Quantum algorithm for simulating the wave equation

We present a quantum algorithm for simulating the wave equation under Dirichlet and Neumann boundary conditions. The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines. It relies on factorizations of discretized Laplacian operators to allow for polynomially improved scaling in truncation errors and improved scaling for state preparation relative to general purpose quantum algorithms for solving linear differential equations. Relative to classical algorithms for simulating the $D$ -dimensional wave equation, our quantum algorithm achieves exponential space savings and achieves a speedup which is polynomial for fixed $D$ and exponential in $D$ . We also consider using Hamiltonian simulation for Klein-Gordon equations and Maxwell's equations.