[Paper Review] Quantum simulation of Maxwell's equations via Schrödingersation

Shi Jin, Nana Liu+1 more|arXiv (Cornell University)|Aug 16, 2023

Quantum Computing Algorithms and ArchitectureComputer Science3 citations

TL;DR

This paper presents quantum algorithms for simulating Maxwell’s equations using the Schrödingerisation method, which transforms non-unitary electromagnetic dynamics into unitary evolution via a warped phase transformation in an extended space. The approach enables accurate simulation of electromagnetic fields with physical boundary and interface conditions, achieving high fidelity in both discrete (Yee and upwind schemes) and continuous-variable quantum frameworks, with numerical results matching reference solutions from QLA and exact analytical solutions.

ABSTRACT

We present quantum algorithms for electromagnetic fields governed by Maxwell's equations. The algorithms are based on the Schrödingersation 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.

Motivation & Objective

• To develop quantum algorithms for Maxwell’s equations that incorporate physical boundary and interface conditions, which are challenging in standard quantum simulation due to non-unitary dynamics.
• To extend the Schrödingerisation method—previously applied to ODEs and PDEs—to electromagnetic field systems governed by Maxwell’s equations.
• To demonstrate the preservation of key physical properties such as divergence-free magnetic fields and total energy in the quantum simulation framework.
• To enable quantum simulation in both discrete (qubit-based) and continuous-variable quantum computing platforms, enhancing accessibility for near-term devices.
• To validate the method through numerical experiments on perfect conductor, impedance, and inhomogeneous media with material interfaces, comparing against analytical and reference numerical solutions.

Proposed method

• The Schrödingerisation method transforms Maxwell’s equations into a unitary evolution problem by introducing an auxiliary space-like dimension through a warped phase transformation, converting non-unitary dynamics into unitary dynamics.
• The method is applied to both the Riemann-Silberstein vector formulation (eight-dimensional matrix representation) and the standard electric/magnetic field formulation.
• For spatial discretization, the Yee scheme and upwind scheme are employed, with the latter showing first-order spatial accuracy but good stability under Schrödingerisation.
• Boundary conditions—periodic, perfect conductor, and impedance—are implemented via unitary transformations on the matrix representation, preserving physical constraints.
• Interface conditions in inhomogeneous media are handled using immersed interface methods (IIM), ensuring accurate treatment of discontinuities in permittivity and permeability.
• The framework is extended to continuous-variable quantum systems, avoiding dense Hamiltonian matrices from spatially varying velocity and enabling analog quantum simulation.

Experimental results

Research questions

• RQ1 Can the Schrödingerisation method be effectively applied to Maxwell’s equations to enable unitary quantum simulation with physical boundary conditions?
• RQ2 How does Schrödingerisation affect the preservation of fundamental physical properties such as divergence-free magnetic fields and total electromagnetic energy in numerical schemes?
• RQ3 To what extent can the method accurately simulate electromagnetic wave propagation across material interfaces with discontinuous permittivity and permeability?
• RQ4 How do the Yee scheme and upwind scheme compare in accuracy and stability when combined with Schrödingerisation for quantum simulation of Maxwell’s equations?
• RQ5 Can the Schrödingerisation approach be adapted to continuous-variable quantum systems to enable analog quantum simulation of electromagnetic fields?

Key findings

• The Schrödingerisation method successfully preserves the divergence-free condition of the magnetic field and the total energy of the electromagnetic field in both Yee and upwind schemes, as validated through numerical experiments.
• Numerical simulations show that the Schrödingerisation approach with the Yee scheme achieves higher accuracy than the upwind scheme at the same mesh size, with the latter showing first-order spatial convergence.
• The method accurately captures electromagnetic wave reflection and transmission at material interfaces, with results closely matching the exact analytical solutions for a dielectric interface with ε₂=2, μ₂=2.
• For a Gaussian pulse propagating from vacuum (ε₁=1, μ₁=1) to a dielectric medium (ε₂=3, μ₂=1), the Schrödingerisation method produces solutions that agree well with those from the quantum lattice algorithm (QLA) using a fine mesh (M=2¹¹).
• The implementation of perfect conductor and impedance boundary conditions via unitary transformations in the Schrödingerised framework maintains physical consistency and matches expected field behaviors.
• In the continuous-variable quantum framework, the method avoids dense Hamiltonian matrices arising from spatially varying wave velocity, enabling more efficient analog quantum simulation of Maxwell’s equations.