Nearly optimal algorithms to learn sparse quantum Hamiltonians in physically motivated distances

2025-09-11

AI Summary

This work addresses the problem of learning sparse quantum Hamiltonians $H$ in the Pauli basis, where existing methods suffer from two key limitations: the absence of physically interpretable error metrics and matching information-theoretic lower bounds. To resolve these, we propose a novel evaluation framework grounded in physical distinguishability, introducing time-constrained distance and temperature-constrained distance as physically meaningful alternatives to conventional mathematical error measures. Methodologically, inspired by the Valiant–Vazirani theorem, we design a Pauli coefficient isolation technique enabling independent querying and recovery of individual coefficients within the unknown support set. Our approach integrates time-evolution sampling, compressed sensing, randomized hashing measurements, and iterative support estimation. Theoretically, our algorithm achieves near-optimal learning under time-constrained distance using only $O(s log(1/varepsilon))$ experiments and $O(s^2/varepsilon)$ total evolution time—substantially improving upon prior results.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the problem of learning Hamiltonians $H$ that are $s$-sparse in the Pauli basis, given access to their time evolution. Although Hamiltonian learning has been extensively investigated, two issues recur in much of the existing literature: the absence of matching lower bounds and the use of mathematically convenient but physically opaque error measures. We address both challenges by introducing two physically motivated distances between Hamiltonians and designing a nearly optimal algorithm with respect to one of these metrics. The first, time-constrained distance, quantifies distinguishability through dynamical evolution up to a bounded time. The second, temperature-constrained distance, captures distinguishability through thermal states at bounded inverse temperatures. We show that $s$-sparse Hamiltonians with bounded operator norm can be learned in both distances with $O(s log(1/ε))$ experiments and $O(s^2/ε)$ evolution time. For the time-constrained distance, we further establish lower bounds of $Ω((s/n)log(1/ε) + s)$ experiments and $Ω(sqrt{s}/ε)$ evolution time, demonstrating near-optimality in the number of experiments. As an intermediate result, we obtain an algorithm that learns every Pauli coefficient of $s$-sparse Hamiltonians up to error $ε$ in $O(slog(1/ε))$ experiments and $O(s/ε)$ evolution time, improving upon several recent results. The source of this improvement is a new isolation technique, inspired by the Valiant-Vazirani theorem (STOC'85), which shows that NP is as easy as detecting unique solutions. This isolation technique allows us to query the time evolution of a single Pauli coefficient of a sparse Hamiltonian--even when the Pauli support of the Hamiltonian is unknown--ultimately enabling us to recover the Pauli support itself.

Problem

• Learning sparse quantum Hamiltonians with optimal efficiency
• Addressing physically meaningful error metrics in Hamiltonian learning
• Establishing nearly matching upper and lower complexity bounds

Innovation

• Introduces time-constrained and temperature-constrained distance metrics
• Uses Valiant-Vazirani inspired isolation technique for coefficient querying
• Achieves O(s log(1/ε)) experiments with O(s²/ε) evolution time