Algorithmic Advances Towards a Realizable Quantum Lattice Boltzmann Method

TL;DR

This work addresses the key barriers to realizing the Quantum Lattice Boltzmann Method on real quantum hardware by introducing a suite of algorithmic and architectural innovations. It combines tensor-network (MPS) loading for efficient initial-state encoding, an LCU-based collision step, a one-hot streaming encoding to reduce circuit depth, and observable-based readout with robust error mitigation to enable hardware demonstrations. The authors validate their approach with a hardware run of a 2D advection-diffusion problem on a $16×16$ grid using IonQ Forte, achieving high fidelity despite shallow circuit depths, and extend the methodology to 3D and non-uniform velocity fields through simulations and generalized circuit constructions. These advances collectively establish a practical pathway toward scalable, quantum-enabled CFD solvers on near-term devices, with potential industrial impact once larger qubit counts and longer coherence times become available. The mathematical core relies on the advection-diffusion PDE $∂_t Φ = D ∇^2 Φ - ∇·(uΦ)$ and the LBM update rules, now implemented via hardware-friendly encodings and probabilistic readouts, all wrapped in a framework capable of sustaining multiple timesteps per circuit while maintaining tractable success probabilities.

Abstract

The Quantum Lattice Boltzmann Method (QLBM) is one of the most promising approaches for realizing the potential of quantum computing in simulating computational fluid dynamics. Many recent works mostly focus on classical simulation, and rely on full state tomography. Several key algorithmic issues like observable readout, data encoding, and impractical circuit depth remain unsolved. As a result, these are not directly realizable on any quantum hardware. We present a series of novel algorithmic advances which allow us to implement the QLBM algorithm, for the first time, on a quantum computer. Hardware results for the time evolution of a 2D Gaussian initial density distribution subject to a uniform advection-diffusion field are presented. Furthermore, 3D simulation results are presented for particular non-uniform advection fields, devised so as to avoid the problem of diminishing probability of success due to repeated post-selection operations required for multiple timesteps. We demonstrate the evolution of an initial quantum state governed by the advection-diffusion equation, accounting for the iterative nature of the explicit QLBM algorithm. A tensor network encoding scheme is used to represent the initial condition supplied to the advection-diffusion equation, significantly reducing the two-qubit gate count affording a shorter circuit depth. Further reductions are made in the collision and streaming operators. Collectively, these advances give a path to realizing more practical, 2D and 3D QLBM applications with non-trivial velocity fields on quantum hardware.

Figures (10)

• Figure 1: An overview of the full QLBM pipeline. The mesoscopic D2Q5 lattice Boltzmann formulation is used to simulate the macroscopic fluid dynamics. The quantum algorithm a smooth density distribution as input (in this case a 2D Gaussian distribution) and applies T iterations of the QLBM quantum circuit. The input distribution loaded into the amplitude of the $\log(N)$ grid qubits, $\ket{0}_G$, using a MPS state preparation circuit. The collision operator is prepared efficiently using a unary state preparation circuit on the $M$ "direction qubits", $\ket{0}_D$, as described in the text. The streaming step acts as a series of operations on the grid qubits which are controlled on the one-hot encoded direction qubits. Additional ancilla qubits, $\ket{0}_A$, are used to reduce the total gate count of the multi-control unitary gates. Additional ancilla qubits are also used to detect certain errors which occur during the circuit execution.
• Figure 2: a) The time evolution of a 2D Gaussian density distribution for 10 time steps using the QLBM algorithm evaluated using an ideal simulation (top), and the error mitigated wave function executed on the IonQ Forte quantum computer (bottom). b) The fidelity, $|\langle \psi | \phi\rangle|$, between the density distribution measured from the QPU and the classical LBM solution (orange curve). This fidelity is further improved by reconstructing a 2D Gaussian distribution using observables measured from the wave function, where over $92\%$ fidelity is achieved at all time steps (blue curve).
• Figure 3: The two-qubit gate count of the streaming operator as a function of lattice grid size for the D2Q5 advection-diffusion model using the prior compilation technique from literature, the improved compilation with ancilla qubits, and using our novel one-hot encoding scheme.
• Figure 4: We show the fidelity of the quantum states obtained from the QLBM circuit using shot based simulator and compare between full state tomography (qst) and observable based reconstruction (obsv). Fidelity is calculated based on the ideal simulation of the above strategies and the output of statevector simulation.
• Figure 5: ( left) The share of total shots measured from the QPU backend for each time-step, which are in: the perpendicular subspace (green), the good subspace but with flagged errors (blue) and the good subspace with no detected errors (red). Black bars show the number of usable shots in the ideal noiseless case. (upper right) The density distribution after one timestep using only the usable shots. (lower right) The difference in the density before and after the flagged errors are removed.