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 Schr\"odinger equation (HSE), which can be promising in simulating three-dimensional turbulent flows in various engineering applications. The HSE is derived by generalizing the Madelung transform to compressible or 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 partial exponential speedup.
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 Schr"odinger equation (HSE), which can be promising in simulating three-dimensional turbulent flows in various engineering applications. The HSE is derived by generalizing the Madelung transform to compressible or 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 partial exponential speedup.
Abstract There is great interest in using quantum computers to efficiently simulate a quantum system’s dynamics as existing classical computers cannot do this. Little attention, however, has been given to quantum simulation of a classical nonlinear continuum system such as a viscous fluid even though this too is hard for classical computers. Such fluids obey the Navier–Stokes nonlinear partial differential equations, whose solution is essential to the aerospace industry, weather forecasting, plasma magneto-hydrodynamics, and astrophysics. Here we present a quantum algorithm for solving the Navier–Stokes equations. We test the algorithm by using it to find the steady-state inviscid, compressible flow through a convergent-divergent nozzle when a shockwave is (is not) present. We find excellent agreement between numerical simulation results and the exact solution, including shockwave capture when present. Finally, we compare the algorithm’s computational cost to deterministic and random classical algorithms and show that a significant speed-up is possible. Our work points to a large new application area for quantum computing with substantial economic impact, including the trillion-dollar aerospace industry, weather-forecasting, and engineered-plasma technologies.
A new algorithm for solving the Navier-Stokes equations (NSE) on a quantum device is presented. The non-linearity of fluid flow equations is managed by using the stream function-vorticity formulation, while the lattice Boltzmann method (LBM) is utilized for solving the corresponding system of equations numerically. Following the nature of the lattice Boltzmann method, the proposed quantum algorithm consists of five major sections: initialization, collision, propagation, boundary condition implementation, and macros calculation. The collision and boundary condition step is quantumly implemented by applying the standard-form encoding approach, while the quantum walk procedure is applied for the propagation step. The algorithm is implemented by using IBM's quantum computing software development framework Qiskit, while for the verification purposes two-dimensional (2D) cavity flow is simulated and compared with classical code.
The coexistence of quantum coherence, nonlinear fluid dynamics, and biomolecular π-fields within biological and nanoscale systems requires a unified mathematical description. Classical hydrodynamics alone fails to capture coherence propagation, while the linear Schrödinger equation cannot model dissipation, viscosity, or turbulence. Here, I introduce the Schrödinger–Navier–Stokes–π Unified Computational Framework, a novel equation set combining: the quantum phase field of the Schrödinger equation, the viscous and nonlinear transport of Navier–Stokes dynamics, the curvature-driven quantum π-potential, and a hybrid Hamiltonian–dissipative evolution capturing both coherence and irreversible flow. This framework predicts quantum-fluid transitions, π-induced tunneling acceleration, coherence-enhanced mixing, nanoscale turbulence, and dynamic switching between classical and quantum transport regimes. It provides a general model applicable to biology, nanotechnology, photonics, and quantum materials. Keywords : Quantum Hydrodynamics Navier–Stokes Schrödinger Equation Unified Computational Framework π-field Dynamics Quantum Fluid Mechanics Hybrid Quantum-Classical Transport Coherence-Based Models Quantum Biology Nanoscale Transport Bio-Quantum Modeling Quantum Potential Quantum Fluid Engineering Biophysical Modeling Nonlinear Dynamics Computational Physics High-Order Numerical Methods Dissipative Quantum Systems Biomolecular Channels Quantum-Inspired Computing
In this paper, a new algorithm for solving the Navier–Stokes equations (NSE) on a quantum device is presented. For the fluid flow equations, the stream function-vorticity formulation is adopted, while the lattice Boltzmann method (LBM) is utilized for solving the corresponding system of equations numerically for one time step. Following the nature of the lattice Boltzmann method, the proposed quantum algorithm consists of five major sections: initialization, collision, propagation, boundary condition implementation and calculation of macroscopic quantities. The collision and boundary condition step is quantumly implemented by applying the standard-form encoding approach, while the quantum walk procedure is applied for the propagation step. The algorithm is implemented by using IBM’s quantum computing software development framework Qiskit, while for the verification purposes, two-dimensional (2D) cavity flow is simulated and compared with classical code.
I present a unified theoretical and numerical framework that couples quantum wave dynamics (Schrödinger) with classical viscous flow (Navier–Stokes) through an emergent quantum-π field (π_q) that encodes phase-topology and coherence. The model reproduces Schrödinger dynamics in the conservative limit, Navier–Stokes turbulence in the dissipative limit, and novel intermediate regimes where quantum coherence and fluid turbulence coexist and interact. I derive the governing equations by (i) applying a Madelung decomposition to a complex field ψ, (ii) introducing a controlled viscous regularization and non-linear coupling terms, and (iii) coupling a dynamically evolving π_q scalar (or tensor) field that modulates local coherence, effective mass, and information flux. I then present a robust hybrid numerical method (QHFVM — Quantum Hydrodynamic Finite-Volume Method) combining split-step spectral propagation for dispersive quantum terms and conservative finite-volume solvers for advective, viscous and pressure dynamics. I validate the approach on a set of canonical problems (quantum vortex shedding in a viscous background, π-phase revival in confined geometries, and turbulence spectra with quantum corrections), show numerical convergence and conservation properties, and outline applications spanning quantum fluids, nanoscale bio-fluids, materials, and hybrid QEC architectures. Flash explanations highlight intuition and immediate experimental tests.
The search for quantum-like wave formulations of the Navier-Stokes (Schrödinger-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.