Parallel Spooky Pebbling Makes Regev Factoring More Practical

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 𝓁 can be pebbled in depth 2𝓁 (which is exactly optimal) using space ≤ 2.47 log 𝓁. 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 𝓁 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 Ekerå and Gärtner for Shor's algorithm (IACR Communications in Cryptology 2025). While the space required for Shor's algorithm is considerably less than any variant of Regev's algorithm including ours, and thus Shor likely remains the best candidate for the first quantum factorization of large integers, our results show that implementations of Regev's algorithm are far from fully optimized, and thus Regev's algorithm may have practical importance in the future. We also believe our pebbling techniques will find applications in quantum cryptanalysis beyond integer factorization, and in quantum circuit compilation more broadly.