Antonio Bastida Zamora
Quantum Algorithms Researcher
Why a new method?
Quantum lattice gas automata have a unitary collision rule and can emulate nonlinear effects through particle collisions. However, they are noisy because they use Boolean variables, and parameters like viscosity, density, and velocity are difficult to control, as these are macroscopic properties while the method operates at a microscopic scale.
In contrast, the lattice Boltzmann method has been widely applied in computational physics, including CFD, electromagnetics, and other areas. However, its quantum version often requires non-unitary collision operators with nonlinear terms, which cannot be directly encoded on a quantum computer over multiple time steps.
Exploring a new hybrid method could preserve the advantages of both approaches. A hybrid has already been investigated in the literature (Wang et al., 2025). While the results were promising, the goal of achieving an efficient and unitary quantum circuit at a mesoscopic scale over several time steps was not reached. This highlighted the need for a completely new algorithm suitable for both classical and quantum devices.
What is Lattice Gas Automata?
Lattice gas automata are cellular automata in which all interactions between particles are local and simultaneous, obeying mass and momentum conservation. They are a microscopic model in which fictitious particles are represented, collide, and propagate through a lattice.
The large number of particles and collisions needed to achieve low viscosities, noise, and limited control over macroscopic parameters make the model computationally expensive and impractical for practical applications.
Several LGA variants have been studied. One of the most widely used is the Frisch-Hasslacher-Pomeau (FHP) model, which is the simplest model capable of simulating the Navier-Stokes equations (Frisch et al., 1986).
What is the lattice Boltzmann method?
The lattice Boltzmann method was originally developed as an alternative to lattice gas automata, which, although promising, was noisy and computationally expensive. Instead of simulating the collision and propagation of each individual particle, this method calculates the ensemble average of all particles. As a result, the model is no longer microscopic but mesoscopic, effectively simulating the Boltzmann equation.
Rather than using Boolean variables to indicate particle positions, probability density distributions with floating-point numbers are assigned to a lattice site and a channel, similar to lattice gas automata. Over the years, several simplifications, optimizations, and studies have been applied to the lattice Boltzmann method, making it a powerful and widely used model in computational physics today.
What is new with Float Lattice Gas Automata?
This work introduces a new algorithm called FLGA. This method builds on the integer lattice gas automata work by Blommel and Wagner (2018), where they expanded lattice gas automata to achieve the same equilibrium density distribution functions as the Lattice Boltzmann method.
This allows lattice gas automata to achieve the same results as the lattice Boltzmann method while still using microscopic collision rules. Building on this, we went one step further. Similar to the process used to derive LBM from LGA in the 1980s, we developed a method using floating-point numbers at a mesoscopic scale with collision rules inspired by particle interactions.
Fig. 1 Scheme of channels for D2Q9. Each number represents the index used to map each channel
Although the method operates at a mesoscopic scale, it does not directly calculate the density distribution functions from the Boltzmann equation. Instead, it generates them through interactions between probabilities at each channel and lattice site.
Additionally, we analyzed how the model’s control parameters influence the final fluid viscosity and how it behaves when varying the number of channels interacting in each collision.
Finally, we constructed a quantum algorithm based on FLGA. This quantum version produces accurate results compared to its classical counterpart, with low depth and unitary operations.
Fig. 2 Quantum circuit scheme for QFLGA. This general diagram uses two-body collisions, but more body collisions can be implemented by adding additional registers |ci>.
Fig. 3 Quantum circuit from collision operator of QFLGA algorithm
Challenges with existing approaches
Despite the advances in state-of-the-art quantum algorithms for lattice gas automata, which enable particles to collide and propagate simultaneously across all lattice sites, several challenges exist.
Firstly, quantum lattice gas automata share the same issues as their classical counterparts. The high noise level in the algorithm requires repeating simulations multiple times to reduce noise in the results. Furthermore, there is limited control over viscosity, a crucial parameter in CFD. The large number of particles required for practical applications demands many channels and collisions, which complicates the method’s usability. Another challenge to the quantum algorithm is the difficulty in simulating multiple time steps before measurements following collision and propagation (Schalkers & Möller, 2024).
The quantum lattice Boltzmann method has proven reliable, efficient, and straightforward to implement on quantum computers. However, two significant issues remain. First, simulating multiple time steps without intermediate measurements is challenging due to the nature of quantum physics. Quantum computing relies on linear algebra, whereas quantum LBM for CFD, particularly the Navier-Stokes equations, involves nonlinear terms that cannot be simulated beyond a single time step. Second, the collision term in LBM is often non-unitary, lowering the probability of obtaining the correct outcome.
What is value added with Quanscient?
At Quanscient Quantum Labs, we have developed a new classical algorithm called FLGA. Classically, this algorithm achieves the same results as LBM, with comparable or, in some cases, improved efficiency with respect to Bhatnagar Gross and Krook (GBK) approximations. Controlling parameters like viscosity is more challenging since it is not directly assigned as in LBM but emerges from interactions between different channels at each lattice site. The method enables overrelaxation, allowing simulation of very low viscosity values, which was not possible with the original integer lattice gas automata.
FLGA represents a significant advancement and opens a new research area in CFD, expected to inspire further studies. In quantum algorithms for CFD, we achieved the goal of developing a quantum algorithm that is both unitary and efficient. However, because the method operates at a mesoscopic scale, nonlinear terms arise, as in LBM, preventing time-step concatenation.
