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Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

Jeongwan Haah, Robin Kothari, Ewin Tang

TL;DR

This work gives a sharp, high-temperature algorithm for learning quantum Hamiltonians from Gibbs states with coefficients bounded in $[-1,1]$, achieving a near-optimal sample complexity of $S=O\left(\dfrac{\log N}{(\beta\varepsilon)^2}\right)$ and time linear in the total samples, $O(SN)$. The key technical advance is a constructive cluster expansion of the log-partition function, enabling efficient computation of expectation-values as polynomials in the unknown coefficients, which in turn empowers a Newton–Raphson inversion to recover the Hamiltonian parameters. The authors also prove matching information-theoretic lower bounds, establishing optimality in the high-temperature regime, and extend the approach to learning from real-time evolution in the appendix. Together, these results close the gap to classical sample-complexity bounds in the high-temperature limit and provide a scalable, time-efficient method for quantum Hamiltonian learning in a broad low-intersection setting.

Abstract

We study the problem of learning a Hamiltonian $H$ to precision $\varepsilon$, supposing we are given copies of its Gibbs state $ρ=\exp(-βH)/\operatorname{Tr}(\exp(-βH))$ at a known inverse temperature $β$. Anshu, Arunachalam, Kuwahara, and Soleimanifar (Nature Physics, 2021, arXiv:2004.07266) recently studied the sample complexity (number of copies of $ρ$ needed) of this problem for geometrically local $N$-qubit Hamiltonians. In the high-temperature (low $β$) regime, their algorithm has sample complexity poly$(N, 1/β,1/\varepsilon)$ 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 $S = O(\log N/(β\varepsilon)^{2})$ 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.

Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

TL;DR

This work gives a sharp, high-temperature algorithm for learning quantum Hamiltonians from Gibbs states with coefficients bounded in , achieving a near-optimal sample complexity of and time linear in the total samples, . The key technical advance is a constructive cluster expansion of the log-partition function, enabling efficient computation of expectation-values as polynomials in the unknown coefficients, which in turn empowers a Newton–Raphson inversion to recover the Hamiltonian parameters. The authors also prove matching information-theoretic lower bounds, establishing optimality in the high-temperature regime, and extend the approach to learning from real-time evolution in the appendix. Together, these results close the gap to classical sample-complexity bounds in the high-temperature limit and provide a scalable, time-efficient method for quantum Hamiltonian learning in a broad low-intersection setting.

Abstract

We study the problem of learning a Hamiltonian to precision , supposing we are given copies of its Gibbs state at a known inverse temperature . Anshu, Arunachalam, Kuwahara, and Soleimanifar (Nature Physics, 2021, arXiv:2004.07266) recently studied the sample complexity (number of copies of needed) of this problem for geometrically local -qubit Hamiltonians. In the high-temperature (low ) regime, their algorithm has sample complexity poly 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 with sample complexity and time complexity linear in the sample size, . 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 from a real-time evolution unitary in a small regime with similar sample and time complexity.

Paper Structure

This paper contains 30 sections, 34 theorems, 152 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Time complexity
  5. Analytic functions and series expansions
  6. Series expansions of expectation values
  7. Overview of the proof.
  8. The logarithmic partition function
  9. Deriving multivariate Taylor series expansions
  10. Counting connected clusters
  11. Enumerating connected clusters
  12. Estimating cluster derivatives
  13. Counting partitions of a cluster
  14. Computing cluster derivatives
  15. Time complexity.
  16. ...and 15 more sections

Key Result

theorem 1.1

Let $H$ be a low-intersection Hamiltonian on $N$ qubits, $\varepsilon>0$, and $\beta < \beta_c$. Then we can learn the coefficients of $H$ with $\ell_\infty$ error $\varepsilon$ and failure probability $\delta$ using $O\bigl(\frac{1}{\beta^2\varepsilon^2}\log\frac{N}{\delta}\bigr)$ samples. Conseque

Theorems & Definitions (68)

  • theorem 1.1: Algorithm
  • theorem 1.2: Lower bound
  • theorem 1.3: Real-time dynamics
  • definition 2.1: Hamiltonian
  • definition 2.2: Dual interaction graph
  • definition 2.3: Gibbs state
  • remark 2.4
  • lemma 2.5
  • proof
  • theorem 3.1
  • ...and 58 more