Quantum Algorithm for Solving the Wave Equation on a Hexagonal Grid

Numerical methods based on finite differences for solving the wave equation are prone to various errors, a challenge that also applies to the semi-discrete case of a recently proposed quantum algorithm that preserves continuous time while discretizing space into a quantum state vector. The inherent limitations of quantum processing, which affect the generality and extensibility of the quantum algorithm compared to classical numerical solvers, are offset by the logarithmic scaling of the number of qubits with the number of grid points. The higher grid density in combination with the higher-order approximation of the continuous Laplacian operator improves the numerical accuracy. In contrast to the Dirichlet boundary condition, however, the Neumann boundary condition in the quantum algorithm cannot exceed a second-order Laplacian approximation. Moreover, the anisotropy error is inherent to the lattice geometry. In this work, we propose a quantum wave equation solver on a hexagonal grid, which exhibits intrinsic grid dispersion comparable to that of a fourth-order accurate method on a rectangular grid and provides a quadratic improvement in the anisotropy measures. We show that the graph Laplacian on a hexagonal grid embedding the Neumann boundary condition is a bidirected graph. To make it suitable for use as a system Hamiltonian, a symmetrization and factorization based on the asymmetric adjacency matrix is proposed. Dispersion relations for quantum algorithms solving the wave equation on different lattices are derived and compared. The algorithms are analyzed as a limiting case of Yee's finite-difference time-domain method in the context of electromagnetic wave propagation.