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.

翻译：1994年，Shor提出了著名的量子算法，可在多项式时间内完成整数因式分解与离散对数计算。2023年，Regev提出了Shor算法的多维版本，所需量子门数量显著减少。该算法依赖于一个数论猜想，即 $(\mathbb{Z}/N\mathbb{Z})^{\times}$ 中的元素可表示为极小素数的短乘积形式。我们利用零密度估计等解析数论工具证明了该猜想的一个版本。由此，我们获得了这一改进量子算法及其后续变体正确性的无条件证明。