Quantum simulation of Maxwell’s equations via Schrödingerisation

¹ Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China
² Ministry of Education, Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, P.R. China
³ Shanghai Artificial Intelligence Laboratory, Shanghai 200240, P.R. China
⁴ School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China
⁵ University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai 200240, P.R. China

^* Corresponding author: chuwenii@sjtu.edu.cn; chuwenii@lsec.cc.ac.cn

Received: 18 September 2023
Accepted: 4 June 2024

Abstract

We present quantum algorithms for electromagnetic fields governed by Maxwell’s equations. The algorithms are based on the Schrödingerisation approach, which transforms any linear PDEs and ODEs with non-unitary dynamics into a system evolving under unitary dynamics, via a warped phase transformation that maps the equation into one higher dimension. In this paper, our quantum algorithms are based on either a direct approximation of Maxwell’s equations combined with Yee’s algorithm, or a matrix representation in terms of Riemann–Silberstein vectors combined with a spectral approach and an upwind scheme. We implement these algorithms with physical boundary conditions, including perfect conductor and impedance boundaries. We also solve Maxwell’s equations for a linear inhomogeneous medium, specifically the interface problem. Several numerical experiments are performed to demonstrate the validity of this approach. In addition, instead of qubits, the quantum algorithms can also be formulated in the continuous variable quantum framework, which allows the quantum simulation of Maxwell’s equations in analog quantum simulation.