Benchmarking One-Dimensional Quantum Lattice Boltzmann Methods on Quantum Hardware

Abstract

Classical computational fluid dynamics (CFD) methods encounter significant computational bottlenecks in large systems and at high Reynolds number flows due to the rapid increase in degrees of freedom, even on flagship supercomputers. As quantum computers emerge, the Quantum Lattice Boltzmann Method (QLBM) has been proposed as a quantum-native alternative for solving fluid dynamics problems. However, until recently, nearly all studies have relied on classical quantum simulators rather than real quantum hardware. This study presents a benchmark of a one-dimensional QLBM implementation executed on three different IBM Quantum hardware. We design and test a quantum circuit for the advection-diffusion equation using amplitude encoding, a Bhatnagar-Gross-Krook (BGK)-type collision operator, and a streaming step with periodic boundary conditions. Our numerical experiments evaluate transpiled two-qubit gate count, total circuit depth, runtime, and accuracy metrics including maximum absolute error ($L_\infty$ norm) and root-mean-square error (RMSE). We find that the present collision and streaming operators dominate the overall cost and become the primary scalability bottleneck on current hardware. To mitigate the prohibitive cost of full quantum state tomography on hardware-based benchmarking exercises, we introduce a sampler-based measurement strategy that reconstructs macroscopic density fields directly from computational-basis measurements. This approach substantially reduces measurement overhead while preserving the expected advection-diffusion dynamics on hardware. These results provide a quantitative assessment of QLBM performance in the noisy intermediate-scale quantum (NISQ) regime and identify the principal algorithmic and experimental limitations that must be addressed for QLBM to become viable for fluid dynamics.

This paper benchmarks a one-dimensional quantum lattice Boltzmann implementation of advection–diffusion on three IBM processors: Eagle-based `ibm_rensselaer` and Heron-based `ibm_kingston` and `ibm_boston`. It examines how circuit resources, execution time, and reconstructed density accuracy change with lattice size and hardware generation. The implementation uses D1Q3 populations, square-root amplitude encoding, an ancilla-assisted linear combination of unitaries for a BGK-type collision step, and velocity-controlled streaming with periodic boundaries. Computational-basis sampling replaces full state tomography; outcomes are post-selected on the collision ancilla and processed into density fields. Experiments use $100{,}000$ shots per case and compare hardware densities with ideal statevector results. Hardware tests cover one timestep on $M=4$–$64$ sites using 5–9 qubits. Transpiled depth rises from 152 to 34,300, while two-qubit gate counts rise from 61 to 14,316, described as approximately $O(M^2)$ scaling. Streaming dominates at smaller sizes, while collision becomes the largest resource contribution at $M=64$. Heron processors generally yield lower errors and shorter runtimes: at $M=64$, execution takes 321–364 seconds versus 2,159 seconds on Eagle. Density RMSE increases from 0.109–0.273 on Heron at $M=4$ to approximately 1.67 at $M=32$. The stated limitations include deep circuits, noise-driven loss of density structure and mass, and errors from post-selection, rescaling, and re-encoding between timesteps. The paper identifies approximately $M=32$ as a practical resolution limit for this implementation. The 150-timestep examples use classical quantum simulation, not hardware; multidimensional and nonlinear extensions remain future work.