The Jacobi Factoring Circuit: Quantum Factoring with Near-Linear Gates and Sublinear Space and Depth

TL;DR

This paper introduces a compact quantum circuit capable of factoring certain large integers efficiently using near-linear gates and sublinear space and depth, advancing quantum factoring techniques.

Contribution

It presents the first polynomial-time quantum circuit with sublinear qubit count for a classically-hard factoring problem, utilizing a novel space-efficient Jacobi symbol algorithm.

Findings

Successfully factors integers of the form P^2 Q with small Q in sublinear space and depth.

Achieves a quantum circuit with O(log Q) depth and O(n) gates for specific integer classes.

Demonstrates potential for efficient, verifiable quantum proofs of quantumness.

Abstract

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, $\tilde{O}(\log Q)$) using $\tilde{O}(n)$ 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.