Hybrid quantum algorithms for flow problems

For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with an end-to-end practical applications -- in this instance, fluid dynamics. We debut here a high performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates $T_{0}$ (a parameter crucial to Hamiltonian simulations) to the condition number $\kappa$ of the simulation matrix, and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for (a) the functional form or sparse quantum state preparation, and (b) \textit{in-situ} quantum post-processing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be $\mathcal{O}(\textrm{polylog} (N/\epsilon)\kappa/\epsilon_{QPP})$, where $N$ is the size of the linear system of equations, $\epsilon$ is the the solution error and $\epsilon_{QPP}$ is the error in post processing. This work suggests a path towards quantum simulation of fluid flows, and highlights the special considerations needed at the gate level implementation of QC.