Learning Trotter Orderings for Heisenberg Hamiltonians with a Ranking Transformer
Abstract
Trotterization approximates quantum time evolution by sequentially applying Hamiltonian terms. Because noncommuting terms introduce ordering-dependent errors, selecting an optimal term ordering is a combinatorial problem over a factorial search space. Prior approaches rely on fixed heuristics or on selection among predefined structured orderings, both of which require simulating candidates before choosing among them. For 1D and 2D Heisenberg-style Hamiltonians, we instead learn the ordering directly with a physics-aware ranking transformer that assigns a scalar score to each Hamiltonian term and predicts an ordering by sorting these scores. Physical structure enters through commutator-biased attention and an anticommutation-weighted ranking loss, and the models are trained on simulated-annealing (SA) reference orderings. We train separate models for first- and second-order Trotterization, each jointly on chains up to 14 qubits and lattices up to 12 qubits, and evaluate them on unseen chains with 16-20 qubits and lattices with 16 and 20 qubits. Predicted orderings are evaluated by the median gap in simulation fidelity to the SA reference. At first order, the model reaches a pooled gap below 10^-4 on chains, 0.0144 and 0.0088 on triangular lattices, and 0.0948 and 0.0823 on rectangular lattices at 16 and 20 qubits; at second order, the gaps are 0.0114, 0.0364 and 0.0244, and 0.1348 and 0.1239. The prediction exceeds the SA reference on 34% of first-order and 8% of second-order chain instances and can match its fidelity even when the two sequences differ because commuting terms may be rearranged without changing the Trotter unitary. The learned models generalize to larger systems across all three geometries, performing best on chains and triangular lattices, and produce an ordering in one forward pass without candidate enumeration, simulated annealing, or fidelity evaluation.
Related
- Variational study of the magnetization plateaus in the spin-1/2 kagome Heisenberg antiferromagnet: An approach from vision transformer neural quantum states
- Learning SYK Hamiltonians
- Operator Score Matching for Learning Quantum Hamiltonians
- Classical Hardness of Learning Functions of Hamiltonians
- Scalably learning one-dimensional quantum many-body Hamiltonians from dynamical data
BibTeX
@misc{aktar2026learning,
title = {{Learning Trotter Orderings for Heisenberg Hamiltonians with a Ranking Transformer}},
author = {Shamminuj Aktar and Reuben Tate and Stephan Eidenbenz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31977},
eprint = {2609.31977},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31977},
}
RIS
TY - UNPB
TI - Learning Trotter Orderings for Heisenberg Hamiltonians with a Ranking Transformer
AU - Shamminuj Aktar
AU - Reuben Tate
AU - Stephan Eidenbenz
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31977
UR - https://arxiv.org/abs/2609.31977
AB - Trotterization approximates quantum time evolution by sequentially applying Hamiltonian terms. Because noncommuting terms introduce ordering-dependent errors, selecting an optimal term ordering is a combinatorial problem over a factorial search space. Prior approaches rely on fixed heuristics or on selection among predefined structured orderings, both of which require simulating candidates before choosing among them. For 1D and 2D Heisenberg-style Hamiltonians, we instead learn the ordering directly with a physics-aware ranking transformer that assigns a scalar score to each Hamiltonian term and predicts an ordering by sorting these scores. Physical structure enters through commutator-biased attention and an anticommutation-weighted ranking loss, and the models are trained on simulated-annealing (SA) reference orderings. We train separate models for first- and second-order Trotterization, each jointly on chains up to 14 qubits and lattices up to 12 qubits, and evaluate them on unseen chains with 16-20 qubits and lattices with 16 and 20 qubits. Predicted orderings are evaluated by the median gap in simulation fidelity to the SA reference. At first order, the model reaches a pooled gap below 10^-4 on chains, 0.0144 and 0.0088 on triangular lattices, and 0.0948 and 0.0823 on rectangular lattices at 16 and 20 qubits; at second order, the gaps are 0.0114, 0.0364 and 0.0244, and 0.1348 and 0.1239. The prediction exceeds the SA reference on 34% of first-order and 8% of second-order chain instances and can match its fidelity even when the two sequences differ because commuting terms may be rearranged without changing the Trotter unitary. The learned models generalize to larger systems across all three geometries, performing best on chains and triangular lattices, and produce an ordering in one forward pass without candidate enumeration, simulated annealing, or fidelity evaluation.
ER -