A Remarkable Application of Zassenhaus Formula to Strongly Correlated Electron Systems

TL;DR

This paper introduces a simplified Zassenhaus decomposition applicable under specific conditions, enabling an exact, Trotterization-free Unitary Coupled Cluster method for strongly correlated electrons on quantum computers.

Contribution

It presents a novel application of the Zassenhaus formula to develop an exact, Trotterization-free ansatz for quantum simulations of strongly correlated electron systems.

Findings

The decomposition simplifies under the no-mixed adjoint property.

The method is exact with a finite number of Givens gates.

Provides insights into when Trotterization yields exact solutions.

Abstract

We show that the Zassenhaus decomposition for the exponential of the sum of two non-commuting operators, simplifies drastically when these operators satisfy a simple condition, called the no-mixed adjoint property. An important application to a Unitary Coupled Cluster method for strongly correlated electron systems is presented. This ansatz requires no Trotterization and is exact on a quantum computer with a finite number of Givens gate equals to the number of free parameters. The formulas obtained in this work also shed light on why and when optimization after Trotterization gives exact solutions in disentangled forms of unitary coupled cluster.