Quantum and Classical Combinatorial Optimizations Applied to Lattice-Based Factorization

TL;DR

This paper critically evaluates quantum and classical lattice-based methods for integer factorization, showing they do not scale well for large numbers and that classical heuristics outperform quantum approaches, despite some promising mathematical challenges.

Contribution

It provides empirical evidence that lattice-based quantum factoring methods are ineffective for large integers and compares their performance to classical heuristics, highlighting future research directions.

Findings

Lattice-based factoring does not scale to larger numbers.

Classical heuristics outperform quantum methods in this context.

Quantum enhancements do not significantly improve factoring success.

Abstract

The availability of working quantum computers has led to several proposals and claims of quantum advantage. In 2023, this has included claims that quantum computers can successfully factor large integers, by optimizing the search for nearby integers whose prime factors are all small. This paper demonstrates that the hope of factoring numbers of commercial significance using these methods is unfounded. Mathematically, this is because the density of smooth numbers (numbers all of whose prime factors are small) decays exponentially as n grows. Our experimental reproductions and analysis show that lattice-based factoring does not scale successfully to larger numbers, that the proposed quantum enhancements do not alter this conclusion, and that other simpler classical optimization heuristics perform much better for lattice-based factoring. However, many topics in this area have interesting applications and mathematical challenges, independently of factoring itself. We consider particular cases of the CVP, and opportunities for applying quantum techniques to other parts of the factorization pipeline, including the solution of linear equations modulo 2. Though the goal of factoring 1000-bit numbers is still out-of-reach, the combinatoric landscape is promising, and warrants further research with more circumspect objectives.