Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. The existing proposals for fault-tolerant quantum algorithms for CFD have almost exclusively been based on the Carleman embedding method, used to encode nonlinearities on a quantum computer. Here, we first identify several key challenges that must be addressed in order to establish quantum advantage using this method, including Carleman convergence, time-stepping requirements, condition-number scaling, and data extraction. Building on this analysis, we develop a novel algorithm for the incompressible lattice Boltzmann equation that addresses them, and provide its detailed analysis, including all potential sources of algorithmic complexity, as well as gate count estimates. We find that for an end-to-end problem, a modest quantum advantage may be preserved for selected observables in the high-error-tolerance regime. We establish a lower bound for the Reynolds number scaling of our quantum algorithm in dimension
D at Kolmogorov microscale resolution with
O(Re43(1+2D)×qM), where
qM is a multiplicative overhead for data extraction with
qM=O(Re83) for the drag force. This upper bounds the scaling improvement over classical algorithms by
O(Re83D). However, our numerical investigations suggest a lower speedup, with a scaling estimate of
O(Re1.936×qM) for
D=2. Our results give robust evidence that small, but nontrivial, quantum advantages can be achieved in the context of CFD, and motivate the need for additional rigorous end-to-end quantum algorithm development.
Quantum algorithms have been identified as a potential means to accelerate computational fluid dynamics (CFD) simulations, with the lattice Boltzmann method (LBM) being a promising candidate for realizing quantum speedups. Here, we extend the recent quantum algorithm for the incompressible LBM to account for realistic fluid dynamics setups by incorporating walls, inlets, outlets, and external forcing. We analyze the associated complexity cost and show that these modifications preserve the asymptotic scaling, and potential quantum advantage, of the original algorithm. Moreover, to support our theoretical analysis, we provide a classical numerical study illustrating the accuracy, complexity, and convergence of the algorithm for representative incompressible-flow cases, including the driven Taylor-Green vortex, the lid-driven cavity flow, and the flow past a cylinder. Our results provide a pathway to accurate quantum simulation of nonlinear fluid dynamics, and a framework for extending quantum LBM to more challenging flow configurations.
Quantum computing has emerged as a powerful potential accelerator for computational fluid dynamics (CFD), but whether this promise can be realized in practice depends on how fluid information is encoded on quantum hardware. This review provides an architecture-agnostic assessment of encoding strategies for quantum-enhanced fluid simulation, focusing on the trade-offs they impose on state preparation, measurement, boundary treatment, nonlinear dynamics, and temporal evolution. We examine the principal encoding paradigms used in the literature and relate them to representative quantum algorithms for fluid simulation. Through these examples, we show that encoding choices fundamentally shape both the algorithm itself and also the practical feasibility of quantum CFD. For example, highly compact encodings can offer attractive asymptotic advantages but might introduce severe bottlenecks in readout, state preparation, and nonlinear processing, whereas less compact representations may simplify interactions and improve compatibility with analog and near-term hardware. No single encoding is universally optimal, rather the most suitable choice depends strongly on the structure of the fluid problem, the computational objective and the constraints of the target quantum platform. We therefore argue that encoding should be treated as a primary design variable in quantum CFD and revisited iteratively throughout the design pipeline, as different algorithmic components interact and influence one another.