Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

We study the problem of learning a Hamiltonian H to precision \varepsilon, supposing we are given copies of its Gibbs state \rho =\exp(-\beta H)/\mathrm{Tr}(\exp(-\beta H)) at a known inverse temperature \beta. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample compl...

Full description

Saved in:
Bibliographic Details
Published in:Proceedings / annual Symposium on Foundations of Computer Science pp. 135 - 146
Main Authors: Haah, Jeongwan, Kothari, Robin, Tang, Ewin
Format: Conference Proceeding
Language:English
Published: IEEE 01.10.2022
Subjects:
Complexity theory
Computer science
Hamiltonian learning
Markov random field
partition function
quantum algorithm
Quantum computing
real-time evolution
Real-time systems
Time complexity
ISSN:2575-8454
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We study the problem of learning a Hamiltonian H to precision \varepsilon, supposing we are given copies of its Gibbs state \rho =\exp(-\beta H)/\mathrm{Tr}(\exp(-\beta H)) at a known inverse temperature \beta. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample complexity (number of copies of \rho needed) of this problem for geometrically local N-qubit Hamiltonians. In the high-temperature (low \beta) regime, their algorithm has sample complexity poly (N, 1/\beta, 1/\varepsilon) and can be implemented with polynomial, but suboptimal, time complexity. In this paper, we study the same question for a more general class of Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error \varepsilon with sample complexity S=O(\log N/(\beta\varepsilon)^{2}) and time complexity linear in the sample size, O(SN). Furthermore, we prove a matching lower bound showing that our algorithm's sample complexity is optimal, and hence our time complexity is also optimal. In the appendix, we show that virtually the same algorithm can be used to learn H from a real-time evolution unitary e^{-i t H} in a small t regime with similar sample and time complexity.
ISSN:2575-8454
DOI:10.1109/FOCS54457.2022.00020