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  • Featured in Physics
  • Open Access

Ansatz-Free Hamiltonian Learning with Heisenberg-Limited Scaling

Hong-Ye Hu1,*,†, Muzhou Ma2,*,‡, Weiyuan Gong3, Qi Ye4,3, Yu Tong5,6,7, Steven T. Flammia8,9, and Susanne F. Yelin1,§

  • *Alphabetical ordering and equal contributions.
  • †Contact author: hongyehu@g.harvard.edu
  • ‡Contact author: muzhouma2002@gmail.com
  • §Contact author: syelin@g.harvard.edu
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PRX Quantum 6, 040315 – Published 22 October, 2025

DOI: https://doi.org/10.1103/j7b8-pb77

Abstract

Learning the unknown interactions that govern a quantum system is crucial for quantum information processing, device benchmarking, and quantum sensing. The problem, known as Hamiltonian learning, is well understood under the assumption that interactions are local, but this assumption may not hold for arbitrary Hamiltonians. Previous methods all require high-order inverse polynomial dependency with precision, unable to surpass the standard quantum limit and reach the gold-standard Heisenberg-limited scaling. Whether Heisenberg-limited Hamiltonian learning is possible without prior assumptions about the interaction structures, a challenge we term ansatz-free Hamiltonian learning, remains an open question. In this work, we present a quantum algorithm to learn arbitrary sparse Hamiltonians without any structure constraints using only black-box queries of the system’s real-time evolution and minimal digital controls to attain Heisenberg-limited scaling in estimation error. Our method is also resilient to state-preparation-and-measurement errors, enhancing its practical feasibility. We numerically demonstrate our ansatz-free protocol for learning physical Hamiltonians and validating analog quantum simulations, benchmarking our performance against the state-of-the-art Heisenberg-limited learning approach. Moreover, we establish a fundamental trade-off between total evolution time and quantum control on learning arbitrary interactions, revealing the intrinsic interplay between controllability and total evolution-time complexity for any learning algorithm. These results pave the way for further exploration into Heisenberg-limited Hamiltonian learning in complex quantum systems under minimal assumptions, potentially enabling new benchmarking and verification protocols.

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Physics Subject Headings (PhySH)

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Quantum Systems Modeled Without Prior Assumptions

Published 22 October, 2025

An improved algorithm for learning the static and dynamic properties of a quantum system could have applications in quantum computing, simulation, and sensing.

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Popular Summary

Characterizing the interactions within a quantum system is a key task in quantum information science, with applications in device benchmarking, quantum simulation verification, and quantum sensing. This process is also known as Hamiltonian learning. Achieving high precision with minimal experimental efforts is the ultimate goal. Fundamentally, the optimal efficiency is set by the Heisenberg limit of quantum mechanics. However, most existing approaches can only reach the optimal Heisenberg limit by making strong assumptions about the system’s internal structure, such as assuming that interactions are local.

In this work, we develop a new quantum algorithm, called ansatz-free Hamiltonian learning, that achieves Heisenberg-limited scaling without relying on any prior knowledge or specific assumptions about the interaction structure of the system. Our approach treats the quantum system as a black box, requiring only access to its time evolution and the ability to apply simple operations using a quantum computer. This makes the method broadly applicable, especially in experimental settings where the interaction structure is unknown. We also identify a fundamental trade-off between the amount of time required for experiments and the complexity of controls needed for learning unknown Hamiltonians.

Our findings establish a new theoretical foundation for learning in quantum systems under minimal assumptions. They also offer practical tools for characterizing and validating quantum devices. Future work may explore how these ideas can be extended to adaptive strategies and ansatz-free learning of quantum channels, paving the way for precise and scalable quantum diagnostics.

Article Text

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PRX Quantum 6, 040315– Published 22 October, 2025
  • Received 31 March 2025
  • Accepted 1 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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