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. Separately, previous work by Blocki et al. studied the benefits of parallelism in pebble games. In this work we define and study parallel spooky pebble games, showing that parallelism and spookiness can yield impressive gains when used together. First, we show by construction that a line graph of length $\ell$ can be pebbled in depth $2\ell$ (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 then show that these techniques can significantly reduce the cost of the arithmetic in Regev's factoring algorithm. 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. 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 Regev's algorithm may have practical importance in the future. We also believe our pebbling techniques are applicable in quantum cryptanalysis beyond integer factorization, and in quantum circuit compilation more broadly.