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Parallel Spooky Pebbling Makes Regev Factoring More Practical

Gregory D. Kahanamoku-Meyer, Seyoon Ragavan, Katherine Van Kirk

TL;DR

This work advances quantum algorithm design for inherently sequential tasks by combining parallelism with spooky mid-circuit measurements in pebble games. The authors define parallel spooky pebbling, prove a near-optimal depth of $2\ell$ using about $2.47\log \ell$ pebbles on a line graph, and supplement with an $A^*$ search to obtain exact optimal schemes for concrete sizes. They then apply these techniques to Regev's factoring circuit, achieving concrete reductions in mod $N$ multiplication depth (e.g., around $157$ for $n=2048$ and $193$ for $n=4096$) with modest qubit counts, improving on prior Regev variants and some Shor benchmarks. The results suggest that Regev-style factoring could become practically relevant with further optimizations, while also offering a general toolkit for quantum cryptanalysis that leverages space-time tradeoffs enabled by mid-circuit measurements and parallelization.

Abstract

"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 Ekerå and Gärtner 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.

Parallel Spooky Pebbling Makes Regev Factoring More Practical

TL;DR

This work advances quantum algorithm design for inherently sequential tasks by combining parallelism with spooky mid-circuit measurements in pebble games. The authors define parallel spooky pebbling, prove a near-optimal depth of using about pebbles on a line graph, and supplement with an search to obtain exact optimal schemes for concrete sizes. They then apply these techniques to Regev's factoring circuit, achieving concrete reductions in mod multiplication depth (e.g., around for and for ) with modest qubit counts, improving on prior Regev variants and some Shor benchmarks. The results suggest that Regev-style factoring could become practically relevant with further optimizations, while also offering a general toolkit for quantum cryptanalysis that leverages space-time tradeoffs enabled by mid-circuit measurements and parallelization.

Abstract

"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 can be pebbled in depth (which is exactly optimal) using space . Then, to explore pebbling schemes using even less space, we use a highly optimized search implemented in Julia to find the lowest-depth parallel spooky pebbling possible for a range of concrete line graph lengths given a constant number of pebbles . 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 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 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.

Paper Structure

This paper contains 62 sections, 12 theorems, 29 equations, 1 figure, 7 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our target application: concretely efficient Regev factoring.
  3. Outlook and caveats.
  4. Other applications.
  5. Organization of the paper.
  6. Technical Overview
  7. Pebbling Games, Spooky and Parallelized
  8. Applying Pebbling Games to Regev Factoring
  9. Why pebbling?
  10. Formulating Regev's circuit as a pebbling game.
  11. Amortizing step \ref{['step:treebasic']} across pebbles.
  12. Ghosting inside step \ref{['step:treebasic']}.
  13. Computing $\prod_{j = 1}^d a_j^{z_j, [i:i+w)}$ in practice.
  14. Not all pebbles are created equal!
  15. Parallel Spooky Pebbling Games
  16. ...and 47 more sections

Key Result

Theorem 2.9

There exist sequential spooky pebbling games achieving any of the following guarantees: Moreover, the third item here is asymptotically tight: for any $s, m$ such that $n > \binom{m+s-2}{s-2}$, a sequential spooky pebbling of length $\ell$ must use at least $\Omega(m\ell)$ steps Kornerup2025tightboundsspooky.

Figures (1)

  • Figure 1: Results of the $A^*$ search for optimal parallel pebble games. In the left figure, we plot the minimum number of pebbles $s$ with which it is possible to achieve the absolute optimal depth $2\ell$, and compare with the space used by the $A$-sequence construction of Theorem \ref{['thm:parallelpebblingmain']}. In the right figure, we plot the depth as a function of $\ell$, for different constant values of the number of pebbles $s$.

Theorems & Definitions (41)

  • Definition 2.1: Pebbling Games and States
  • Definition 2.2: Pebbling Moves
  • Definition 2.3: Sequential and Parallel Pebbling Games
  • Definition 2.4: Pebbling Cost and Depth
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7: Optimal-Depth Pebbling
  • Definition 2.8: Pebbling Space
  • Theorem 2.9: Sequential Spooky Pebbling
  • Theorem 2.10
  • ...and 31 more