Certifying and learning quantum Ising Hamiltonians

2025-09-12

AI Summary

This work systematically investigates the certification and learning of quantum Ising Hamiltonians and their Gibbs states. For four core tasks—Hamiltonian certification, Gibbs state learning, Gibbs state certification, and generalization to arbitrary constant-$k$-local Hamiltonians—we propose a unified framework based on time-evolution access and Fourier analysis, particularly leveraging the Bonami–Beckner inequality. Our method achieves near-optimal Hamiltonian property testing in Frobenius norm, with certification complexity $ ilde{O}(1/varepsilon)$ queries—reaching the Heisenberg limit for the first time. We design sample-optimal algorithms for both Gibbs state learning and certification, resolving a long-standing open problem. Furthermore, we extend these results to general constant-$k$-local Hamiltonians. Collectively, our approach breaks the exponential sample-complexity barrier inherent in conventional methods, significantly improving the efficiency and accuracy of verification and reconstruction for quantum many-body systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

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

Problem

• Certifying quantum Ising Hamiltonians efficiently in Frobenius norm
• Learning Ising Gibbs states with optimal sample complexity
• Extending certification methods to general k-local Hamiltonians

Innovation

• Certification via time-evolution operator access
• Sample-efficient Gibbs state learning algorithm
• Efficient trace norm certification method