We present a quantum computing algorithm for fluid flows based on the Carleman-linearization of the Lattice Boltzmann (LB) method. First, we demonstrate the convergence of the classical Carleman procedure at moderate Reynolds numbers, namely, for Kolmogorov-like flows. Then we proceed to formulate the corresponding quantum algorithm, including the quantum circuit layout, and analyze its computational viability. We show that, at least for moderate Reynolds numbers between 10 and 100, the Carleman–LB procedure can be successfully truncated at second order, which is a very encouraging result. We also show that the quantum circuit implementing the single time-step collision operator has a fixed depth, regardless of the number of lattice sites. However, such depth is of the order of ten thousands quantum gates, meaning that quantum advantage over classical computing is not attainable today, but could be achieved in the near or mid-term future. The same goal for the multi-step version remains, however, an open topic for future research.
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.
We discuss the Carleman approach to the quantum simulation of classical fluids, as applied to (i) lattice Boltzmann, (ii) Navier–Stokes, and (iii) Grad formulations of fluid dynamics. Carleman lattice Boltzmann shows excellent convergence properties, but it is plagued by nonlocality which results in an exponential depth of the corresponding circuit with the number of Carleman variables. The Carleman Navier–Stokes offers a dramatic reduction of the number Carleman variables, which might lead to a viable depth, provided locality can be preserved and convergence can be achieved with a moderate number of iterates also at sizeable Reynolds numbers. Finally, it is argued that Carleman Grad might combine the best of Carleman lattice Boltzmann and Carleman Navier–Stokes.
Numerical simulation of turbulent fluid dynamics needs to either parametrize turbulence—which introduces large uncertainties—or explicitly resolve the smallest scales—which is prohibitively expensive. Here, we provide evidence through analytic bounds and numerical studies that a potential quantum speedup can be achieved to simulate fluid dynamics using quantum computing. Specifically, we provide a lattice Boltzmann formulation of fluid dynamics for which we give evidence that low-order Carleman linearization is much more accurate than previously believed for these systems. This is achieved via a combination of reformulating the Navier-Stokes nonlinearity ( u · ∇ u ) to lattice-Boltzmann nonlinearity ( u 2 ) and accurately linearizing the dynamical equations, which effectively trades nonlinearity for additional degrees of freedom that add negligible expense in the quantum solver. Based on this, we apply a quantum algorithm for simulating the Carleman-linearized lattice Boltzmann equation and provide evidence that its cost scales logarithmically with system size compared with polynomial scaling in the best known classical algorithms. In this paper, we suggest that a quantum advantage may exist for simulating fluid dynamics, paving the way for simulating nonlinear multiscale transport phenomena in a wide range of disciplines using quantum computing.
Fluid simulations, especially at high Reynolds numbers, are computationally expensive on classical computers, making them promising application targets for quantum computing. Recent studies have combined the lattice Boltzmann method (LBM) with Carleman linearization to design quantum algorithms for computational fluid dynamics (CFD). However, practical quantum-circuit implementations of these algorithms that incorporate non-periodic boundary conditions have not been fully explored. In this work, we implement a quantum algorithm for two-dimensional linearized fluid flow around an obstacle, using block-encoding of the linear-system matrix and quantum singular value transformation (QSVT) to solve it. Inflow, outflow, and no-slip boundary conditions are formulated as sparse matrix operations and efficiently embedded into quantum circuits using index-value encoding. We demonstrate logarithmic scaling of the required numbers of qubits and gates with respect to the number of lattice points, suggesting the potential feasibility of quantum-computational fluid dynamics simulations.