Quantum Simulation of Stokes Flow via Schr\"odingerisation and Artificial Compressibility

Simulating incompressible Stokes flow is essential for studies in microfluidics and low-Reynolds number hydrodynamics.

However, the computational cost of resolving the associated saddle-point problem grows prohibitively with the dimensionality of the problem.

In this work, we present a quantum algorithm based on the Schrödingerisation technique for the Stokes equations, incorporating an artificial compressibility regularization.

The core of our approach is the design of an explicit quantum circuit that encodes the resulting regularized system.

The artificial compressibility formulation provides a unified framework for the system, which is then efficiently mapped to a quantum circuit via the Schrödingerisation procedure.

A rigorous complexity analysis demonstrates the quantum computational advantage of our algorithms in high-dimensional settings, notably an exponential speedup in problem dimensionality.

The validity and scalability of the proposed method are corroborated by numerical simulations performed on Qiskit.