# Optimal Ansatz-free Hamiltonian Learning In Situ

arXiv: https://arxiv.org/abs/2606.19486 · HTML: https://mlsanity.com/paper/2606.19486/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 55 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [Scalably learning quantum many-body Hamiltonians from dynamical data](https://mlsanity.com/paper/2209.14328/) (2022)
- [Learning k-body Hamiltonians via compressed sensing](https://mlsanity.com/paper/2410.18928/) (2024)
- [Nearly optimal algorithms to learn sparse quantum Hamiltonians in physically motivated distances](https://mlsanity.com/paper/2509.09813/) (2025)
- [Quantum Probe Tomography](https://mlsanity.com/paper/2510.08499/) (2025)
- [Hamiltonian Locality Testing via Trotterized Postselection](https://mlsanity.com/paper/2505.06478/) (2025)
- [Certifying and learning local quantum Hamiltonians](https://mlsanity.com/paper/2603.29809/) (2026)

## Built on by (most cited first, 3 of 3)

- [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)
- [Out-of-control Hamiltonian Learning](https://mlsanity.com/paper/2610.06709/) (2026)

## Abstract (arXiv)

Characterizing the features of a Hamiltonian that governs a quantum system serves as a fundamental subroutine of quantum device calibration, signal sensing, and error correction. Recent works have proposed protocols achieving the optimal Heisenberg-limited scaling learning ansatz-free Hamiltonians from their real-time evolutions without fully specifying interaction structures. However, these protocols rely on both deep circuits with interleaving probes and control, and extremely short time resolution, making them difficult to implement on near- and intermediate-term in situ quantum experiments. In this work, we propose a computationally efficient, control-free, and ancilla-free algorithm that uses only Pauli product state preparation and measurement, and learns an ansatz-free Hamiltonian H with ||H||≤Lambda in total evolution time of Theta(fracLambdaε2log(fracLambdaε)). The evolution time cost of our algorithm is optimal for any control-free protocols as we further prove a lower bound of Omega(fracLambdaε2log(fracLambdaε)). Technically, our method introduces a randomized-sampling framework that combines band-limited kernel-based time sampling with a displacement sieve for Hamiltonian structure learning. The characteristic probe time resolution depends only on Lambda instead of varepsilon, which makes our protocol especially appealing in the high-precision regime for sensing and calibration applications. We also show that the algorithm maintains the same asymptotic total evolution time in the presence of state-preparation-and-measurement (SPAM) noise when the Hamiltonian is local after calibration. Our results demonstrate the fundamental cost of experimentally friendly Hamiltonian learning and provide a practical route to rigorous in situ characterization of near-term quantum platforms.