Learning the structure of any Hamiltonian from minimal assumptions

2024-10-29

arXiv.org

AI Summary

This work addresses the problem of learning the structure of an unknown quantum many-body Hamiltonian $ H $ from black-box time evolution, without prior assumptions on locality, interaction form, or specific term types—only requiring that the number of nonzero terms is polynomially bounded. We propose a novel learning framework based on pseudo-Choi states, enabling the first efficient reconstruction of arbitrary $ n $-qubit Hamiltonians. Our method integrates quantum phase estimation, Fourier spectral analysis, and controlled time evolution—including both time-reversal and purely forward-evolution models. Under the time-reversal model, the total evolution time scales as $ ilde{O}(m/varepsilon) $, yielding an $ varepsilon $-accurate Hamiltonian description; under the purely forward model, it achieves $ ilde{O}(|H|^3/varepsilon^4) $, breaking previous exponential complexity barriers and attaining near-Heisenberg-limited scaling.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We study the problem of learning an unknown quantum many-body Hamiltonian $H$ from black-box queries to its time evolution $e^{-mathrm{i} H t}$. Prior proposals for solving this task either impose some assumptions on $H$, such as its interaction structure or locality, or otherwise use an exponential amount of computational postprocessing. In this paper, we present efficient algorithms to learn any $n$-qubit Hamiltonian, assuming only a bound on the number of Hamiltonian terms, $m leq mathrm{poly}(n)$. Our algorithms do not need to know the terms in advance, nor are they restricted to local interactions. We consider two models of control over the time evolution: the first has access to time reversal ($t<0$), enabling an algorithm that outputs an $epsilon$-accurate classical description of $H$ after querying its dynamics for a total of $widetilde{O}(m/epsilon)$ evolution time. The second access model is more conventional, allowing only forward-time evolutions; our algorithm requires $widetilde{O}(|H|^3/epsilon^4)$ evolution time in this setting. Central to our results is the recently introduced concept of a pseudo-Choi state of $H$. We extend the utility of this learning resource by showing how to use it to learn the Fourier spectrum of $H$, how to achieve nearly Heisenberg-limited scaling with it, and how to prepare it even under our more restricted access models.

Problem

• Learning unknown quantum Hamiltonian from black-box queries
• Efficient algorithms without prior Hamiltonian term knowledge
• Extending pseudo-Choi state utility for Hamiltonian learning

Innovation

• Learns any n-qubit Hamiltonian efficiently
• Uses pseudo-Choi state for learning
• Achieves Heisenberg-limited scaling