Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

TL;DR

This paper develops high-order Trotterization formulas for time-dependent Hamiltonians, minimizing the number of exponentials needed, and demonstrates their effectiveness through numerical benchmarks in quantum simulations.

Contribution

The authors derive new minimal-exponential high-order Trotterization formulas for time-dependent operators, improving efficiency over existing methods.

Findings

Fourth-order and sixth-order formulas with fewer exponentials.

The 9-exponential formula has smaller errors than Suzuki's in quantum Ising chain simulations.

Numerical benchmarks confirm the effectiveness of the new formulas.

Abstract

When a time propagator $e^{\delta t A}$ for duration $\delta t$ consists of two noncommuting parts $A=X+Y$, Trotterization approximately decomposes the propagator into a product of exponentials of $X$ and $Y$. Various Trotterization formulas have been utilized in quantum and classical computers, but much less is known for the Trotterization with the time-dependent generator $A(t)$. Here, for $A(t)$ given by the sum of two operators $X$ and $Y$ with time-dependent coefficients $A(t) = x(t) X + y(t) Y$, we develop a systematic approach to derive high-order Trotterization formulas with minimum possible exponentials. In particular, we obtain fourth-order and sixth-order Trotterization formulas involving seven and fifteen exponentials, respectively, which are no more than those for time-independent generators. We also construct another fourth-order formula consisting of nine exponentials having a smaller error coefficient. Finally, we numerically benchmark the fourth-order formulas in a Hamiltonian simulation for a quantum Ising chain, showing that the 9-exponential formula accompanies smaller errors per local quantum gate than the well-known Suzuki formula.