A Rigorous Introduction to Hamiltonian Simulation via High-Order Product Formulas

TL;DR

This paper offers a detailed, mathematically rigorous overview of high-order product formulas for Hamiltonian simulation in quantum computing, emphasizing their theoretical foundations, advantages, and limitations.

Contribution

It provides a comprehensive, self-contained introduction to high-order product formulas, including Suzuki's recursive method, for efficient quantum Hamiltonian simulation.

Findings

Higher-order product formulas improve error scaling in Hamiltonian simulation.

Suzuki's recursive method enhances the efficiency of product formulas.

Discussion of applications to $k$-local Hamiltonians and open challenges.

Abstract

This work provides a rigorous and self-contained introduction to numerical methods for Hamiltonian simulation in quantum computing, with a focus on high-order product formulas for efficiently approximating the time evolution of quantum systems. Aimed at students and researchers seeking a clear mathematical treatment, the study begins with the foundational principles of quantum mechanics and quantum computation before presenting the Lie-Trotter product formula and its higher-order generalizations. In particular, Suzuki's recursive method is explored to achieve improved error scaling. Through theoretical analysis and illustrative examples, the advantages and limitations of these techniques are discussed, with an emphasis on their application to $k$-local Hamiltonians and their role in overcoming classical computational bottlenecks. The work concludes with a brief overview of current advances and open challenges in Hamiltonian simulation.