We presented an alternative method based on approximating channels in neighboring sites, enabling accurate results over two or three time steps. Nonetheless, developing a general algorithm for an arbitrary number of time steps remains future work. However, Quanscient has already demonstrated particular models that can also efficiently concatenate time-steps, as demonstrated in our project with Airbus and Oxford Ionics.
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Case examples
1D shockwave propagation
Simulation objective
Validating the accuracy of FLGA and QFLGA in 1D.
The model
FLGA and LBM and QFLGA vs. FLGA.
Key results
The simulation is in 1D with two walls (right and left) and two different densities that create a shockwave. FLGA shows excellent accuracy compared to LBM, obtaining the same results after several time steps at different viscosities. FLGA is then compared with QFLGA, also obtaining the same results, validating the quantum algorithm with FLGA and therefore matching LBM results.
Fig. 4 Density ρ (upper panel) and velocity ux (lower panel) for a D1Q3 shockwave model using FLGA with T = 1280 and averaging per lattice site of 10.
Fig. 5 Comparison between QFLGA and FLGA for a shockwave in 1D using L= 128 and C = 0.5
Taylor-Green vortex decay
Simulation objective
Validating the accuracy of FLGA in 2D.
The model
FLGA vs. LBM for D2Q9 (2 dimensions and 9 channels).
Key results
The simulation is in 2D, using a particular trigonometric initialization widely used in CFD benchmarks. It has no boundary conditions, an analytical solution, and is very dependent on viscosity. The simulation again shows high accuracy between LBM and FLGA.
Fig. 6 Taylor-Green vortex simulation at different time steps comparing FLGA with the theoretical expression. The figures show the density ρ, X-velocity ux and total velocity |u|= u2 x + u2 y from left to right. The simulation relaxation time τ = 12 was obtained with C = 0.3.
Lid-driven cavity flow
Simulation objective
Validating the accuracy of FLGA using three interacting channels in 2D in complex cases with boundary conditions, where setting up the same viscosity in FLGA and LBM is harder.
The model
FLGA (with three interacting channels, previously we had two) vs. LBM for D2Q9.
Key results
The simulation is in 2D, with left, bottom, and right walls and a lid on top that adds velocity to the particles above. This is a common benchmark in CFD due to the nonlinear dynamics produced and its simple yet multiple boundary conditions. In this case, we tested FLGA vs. LBM for very low viscosities (high Reynolds number), using three interacting channels in the FLGA collisions. Despite obtaining similar results, we observe some discrepancies. These differences arise from the different viscosities in LBM with the BGK simplification used in the simulations, compared to the generated viscosities by FLGA.
Fig. 7 Lid-driven cavity test using FLGA, where density ρ, x-velocity ux, and velocity field streamlines are shown compared with LBM. The simulation for FLGA time is T = 20000 with a factor C2 = 0.2 and C3 = 1.23, lattice sites L= 100, and ux = 0.2 for the upper wall. For LBM, similar parameters are chosen, setting the Reynolds number to Re= 1100.
Insights of the model
Additional simulations were conducted during this work. These focused on understanding how the FLGA control parameter, which governs interaction strength, relates to viscosity. We also examined how viscosity dependence varies when increasing the number of particle interactions. These studies compared known LBM viscosities with FLGA results using a Taylor-Green vortex decay benchmark. The approach was entirely heuristic.
Key benefits of the model
The model demonstrated high accuracy compared to LBM, greater flexibility than LGA in generating various viscosities, and efficiency comparable to LBM. It also revealed how viscosity depends on the model’s control parameter.
The quantum algorithm achieved unitary collision rules, low circuit depth, and accurate results.
QFLGA shows strong potential to simulate multiple time steps using approximations or alternative methods like Carleman linearization.
Conclusion
We have developed a new classical algorithm as an intermediary step between LGA and LBM. Following previous work on integer lattice gas automata, we expanded the method to a mesoscopic scale by using an ensemble average with floating-point numbers.
Unlike previous work, the new method is faster, more efficient, and noiseless. We validated the results compared with LBM, obtaining high accuracy.
Following a heuristic approach, we studied how the viscosity of the model differs when changing the interaction strength (effective collision rate) between probability distribution functions. After developing the classical algorithm, we constructed a new unitary quantum circuit based on FLGA for 1D with low depth. We validated the algorithm compared to classical FLGA for a shockwave simulation, obtaining accurate results.
References
Blommel, T., & Wagner, A. J. (2018). Physical Review E, 97, 023310. https://doi.org/10.1103/PhysRevE.97.023310
Frisch, U., Hasslacher, B., & Pomeau, Y. (1986). Lattice-gas automata for the Navier-Stokes equation. Physical Review Letters, 56(14), 1505. https://doi.org/10.1103/PhysRevLett.56.1505
Schalkers, M. A., & Möller, M. (2024). On the importance of data encoding in quantum Boltzmann methods. Quantum Information Processing. https://arxiv.org/abs/2302.05305
Wang, B., Meng, Z., Zhao, Y., & Yang, Y. (2025). Quantum lattice Boltzmann method for simulating nonlinear fluid dynamics [Preprint]. arXiv. https://arxiv.org/abs/2502.16568
