Quantum Algorithms for Heterogeneous PDEs: The Neutron Diffusion Eigenvalue Problem

Abstract

We develop a quantum algorithm to solve a type of linear reaction-diffusion equation, the neutron diffusion (generalized) k-eigenvalue problem that establishes nuclear criticality. The algorithm handles an equation with piecewise constant coefficients, describing a problem in a heterogeneous medium. We apply uniform finite elements and show that the quantum algorithm provides significant polynomial end-to-end speedup over classical uniform finite element methods in the cases tested, although nonuniform classical methods can perform much better than uniform ones. Our work leverages recent advances in quantum linear systems--fast inversion and quantum preconditioning--and uses Hamiltonian simulation as a subroutine. Our results suggest that quantum algorithms may provide speedups for heterogeneous PDEs, though the extent of this advantage over the fastest classical algorithm depends on the effectiveness of other classical approaches such as nonuniform or adaptive meshing for a given problem instance.