Quantum Boundary Integral Algorithm for linear and nonlinear PDEs in Quantum Computation of Fluid Dynamics

The advent of quantum computing (QC) and its potential in solving certain classes of problems with large computational speed-ups compared to its classical counterparts, kindles one to try utilizing such abilities to solve fluid flow problems. Though there has been an influx of new and improved quantum algorithms, recent efforts in this direction also point to critical caveats that need addressing, especially to be able to simulate fluid flows. For instance, the class of Quantum Linear System Algorithms (QLSA) has evolved greatly over time with its origins stemming from the HHL (Harrow-Hassidim-Lloyd) algorithm, that aims at solving linear system of equations with a significant speedup, which inherently also reflects the linearity of quantum mechanics itself. However fluid flows are mostly nonlinear. The aim of this work is to explore how one could reformulate the problem to leverage the advantage of QLSA to encode the nonlinearities, while also preserving the speed-up.

In this setting, and with a view to permit end-to-end solutions, we have developed a high performance quantum simulator called QuON; QuON is specifically designed for simulating fluid flows using QC. Using QuOn, the time-dependent Poiseuille and Couette flows are solved as initial examples to demonstrate, for the first time, a high precision, fully gate level implementation of a QLSA method for a CFD simulation. Next, we propose a Quantum Boundary Integral Algorithm (QBIA) formulation that seeks to solve nonlinear partial differential equations by combining the Homotopy Analysis Method (HAM) with QLSA. We expect that our integral formulation is better suited for QC than the usual methods that focus on discretizing the strong differential form of the governing dynamics, both in terms of accuracy and efficiency. We discuss some preliminary efforts to solve a 1D Burgers flow using this approach. Along with this we also aim at evaluating the minimum requirements as well as the potential and obstacles for transitioning from computational fluid dynamics (CFD) to Quantum Computation of Fluid Dynamics (QCFD).