Quantum Lattice Boltzmann with Denoising Collision Operators

The method is a quantum Lattice Boltzmann algorithm based on amplitude encoding of the particle distribution functions. The quantum state is defined over a spatial register and a velocity register. At each lattice site, the local velocity populations are encoded through square-root amplitudes, so that measurement probabilities reproduce the classical LBM populations. We use a one-hot encoding for the velocity register, which simplifies the control logic required for streaming and boundary operations.

The main methodological contribution is the denoising collision operator. Instead of computing the nonlinear equilibrium distribution through quantum arithmetic or classical processing, we reformulate the collision step geometrically as an orthogonal projection onto a local linearization of the equilibrium manifold around a reference velocity. This projector removes non-equilibrium components while preserving the relevant hydrodynamic structure. The construction depends on a selected reference velocity, and the resulting collision error is controlled by the mismatch between this reference velocity and the actual macroscopic velocity.

Since the denoising collision operator is non-unitary, it is embedded into a unitary operator using block encoding with an ancilla. The collision circuit is then combined with quantum streaming, implemented as velocity-controlled shift operations on the spatial register. The bounce-back boundary conditions are implemented by integrating collision, streaming, solid-node queries, and velocity reversal into a single LBM update. Physical quantities, such as hydrodynamic force on a solid body, are extracted by measuring position-velocity observables and averaging the corresponding sampled values.

We provide a full circuit implementation of the quantum LBM algorithm with gate-level constructions for collision, streaming, and bounce-back boundary conditions. These constructions are decomposed into single- and two-qubit basis gates executable on standard quantum simulators.

Numerically, the algorithm is tested on four representative problems: one-dimensional advection-diffusion with a Fourier-mode velocity, two-dimensional advection-diffusion of a Gaussian hill, Taylor-Green vortex decay, and flow around a circular cylinder. The simulations are compared against classical LBM and, where available, analytical solutions. All experiments are run for up to 10,000 timesteps, and accuracy is quantified using the relative error in the macroscopic fields

The advection-diffusion tests show that the method captures advective transport accurately, although diffusion can be underdamped in diffusion-dominated regimes. The Taylor-Green vortex test demonstrates that the method reproduces the temporal decay trend, but the error is sensitive to the chosen reference velocity. The cylinder-flow experiment further confirms this point: using an empirically motivated reference velocity produces results comparable to classical LBM and improves both accuracy and stability. These demonstrations support the practical validity of the denoising collision mechanism while identifying the reference velocity as a key parameter controlling performance.

The proposed quantum LBM offers a clear scaling motivation. For a lattice with N nodes, the spatial domain is encoded in a logarithmic number of qubits, O(logN), while the velocity register has size determined only by the chosen lattice stencil. In standard LBM configurations this is a small constant: for example, D3Q27 requires 27 velocity qubits in the one-hot encoding. Thus, increasing the spatial resolution  enlarges the spatial register logarithmically, rather than requiring one degree of freedom per grid point as in a direct classical representation.

The approach is not tied to a specific benchmark geometry or problem setting. It applies to advection-diffusion and hydrodynamic flow problems in one, two, and three spatial dimensions, provided an appropriate LBM lattice is chosen. From the circuit perspective, the collision cost scales quadratically with the number q of discrete velocities. Since q is fixed for standard lattices, this remains independent of the number of spatial nodes. The streaming step is implemented by controlled shifts on the spatial register, so its cost scales mainly with the number of spatial qubits, i.e., logarithmically in the number of lattice nodes for standard quantum adders.

A realistic route to scaling is a hybrid quantum-classical strategy. The accuracy of the denoising collision depends on the reference velocity, and this velocity may vary across space and time. Instead of choosing a single global reference, a classical solver could provide a coarse space-time-resolved velocity field at low spatial and temporal resolution. This reference field could then be interpolated and supplied to the quantum LBM simulation, while the quantum circuit evolves the high-resolution distribution state. This would use classical hardware for inexpensive macroscopic guidance and quantum hardware for the fine-scale lattice evolution. This makes the method a promising framework for scalable quantum CFD, although a full advantage claim still requires further work on dynamic reference selection and end-to-end resource estimates.