# Hamiltonian Property Testing

arXiv: https://arxiv.org/abs/2403.02968 · HTML: https://mlsanity.com/paper/2403.02968/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 56 references that are in the map.
- First posted on arXiv 5 Mar 2024.

## Builds on (most cited first, 10 of 10)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [Out-of-distribution generalization for learning quantum dynamics](https://mlsanity.com/paper/2204.10268/) (2022)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [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)
- [Learning shallow quantum circuits](https://mlsanity.com/paper/2401.10095/) (2024)
- [Practical Black Box Hamiltonian Learning](https://mlsanity.com/paper/2206.15464/) (2022)
- [Improved Stabilizer Estimation via Bell Difference Sampling](https://mlsanity.com/paper/2304.13915/) (2023)
- [Lower Bounds for Learning Quantum States with Single-Copy Measurements](https://mlsanity.com/paper/2207.14438/) (2022)

## Built on by (most cited first, 6 of 6)

- [Learning k-body Hamiltonians via compressed sensing](https://mlsanity.com/paper/2410.18928/) (2024)
- [Learning the structure of any Hamiltonian from minimal assumptions](https://mlsanity.com/paper/2410.21635/) (2024)
- [Testing classical properties from quantum data](https://mlsanity.com/paper/2411.12730/) (2024)
- [Instance-Optimal Quantum State Certification with Entangled Measurements](https://mlsanity.com/paper/2507.06010/) (2025)
- [Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography](https://mlsanity.com/paper/2609.04165/) (2026)
- [Hamiltonian locality testing and certification do not achieve the Heisenberg limit](https://mlsanity.com/paper/2610.03205/) (2026)

## Abstract (arXiv)

Locality is a fundamental feature of many physical time evolutions. Assumptions on locality and related structural properties also underlie recently proposed procedures for learning an unknown Hamiltonian from access to the induced time evolution. However, no protocols to rigorously test whether an unknown Hamiltonian is local were known. We investigate Hamiltonian locality testing as a property testing problem, where the task is to determine whether an unknown n-qubit Hamiltonian H is k-local or varepsilon-far from all k-local Hamiltonians, given access to the time evolution along H. First, we emphasize the importance of the chosen distance measure: With respect to the operator norm, a worst-case distance measure, incoherent quantum locality testers require tildeOmega(2n) many time evolution queries and an expected total evolution time of tildeOmega(2n / varepsilon), and even coherent testers need Omega(2n/2) many queries and Omega(2n/2/varepsilon) total evolution time. In contrast, when distances are measured according to the normalized Frobenius norm, corresponding to an average-case distance, we give a sample-, time-, and computationally efficient incoherent Hamiltonian locality testing algorithm based on randomized measurements. In fact, our procedure can be used to simultaneously test a wide class of Hamiltonian properties beyond locality. Finally, we prove that learning a general Hamiltonian remains exponentially hard with this average-case distance, thereby establishing an exponential separation between Hamiltonian testing and learning. Our work initiates the study of property testing for quantum Hamiltonians, demonstrating that a broad class of Hamiltonian properties is efficiently testable even with limited quantum capabilities, and positioning Hamiltonian testing as an independent area of research alongside Hamiltonian learning.