# Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix

arXiv: https://arxiv.org/abs/2212.04471 · HTML: https://mlsanity.com/paper/2212.04471/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 81 references that are in the map.
- First posted on arXiv 8 Dec 2022.

## Builds on (most cited first, 27 of 27)

- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Encoding-dependent generalization bounds for parametrized quantum circuits](https://mlsanity.com/paper/2106.03880/) (2021)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Exponential separations between learning with and without quantum memory](https://mlsanity.com/paper/2111.05881/) (2021)
- [Out-of-distribution generalization for learning quantum dynamics](https://mlsanity.com/paper/2204.10268/) (2022)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [Pseudo-dimension of quantum circuits](https://mlsanity.com/paper/2002.01490/) (2020)
- [On the statistical complexity of quantum circuits](https://mlsanity.com/paper/2101.06154/) (2021)
- [Structural risk minimization for quantum linear classifiers](https://mlsanity.com/paper/2105.05566/) (2021)
- [Learning many-body Hamiltonians with Heisenberg-limited scaling](https://mlsanity.com/paper/2210.03030/) (2022)
- [Sample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem](https://mlsanity.com/paper/1810.10938/) (2018)
- [A Hierarchy for Replica Quantum Advantage](https://mlsanity.com/paper/2111.05874/) (2021)
- [Practical Black Box Hamiltonian Learning](https://mlsanity.com/paper/2206.15464/) (2022)
- [Sample Complexity of Learning Parametric Quantum Circuits](https://mlsanity.com/paper/2107.09078/) (2021)
- [The Complexity of NISQ](https://mlsanity.com/paper/2210.07234/) (2022)
- [Foundations for learning from noisy quantum experiments](https://mlsanity.com/paper/2204.13691/) (2022)
- [Adaptive Online Learning of Quantum States](https://mlsanity.com/paper/2206.00220/) (2022)
- [Effects of quantum resources on the statistical complexity of quantum circuits](https://mlsanity.com/paper/2102.03282/) (2021)
- [Scalably learning quantum many-body Hamiltonians from dynamical data](https://mlsanity.com/paper/2209.14328/) (2022)
- [Binary Classification with Classical Instances and Quantum Labels](https://mlsanity.com/paper/2006.06005/) (2020)

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

- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning quantum states and unitaries of bounded gate complexity](https://mlsanity.com/paper/2310.19882/) (2023)
- [The power and limitations of learning quantum dynamics incoherently](https://mlsanity.com/paper/2303.12834/) (2023)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [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)
- [Tight bounds on Pauli channel learning without entanglement](https://mlsanity.com/paper/2309.13461/) (2023)
- [Classical Verification of Quantum Learning](https://mlsanity.com/paper/2306.04843/) (2023)
- [Hamiltonian Property Testing](https://mlsanity.com/paper/2403.02968/) (2024)
- [Transition Role of Entangled Data in Quantum Machine Learning](https://mlsanity.com/paper/2306.03481/) (2023)
- [Learning unitaries with quantum statistical queries](https://mlsanity.com/paper/2310.02254/) (2023)
- [Learning the structure of any Hamiltonian from minimal assumptions](https://mlsanity.com/paper/2410.21635/) (2024)
- [Learning Quantum Processes with Quantum Statistical Queries](https://mlsanity.com/paper/2310.02075/) (2023)
- [Ansatz-free Hamiltonian learning with Heisenberg-limited scaling](https://mlsanity.com/paper/2502.11900/) (2025)
- [An optimal tradeoff between entanglement and copy complexity for state tomography](https://mlsanity.com/paper/2402.16353/) (2024)
- [Simple algorithms to test and learn local Hamiltonians](https://mlsanity.com/paper/2404.06282/) (2024)
- [Testing classical properties from quantum data](https://mlsanity.com/paper/2411.12730/) (2024)
- [A PAC-Bayesian approach to generalization for quantum models](https://mlsanity.com/paper/2603.22964/) (2026)
- [Optimal certification of constant-local Hamiltonians](https://mlsanity.com/paper/2512.09778/) (2025)
- [Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach](https://mlsanity.com/paper/2602.18377/) (2026)
- [Efficient information recovery from Pauli noise via classical shadow](https://mlsanity.com/paper/2305.04148/) (2023)
- [Adaptivity can help exponentially for shadow tomography](https://mlsanity.com/paper/2412.19022/) (2024)
- [Hamiltonian locality testing and certification do not achieve the Heisenberg limit](https://mlsanity.com/paper/2610.03205/) (2026)

## Abstract (arXiv)

Learning about physical systems from quantum-enhanced experiments, relying on a quantum memory and quantum processing, can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for a variety of state learning tasks, quantum process learning allows for comparable advantages only with a careful problem formulation and is less understood. We establish an exponential quantum advantage for learning an unknown n-qubit quantum process N. We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix of an arbitrary N, (b) predicting expectation values of bounded Pauli-sparse observables measured on the output of an arbitrary N upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary bounded observables measured on the output of an unknown N with sparse Pauli transfer matrix upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearly-in-n many copies of the Choi state of N, and even time-efficiently in the case of (b). In contrast, any learner without quantum memory requires exponentially-in-n many queries, even when querying N on subsystems of adaptively chosen states and performing adaptively chosen measurements. In proving this separation, we extend existing shadow tomography upper and lower bounds from states to channels via the Choi-Jamiolkowski isomorphism. Moreover, we combine Pauli transfer matrix learning with polynomial interpolation techniques to develop a procedure for learning arbitrary Hamiltonians, which may have non-local all-to-all interactions, from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.