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

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.

Minor typos corrected. 25 pages

Number Theory (math.NT); Computational Complexity (cs.CC); Quantum Physics (quant-ph)

arXiv:2404.16450 [math.NT]

(or arXiv:2404.16450v3 [math.NT] for this version)

https://doi.org/10.48550/arXiv.2404.16450 Focus to learn more arXiv-issued DOI via DataCite

https://doi.org/10.1017/fmp.2025.10023 Focus to learn more DOI(s) linking to related resources

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