发表机构
Google Quantum AI; Harvard（Google量子AI; 哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决Regev量子因式分解算法的流式问题，通过叠加掩码和斐波那契幂相位估计，将空间占用降至$n+M(n)+O(\sqrt{n})$，实现最低空间实例化。
AI 中文摘要
Regev因式分解算法的一个主要实际障碍是其巨大的空间占用。特别是，该算法曾被认为与“量子比特回收”不兼容，后者在相位估计中流式处理控制量子比特，而非同时存储所有控制量子比特。我们展示了如何解决这一流式问题。我们基于Ragavan和Vaikuntanathan的构造，该构造使用斐波那契累加器将量子比特数从$O(n^{1.5})$降至$11.32n$。我们做出了两项关键改进。首先，我们使用叠加掩码来防止不需要的信息进入累加器，从而消除了反计算的需要。其次，我们展示了如何使用幺正操作的斐波那契幂而非二的幂来执行相位估计。这避免了Ragavan和Vaikuntanathan构造所需的各种编码转换。经过这些改进，相位估计量子比特在电路中仅使用一次，从而允许它们被流式处理。我们的改进将Regev因式分解算法的空间占用降至$n+M(n)+O(\sqrt{n})$，其中$M(n)$是就地乘法所需的空间。例如，使用Luo等人（2026）的就地乘法器，Regev算法可以用$4n+O(\sqrt{n})$个量子比特分解一个$n$位整数。尽管我们的构造仍达不到领先的Shor算法实现所需的$n/2+\epsilon n$个量子比特，但它是迄今为止Regev算法的最低空间实例化，并表明Regev算法仍远未完全优化。
英文摘要
A major practical obstacle to Regev's factoring algorithm is its large space usage. In particular, the algorithm was believed to be incompatible with "qubit recycling," which streams the control qubits in phase estimation rather than storing them all simultaneously. We show how to fix this streaming problem. We build on the construction of Ragavan and Vaikuntanathan, which used Fibonacci accumulators to reduce the qubit count from $O(n^{1.5})$ to $11.32n$. We make two key improvements. First, we use superposition masking to prevent unwanted information from entering the accumulators, removing the need for uncomputation. Second, we show how to perform phase estimation using Fibonacci powers of a unitary operation rather than powers of two. This avoids various encoding conversions required by the construction of Ragavan and Vaikuntanathan. After these improvements, the phase-estimation qubits are used only once in the circuit, allowing them to be streamed. Our improvements reduce the space usage of Regev's factoring algorithm to $n+M(n)+O(\sqrt{n})$, where $M(n)$ is the space required for an in-place multiplication. For example, using the in-place multiplier of Luo et al. (2026), Regev's algorithm can factor an $n$-bit integer using $4n+O(\sqrt{n})$ qubits. Although our construction still falls short of the $n/2+εn$ qubits sufficient for leading implementations of Shor's algorithm, it is the lowest-space instantiation of Regev's algorithm to date and suggests that Regev's algorithm is still far from fully optimized.
Comments31+22 pages, 3 figures
