Katherine Van Kirk

We present a compact quantum circuit for factoring a large class of integers, including some whose classical hardness is expected to be equivalent to RSA (but not including RSA integers themselves). Most notably, we factor $n$-bit integers of the form $P^2 Q$ with $\log Q = \Theta(n^a)$ for $a \in (2/3, 1)$ in space and depth sublinear in n (specifically, $\widetilde{O}(\log Q)$) using $\widetilde{O}(n)$ quantum gates; for these integers, no known classical algorithms exploit the relatively small size of $Q$ to run asymptotically faster than general-purpose factoring algorithms. To our knowledge, this is the first polynomial-time circuit to achieve sublinear qubit count for a classically-hard factoring problem. We thus believe that factoring such numbers has potential to be the most concretely efficient classically-verifiable proof of quantumness currently known. Our circuit builds on the quantum algorithm for squarefree decomposition discovered by Li, Peng, Du, and Suter (Nature Scientific Reports 2012), which relies on computing the Jacobi symbol in quantum superposition. The technical core of our contribution is a new space-efficient quantum algorithm to compute the Jacobi symbol of $A$ mod $B$, in the regime where $B$ is classical and much larger than $A$. Our circuit for computing the Jacobi symbol generalizes to related problems such as computing the greatest common divisor and modular inverses, and thus could be of independent interest.

We define and study \emph{parallel spooky pebble games} in the context of carrying out an inherently sequential computation on a quantum computer, where reversibility is a concern. Previous work by Blocki, Holman, and Lee (TCC 2022) and Gidney studied the benefits offered by either parallelism or spookiness respectively; we show that these two resources can enable even more efficient guarantees when utilized in tandem. Specifically, we show that with parallelism and spookiness, a line graph of length $\ell$ can be pebbled in time $2\ell$ (which is exactly optimal) and space $\leq 2.47\log \ell$. We show that these techniques can be applied to Regev's factoring algorithm (Journal of the ACM 2025) to obtain significant concrete improvements on previous implementations of the underlying arithmetic (Ragavan and Vaikuntanathan, CRYPTO 2024). For example, we can factor 4096-bit integers $N$ using a mod $N$ multiplication depth of 328, which outperforms the 680 required of previous variants of Regev and the 444 reported by Eker{\aa} and G{\"a}rtner (IACR Communications in Cryptology 2025) for Shor's algorithm. We believe our pebbling techniques will find applications in quantum cryptanalysis beyond integer factorization.