# Learning many-body Hamiltonians with Heisenberg-limited scaling

arXiv: https://arxiv.org/abs/2210.03030 · HTML: https://mlsanity.com/paper/2210.03030/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 90 references that are in the map.
- First posted on arXiv 6 Oct 2022.

## 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)
- [Exponential separations between learning with and without quantum memory](https://mlsanity.com/paper/2111.05881/) (2021)
- [Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm](https://mlsanity.com/paper/1910.02071/) (2019)
- [Practical Black Box Hamiltonian Learning](https://mlsanity.com/paper/2206.15464/) (2022)
- [Foundations for learning from noisy quantum experiments](https://mlsanity.com/paper/2204.13691/) (2022)
- [Time-dependent Hamiltonian simulation with L1-norm scaling](https://mlsanity.com/paper/1906.07115/) (2019)
- [Nearly tight Trotterization of interacting electrons](https://mlsanity.com/paper/2012.09194/) (2020)
- [Improved approximation algorithms for bounded-degree local Hamiltonians](https://mlsanity.com/paper/2105.01193/) (2021)
- [Provably accurate simulation of gauge theories and bosonic systems](https://mlsanity.com/paper/2110.06942/) (2021)

## Built on by (most cited first, 23 of 23)

- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Exponential separations between classical and quantum learners](https://mlsanity.com/paper/2306.16028/) (2023)
- [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)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Generative quantum advantage for classical and quantum problems](https://mlsanity.com/paper/2509.09033/) (2025)
- [ShadowNet for Data-Centric Quantum System Learning](https://mlsanity.com/paper/2308.11290/) (2023)
- [Learning k-body Hamiltonians via compressed sensing](https://mlsanity.com/paper/2410.18928/) (2024)
- [Solving The Quantum Many-Body Hamiltonian Learning Problem with Neural Differential Equations](https://mlsanity.com/paper/2408.08639/) (2024)
- [On the Role of Entanglement and Statistics in Learning](https://mlsanity.com/paper/2306.03161/) (2023)
- [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)
- [Sample-efficient Model-based Reinforcement Learning for Quantum Control](https://mlsanity.com/paper/2304.09718/) (2023)
- [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)
- [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)
- [Quantum Scrambling Born Machine](https://mlsanity.com/paper/2602.17281/) (2026)
- [Four Generations of Quantum Biomedical Sensors](https://mlsanity.com/paper/2603.29944/) (2026)
- [Generative Learning for Quantum Measurement Design](https://mlsanity.com/paper/2608.11396/) (2026)
- [Hamiltonian locality testing and certification do not achieve the Heisenberg limit](https://mlsanity.com/paper/2610.03205/) (2026)

## Abstract (arXiv)

Learning a many-body Hamiltonian from its dynamics is a fundamental problem in physics. In this work, we propose the first algorithm to achieve the Heisenberg limit for learning an interacting N-qubit local Hamiltonian. After a total evolution time of O(ε-1), the proposed algorithm can efficiently estimate any parameter in the N-qubit Hamiltonian to ε-error with high probability. The proposed algorithm is robust against state preparation and measurement error, does not require eigenstates or thermal states, and only uses polylog(ε-1) experiments. In contrast, the best previous algorithms, such as recent works using gradient-based optimization or polynomial interpolation, require a total evolution time of O(ε-2) and O(ε-2) experiments. Our algorithm uses ideas from quantum simulation to decouple the unknown N-qubit Hamiltonian H into noninteracting patches, and learns H using a quantum-enhanced divide-and-conquer approach. We prove a matching lower bound to establish the asymptotic optimality of our algorithm.