Schr\"odinger-Navier-Stokes Equation for the Quantum Simulation of Navier-Stokes Flows

TL;DR

This paper develops a quantum algorithm for simulating Navier-Stokes flows using a Schr"odinger-like wave formulation, addressing computational challenges and demonstrating classical emulation for specific fluid flows.

Contribution

It introduces a novel quantum algorithm based on tensor-network Carleman embedding of the Hamilton-Jacobi formulation for Navier-Stokes equations, including pressure and vorticity.

Findings

Quantum dissipator challenges clarified and addressed

Tensor-network CHJ algorithm developed with memory efficiency

Classical emulation shows convergence and accuracy for Kolmogorov-like flows

Abstract

The search for quantum-like wave formulations of the Navier-Stokes (Schr\"odinger-Navier-Stokes, SNS for short) equations describing classical dissipative fluids has met with increasing attention in the recent years, due to the large portfolio of potential applications in science and engineering. A SNS formulation of classical fluids was first presented in a largely un-noticed paper by Dietrich and Vautherin back in 1985(Journal de Physique). In this paper, we revisit this specific SNS approach and assess its viability for quantum implementations based on Carleman embedding/linearization techniques. Specifically, we i) Clarify in full mathematical detail why the SNS dissipator presents a steep challenge for quantum computers and propose a way out strategy based on the Hamilton-Jacobi (HJ) formulation of fluid dynamics; ii) Develop a corresponding quantum algorithm using a new technique based on a tensor-network representation of Carleman embedding of the HJ equations (CHJ) which permits substantial memory savings; iii) Emulate the CHJ quantum algorithm on a classical computer and analyse its convergence and accuracy for the specific case of Kolmogorov-like flows at moderate Reynolds numbers. To the best of our knowledge, this is the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity.