Structure-Aware Transformers for Learning Near-Optimal Trotter Orderings with System-Size Generalization in 1D Heisenberg Hamiltonians

TL;DR

This paper introduces a transformer-based model that predicts near-optimal Trotter orderings for 1D Heisenberg Hamiltonians, enabling system-size generalization and reducing computational costs in quantum simulations.

Contribution

It presents the first learned model for Trotter ordering that generalizes across system sizes, trained on small systems to predict orderings for larger ones without fidelity evaluation.

Findings

Model achieves a mean fidelity gap of 0.00115 on out-of-range systems.

Generalization improves when training includes systems up to 8 qubits.

First application of machine learning to Trotter ordering with cross-system size generalization.

Abstract

Trotterization is a standard approach for simulating quantum time evolution on quantum computers, where the Hamiltonian is split into local terms and each term is applied in sequence. The order of these terms affects the fidelity of the simulation when they do not commute, so the choice of ordering directly impacts the accuracy of the simulation. We study this problem for one-dimensional XXZ Heisenberg Hamiltonians using a structured set of 24 candidate orderings derived from colorings of the Hamiltonian's commutation graph and their group permutations. Finding the best candidate for large systems becomes prohibitive because fidelity evaluation is computationally expensive. In this work, we train a transformer encoder on smaller systems to predict the best candidate ordering for larger systems directly from Hamiltonian and Trotter-configuration features, without computing candidate fidelities at inference time. The model is trained on in-range systems of 3 to 14 qubits with 15-qubit systems held out for validation. Experimental results show that the model reaches a mean test fidelity gap of 0.00115 relative to the best of the 24 candidates on out-of-range systems of 16 to 20 qubits. A training-size sweep further shows that generalization emerges once training includes systems up to L=8 qubits, with validation at L=9, and the gap continues to decrease as the training range grows. To our knowledge, this is the first application of a learned model to Trotter ordering, and it motivates future work on AI-guided Trotter ordering with generalization across Hamiltonian families and system types.