Certifying and learning local quantum Hamiltonians

2026-03-31

AI Summary

This work addresses the problem of efficiently certifying constant-local quantum Hamiltonians and their Gibbs states, overcoming bottlenecks in sample and time complexity inherent in existing approaches. By leveraging access to the Hamiltonian’s time evolution operator, the authors propose the first optimal certification algorithm that achieves the Heisenberg-limit lower bound, requiring only $O(1/\varepsilon)$ evolution time. Moreover, bypassing the need to directly learn the Hamiltonian itself, they instead develop a method to directly and efficiently learn and certify the Gibbs state in trace distance, thereby resolving an open question posed by Anshu. The approach combines Fourier-analytic tools—such as the Bonami hypercontractivity lemma—to yield an algorithm that is simultaneously sample- and time-efficient in all relevant parameters.

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).

Problem

• quantum Hamiltonians
• certification
• learning
• Gibbs states
• local Hamiltonians

Innovation

• local Hamiltonian certification
• Heisenberg scaling
• Gibbs state learning
• sample efficiency
• Bonami hypercontractivity