Learning Quantum Hamiltonians at Any Temperature in Polynomial Time
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3618260.3649619.pdf
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Author(s)
Bakshi, Ainesh
Liu, Allen
Moitra, Ankur
Tang, Ewin
Date Issued
June 10, 2024
Publisher
ACM STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Bakshi, Ainesh, Liu, Allen, Moitra, Ankur and Tang, Ewin. 2024. "Learning Quantum Hamiltonians at Any Temperature in Polynomial Time."
Version
Final published version
Abstract
We study the problem of learning a local quantum Hamiltonian
given copies of its Gibbs state =
− /tr(
− ) at a known
inverse temperature > 0. Anshu, Arunachalam, Kuwahara, and
Soleimanifar gave an algorithm to learn a Hamiltonian on qubits
to precision with only polynomially many copies of the Gibbs
state, but which takes exponential time. Obtaining a computationally e cient algorithm has been a major open problem, with prior
work only resolving this in the limited cases of high temperature or
commuting terms. We fully resolve this problem, giving a polynomial time algorithm for learning to precision from polynomially
many copies of the Gibbs state at any constant > 0.
Our main technical contribution is a new at polynomial approximation to the exponential function, and a translation between
multi-variate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system.
We then show that solving a low-degree sum-of-squares relaxation
of this polynomial system su ces to accurately learn the Hamiltonian.
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1145/3618260.3649619
