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.

翻译：可工作的量子计算机的出现催生了若干关于量子优势的提议与宣称。2023年，有声称称量子计算机能够通过优化搜索其所有素因子均为小整数的邻近整数，成功分解大整数。本文证明，利用这些方法分解具有商业价值的数字的期望是毫无根据的。从数学上看，这是因为光滑数（所有素因子均为小整数的数）的密度随着n的增长呈指数衰减。我们的实验复现与分析表明，基于格的分解无法成功扩展到更大的数字，所提出的量子增强并未改变这一结论，而其他更简单的经典优化启发式方法在基于格分解上表现更优。然而，该领域的许多主题具有有趣的应用和数学挑战，独立于分解本身。我们考虑了CVP的特殊情形，以及将量子技术应用于分解流程其他部分（包括模2线性方程求解）的机会。尽管分解1000比特数字的目标仍遥不可及，但组合优化前景广阔，值得以更审慎的目标开展进一步研究。