Schrödinger–Navier–Stokes Equation for the Quantum Simulation of Navier–Stokes Flows

Abstract The search for quantum-like wave formulations of the Navier–Stokes equations (NSEs), here referred to as Schrödinger–Navier–Stokes (SNS) equation, has attracted increasing attention in recent years because of its potential application in the simulation of classical dissipative fluids on quantum computers. An SNS formulation of classical fluids was first presented in a largely unnoticed paper by Dietrich and Vautherin in 1985 [Sur l’équivalence entre des types particuliers des équations de Navier–Stokes et de Schrödinger non linéaire J. Phys. 46 313–6]. In this paper, we revisit this SNS formulation and assess its suitability for quantum implementation based on Carleman linearization. Specifically, we (i) clarify why the non-polynomial dissipative and quantum-pressure terms of the SNS equation obstruct a direct Carleman treatment and reformulate the dynamics as a Navier–Stokes–Hamilton–Jacobi (NSHJ) system; (ii) develop a corresponding quantum algorithm based on Carleman linearization of the NSHJ equations, referred to as Carleman–Hamilton–Jacobi (CHJ), together with a tensor-network representation that substantially reduces the memory requirements of its classical emulation; and (iii) emulate the CHJ dynamics on a classical computer and analyze its convergence and accuracy for 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 full NSE, including pressure, dissipation and vorticity.