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Rotation Collision based Quantum Lattice Boltzmann Methods

This paper proposes a novel unitary quantum lattice Boltzmann method that resolves the fundamental challenge of the non-unitary BGK collision operator by introducing a rotation-based collision operator, for which gate-level quantum circuits are constructed and validated for both convection-diffusion and incompressible Navier-Stokes equations.

Original authors: Kangyang Zeng, Changshen Huang, Xi Liu, Xiaodong Niu, Zhenhua Chai

Published 2026-09-16

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CC BY 4.0

Original authors: Kangyang Zeng, Changshen Huang, Xi Liu, Xiaodong Niu, Zhenhua Chai

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, scientists have relied on a powerful mathematical tool called the lattice Boltzmann method to simulate how fluids move and heat spreads. Imagine trying to predict how smoke swirls around a building or how blood flows through a tiny capillary; this method breaks the fluid down into a grid of tiny points, tracking how particles at each point bump into their neighbors and move to the next spot. It is a middle-ground approach, sitting between the chaotic motion of individual molecules and the smooth flow of a river, making it ideal for complex engineering problems. However, as computers have grown more powerful, researchers have begun looking toward quantum computers, machines that use the strange rules of quantum mechanics to solve problems far faster than today's supercomputers. The challenge has been that the standard way these simulations handle particle collisions does not fit the rigid rules of quantum mechanics, which require every step to be perfectly reversible and preserve a specific kind of mathematical balance.

A team of researchers from Huazhong University of Science and Technology and Shantou University in China has now developed a new way to bridge this gap. They created a version of the lattice Boltzmann method that works naturally on quantum computers by replacing the standard collision step with a "rotation." In their new framework, instead of forcing the fluid particles to relax toward a calm state in a way that loses information, the quantum computer gently rotates the state of the particles toward that calm state. This rotation is a perfect fit for quantum mechanics because it is a reversible operation that keeps the system balanced. The researchers tested this idea using two different models: one for simpler heat and mass transfer problems and another for the more complex flow of incompressible fluids, like water moving through a pipe. Their work shows that this new method can accurately mimic the behavior of real fluids while obeying the strict laws of quantum physics.

The core problem the team tackled was a fundamental mismatch between how fluids behave and how quantum computers operate. In the traditional method, when particles collide, they lose a bit of energy and settle down, a process that is not reversible. Quantum computers, however, cannot perform operations that lose information; every step must be a unitary transformation, meaning the total amount of information stays constant, like a perfect shuffle of a deck of cards where no cards are ever lost. Previous attempts to fix this involved complicated workarounds, such as using extra helper bits that often failed or required removing the collision step entirely, which changed the physics of the simulation. The new approach proposed by the Chinese team avoids these pitfalls by treating the collision not as a loss of energy, but as a rotation in a mathematical space. They showed that by rotating the state of the fluid particles just the right amount, the system behaves almost exactly like the traditional method, but without breaking the rules of quantum mechanics.

To prove their idea works, the researchers ran detailed numerical experiments on a classical computer to simulate how their new quantum method would perform. They first tested it on a classic fluid dynamics problem known as the Taylor–Green vortex, a swirling pattern of fluid that decays over time due to friction. They compared their rotation-based method against the standard classical method and found that the results were nearly identical. The swirling patterns of the fluid, including the location and strength of the vortices, matched perfectly. They also tested a flow known as Poiseuille flow, which describes how fluid moves through a channel, and again, the new method produced results that aligned closely with the known mathematical solutions. The team found that the small differences between their new method and the old one were so tiny that they only appeared when the time steps used in the calculation were very large, confirming that the new approach is highly accurate.

A particularly clever feature of this new method is how it handles different types of fluid problems. For the simpler model used to study heat and diffusion, the researchers discovered that the "target" state the fluid is trying to reach is fixed and does not change as the simulation runs. This means that the part of the quantum computer circuit responsible for preparing this target state can be built once and reused over and over again, saving a significant amount of computational effort. However, for the more complex model used to simulate water flow, the target state changes depending on how fast the fluid is moving at that moment. In this case, the researchers showed that the system can still work by measuring the fluid's speed at each step and updating the circuit accordingly, though this requires a slightly more complex process.

The study concludes that this rotation-based framework provides a solid, mathematically sound way to run fluid simulations on quantum computers. It solves the long-standing problem of how to make the collision step reversible without sacrificing accuracy or requiring impossible amounts of computing power. While the current results are simulations run on classical computers, the underlying logic is designed specifically for quantum hardware. The researchers note that future work will need to address how to implement the movement of particles and the boundaries of the simulation entirely within a quantum framework, but this new method establishes the crucial collision step as a viable, unitary process. By turning a dissipative collision into a precise rotation, the team has opened a clear path for bringing one of the most important tools in fluid dynamics into the quantum era.

Technical Summary: Rotation Collision Based Quantum Lattice Boltzmann Methods

Problem Statement
The Lattice Boltzmann Method (LBM) is a widely used mesoscopic approach for computational fluid dynamics (CFD), characterized by a collision-streaming structure well-suited for parallel computing. However, translating LBM to quantum computing faces two fundamental obstacles:

Unitarity: Quantum evolution requires unitary operators, whereas the standard BGK collision operator in LBM is dissipative and non-unitary.
Nonlinearity: The equilibrium distribution function in models like D2Q9 contains quadratic terms of fluid velocity. Directly encoding these into quantum circuits without violating the no-cloning theorem is challenging.

