Quantum computing of fluid dynamics using the hydrodynamic Schrödinger equation

Abstract

Simulating fluid dynamics on a quantum computer is intrinsically difficult due to the nonlinear and non-Hamiltonian nature of the Navier-Stokes equation (NSE). We propose a framework for quantum computing of fluid dynamics based on the hydrodynamic Schrödinger equation (HSE), which can be promising in simulating threedimensional turbulent flows in various engineering applications. The HSE is derived by generalizing the Madelung transform to compressible/incompressible flows with finite vorticity and dissipation. Since the HSE is expressed as a unitary operator on a two-component wave function, it is more suitable than the NSE for quantum computing. The flow governed by the HSE can resemble a turbulent flow consisting of tangled vortex tubes with the five-thirds scaling of energy spectrum. We develop a prediction-correction quantum algorithm to solve the HSE. This algorithm is implemented for simple flows on the quantum simulator Qiskit with exponential speedup. I. INTRODUCTIONQuantum computing has emerged to be the next disruptive technology since Feynman pointed out the enormous potential of quantum simulation [1]. Compared to conventional digital computing, quantum computing can dramatically reduce the execution time, memory usage, and energy consumption [2]. There are various hardware techniques for quantum logic gates [3][4][5][6][7][8], quantum algorithms for specific tasks [9][10][11][12], and applications [13][14][15][16]

Quantum computing for fluids: Where do we stand?

“…The major caveat is that the Madelung fluid is a far distant relative of classical Navier-Stokes fluids, the main points of departure being i) dissipation, ii) non-gradient flow, iii) absence of quantum potential. Very recent work has shown how to mend points ii) and iii) using quaternions, but still leaves item i) open [37]. Encouraging work to solve item i) as well has just appeared [38].…”

Quantum Computing CFD Simulations for Aeronautical Heat Exchangers

“…Indeed, several approaches have been proposed for the solutions of CFD problems with the use of quantum algorithms and / or quantum computers, as summarized in [6], being the rationale of the development the estimated scalability of quantum computing [7] and the analogy between Navier Stokes equations (NSE) with the Schrödinger equation through the Madelung transform, i.e., the Hydrodynamic Schrödinger equation (HSE) [8].…”