Simulating Time-dependent Hamiltonian Based On High Order Runge-Kutta and Forward Euler Method

TL;DR

This paper introduces a quantum simulation method for specific time-dependent Hamiltonians using high-order Runge-Kutta and forward Euler techniques, leveraging quantum singular value transformation for efficient evolution operator construction.

Contribution

It develops a novel quantum simulation approach combining high-order Runge-Kutta and forward Euler methods for time-dependent Hamiltonians with bounded derivatives.

Findings

Efficient quantum algorithms for simulating certain time-dependent Hamiltonians.

Utilizes quantum singular value transformation to iteratively build evolution operators.

Applicable to Hamiltonians with bounded, computable time-dependent coefficients.

Abstract

We propose a new method for simulating certain type of time-dependent Hamiltonian $H(t) = \sum_{i=1}^m \gamma_i(t) H_i$ where $\gamma_i(t)$ (and its higher order derivatives) is bounded, computable function of time $t$, and each $H_i$ is time-independent, and could be efficiently simulated. Our quantum algorithms are based on high-order Runge-Kutta method and forward Euler method, where the time interval is divided into subintervals. Then in an iterative manner, the evolution operator at given time step is built upon the evolution operator at previous time step, utilizing algorithmic operations from the recently introduced quantum singular value transformation framework.