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  • Open Access

Heisenberg-Limited Hamiltonian Learning Without Short-Time Control

Myeongjin Shin1,*, Junseo Lee2,†,§, and Changhun Oh3,‡

  • *Contact author: hanwoolmj@kaist.ac.kr
  • †Contact author: junseolee@fas.harvard.edu
  • ‡Contact author: changhun0218@gmail.com
  • §Present address: Harvard University.
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PRX Quantum 7, 043011 – Published 16 October, 2026

DOI: https://doi.org/10.1103/kjvf-jgqj

Abstract

Characterizing quantum systems by learning their underlying Hamiltonians is a central task in quantum information science. While recent algorithmic advances have achieved near-optimal efficiency in this task, they critically rely on accessing arbitrarily short-time dynamics. This reliance poses severe experimental challenges due to finite control bandwidth and transient pulse errors. In this work, we demonstrate that Heisenberg-limited Hamiltonian learning can be achieved without short-time control. We introduce a framework in which every query to the unknown dynamics has duration at least a prescribed minimum time T, and show that this restriction does not preclude Heisenberg-limited scaling. The key ingredient is a method for emulating the continuous quantum control required by iterative learning algorithms using only such lower-bounded evolution times. This reduces the learning task to sparse pure-state tomography. Notably, for logarithmically sparse Hamiltonians, our algorithm achieves the information-theoretically optimal 1/ε scaling in total evolution time for any arbitrary constant minimum evolution time T. For many-body (polynomially sparse) systems, we uncover a rigorous quantitative tradeoff, showing that the minimum required evolution time can be significantly relaxed from the standard limit at a polynomial cost in total evolution time. Our results affirmatively resolve a prominent open problem in the field and reveal that high-bandwidth, ultra-short pulses are not fundamentally necessary for optimal quantum learning.

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Popular Summary

Accurate knowledge of a quantum system’s interactions is essential for predicting, controlling its dynamics and benchmarking quantum devices. Existing methods for learning these interactions with optimal precision often rely on very short evolutions, which can be difficult to implement when experimental controls require finite switching times. We develop a Hamiltonian learning algorithm that replaces short evolutions with longer ones. For systems with sufficiently few interaction terms, the algorithm retains Heisenberg-limited scaling, in which the estimation error decreases essentially inversely with the total evolution time, despite a fixed lower bound on the duration of every evolution. For systems with more interaction terms, we quantify the trade-off between this minimum duration and the resources required for learning. Our results show that ultrashort evolution is not a prerequisite for Heisenberg-limited Hamiltonian learning and identify the control resources needed to achieve this precision under experimentally motivated timing constraints.

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Outline

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PRX Quantum 7, 043011– Published 16 October, 2026
  • Received 30 April 2026
  • Revised 23 July 2026
  • Accepted 17 September 2026
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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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