Simulating unsteady flows on near-term quantum computers

Quantum computers are favored to overtake classical devices in solving certain tasks, with substantial gains in speed and memory. Despite fervent efforts on theoretical and experimental fronts, the gap between them precludes utilitarian quantum algorithms to solve practical problems. In particular, we are interested in solving nonlinear fluid flow problems. To this end, we propose here a quantum algorithm based on Linear combination of Unitaries, consisting of Time Marching Compact Quantum Circuits to solve unsteady PDEs. The present algorithm has a time complexity that is near-optimal (logarithmic) compared to existing proposals, along with a qubit complexity that is logarithmic in the problem size. We outline specific solutions to bottlenecks such as the quantum state preparation, quantum measurements, as well as noise and decoherence from real quantum devices, which tend to diminish quantum advantage. In that sense, our algorithm is end-to-end, preserving potential quantum advantage.  To assess its performance, we simulate the well-known one-dimensional, linear advection-diffusion problem by implementing the algorithm on QFlowS (an in-house quantum simulator) and by performing experiments on an actual quantum computer (IBM Cairo). Our results show that the proposed quantum algorithm successfully captures the flow physics qualitatively and quantitatively. We make a case that the present approach is amenable for near-term devices, and highlight that the proposed algorithm offers an efficient way for performing iterative matrix operations such as inversions and matrix-vector products, thus broadening its applicability well beyond fluid dynamics.