Quantum Algorithms for Heterogeneous PDEs: The Neutron Diffusion Eigenvalue Problem

TL;DR

This paper presents a hybrid classical-quantum algorithm for solving heterogeneous neutron diffusion eigenvalue problems, demonstrating polynomial speedup over classical methods by leveraging recent quantum linear system techniques.

Contribution

It introduces a quantum algorithm tailored for piecewise constant coefficient PDEs in heterogeneous media, combining finite elements with quantum linear system solutions.

Findings

Quantum algorithm achieves polynomial speedup over classical methods.

Utilizes quantum linear systems, Hamiltonian simulation, and preconditioning.

Speedup depends on classical approach effectiveness like adaptive meshing.

Abstract

We develop a hybrid classical-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 its classical counterparts. This speedup 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.