Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

TL;DR

This paper presents an optimal method for learning quantum Hamiltonians from high-temperature Gibbs states, achieving minimal sample complexity and demonstrating the optimality of the approach, with extensions to real-time evolution.

Contribution

It introduces a new algorithm for Hamiltonian learning with optimal sample complexity and time, extending previous work to more general Hamiltonians and real-time evolutions.

Findings

Sample complexity is $O(rac{ ext{log} N}{eta ext{ } ext{error}^2})$.

The algorithm's time complexity is linear in the sample size, $O(S N)$.

The sample complexity is proven to be optimal via a matching lower bound.

Abstract

We study the problem of learning a Hamiltonian $H$ to precision $\varepsilon$, supposing we are given copies of its Gibbs state $\rho=\exp(-\beta H)/\operatorname{Tr}(\exp(-\beta H))$ at a known inverse temperature $\beta$. Anshu, Arunachalam, Kuwahara, and Soleimanifar (Nature Physics, 2021, arXiv:2004.07266) 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, $O(S N)$. 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 $e^{-it H}$ in a small $t$ regime with similar sample and time complexity.