Existing strategies to address these issues have significant limitations. Some methods decompose non-unitary operators into linear combinations of unitary operators, resulting in success probabilities that decrease exponentially with system size. Others avoid the collision term entirely, restricting physical scope, or linearize the operator, introducing approximation errors that deviate from the original physical model. Furthermore, approaches that represent the collision operator as a state-dependent unitary matrix often require recomputation at every time step with gate complexity growing exponentially. Crucially, a systematically constructed unitary alternative that reproduces the BGK relaxation structure to leading order from first principles has not been established.

Methodology
This work proposes a novel Quantum Lattice Boltzmann Method (QLBM) based on a rotation collision operator. The core methodology involves mapping the non-unitary BGK relaxation into a unitary rotation within the amplitude space of the distribution function.

Amplitude Encoding: The distribution function $f_i$ is encoded into a quantum state $|\psi\rangle = \frac{1}{\sqrt{\rho}} \sum_i \sqrt{f_i} |i\rangle$ . In this space, the BGK collision reduces the deviation from equilibrium by a factor of $(1 - 1/\tau)$ .
Rotation Collision Operator: The authors define the angle $\phi$ between the current state $|\psi\rangle$ and the equilibrium state $|\psi_{eq}\rangle$ . The BGK relaxation is mimicked by a unitary rotation $U_{col}$ that rotates $|\psi\rangle$ toward $|\psi_{eq}\rangle$ by an angle $\theta = \phi/\tau$ . This operator is constructed using generalized phase reflections and state preparation circuits (specifically the Shende–Bullock–Markov scheme).
Theoretical Consistency: The paper demonstrates that while the rotation is exact in the Hilbert space, mapping it back to the distribution function space yields a result that agrees with the classical BGK collision up to first order in the time step $\delta t$ . The discrepancy is shown to be of order $O(\delta t^2)$ , consistent with the truncation error of standard LBM.
Model Implementations:
- D2Q5 (Convection-Diffusion): For this model, the equilibrium distribution is linear in the macroscopic variable $\phi$ . Consequently, the normalized equilibrium state is a fixed vector independent of the instantaneous distribution. This allows the equilibrium-state preparation circuit to be precompiled and reused throughout the simulation.
- D2Q9 (Incompressible Navier-Stokes): Here, the equilibrium distribution depends nonlinearly on the local velocity. The equilibrium state and collision parameters vary with the current state. In the proposed implementation, macroscopic variables are reconstructed via measurement at each time step to update the collision operator, though the framework supports sequential concatenation of unitary blocks once parameters are specified.

Key Contributions

Unitary Formulation: The paper establishes a unitary formulation of the collision process that replaces the non-unitary BGK relaxation with a rotation in amplitude space, addressing the unitarity problem from first principles without resorting to probabilistic post-selection or collisionless approximations.
Gate-Level Circuits: The authors construct explicit gate-level quantum circuits for both D2Q5 and D2Q9 models.
- For D2Q5, the fixed nature of the equilibrium state allows for significant circuit reuse, where the equilibrium preparation circuit is compiled once.
- For D2Q9, the framework details how to handle state-dependent updates, comparing different circuit synthesis strategies (e.g., direct unitary matrix application vs. eigenbasis encoding).
Sequential Concatenation: The work outlines a "sequential circuit concatenation" strategy where collision-streaming blocks for successive time steps are connected without intermediate projective measurements solely for composition, preserving quantum coherence during the unitary evolution (assuming state-dependent parameters are determined coherently or via hybrid classical-quantum updates).

Numerical Results
The developed QLBM was validated through numerical experiments on classical simulators:

Taylor–Green Vortex (D2Q9): Simulations of the 2D Taylor–Green vortex at $Re=100$ showed that the vorticity fields obtained via the rotation collision operator exhibit nearly identical spatial patterns to those from the classical BGK operator. Minor differences in magnitude were attributed to the nonlinear mapping between amplitude evolution and distribution reconstruction.
Poiseuille Flow (D2Q9): Simulations at $Re=10, 40,$ and $100$ demonstrated that the QLBM velocity profiles agree well with analytical solutions and classical lattice BGK results.
Error Scaling: Parameter sweeps confirmed that the discrepancy between the rotation collision and the classical BGK collision scales as $O(\delta t^2)$ , validating the theoretical analysis. The error behavior matches that of the classical method regarding time step and relaxation time variations.

Significance and Claims
The paper claims to provide a unitary and gate-level realizable formulation of the Lattice Boltzmann method that resolves the fundamental unitarity challenge of the BGK operator. A primary theoretical contribution is the identification of a reusable equilibrium-state circuit structure for the D2Q5 model, which offers a distinct advantage for quantum implementation by reducing the need for dynamic recompilation.

The authors modestly note that while the collision operator is unitary, the current implementation for D2Q9 relies on a hybrid approach where macroscopic variables are reconstructed classically to update the collision parameters. They acknowledge that a fully coherent determination of these state-dependent quantities remains an open algorithmic problem. Additionally, the study is limited to classical simulations; the impact of hardware noise on real quantum devices and the full cost of state preparation and measurement (which scale as $O(N)$ ) are identified as areas for future investigation. The work does not claim to solve the nonlinearity issue of D2Q9 entirely within a purely quantum framework without measurement but provides a systematic unitary alternative to the BGK operator that preserves the physical relaxation structure.

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