S2 - Computational Fluid Dynamics I

The lattice Boltzmann method (LBM) is an established explicit time-marching scheme in computational fluid dynamics (CFD). Its success is rooted to the fact that it is relatively simple, resulting in flexibility for the implementation and the advantage that it is relatively easy to parallelize on classical hardware. Also in particular because of this, it is regarded as a promising method for an implementation via quantum computing (QC) with the aim of achieving faster algorithms [1, 2].
However, the treatment of non-linear problems like the BGK collision operator in the LBM with an equilibrium distribution function that is non-linear w.r.t. the fluid velocity is challenging for an implementation as a quantum circuit, which is why the mentioned example could be realized in previous quantum algorithms for the LBM only with relatively high computational efforts [3]. In particular, for the use of the common amplitude encoding, the circuit depth scales in general exponentially in the number of involved qubits, i.e. linearly in the number of grid points [4].
Our contribution presents an alternative encoding concept for the aim to implement the LBM via QC for generic and thus also non-linear problems.

High-fidelity simulation of complex transport phenomena remains a central challenge in computational fluid dynamics. Direct Numerical Simulation (DNS) can resolve fluid behaviour with high accuracy, but its extremely large computational and memory requirements make it impractical for large-scale or industrial applications on classical hardware.
Quantum Computing is a potential way to reduce computational cost while keeping accuracy. However, quantum-assisted fluid dynamics still faces hardware limits, shallow circuit depths, and difficulty in representing nonlinear physical effects within quantum circuits. Consequently, many current methods must simplify physics, use fixed or constrained parameters, or remain limited to low-complexity flow regimes, which reduces their practical applicability.
The Lattice Boltzmann Method (LBM) is promising for quantum implementation due to its structured, locally defined update rules. Despite this, a major bottleneck arises with the collision operator. This component is inherently nonlinear and non-unitary, making it hard to implement directly on gate-based quantum computers.
The core challenge is both computational and physical. Efficiently and accurately capturing nonlinear fluid interactions at scale within a quantum framework is very difficult. Addressing this challenge has significant practical value: it would enable more efficient and accurate simulation of complex flows in engineering and science. For this reason, quantum-assisted formulations are motivated by the need to overcome the limits of classical computation while remaining compatible with near-term quantum hardware. This represents a key step toward practical quantum advantage in fluid dynamics simulation.

Accurate treatment of boundary conditions is a central requirement in computational fluid dynamics (CFD). For flows at the macroscale, the no-slip condition at the wall is well established. However, this assumption breaks down in microscale or rarefied flow regimes, where the mean free path of fluid molecules becomes comparable to the characteristic length of the flow domain. The degree of flow rarefaction is described by a non-negligible Knudsen number, Kn = λ/L, where λ is the molecular mean free path and L is the characteristic flow length. In such regimes, a velocity slip occurs at the boundary, and the first-order Maxwell slip boundary condition must be applied: u_wall = U_ref + α · λ · (∂u/∂y)|_wall, where U_ref is the prescribed reference wall velocity, α is the tangential momentum accommodation coefficient, λ is the mean free path, and ∂u/∂y is the wall-normal velocity gradient evaluated at the boundary. This boundary condition is widely used in microfluidics and flow in porous media, where slip effects are physically significant and must be correctly captured for accurate simulation results.
The Quantum Lattice Boltzmann Method (QLBM) is a promising approach to quantum CFD that maps the classical Lattice Boltzmann Method (LBM) onto a quantum computer. The LBM simulates fluid flows by tracking the evolution of discrete particle distribution functions on a regular lattice through alternating collision and streaming steps. Its inherently linear structure in the low-Reynolds-number regime makes it well suited for quantum implementation. Our previous work established a QLBM framework based on linear equilibrium distribution functions, where the collision operator was expressed as a matrix multiplication and decomposed using Singular Value Decomposition (SVD). The no-slip bounce-back boundary condition was incorporated directly into the collision matrix, preserving the unitarity of the streaming step.
Extending this framework to support slip boundary is an important step toward quantum simulation of practical microfluidic flows. The core challenge is that the slip condition introduces a dependence on the local wall-normal velocity gradient, which must be expressed consistently within the linear algebraic structure of the quantum circuit. This work addresses that challenge by integrating the first-order Maxwell slip condition into the collision matrix of the QLBM, thereby preserving the overall structure of the quantum algorithm while extending its physical applicability to near-rarefied flow regimes.

Quantum lattice Boltzmann methods (QLBM) offer a promising framework for simulating Computational Fluid Dynamics (CFD) using quantum circuits. Recent work has introduced a zone-agnostic (ZA) method for implementing boundary conditions for complex shapes, without iterating through each segment of the boundary individually. Despite claims of improved asymptotic scaling, that work has not been proven for any boundaries complex enough to be of any relevance to practical applications. This becomes a bottleneck when dealing with irregular, data-defined, or evolving boundaries commonly encountered in engineering applications. This paper, therefore, explores replacing explicitly constructed geometric oracles with data-driven models. By embedding learned decision functions into quantum circuits, we aim to reduce computation overhead and enable flexible handling of complex geometries. The approach is relevant for scaling QLBM to real-world applications in cases where analytical descriptions of the geometry are either undefined or too expensive to implement by means of quantum arithmetic (e.g., [1, 2, 3]).