Despite advances in high-performance computing, accurately simulating highly turbulent flows, critical for applications like aircraft design, remains computationally infeasible, often forcing engineers to rely on less precise models. Researchers are now exploring quantum computing as a potential solution, using block-encoding techniques to overcome limitations initially restricting quantum operations to unitary transformations.

This approach reformulates nonlinear equations into linear representations suitable for quantum linear solvers, potentially offering an exponential speedup; however, significant algorithmic challenges must be addressed to realize a true quantum advantage in computational fluid dynamics. The work analyzes recent quantum linear solver approaches and develops a new quantum algorithm designed to overcome these hurdles.

Turbulence Simulation Limits of Classical Methods

Direct Numerical Simulation, a method for solving three-dimensional Navier-Stokes equations, frequently proves computationally infeasible when applied to highly turbulent flows critical for aircraft design and other industrial applications. Despite advancements in high-performance computing, simulating turbulence across vast lengths with the necessary fine resolution remains a significant challenge; practitioners often rely on less accurate models like Reynolds-averaged Navier-Stokes or large eddy simulations as a consequence. The inability to achieve large-scale DNS solutions stems from the sheer number of spatial discretization points needed to capture all scales of complexity within the flow.

A classical simulation requires storing local lattice state vectors at each site, resulting in a memory requirement that scales with the number of lattice sites and discretization points. The total number of data qubits required by a new quantum algorithm, however, scales differently, offering a potential pathway beyond these limitations.

The algorithm’s memory footprint, determined by equations within the research, depends on the logarithm of the inverse error, rather than a direct relationship with the number of discretization points. This distinction is important because it suggests a pathway to reduce the memory burden as error correction improves, a factor not present in classical simulations. Currently, the complexity of the best quantum linear solver scales linearly with a parameter denoted as κ, leading to a quantum algorithm with complexity mirroring classical algorithms in terms of spatial discretization points and simulation time.

Researchers found that if the inefficient data extraction problem cannot be mitigated, no quantum advantage for the simulation of nonlinear dynamics is possible. The study concludes that while a quantum advantage is not guaranteed, the potential for reduced memory requirements, coupled with advancements in quantum linear solvers, warrants further investigation into quantum computing as a means of simulating complex turbulent flows.

Navier-Stokes Equations and Direct Numerical Simulation

Direct Numerical Simulation (DNS) relies on resolving the Kolmogorov microscale, the smallest turbulent eddies, with a spatial discretization parameter that dictates the necessary resolution to accurately capture the physics of fluid flow. The number of grid points required to achieve this resolution scales with the Reynolds number, a dimensionless quantity characterizing the ratio of inertial to viscous forces. This parameter selection ensures the simulation accurately represents the turbulence, but also highlights a fundamental limitation of DNS: the computational cost increases dramatically with higher Reynolds numbers and finer resolutions.

The lattice Boltzmann equation provides an alternative approach to numerically solving the Navier-Stokes equations, employing a discrete velocity space and a relaxation time to model fluid behavior. Using lattice units, first and second-order accurate lattice Boltzmann equations can be formulated, allowing for conversion to physical units through parameters like total evolution time, spatial dimension, and lattice velocity.

A key relationship emerges between physical kinematic viscosity, simulation parameters, and the lattice relaxation time τ; this allows researchers flexibility in choosing two of three model parameters, $τ^{}$ ⋆, Δx, and Δt, to accurately model a given physical scenario. This flexibility is helpful because it allows for optimization of computational resources, but also introduces considerations regarding accuracy and stability.

The kinematic shear viscosity ν is directly linked to the LBE relaxation time τ, a connection that dictates how well the simulation captures the fluid’s resistance to flow. The work demonstrates that careful selection of these parameters is essential for balancing computational efficiency with the need for accurate representation of turbulent dynamics.

Quantum Linear Solvers for Fluid Dynamics

The discrete Carleman embedding transforms a nonlinear fluid dynamics problem into a form suitable for quantum computation, establishing a well-defined problem for simulation via unitaries $U_{A}$ and $U_{b}$ , according to the work. This process begins with spatial discretization of the discrete Boltzmann equation, converting it from a partial differential equation into an ordinary differential equation, a necessary step for quantum processing. The resulting nonlinear ordinary differential equation then undergoes Carleman linear embedding, generating a linear equation representing fluid dynamics within an expanded linear space.

Each query to the unitary operator $U_{b}$ requires precisely $N_{C}$ queries to ${U_{ψ}}_{i} n i$ , eliminating hidden constant prefactors and streamlining the quantum computation, the researchers report. This efficiency stems from the careful construction of the linear representation, allowing for a direct mapping onto quantum algorithms designed to solve linear ordinary differential equations.

The resulting quantum state, coherently encoding the solution, can then be measured to extract information about the fluid system being modeled. To solve this linear equation, the team employed a quantum linear solver from a 2023 reference, selecting it because it currently provides the lowest rigorously guaranteed upper bound for query complexity, $q_{Q}$ .

