Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits

TL;DR

This paper introduces a scalable quantum circuit method for simulating hyperbolic partial differential equations, significantly reducing computational complexity and demonstrating practical feasibility through experiments.

Contribution

It provides an explicit gate construction for Hamiltonian simulation of PDEs, enabling efficient quantum algorithms with reduced resource requirements.

Findings

Quantum circuits have exponentially smaller space and time complexities than classical algorithms.

Numerical experiments validate the effectiveness of the proposed quantum simulation method.

Real-device experiments confirm practical applicability for wave equations.

Abstract

Solving partial differential equations for extremely large-scale systems within a feasible computation time serves in accelerating engineering developments. Quantum computing algorithms, particularly the Hamiltonian simulations, present a potential and promising approach to achieve this purpose. Actually, there are several oracle-based Hamiltonian simulations with potential quantum speedup, but their detailed implementations and accordingly the detailed computational complexities are all unclear. This paper presents a method that enables us to explicitly implement the quantum circuit for Hamiltonian simulation; the key technique is the explicit gate construction of differential operators contained in the target partial differential equation discretized by the finite difference method. Moreover, we show that the space and time complexities of the constructed circuit are exponentially smaller than those of conventional classical algorithms. We also provide numerical experiments and an experiment on a real device for the wave equation to demonstrate the validity of our proposed method.