Suzuki Type Estimates for Exponentiated Sums and Generalized Lie-Trotter Formulas in Banach Algebras
Zhenhua Wang
TL;DR
This work develops Suzuki-type error estimates for Lie-Trotter decompositions in Banach algebras using Jordan products, extending Lie-Trotter theory to Jordan-Banach and JB-algebras in quantum contexts. It proves two $1/n^2$-order error bounds for second-order Jordan-Trotter formulas and extends the framework to Jordan-Trotter triple-product decompositions, yielding two generalized Lie-Trotter formulas with explicit bounds. A central corollary shows that the Suzuki symmetrized approximation emerges directly from these results, and that for any finite family $A_1,\
Abstract
The Lie-Trotter formula has been a fundamental tool in quantum mechanics, quantum computing, and quantum simulations. The error estimations for the Lie-Trotter product formula play a crucial role in achieving scalability and computational efficiency. In this note, we present two error estimates of Lie-Trotter product formulas, utilizing Jordan product within Banach algebras. Additionally, we introduce two generalized Lie-Trotter formula and provide two explicit estimation formulas. Consequently, the renowned Suzuki symmetrized approximation for the exponentiated sums follows directly from our main Theorem.