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

Journal article

In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multidimensional version of Shor’s algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in 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.

Authors: Pilatte, C

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press
• Journal: Forum of Mathematics, Pi
• Volume: 14
• Article number: e5
• Publication date: 2026-02-09
• Acceptance date: 2025-10-08
• DOI: 10.1017/fmp.2025.10023
• EISSN: 2050-5086
• Language: English
• Keywords: 68Q12; 11L40; 11Y16; 11H06; 11M26