We discuss the Carleman linearization approach to the quantum simulation of classical fluids based on Grad's generalized hydrodynamics and compare it to previous investigations based on lattice Boltzmann and Navier-Stokes formulations. We show that the Carleman-Grad procedure exhibits intermediate properties between the two. Namely, convergence of the Carleman iteration over a few tens of timesteps and a potentially viable quantum circuit implementation using quantum linear algebra solvers. However, both features still need substantial improvements to yield a viable quantum algorithm for fluid flows.
We discuss the Carleman linearization approach to the quantum simulation of classical fluids based on Grad's generalized hydrodynamics and compare it to previous investigations based on lattice Boltzmann and Navier-Stokes formulations. We show that the Carleman-Grad procedure exhibits intermediate properties between the two. Namely, convergence of the Carleman iteration over a few tens of timesteps and a potentially viable quantum circuit implementation using quantum linear algebra solvers. However, both features still need substantial improvements to yield a viable quantum algorithm for fluid flows.
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.
The search for quantum-like wave formulations of the Navier-Stokes (Schrödinger-Navier-Stokes, SNS for short) equations describing classical dissipative fluids has met with increasing attention in the recent years, due to the large portfolio of potential applications in science and engineering. A SNS formulation of classical fluids was first presented in a largely un-noticed paper by Dietrich and Vautherin back in 1985(Journal de Physique). In this paper, we revisit this specific SNS approach and assess its viability for quantum implementations based on Carleman embedding/linearization techniques. Specifically, we i) Clarify in full mathematical detail why the SNS dissipator presents a steep challenge for quantum computers and propose a way out strategy based on the Hamilton-Jacobi (HJ) formulation of fluid dynamics; ii) Develop a corresponding quantum algorithm using a new technique based on a tensor-network representation of Carleman embedding of the HJ equations (CHJ) which permits substantial memory savings; iii) Emulate the CHJ quantum algorithm on a classical computer and analyse its convergence and accuracy for the specific case of Kolmogorov-like flows at moderate Reynolds numbers.
To the best of our knowledge, this is the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity.
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.
The implementation of quantum algorithms for the simulation of classical fluid dynamics poses a fundamental challenge due to the nonlinearity of the fluid equations. In this work, we provide a pedagogical introduction to quantum computing algorithms for simulating classical fluids, with a special focus on the Carleman-Lattice Boltzmann algorithm, which has captured significant attention in the last couple of years. While this algorithm demonstrates satisfactory convergence to analytical solutions for systems at low-to-moderate Reynolds numbers, it also shows an exponential depth of the corresponding quantum circuit. As a result much further analysis is needed to assess the availability of the Carleman-Lattice Boltzmann method on a quantum computer.
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.
Quantum computation of fluid dynamics has attracted growing attention as a key application of fault-tolerant quantum computers anticipated in the coming decade, with lattice Boltzmann methods emerging as a particularly promising approach. Explicit and efficient elementary-gate-level circuit simulations, however, have so far been demonstrated only in the linear case. Here we include the leading nonlinearity through second-order Carleman linearization of the one-dimensional Boltzmann equation, and demonstrate, via explicit quantum-circuit simulation, the preparation of the final-time state using a Taylor-expansion-based ODE solver based on the quantum singular value transformation. With this construction, we analyze the gate and qubit complexities, which scale logarithmically with the grid size, the nonlinearity captured by the higher-order Carleman linearization, and the practical utility of higher-order expansions in the Taylor ODE solver. The construction provides a concrete baseline for computational cost reduction and further developments such as extensions to higher dimensions, complex geometries, and the extraction of physical quantities, towards industrially useful quantum CFD.