Quantum Lattice Boltzmann Solutions for Transport under 3D Spatially Varying Advection on Trapped Ion Hardware

Abstract: The Quantum Lattice Boltzmann Method (QLBM) has emerged as one of the most promising quantum computing approaches for the numerical simulation of problems in computational fluid dynamics (CFD). The dynamics is formulated in terms of mesoscopic particle distribution functions governed by a discrete Boltzmann transport equation, comprising local streaming and collision operations. In this work, the resulting macroscopic behavior corresponds to the advection-diffusion equation, which we adopt as a canonical model problem for transport phenomena. Building upon recent progress in QLBM implementations, we advance towards more realistic problem settings that better reflect conventional CFD requirements. We address, for the first time, transport under the action of non uniform velocity fields on quantum hardware. We implement our demonstration using IonQ's trapped-ion systems including Forte generation systems and a 64-qubit Barium development system similar to the forthcoming IonQ Tempo line. We identify the density readout and subsequent reloading of the fluid density as a potential bottleneck of the current algorithm and discuss several approaches to mitigate this bottleneck. We identify the use of MPS shadow tomography as a promising method to efficiently scale the readout to large system with complex density distributions. Lastly, we introduce and simulate a novel method to implement wall boundaries for advection-diffusion in QLBM, and discuss the prospects of scaling to higher-complexity problems.

• The paper implements the Quantum Lattice Boltzmann Method on trapped-ion hardware to simulate 3D advection-diffusion with single-step fidelities up to 0.97 on 512-grid sites.
• It introduces modular quantum circuits for streaming, collisional relaxation, and error detection, combined with MPS smoothing and shadow tomography for efficient state reconstruction.
• The approach exhibits inherent noise resilience and scalable boundary handling, paving the way for hybrid quantum-classical CFD applications.

Quantum Lattice Boltzmann Simulations of 3D Advection-Diffusion on Trapped-Ion Quantum Hardware

Introduction and Context

The paper "Quantum Lattice Boltzmann Solutions for Transport under 3D Spatially Varying Advection on Trapped Ion Hardware" (2604.28121) addresses the quantum simulation of transport phenomena governed by the advection-diffusion equation, applied to three-dimensional, spatially varying flow fields. The work leverages the Quantum Lattice Boltzmann Method (QLBM), a mesoscopic approach where local streaming and collision rules discretize transport equations, encoding them efficiently in quantum circuits. Unlike quantum algorithms for linear systems or variational PDE solvers, QLBM is inherently suited for grid-based problems and demonstrates potential logarithmic scaling in quantum resources with system size.

This study presents the first implementation of QLBM for 3D transport under nonuniform velocity fields on actual quantum hardware, specifically IonQ’s state-of-the-art trapped-ion platforms, and discusses algorithmic and implementation innovations required for high-fidelity, scalable simulation.

Algorithmic Advances and Circuit Construction

The QLBM framework was realized for both D2Q5 and D3Q7 Boltzmann models, targeting advection-diffusion under divergence-free velocity fields and using BGK-type collision operators ($\tau = 1$). The quantum circuit design is modular: the "grid" and "direction" registers enable efficient mapping of position and discrete velocity states. The core operations involve:

• Streaming Operator ($U_s$): Encodes translation of particle distributions between adjacent lattice sites for each velocity direction.
• PREP/UNPREP Operators: Handle collisional relaxation towards equilibrium using log-depth quantum circuits (mainly via MPS-based state preparation).
• Error Detection Gadget: Ancilla-assisted checks mitigate the impact of hardware-induced errors, especially bit-flips in the direction register.

Crucially, the circuits exploit the regularity and locality of LBM update rules, allowing relatively shallow quantum depths for nontrivial system sizes. For each explicit time step, the quantum state is measured, post-processed, and reloaded—a procedure central to both current NISQ-era hardware limitations and iterative evolution.

Efficient Measurement, Readout, and Scaling

A principal challenge in NISQ QLBM implementations is the iterative requirement for high-fidelity state reconstruction from projective measurements, which rapidly becomes resource-intensive as the system size increases. The paper introduces a hybrid chain of classical postprocessing techniques:

• Kernel Density Estimation (KDE): Smooths measurement noise and statistical fluctuations in sampled density distributions.
• Matrix Product State (MPS) Smoothing: Compresses and denoises quantum data, exploiting the empirical low entanglement of fluid transport solutions over moderate time horizons.
• Classical Shadow MPS Tomography: Extends state reconstruction to larger lattices by combining randomized measurement bases with tensor-network fitting, enabling robust learning of MPS state representations even with limited shot counts.

Strong numerical results include single-time-step fidelities up to $0.97$ for 3D simulations on $8 \times 8 \times 8$ grids ($N=512$) and persistent fidelities above $0.88$ after six time steps, despite the use of circuits with over 300 two-qubit gates on 21 qubits. Employing shadow-MPS tomography, the authors demonstrate that even with $N=4096$ lattice sites, accurate density reconstruction (fidelity $\gtrsim 0.89$) is feasible with $20\,000$ shots, a significant reduction in resource demands compared to direct computational-basis tomography.

Hardware Demonstration and Noise Robustness

The work provides a detailed experimental methodology, reporting on simulations carried out using IonQ's Yb$^{+}$ Forte and prototype $U_s$0-qubit Barium systems (precursors to the IonQ Tempo architecture). These architectures feature individualized optical addressing and sophisticated error-mitigation schemes, enabling extended circuit depths compatible with QLBM workflows.

A notable finding is the inherent noise resilience of the QLBM algorithm, underpinned by the specific one-hot encoding and logical subspace structure: most bit-flip errors in the direction register are projected into orthogonal subspaces and filtered during the measurement post-selection. This error structure, combined with tensor-network-based state preparation and readout, yields robustness even in the presence of hardware noise and shot limitations.

Algorithmic Extensions: General Velocity Fields and Wall Boundaries

To generalize beyond the demonstrated swirl flow, the authors introduce efficient procedures for encoding arbitrary divergence-free velocity fields using sparse Walsh-Hadamard (WHT) transforms and linearly interpolated QPIXL quantum circuits. These enable O$U_s$1 complexity for grid coarsening factor $U_s$2 and system size $U_s$3, suitable for smooth fields.

Furthermore, the paper presents a method for enforcing wall boundary conditions critical for practical CFD applications. By augmenting the circuit with a "wall register" and controlled rotations, the algorithm efficiently blocks streaming in directions adjacent to solid objects, enabling simulation of flows around complex geometries (e.g., species transport from a suspended cube in a pipe).

Numerical and Fidelity Benchmarks

Key numerical benchmarks and claims include:

• High single-step fidelities: Up to $U_s$4 on quantum hardware with $U_s$5 for QLBM advection-diffusion.
• Scalability: Shadow-MPS tomography enables stable evolution and reconstruction up to $U_s$6 lattice sites with fidelities $U_s$7 using $U_s$8 random basis settings, outperforming direct histogram-based methods, especially as lattice size or evolution time increases.
• Boundary handling: Wall conditions are enforced in a manner that does not increase quantum resource scaling with domain size; operations depend only on local wall information and directionality.

The paper highlights that, for the studied class of flows, density fields remain well compressed by fixed-bond-dimension MPS representations throughout the simulation window, supporting efficient quantum-classical workflows.

Implications and Future Directions

This work presents a comprehensive advancement towards practical quantum simulation of classical fluid transport processes under realistic conditions. The integration of hardware demonstrations, advanced state reconstruction, and boundary handling sets a foundation for hybrid quantum-classical CFD tools.

Practical implications include:

• Near-term feasibility: The demonstrated robustness and scaling suggest that NISQ devices can meaningfully contribute to CFD for moderate problem sizes, especially in parameter studies or scenarios where classical simulation is bottlenecked by memory or circuit depth.
• Workflow integration: The modular construction and explicit treatment of boundary and measurement protocols lay groundwork for incorporating quantum algorithms into complete CFD pipelines, with quantum processing handling the computational core and classical routines managing data movement and pre-/post-processing.

Theoretically, the demonstrated error resilience and data compression via MPS suggest that QLBM could be extended to address non-equilibrium, multi-phase, or turbulent phenomena, especially as quantum hardware scales and error correction matures.

Conclusion

The paper establishes the first quantum hardware realization of 3D QLBM for advection-diffusion in spatially varying flows, with innovations in measurement-driven evolution and scalable state reconstruction. The demonstrated methods—MPS-based loading, shadow tomography, and quantum circuit techniques for complex boundaries—address key bottlenecks in existing quantum CFD proposals. The results mark a significant technical step towards quantum-enabled simulation of fluid transport, informing both future algorithmic research and the practical design of hybrid classical-quantum engineering workflows (2604.28121).