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

TL;DR

This work extends Trotter-Suzuki decompositions to Jordan-Banach algebras, addressing the existence of second-order error estimates and proving them with explicit bounds. It then develops third-order and general higher-order Jordan-Trotter product formulas, including symmetric and non-symmetric constructions, and provides rigorous error analyses in both analytic and fidelity terms. The authors apply these formulas to simulate Trotter-factorized spins, demonstrating improved accuracy and broader regime performance compared to traditional methods, and present visualization resources (contour plots, state-fidelity plots) to illustrate practical benefits. By integrating JB*-algebras and non-associative operator dynamics into quantum-time evolution approximations, the paper broadens the mathematical toolkit for quantum simulation under symmetry constraints and non-associative settings.

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.

Figures (2)

• Figure 1: Comparison of the error associated with the third-order Trotter-Suzuki decomposition and the Jordan-Trotter product formula for the Hamiltonian $H= d_1 X + d_2 Y$. The left and middle panels show contour plots of the Frobenius norm of the error as a function of $td_1$ and $td_2$ for the third-order Trotter-Suzuki decomposition and the Jordan-Trotter product formula, respectively. The rightmost panel provides a line plot comparing the errors of both methods along the diagonal $td_1 = td_2$.
• Figure 2: Comparison of the time evolution accuracy for the first-order(red), second-order(cyan), and third-order(gray) Trotter-Suzuki decompositions, along with the Jordan-Trotter product formula(darker blue), against the exact result for $H = Z + X$. The plot shows the deviation from the exact solution as a function of evolution time $t$, highlighting the improved accuracy of higher-order approximations.

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Definition 2.6
• Theorem 3.1
• proof
• Proposition 3.2
• proof