Learning the Closest Product State
Name
3717823.3718207.pdf
Size
699.13 KB
Format
Adobe PDF
Checksum (MD5)
aa627b1510ca403019dff2dbb611bec8
Author(s)
Bakshi, Ainesh
Bostanci, John
Kretschmer, William
Landau, Zeph
Li, Jerry
Liu, Allen
O'Donnell, Ryan
Tang, Ewin
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Ainesh Bakshi, John Bostanci, William Kretschmer, Zeph Landau, Jerry Li, Allen Liu, Ryan O'Donnell, and Ewin Tang. 2025. Learning the Closest Product State. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 1212–1221.
Version
Final published version
Abstract
We study the problem of finding a product state with optimal fidelity to an unknown n-qubit quantum state ρ, given copies of ρ. This is a basic instance of a fundamental question in quantum learning: is it possible to efficiently learn a simple approximation to an arbitrary state? We give an algorithm which finds a product state with fidelity ε-close to optimal, using N = npoly(1/ε) copies of ρ and poly(N) classical overhead. We further show that estimating the optimal fidelity is NP-hard for error ε = 1/poly(n), showing that the error dependence cannot be significantly improved. For our algorithm, we build a carefully-defined cover over candidate product states, qubit by qubit, and then demonstrate that extending the cover can be reduced to approximate constrained polynomial optimization. For our proof of hardness, we give a formal reduction from polynomial optimization to finding the closest product state. Together, these results demonstrate a fundamental connection between these two seemingly unrelated questions. Building on our general approach, we also develop more efficient algorithms in three simpler settings: when the optimal fidelity exceeds 5/6; when we restrict ourselves to a discrete class of product states; and when we are allowed to output a matrix product state.
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/10.1145/3717823.3718207
