# Simple algorithms to test and learn local Hamiltonians

arXiv: https://arxiv.org/abs/2404.06282 · HTML: https://mlsanity.com/paper/2404.06282/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 23 references that are in the map.
- First posted on arXiv 9 Apr 2024.

## Builds on (most cited first, 7 of 7)

- [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)
- [Practical Black Box Hamiltonian Learning](https://mlsanity.com/paper/2206.15464/) (2022)
- [Learning low-degree functions from a logarithmic number of random queries](https://mlsanity.com/paper/2109.10162/) (2021)
- [Scalably learning quantum many-body Hamiltonians from dynamical data](https://mlsanity.com/paper/2209.14328/) (2022)

## Built on by (most cited first, 2 of 2)

- [Learning the structure of any Hamiltonian from minimal assumptions](https://mlsanity.com/paper/2410.21635/) (2024)
- [Optimal certification of constant-local Hamiltonians](https://mlsanity.com/paper/2512.09778/) (2025)

## Abstract (arXiv)

We consider the problems of testing and learning an n-qubit k-local Hamiltonian from queries to its evolution operator with respect the 2-norm of the Pauli spectrum, or equivalently, the normalized Frobenius norm. For testing whether a Hamiltonian is ε1-close to k-local or ε2-far from k-local, we show that O(1/(ε2-ε1)8) queries suffice. This solves two questions posed in a recent work by Bluhm, Caro and Oufkir. For learning up to error ε, we show that exp(O(k2+klog(1/ε))) queries suffice. Our proofs are simple, concise and based on Pauli-analytic techniques.