A quantum-compatible linearized collision model for lattice Boltzmann method

In this study, we propose a novel linearized collision model for the lattice Boltzmann method that is explicitly designed for compatibility with quantum computation. Unlike prior work that applies linearization directly to the discrete collision operator, our approach starts from the underlying kinetic equations and derives a physically consistent linear approximation that can be embedded into a quantum algorithmic framework.

The resulting model reformulates the evolution of distribution functions into a linear operator form, which can be mapped onto unitary or block-encoded operators acting on quantum registers. Time evolution is implemented through a sequence of linear transformations corresponding to streaming and collision processes, where the collision step is represented by the proposed linearized operator.

At this stage, we focus on validating the formulation using classical computation. Specifically, we implement the proposed model as a classical algorithm to verify consistency with known LBM behavior. The proposed LBM model, in its quantum algorithm formulation, is also simulated on classical hardware.

The capability of the proposed quantum-compatible LBM formulation is assessed through a combination of classical numerical experiments and quantum simulations, providing a consistent framework for both physical validation and algorithmic verification.

First, we implement the linearized collision model as a classical algorithm and systematically compare its behavior against standard LBM benchmarks for representative flow problems. This step ensures that the proposed linearization preserves essential physical characteristics, including stability, conservation properties, and qualitative flow dynamics, while maintaining consistency with established LBM formulations.

Next, the proposed linearized LBM is reformulated as a quantum algorithm and evaluated via quantum simulations on classical hardware. These simulations confirm that the quantum algorithm accurately reproduces the expected system evolution within numerical accuracy and retains the key dynamical features observed in the classical formulation.

From a complexity perspective, the linearization of the collision operator may enable the use of existing quantum linear algebra techniques (e.g., block encoding) which can offer exponential or polynomial speedups in certain regimes.

In addition, unlike existing approaches that rely on ad hoc linearization applied directly to the discrete collision term, the present model is derived from a physically grounded reformulation of the underlying kinetic equations. As a result, the discretized system is expected to exhibit improved numerical stability and consistency with the original governing equations. This enhanced stability is particularly advantageous when scaling to larger problem sizes, as it can mitigate the accumulation of numerical errors and relax constraints on the time step or resolution, thereby enabling more robust large-scale simulations.

While the current work focuses on proof-of-concept validation using classical simulations, the formulation is designed to be directly transferable to quantum hardware as it matures. The explicit construction of linear operators compatible with quantum circuits provides a clear pathway for implementation on future fault-tolerant quantum computers. Furthermore, the improved stability and physical consistency of the proposed model, combined with its compatibility with quantum linear algebra techniques, suggest favorable scaling properties compared to existing methods.