Operator Score Matching for Learning Quantum Hamiltonians
Abstract
Learning quantum Hamiltonians from low-temperature thermal state measurements is a fundamental problem in quantum physics. Scalability of existing methods is limited by the complexity of semidefinite optimization problems or partition function computation. Here, we develop a quantum analog of classical score matching method that exploits generalized notions of derivatives and integration by parts in operator algebras. We introduce a new \emph{Operator Score Matching} loss function that recovers Hamiltonian parameters by a simple gradient descent without the need to compute intractable normalization constants. Numerical experiments on the Gibbs states of the transverse-field Ising model, XXZ spin chain, Fermi-Hubbard model, and lattice ϕ4 theory demonstrate efficient parameter recovery using finite-depth truncations of nested commutator expansions. Our results establish Operator Score Matching as a practical, partition-function-free framework for quantum Hamiltonian learning.
Related
- Quantum score matching with applications to learning thermal states
- Scalably learning one-dimensional quantum many-body Hamiltonians from dynamical data
- Linear-Scaling Quantum Transport from Machine-Learning Density Functional Theory Hamiltonians
- Dissipation-Enabled Operator-Norm Locality Bounds for Bose-Hubbard Hamiltonians
- Operator Learning with Variational Quantum Circuits
BibTeX
@misc{shukla2026operator,
title = {{Operator Score Matching for Learning Quantum Hamiltonians}},
author = {Shreya Shukla and Abhijith Jayakumar and Andrey Y. Lokhov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25246},
eprint = {2609.25246},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25246},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Operator Score Matching for Learning Quantum Hamiltonians
AU - Shreya Shukla
AU - Abhijith Jayakumar
AU - Andrey Y. Lokhov
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25246
UR - https://arxiv.org/abs/2609.25246
AB - Learning quantum Hamiltonians from low-temperature thermal state measurements is a fundamental problem in quantum physics. Scalability of existing methods is limited by the complexity of semidefinite optimization problems or partition function computation. Here, we develop a quantum analog of classical score matching method that exploits generalized notions of derivatives and integration by parts in operator algebras. We introduce a new \emph{Operator Score Matching} loss function that recovers Hamiltonian parameters by a simple gradient descent without the need to compute intractable normalization constants. Numerical experiments on the Gibbs states of the transverse-field Ising model, XXZ spin chain, Fermi-Hubbard model, and lattice ϕ^4 theory demonstrate efficient parameter recovery using finite-depth truncations of nested commutator expansions. Our results establish Operator Score Matching as a practical, partition-function-free framework for quantum Hamiltonian learning.
ER -