S4 - Computational Fluid Dynamics II

Aeroacoustic noise prediction is a standard industrial computation. Jet noise, airframe noise, fan noise, and combustion noise are all predicted within the Lighthill framework. A separately computed turbulent flow acts as a prescribed monopole, dipole, or quadrupole source, and the radiated sound propagates through the linearized acoustic Euler equations. Statistical formulations of this propagation problem evolve an ensemble of acoustic states simultaneously on a phase-space grid. The storage of that grid grows as the number of grid points per axis raised to the phase-space dimension. At the resolution our bounds establish, the classical state vector grows from gigabytes in six phase-space dimensions to exabytes in eight. A quantum register holding the same information needs only tens of qubits up to dimension ten.


The Koopman–von Neumann (KvN) framework makes this register representation possible. It lifts the classical dynamics to a unitary wavefunction evolution that a quantum computer can execute. The concrete problem we solve is the missing quantitative link in that pipeline. The engineering observables, namely the mean radiated field, the acoustic intensity, the energy, and the root-mean-square amplitudes, are all built from the first and second moments of the wavefunction. We ask how fine the phase-space mesh must be, and therefore how many qubits are needed, to compute these moments to a prescribed accuracy. Without rigorous discretization-error bounds, every hardware estimate for this class of quantum simulation rests on guesswork. We replace the guesswork with closed-form bounds, a design rule, and an executable circuit whose measured error matches the analysis.

In this work, we introduce a reduced-basis algorithm (RBA) for solving polynomial nonlinear differential equations on quantum computers. Solving these types of equations remains one of the most challenging tasks in developing quantum algorithms that can be useful for real-world applications. This is because quantum evolution is governed by linear dynamics, while many problems of interest in the physical world, including fluid flow, transport, reaction-diffusion processes, population dynamics, and plasma physics, are inherently nonlinear. Therefore, if we want to simulate these kinds of systems using quantum computers, we need algorithms that can handle nonlinearity directly. Our algorithm is designed to simulate discretized polynomial nonlinear ODEs and PDEs exactly. More precisely, the nonlinear dynamics of the chosen discrete scheme are recovered exactly, so the only error comes from the discretization itself, as it would in a classical numerical simulation. The relevance of this algorithm is therefore broad, since it can be applied to many different types of nonlinear ODEs and PDEs. The method is especially relevant for fluid dynamics, where nonlinear PDEs such as the Navier-Stokes equations, Burgers-type equations, and lattice Boltzmann-type formulations often involve local spatial discretization. This locality can be exploited to reduce the resource overhead of the algorithm, since the evolution at each grid point depends only on a local stencil. This makes the approach promising for future quantum-assisted simulations of nonlinear fluid and transport problems.