Quantum algorithms for solving a drift-diffusion equation: A complexity analysis

TL;DR

This paper introduces four quantum algorithms for solving multidimensional drift-diffusion equations, demonstrating potential quantum advantages over classical methods in computational complexity.

Contribution

It presents novel quantum algorithms utilizing quantum linear system solvers, Hamiltonian simulation, quantum walks, and Fourier transforms for PDEs, with complexity analysis showing advantages.

Findings

Quantum Fourier transform-based diagonalization offers computational benefits.

Quantum algorithms can efficiently extract probability distributions.

Classical methods are outperformed in complexity by quantum approaches for certain PDEs.

Abstract

We present four quantum algorithms for solving a multidimensional drift-diffusion equation. They rely on a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. We compare the complexities of these methods to their classical counterparts, finding that diagonalization via the quantum Fourier transform offers a quantum computational advantage for solving linear partial differential equations at a fixed final time. We employ a multidimensional amplitude estimation process to extract the full probability distribution from the quantum computer.