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Certifying and learning local quantum Hamiltonians

Andreas Bluhm, Matthias C. Caro, Francisco Escudero Gutiérrez, Junseo Lee, Aadil Oufkir, Cambyse Rouzé, Myeongjin Shin

Abstract

In this work, we study the problems of certifying and learning quantum $k$-local Hamiltonians, for a constant $k$. Our main contributions are as follows: - Certification of Hamiltonians. We show that certifying a local Hamiltonian in normalized Frobenius norm via access to its time-evolution operator can be achieved with only $O(1/\varepsilon)$ evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of $Ω(1/\varepsilon)$. To our knowledge, this is the first optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Hypercontractivity Lemma from Fourier analysis. - Learning Gibbs states. We design an algorithm for learning Gibbs states of local Hamiltonians in trace norm that is sample-efficient in all relevant parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state), and thus inevitably suffered from exponential sample complexity scaling in the inverse temperature. - Certification of Gibbs states. We give an algorithm for certifying Gibbs states of local Hamiltonians in trace norm that is both sample and time-efficient in all relevant parameters, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022).

Certifying and learning local quantum Hamiltonians

Abstract

In this work, we study the problems of certifying and learning quantum -local Hamiltonians, for a constant . Our main contributions are as follows: - Certification of Hamiltonians. We show that certifying a local Hamiltonian in normalized Frobenius norm via access to its time-evolution operator can be achieved with only evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of . To our knowledge, this is the first optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Hypercontractivity Lemma from Fourier analysis. - Learning Gibbs states. We design an algorithm for learning Gibbs states of local Hamiltonians in trace norm that is sample-efficient in all relevant parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state), and thus inevitably suffered from exponential sample complexity scaling in the inverse temperature. - Certification of Gibbs states. We give an algorithm for certifying Gibbs states of local Hamiltonians in trace norm that is both sample and time-efficient in all relevant parameters, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022).

Paper Structure

This paper contains 35 sections, 14 theorems, 47 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Results and Technical Overview
  3. Certification via access to the dynamics
  4. The intolerant algorithm.
  5. Other complexity measures and further remarks about our algorithms.
  6. Learning thermal states
  7. Certifying thermal states
  8. Related work
  9. Hamiltonian learning.
  10. Hamiltonian property testing.
  11. Certification in other settings.
  12. Improved Bell sampling bounds.
  13. Discussion and open problems
  14. Optimal certification from dynamics without locality assumptions.
  15. Time-efficient Gibbs state learning.
  16. ...and 20 more sections

Key Result

Lemma 4

Let $\rho(\beta)$ and $\rho'(\beta)$ be Gibbs states of two $k$-local Hamiltonians $H$ and $H'$ acting on $n$ qubits. Then, In particular, Furthermore, if $|h_P|,|h'_P|\leq 1$ for every $P\in\{I,X,Y,Z\}^{\otimes n}$, then

Theorems & Definitions (14)

  • Lemma 4
  • Theorem 5: childs2021theory
  • Lemma 6
  • Theorem 7: Clifford shadows huang2020predicting
  • Theorem 8
  • Lemma 9: Paley--Zygmund inequality
  • Lemma 10: Bounding $I(t)$ in terms of $\Lambda$
  • Lemma 11
  • Lemma 12
  • Theorem 13: Tolerant certification of $k$-local Hamiltonians
  • ...and 4 more