Trotterization in Quantum Theory
Physics Claire Kluber
TL;DR
The paper addresses computing exponentials of sums of noncommuting operators via Trotterization in quantum theory. It provides a rigorous mathematical formulation and proof of the Trotter Product Formula for (potentially unbounded) self-adjoint operators, guided by unitary evolution groups and Stone's Theorem, including the strong convergence $\mathrm{s-lim}_{n\to\infty} (e^{-i t S/n} e^{-i t T/n})^n = e^{-i t (S+T)}$. It discusses practical implications for quantum circuits, showing how $(e^{A/N} e^{B/N})^N$ approximates $e^{A+B}$ with potential computational advantages, and covers application to Hamiltonians like $H = \frac{\hat{p}^2}{2m} + \hat{x}$. The work also notes special cases, such as anti-commuting operators, where simplified, $q$-commutative generalizations of the Binomial Theorem apply, with references to Scurlock (2020) and Zhao & Yuan (2021).
Abstract
Trotterization in quantum mechanics is an important theoretical concept in handling the exponential of noncommutative operators. In this communication, we give a mathematical formulation of the Trotter Product Formula, and apply it to basic examples in which the utility of Trotterization is evident. Originally, this article was completed in December 2020 as a report under the mentorship of Esteban Cárdenas for the University of Texas at Austin Mathematics Directed Reading Program (DRP). However, the relevance of Trotterization in reducing quantum circuit complexity has warranted the release of a revised and more formal version of the original. Thus, we present a mathematical perspective on Trotterization, including a detailed sketch of a formal proof of the Trotter Product Formula.