Slip Boundary Condition Implementation for Linear Quantum Lattice Boltzmann Method

The proposed approach extends the QLBM framework of our prior work by implementing the first-order Maxwell slip boundary condition within the collision matrix. The quantum circuit architecture remains unchanged: four quantum registers encode the distribution functions of the D2Q9 lattice model, comprising position registers qx and qy (each requiring log2(n) qubits), a register qf (4 qubits for 9 distribution functions and 2 boundary velocities), and a single ancilla qubit for the LCU implementation. The total qubit count is log2(nx · ny) + 5.

In the classical LBM, the bounce-back boundary condition with wall velocity correction is: f̄_ī(x_b, t) = f*_i(x_b, t) − 2w_i ρ_w (c_i · u_w) / c_s², where x_b is a boundary node, ī denotes the direction opposite to i, ρ_w is the wall density, u_w is the wall velocity, w_i are the lattice weights, and c_s is the lattice speed of sound. To implement the Maxwell slip condition, the effective wall velocity u_w is replaced by a slip-corrected velocity: u_slip = U_ref + α · λ · (∂u/∂y)|_wall.

The wall-normal velocity gradient ∂u/∂y is approximated using a first-order backward finite difference scheme. At the bottom wall this gives (u_1 − u_0)/Δx, and at the top wall (u_{ny} − u_{ny−1})/Δx. In the quantum computing algorithm, rather than computing this gradient explicitly from reconstructed macroscopic velocities, it is expressed implicitly in terms of the distribution functions. Following the same approach used to derive the equilibrium distribution in matrix form, the macroscopic velocity at each node is written as a linear combination of the local distribution functions via the first-moment relation u = (1/ρ₀) Σ_i f_i c_i. Substituting this expression at the two nodes adjacent to each wall gives the velocity gradient directly as a linear function of the distribution functions. As a result, the slip correction term remains linear in the distribution functions and can be incorporated into the collision matrix in the same manner as the wall-velocity correction in the no-slip case. The slip boundary condition is applied at both the top and bottom walls.

The modified collision matrix is assembled once prior to the simulation. It is then decomposed via SVD as A = UΣV†, where U and V are unitary matrices and Σ is a diagonal matrix of singular values. The diagonal factor Σ is implemented using the LCU method: Σ = (1/2)(B1 + B2), where B1 = Σ + i√(I − Σ²) and B2 = Σ − i√(I − Σ²) are both unitary. The streaming step is implemented as conditional cyclic shift operations on qx and qy, controlled by velocity-direction states encoded in qf, and remains fully unitary since all boundary treatment resides in the collision matrix.

The quantum circuit is implemented using Qiskit (version 1.4.2) and executed on the StatevectorSimulator, which deterministically evolves the quantum state without noise. This corresponds to option (b) of the demonstration criteria: implementing and running the quantum algorithm on a quantum simulator.

To validate the slip boundary implementation, we consider plane Couette flow with slip at the top wall. The computational domain consists of nx × ny = 8 × 8 lattice nodes, with periodic boundary conditions in the x-direction. The top wall moves at velocity u_w = 0.02 and the bottom wall is stationary. A prescribed slip length α · λ is applied at both boundaries, and the resulting velocity profile is compared against the analytical solution for Couette flow with slip, which takes the form of a linear velocity distribution shifted by the slip contribution.

The simulation is run for 300 time steps with relaxation time τ = 0.9, grid spacing Δx = 1, and time step Δt = 1. The distribution functions are initialized as f_i = w_i, corresponding to a static flow field with zero velocity and unit density. Because the slip correction is expressed implicitly as a linear function of the distribution functions, the collision matrix is assembled once before the simulation begins and does not require updating at each time step. At each step, the real part of the quantum state vector is extracted and used to reinitialize the quantum state for the subsequent iteration, following the same procedure as in the no-slip case.

The QLBM results agree closely with the analytical solution. The results are also consistent with those obtained from a classical LBM simulation using the same linear equilibrium distribution, confirming that the quantum implementation correctly reproduces the slip velocity profile.

The present QLBM framework demonstrates correct algorithmic behaviour for low-Reynolds-number flows with slip boundary conditions on a state vector simulator. To assess scaling potential, we consider both qubit complexity and circuit depth as functions of problem size.

The total number of qubits required is log2(nx · ny) + 5, reflecting the logarithmic scaling in the number of spatial nodes. This is a key structural advantage over classical methods, which require memory proportional to the number of nodes. For example, a domain of 1024 × 1024 nodes would require only 25 qubits in this formulation, compared to storing over one million distribution values classically.

The circuit depth and gate count scale less favourably than the qubit count. In the no-slip formulation, the circuit depth is approximately 5164 for an 8 × 8 grid (11 qubits) and increases to approximately 20534 for a 16 × 16 grid (13 qubits), with the dominant cost arising from the LCU implementation of the collision operator. Incorporating the slip boundary condition modifies the collision matrix by introducing additional non-zero entries that encode the velocity gradient correction. This increases the complexity of the collision matrix prior to SVD decomposition, and is therefore expected to result in larger singular value spreads and additional gate operations in the LCU circuit compared to the no-slip case. A detailed circuit depth analysis for the slip boundary formulation is left for future work. Importantly, however, the slip correction adds no new quantum operations beyond those already required for the collision step , so the structural overhead is contained.

A credible path toward practical scaling lies in a hybrid quantum–classical strategy, in which the collision and streaming steps are executed on quantum hardware while field reconstruction and reinitialization are handled classically. The current formulation is already structured this way. Replacing classical state vector extraction with mid-circuit measurement and post-selection would reduce classical overhead, though at the cost of decreasing success probability over successive iterations. Amplitude amplification techniques could be applied to counteract this effect.

For large-scale three-dimensional simulations with slip boundaries, the logarithmic qubit scaling of QLBM is expected to provide a meaningful memory advantage over classical solvers once fault-tolerant quantum hardware matures. The present work provides a sound foundation for extending the framework to three-dimensional lattice models such as D3Q19 or D3Q27, higher-order slip models, and non-uniform accommodation coefficients relevant to transitional Knudsen number regimes.