Seyoon Ragavan

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.

In this work, we introduce a new toolkit for analyzing \emph{cloning games}, a notion that captures stronger and more quantitative versions of the celebrated quantum no-cloning theorem. This framework allows us to analyze a new cloning game based on \emph{binary phase states}. Our results provide evidence that these games may be able to overcome important limitations of previous candidates based on BB84 states and subspace coset states: in a model where the adversaries are restricted to making a single oracle query, we show that the binary phase variant is $t$-copy secure when $t=o(n/\log n)$. Moreover, for constant $t$, we obtain the \emph{first} optimal bounds of $O(2^{-n})$, asymptotically matching the value attained by a trivial adversarial strategy. We also show a worst-case to average-case reduction which allows us to show the same quantitative results for the new and natural notion of \emph{Haar cloning games}. Our analytic toolkit, which we believe will find further applications, is based on binary subtypes and uses novel bounds on the operator norms of block-wise tensor products of matrices. To illustrate the effectiveness of these new techniques, we present two applications: first, in black-hole physics, where our asymptotically optimal bound offers quantitative insights into information scrambling in idealized models of black holes; and second, in unclonable cryptography, where we (a) construct succinct unclonable encryption schemes from the existence of pseudorandom unitaries, and (b) propose and provide evidence for the security of multi-copy unclonable encryption schemes.

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.