Quantum computing of nonlinear flow problems with a homotopy analysis algorithm

Quantum Computing (QC) has many nominal advantages over classical computing, but has yet to solve important practical problems efficiently. Simulating nonlinear flow physics is an excellent candidate in the latter class, given its computational onerosity and the wide range of applications. However, the linear underpinnings of quantum mechanics itself and the unitarity of fundamental QC operations blockade tractability of nonlinear PDEs. In this work we develop a hybrid quantum algorithm based on an integral formulation of a Homotopy Analysis Algorithm (HAA) to solve the 1D Burgers equation. We do so in an end-to-end manner by addressing the challenges of data encoding and also measurements in QC by outlining a quantum post-processing algorithm to compute the mean viscous dissipation rate. The method involves bespoke linearization of the problem with HAA, followed by a Quantum Linear System Algorithm (QLSA) to solve the system of equations. All the simulations are performed on an in-house high performance quantum simulator which we term QFlowS, designed specifically to simulate fluid flows with QC. We elucidate the performance of the algorithm and prescribe the algorithmic criteria for physically accurate simulations.