# Optimal certification of constant-local Hamiltonians

arXiv: https://arxiv.org/abs/2512.09778 · HTML: https://mlsanity.com/paper/2512.09778/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 50 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Builds on (most cited first, 15 of 15)

- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [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)
- [Structure learning of Hamiltonians from real-time evolution](https://mlsanity.com/paper/2405.00082/) (2024)
- [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)
- [Nearly optimal algorithms to learn sparse quantum Hamiltonians in physically motivated distances](https://mlsanity.com/paper/2509.09813/) (2025)
- [Testing and learning structured quantum Hamiltonians](https://mlsanity.com/paper/2411.00082/) (2024)
- [Ansatz-free Hamiltonian learning with Heisenberg-limited scaling](https://mlsanity.com/paper/2502.11900/) (2025)
- [Hamiltonian Locality Testing via Trotterized Postselection](https://mlsanity.com/paper/2505.06478/) (2025)
- [Simple algorithms to test and learn local Hamiltonians](https://mlsanity.com/paper/2404.06282/) (2024)
- [Efficient Hamiltonian, structure and trace distance learning of Gaussian states](https://mlsanity.com/paper/2411.03163/) (2024)
- [Certifying and learning quantum Ising Hamiltonians](https://mlsanity.com/paper/2509.10239/) (2025)

## Built on by (most cited first, 1 of 1)

- [Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography](https://mlsanity.com/paper/2609.04165/) (2026)

## Abstract (arXiv)

We study the problem of certifying local Hamiltonians from real-time access to their dynamics. Given oracle access to e-itH for an unknown k-local Hamiltonian H and a fully specified target Hamiltonian H0, the goal is to decide whether H is exactly equal to H0 or differs from H0 by at least varepsilon in normalized Frobenius norm, while minimizing the total evolution time. We introduce the first intolerant Hamiltonian certification protocol that achieves optimal performance for all constant-locality Hamiltonians. For general n-qubit, k-local, traceless Hamiltonians, our procedure uses O(ck/varepsilon) total evolution time for a universal constant c, and succeeds with high probability. In particular, for O(1)-local Hamiltonians, the total evolution time becomes Theta(1/varepsilon), matching the known Omega(1/varepsilon) lower bounds and achieving the gold-standard Heisenberg-limit scaling. Prior certification methods either relied on implementing inverse evolution of H, required controlled access to e-itH, or achieved near-optimal guarantees only in restricted settings such as the Ising case (k=2). In contrast, our algorithm requires neither inverse evolution nor controlled operations: it uses only forward real-time dynamics and achieves optimal intolerant certification for all constant-locality Hamiltonians.