"Pebble games," an abstraction from classical reversible computing, have found use in the design of quantum circuits for inherently sequential tasks. Gidney showed that allowing Hadamard basis measurements during pebble games can dramatically improve costs -- an extension termed "spooky pebble games" because the measurements leave temporary phase errors called ghosts. In this work, we define and study parallel spooky pebble games. Previous work by Blocki, Holman, and Lee (TCC 2022) and Gidney studied the benefits offered by either parallelism or spookiness individually; here we show that these resources can yield impressive gains when used together. First, we show by construction that a line graph of length $\ell$ can be pebbled in depth $2\ell$ (which is exactly optimal) using space $\leq 2.47\log \ell$. Then, to explore pebbling schemes using even less space, we use a highly optimized $A^*$ search implemented in Julia to find the lowest-depth parallel spooky pebbling possible for a range of concrete line graph lengths $\ell$ given a constant number of pebbles $s$. We show that these techniques can be applied to Regev's factoring algorithm (Journal of the ACM 2025) to significantly reduce the cost of its arithmetic. For example, we find that 4096-bit integers $N$ can be factored in multiplication depth 193, which outperforms the 680 required of previous variants of Regev and the 444 reported by Eker{\aa} and G\"artner for Shor's algorithm (IACR Communications in Cryptology 2025). While space-optimized implementations of Shor's algorithm remain likely the best candidates for first quantum factorization of large integers, our results show that Regev's algorithm may have practical importance in the future, especially given the possibility of further optimization. Finally, we believe our pebbling techniques will find applications in quantum cryptanalysis beyond integer factorization.

翻译："卵石游戏"这一源于经典可逆计算的抽象概念，已在量子电路设计中用于处理固有顺序性任务。Gidney的研究表明，在卵石游戏中允许进行Hadamard基测量能显著降低计算成本——这一扩展被称为"幽灵卵石游戏"，因为测量会留下称为"幽灵"的临时相位误差。本文定义并研究了并行幽灵卵石游戏。Blocki、Holman和Lee（TCC 2022）以及Gidney的前期工作分别研究了并行性或幽灵特性单独带来的优势；本文则证明当这两种资源结合使用时能产生显著增益。首先，我们通过构造证明：长度为$\ell$的线图可在深度$2\ell$（达到理论最优）下使用空间$\leq 2.47\log \ell$完成卵石放置。随后，为探索使用更少空间的卵石放置方案，我们采用Julia语言实现的高度优化$A^*$搜索算法，针对给定固定卵石数$s$的不同具体线图长度$\ell$，寻找可能的最低深度并行幽灵卵石放置方案。研究表明，这些技术可应用于Regev因式分解算法（Journal of the ACM 2025），显著降低其算术运算成本。例如，我们发现4096位整数$N$可在乘法深度193下完成因式分解，这优于Regev先前变体所需的680深度，也优于Ekerå和Gärtner为Shor算法报告的444深度（IACR Communications in Cryptology 2025）。虽然空间优化的Shor算法实现仍可能是大整数首次量子因式分解的最佳候选方案，但我们的结果表明Regev算法未来可能具有实际重要性，特别是考虑到进一步优化的可能性。最后，我们相信本文的卵石放置技术将在整数因式分解之外的量子密码分析领域获得应用。