The Jacobi Factoring Circuit: Quantum Factoring with Near-Linear Gates and Sublinear Space and Depth
Name
3717823.3718273.pdf
Size
879.91 KB
Format
Adobe PDF
Checksum (MD5)
8fd08b4c99786382c223dc62baa3b2ad
Author(s)
Kahanamoku-Meyer, Gregory D.
Ragavan, Seyoon
Vaikuntanathan, Vinod
Van Kirk, Katherine
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Gregory D. Kahanamoku-Meyer, Seyoon Ragavan, Vinod Vaikuntanathan, and Katherine Van Kirk. 2025. The Jacobi Factoring Circuit: Quantum Factoring with Near-Linear Gates and Sublinear Space and Depth. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 1496–1507.
Version
Final published version
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 P2 Q with logQ = Θ(na) for a ∈ (2/3, 1) in space and depth sublinear in n (specifically, O(logQ)) using 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.
Description
STOC ’25, Prague, Czechia
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3717823.3718273
