# Learning the structure of any Hamiltonian from minimal assumptions

arXiv: https://arxiv.org/abs/2410.21635 · HTML: https://mlsanity.com/paper/2410.21635/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 92 references that are in the map.
- First posted on arXiv 29 Oct 2024.

## Builds on (most cited first, 16 of 16)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Exponential separations between learning with and without quantum memory](https://mlsanity.com/paper/2111.05881/) (2021)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [Learning many-body Hamiltonians with Heisenberg-limited scaling](https://mlsanity.com/paper/2210.03030/) (2022)
- [Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix](https://mlsanity.com/paper/2212.04471/) (2022)
- [Learning quantum Hamiltonians at any temperature in polynomial time](https://mlsanity.com/paper/2310.02243/) (2023)
- [Practical Black Box Hamiltonian Learning](https://mlsanity.com/paper/2206.15464/) (2022)
- [Learning quantum states and unitaries of bounded gate complexity](https://mlsanity.com/paper/2310.19882/) (2023)
- [Structure learning of Hamiltonians from real-time evolution](https://mlsanity.com/paper/2405.00082/) (2024)
- [Optimal learning of quantum Hamiltonians from high-temperature Gibbs states](https://mlsanity.com/paper/2108.04842/) (2021)
- [Query-optimal estimation of unitary channels in diamond distance](https://mlsanity.com/paper/2302.14066/) (2023)
- [Learning k-body Hamiltonians via compressed sensing](https://mlsanity.com/paper/2410.18928/) (2024)
- [Hamiltonian Property Testing](https://mlsanity.com/paper/2403.02968/) (2024)
- [Learning unitaries with quantum statistical queries](https://mlsanity.com/paper/2310.02254/) (2023)
- [Simple algorithms to test and learn local Hamiltonians](https://mlsanity.com/paper/2404.06282/) (2024)

## Built on by (most cited first, 5 of 5)

- [Ansatz-free Hamiltonian learning with Heisenberg-limited scaling](https://mlsanity.com/paper/2502.11900/) (2025)
- [TCS-BENCH: Benchmarking State-of-the-Art Generative AI Theoretical Computer Science Research Ability](https://mlsanity.com/paper/2608.09538/) (2026)
- [Optimal certification of constant-local Hamiltonians](https://mlsanity.com/paper/2512.09778/) (2025)
- [Characterizing Arbitrary Lindbladian Dynamics with a Few Pauli Measurements](https://mlsanity.com/paper/2607.23044/) (2026)
- [Hamiltonian locality testing and certification do not achieve the Heisenberg limit](https://mlsanity.com/paper/2610.03205/) (2026)

## Abstract (arXiv)

We study the problem of learning an unknown quantum many-body Hamiltonian H from black-box queries to its time evolution e-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 algorithms to learn any n-qubit Hamiltonian, which do not need to know the Hamiltonian terms in advance, nor are they restricted to local interactions. Our algorithms are efficient as long as the number of terms m is polynomially bounded in the system size n. 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 ε-accurate classical description of H after querying its dynamics for a total of widetildeO(m/ε) evolution time. The second access model is more conventional, allowing only forward-time evolutions;~our algorithm requires widetildeO(|H|3/ε4) 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.