# Structure learning of Hamiltonians from real-time evolution

arXiv: https://arxiv.org/abs/2405.00082 · HTML: https://mlsanity.com/paper/2405.00082/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 38 references that are in the map.
- First posted on arXiv 30 Apr 2024.

## Builds on (most cited first, 16 of 16)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Gradient Descent Learns Linear Dynamical Systems](https://mlsanity.com/paper/1609.05191/) (2016)
- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [Efficiently learning Ising models on arbitrary graphs](https://mlsanity.com/paper/1411.6156/) (2014)
- [Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models](https://mlsanity.com/paper/1605.07252/) (2016)
- [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)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Learning quantum Hamiltonians at any temperature in polynomial time](https://mlsanity.com/paper/2310.02243/) (2023)
- [Information Theoretic Properties of Markov Random Fields, and their Algorithmic Applications](https://mlsanity.com/paper/1705.11107/) (2017)
- [Learning models of quantum systems from experiments](https://mlsanity.com/paper/2002.06169/) (2020)
- [A Unified Approach to Learning Ising Models: Beyond Independence and Bounded Width](https://mlsanity.com/paper/2311.09197/) (2023)
- [Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2307.06538/) (2023)
- [Optimal learning of quantum Hamiltonians from high-temperature Gibbs states](https://mlsanity.com/paper/2108.04842/) (2021)
- [High-Temperature Gibbs States are Unentangled and Efficiently Preparable](https://mlsanity.com/paper/2403.16850/) (2024)
- [Quantum Model Learning Agent: characterisation of quantum systems through machine learning](https://mlsanity.com/paper/2112.08409/) (2021)

## Built on by (most cited first, 9 of 9)

- [Bypassing the Noisy Parity Barrier: Learning Higher-Order Markov Random Fields from Dynamics](https://mlsanity.com/paper/2409.05284/) (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)
- [Ansatz-free Hamiltonian learning with Heisenberg-limited scaling](https://mlsanity.com/paper/2502.11900/) (2025)
- [Efficient Hamiltonian, structure and trace distance learning of Gaussian states](https://mlsanity.com/paper/2411.03163/) (2024)
- [Score-matching-based Structure Learning for Temporal Data on Networks](https://mlsanity.com/paper/2412.07469/) (2024)
- [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)

We study the problem of Hamiltonian structure learning from real-time evolution: given the ability to apply e-i Ht for an unknown local Hamiltonian H = suma = 1m λa Ea on n qubits, the goal is to recover H. This problem is already well-understood under the assumption that the interaction terms, Ea, are given, and only the interaction strengths, λa, are unknown. But how efficiently can we learn a local Hamiltonian without prior knowledge of its interaction structure? We present a new, general approach to Hamiltonian learning that not only solves the challenging structure learning variant, but also resolves other open questions in the area, all while achieving the gold standard of Heisenberg-limited scaling. In particular, our algorithm recovers the Hamiltonian to varepsilon error with total evolution time O(log (n)/varepsilon), and has the following appealing properties: (1) it does not need to know the Hamiltonian terms; (2) it works beyond the short-range setting, extending to any Hamiltonian H where the sum of terms interacting with a qubit has bounded norm; (3) it evolves according to H in constant time t increments, thus achieving constant time resolution. As an application, we can also learn Hamiltonians exhibiting power-law decay up to accuracy varepsilon with total evolution time beating the standard limit of 1/varepsilon2.