Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems

TL;DR

This paper introduces a near-optimal quantum algorithm for solving nonlinear PDEs related to flow problems, leveraging quantum homotopy analysis to improve scalability, stability, and accuracy over existing methods.

Contribution

It presents a novel quantum homotopy analysis approach for nonlinear PDEs, including complexity bounds and a practical embedding strategy for flow problems.

Findings

Demonstrated numerical simulation of a 1D Burgers problem

Improved complexity estimates over previous quantum PDE algorithms

Showed potential for practical nonlinear flow simulations on quantum devices

Abstract

Quantum algorithms to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present here a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, and nonlinear PDEs. We embed the PDEs in a truncated, high dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods that use a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of scaling of matrix operator norms, condition number, simulation time, and accuracy. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts and query complexity. A physically motivated measure of nonlinearity is connected to a parameter that is similar to the flow Reynolds number $Re_{\textrm{H}}$, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of a one-dimensional Burgers problem. This work shows the potential of the hybrid quantum algorithm for simulating practical and nonlinear phenomena on near-term and fault-tolerant quantum devices.