An Analysis of Commutation-Based Trotter Ordering Strategies on Heisenberg-Style Hamiltonians

TL;DR

This paper investigates how different commutation-based ordering strategies affect Trotterization error in simulating Heisenberg Hamiltonians, combining theoretical analysis with empirical error calculations.

Contribution

It introduces commutation-aware ordering strategies using graph coloring of the commutation graph and analyzes their properties for specific Hamiltonian classes.

Findings

Commutation-based orderings can reduce Trotter error compared to baseline strategies.

Graph coloring techniques effectively identify commuting groups within Hamiltonians.

Empirical results show improved accuracy in 1D and 2D Heisenberg systems.

Abstract

Trotterization is a technique that allows one to approximate a time evolution of a Hamiltonian by repeatedly evolving the individual terms of the Hamiltonian one-at-a-time for small time durations. Bounds on the error of this approximation exist; however, they are typically loose and moreover, it is known that the true error can be greatly influenced by the order in which the terms of the Hamiltonian are evolved. In this work, we consider various ordering strategies that exploit the commutation structure of the Hamiltonian, in addition to a few other baseline ordering strategies. These commutation-based strategies involve dividing the terms of the Hamiltonian into groups where all the terms within each group commute with one another. These groupings can be obtained by using graph coloring techniques on what we call the "commutation graph" of the Hamiltonian. We prove various results regarding the structure and properties of such commutation graphs for certain classes of Hamiltonians. We also empirically calculate the (true) Trotter error using these ordering strategies on various 1D and 2D Heisenberg-style systems.