# Ansatz-free Hamiltonian learning with Heisenberg-limited scaling

arXiv: https://arxiv.org/abs/2502.11900 · HTML: https://mlsanity.com/paper/2502.11900/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 71 references that are in the map.
- First posted on arXiv 17 Feb 2025.

## Builds on (most cited first, 12 of 12)

- [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 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)
- [The Complexity of NISQ](https://mlsanity.com/paper/2210.07234/) (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)
- [Efficient Pauli channel estimation with logarithmic quantum memory](https://mlsanity.com/paper/2309.14326/) (2023)
- [Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows](https://mlsanity.com/paper/2402.17911/) (2024)
- [Learning the structure of any Hamiltonian from minimal assumptions](https://mlsanity.com/paper/2410.21635/) (2024)

## Built on by (most cited first, 4 of 4)

- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [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)

Learning the unknown interactions that govern a quantum system is crucial for quantum information processing, device benchmarking, and quantum sensing. The problem, known as Hamiltonian learning, is well understood under the assumption that interactions are local, but this assumption may not hold for arbitrary Hamiltonians. Previous methods all require high-order inverse polynomial dependency with precision, unable to surpass the standard quantum limit and reach the gold standard Heisenberg-limited scaling. Whether Heisenberg-limited Hamiltonian learning is possible without prior assumptions about the interaction structures, a challenge we term ansatz-free Hamiltonian learning, remains an open question. In this work, we present a quantum algorithm to learn arbitrary sparse Hamiltonians without any structure constraints using only black-box queries of the system's real-time evolution and minimal digital controls to attain Heisenberg-limited scaling in estimation error. Our method is also resilient to state-preparation-and-measurement errors, enhancing its practical feasibility. We numerically demonstrate our ansatz-free protocol for learning physical Hamiltonians and validating analog quantum simulations, benchmarking our performance against the state-of-the-art Heisenberg-limited learning approach. Moreover, we establish a fundamental trade-off between total evolution time and quantum control on learning arbitrary interactions, revealing the intrinsic interplay between controllability and total evolution time complexity for any learning algorithm. These results pave the way for further exploration into Heisenberg-limited Hamiltonian learning in complex quantum systems under minimal assumptions, potentially enabling new benchmarking and verification protocols.