S8 - Computational Fluid Dynamics IV

Due to the fact that quantum computers have the potential to produce a significant gain in computational performance, their features have been used to develop several solution strategies for partial differential equations (PDEs). Among the different types of PDEs, the advection equation is of particular interest, since it can be used to describe the transport of a substance in advection-dominated flow. Some of the approaches that have been developed recently try to compute the concentration of a substance using Hamiltonian simulations [1] and variational quantum algorithms [2]. In this talk, we consider the Quantum Lattice Boltzmann Method (QLBM) to solve the advection equation. The QLBM is a promising method for handling the advection equation, since it approximates the solution of a PDE (the Boltzmann equation) containing the advection operator and whose solution variable is a distribution function. Thus, the solution variable can be used to compute a concentration of a certain substance. Moreover, the solution variable can be encoded within the amplitudes of a quantum state. This allows us to represent the solution values stored in the cell centers of high-dimensional grids in a very efficient way [3]. Despite this feature, there are still many challenges in maintaining the potential advantages of this solution strategy. One of them is to determine the concentration values or the particle densities in a computational domain or sub-regions of a computational domain. If the absolute values of amplitudes occurring in a quantum state are relatively low, one might have to perform many measurements to recover the corresponding concentrations or particle densities with sufficient precision. A main objective of this talk is to present an efficient method that can be used to recover particle densities in a given sub-region of the computational domain.

The lattice Boltzmann method (LBM) is widely used for simulating fluid dynamics due to its simplicity and strong parallelizability. However, its collision operator inherently contains nonlinear terms, which pose a fundamental obstacle to efficient implementation on quantum computers. Quantum computation frameworks are naturally suited to linear operations described by unitary transformations, making direct encoding of nonlinear dynamics highly nontrivial. Existing approaches have attempted to circumvent this issue by introducing ad hoc linearizations of the collision term, often applied at the level of numerical schemes rather than derived from the underlying physical model. As a result, these methods may compromise physical interpretability, numerical stability, or extensibility to more complex systems.


The problem addressed in this work is the construction of a collision model for LBM that is intrinsically compatible with quantum computation, while preserving the essential physical structure of the original kinetic formulation. By revisiting the physical foundations of LBM and reformulating its nonlinear components into a linearized representation suitable for quantum algorithms, this work aims to bridge a critical gap between classical computational fluid dynamics and emerging quantum-assisted computing technologies.

We propose a quantum algorithm for fluid simulation based on the Lattice Boltzmann method (LBM). LBM is widely used in computational fluid dynamics because it evolves particle distribution functions on a discrete lattice through local collision and streaming steps. This structure combines simple update rules with high parallelism and geometric flexibility, making LBM attractive for large-scale fluid solvers. However, its quantum implementation remains challenging: while the streaming step can be represented naturally by unitary operators, the collision step is nonlinear and irreversible, and therefore is not compatible with coherent quantum operations.
Existing quantum LBM formulations often address this difficulty using tomography and repeated state preparation at each timestep. These procedures break coherence and introduce substantial overhead, making scalable multi-timestep quantum fluid simulation difficult. The concrete problem addressed in this work is therefore to construct a quantum-compatible collision mechanism that avoids tomography-based updates and repeated re-preparation, while still preserving the essential nonlinear structure needed to approximate fluid dynamics.
Our approach introduces a collision model designed for coherent quantum implementation and applies it to both hydrodynamic flow and advection-diffusion problems. This provides a step toward fully quantum LBM schemes in which collision, streaming, and boundary treatment can be combined into a circuit-level framework.