(\mathbb{Z}/N\mathbb{Z})^{\times}$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.