Error Estimates and Higher Order Trotter Product Formulas in Jordan-Banach Algebras

TL;DR

This paper extends Trotter-Suzuki approximation error analysis to Jordan-Banach algebras, including higher orders, solving an open problem and demonstrating improved quantum simulation methods.

Contribution

It introduces higher-order Trotter error estimates in Jordan-Banach algebras and addresses an open problem from previous work.

Findings

Successful application to spin system simulations

Improved approximation accuracy demonstrated

Extension of Trotter formulas to non-associative algebras

Abstract

In quantum computing, Trotter estimates are critical for enabling efficient simulation of quantum systems and quantum dynamics, help implement complex quantum algorithms, and provide a systematic way to control approximate errors. In this paper, we extend the analysis of Trotter-Suzuki approximations, including third and higher orders, to Jordan-Banach algebras. We solve an open problem in our earlier paper on the existence of second-order Trotter formula error estimation in Jordan-Banach algebras. To illustrate our work, we apply our formula to simulate Trotter-factorized spins, and show improvements in the approximations. Our approach demonstrates the adaptability of Trotter product formulas and estimates to non-associative settings, which offers new insights into the applications of Jordan algebra theory to operator dynamics.