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arXiv · 2404.16450

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

Cédric Pilatte
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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.

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Cédric Pilatte. 2024-04-25. Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms. https://doi.org/10.1017/fmp.2025.10023

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