This choice minimizes the number of operations needed on the quantum computer, a critical factor in overcoming limitations of current quantum hardware and enabling simulations of increasingly complex fluid dynamics scenarios. The approach offers a pathway to use quantum computers for problems currently intractable for even the most powerful classical supercomputers.

Block-Encoding and Unitary Operations in Quantum Computing

Block-encoding provides a method for representing general matrices within the constraints of quantum computation, which fundamentally relies on unitary operations. This technique allows researchers to encode matrices inside blocks of unitary operations, effectively expanding the range of problems solvable on quantum hardware. By combining these block-encodings with established quantum algorithms like linear combination of unitaries and quantum signal processing, complex linear algebra operations become accessible on a quantum computer.

The algorithm detailed in the work begins with encoding data into quantum registers, a necessary step for interfacing with a quantum linear solver. This process involves constructing a unitary block-encoding, denoted $U_{A}$ , to represent the matrix A, and a unitary state preparation, $U_{b}$ , to represent the vector b. These unitaries are then used as inputs to the quantum linear solver, which aims to find a quantum state representing a solution to the fluid dynamics problem, with a guaranteed level of accuracy.

The team selected a quantum linear solver offering, minimizing the computational demands on the quantum system. Constructing these block-encodings requires decomposing matrices into components that can be efficiently represented within a unitary framework. Specifically, the matrix $F_{2}$ is decomposed into unitary matrices $L_{2}$ and $R_{2}$ , along with a matrix $Σ_{2}$ designed to have the smallest possible rank. This decomposition yields a block-encoding of $F_{2}$ with a specific prefactor, ultimately contributing to the overall efficiency of the quantum algorithm.

Carleman Linearization for Nonlinear CFD

Carleman linearization, a technique transforming nonlinear dynamics into linear representations, recently underwent investigation as a potential pathway for quantum algorithms designed to solve nonlinear differential equations. Authors explored its application to simulating the lattice Boltzmann equation (LBE), a common method in computational fluid dynamics, revealing errors around $10^{- 3}$ for a lid-driven cavity simulation that saturated over time. However, the study found that pure linearization, simply dropping nonlinear terms, outperformed the more complex Carleman truncation in the scenarios tested.

This outcome raises questions about the ability of the Carleman procedure to accurately capture genuinely nonlinear effects, as the observed flows exhibited weak nonlinearities and the pure linearization errors were already small. Increasing the truncation level to address this would introduce an additional computational cost, scaling with O(N^N_C−k), potentially negating any quantum advantage over classical algorithms unless specific conditions are met.

Specifically, the dependence on N, the system size, would need to disappear, or a quadratic improvement from classical O(N) to O(N) would be required to justify the added complexity. The researchers note that a simpler, direct linearization of the original differential equation could have yielded the same results without the need for the Carleman procedure.

The limitations of Carleman linearization stem from its foundation as an expansion around a single equilibrium point, in this case, a state with zero velocity. As dynamics become more complex, featuring multiple fixed points or chaotic behavior, the Carleman embedding approximation is expected to fail.

The observed loss of convergence at higher Reynolds numbers, therefore, isn’t unique to the shifted LBE formulation but rather reflects a known constraint of the method when applied to increasingly nonlinear dynamics. “Since it is known that the Carleman embedding method is just an expansion around one fixed equilibrium point,” the paper states. This suggests that while Carleman linearization may offer a route to quantum simulation, its effectiveness is heavily dependent on the nature of the fluid flow being modeled and its inherent level of nonlinearity.

Lattice Boltzmann Equation as a Kinetic Model

Employing a trapezoidal discretization addresses first-order accuracy issues inherent in approximating integrals important to the lattice Boltzmann equation, allowing for a second-order accurate model without restricting the time step to be less than twice the relaxation time. This advancement builds upon earlier work by introducing a new variable set and a shifted relaxation time, ultimately arriving at a widely used formulation of the lattice Boltzmann equation.

The implicit second-order discretization provides a more stable and accurate representation of fluid dynamics than previous methods. Extracting classical information from the coherent encoding of dynamics remains a critical challenge in applying quantum algorithms to the lattice Boltzmann equation, with most existing literature focusing solely on outputting Carleman states. Two studies detail data extraction, yet they overlook a key ratio, potentially limiting the scalability of these approaches.

This highlights a bottleneck in translating quantum computations into usable fluid flow data, demanding further investigation into efficient readout strategies. Analysis of the incompressible lattice Boltzmann formulation reveals that the term δρ scales as, while other terms exhibit explicit dependence on u^⋆, indicating a relationship between perturbation magnitude and computational complexity. The Chapman-Enskog expansion confirms that non-equilibrium components of the distribution function are of the same order as hydrodynamic velocity perturbations in low-Knudsen regimes.

This suggests that maintaining proximity to the local equilibrium manifold is essential for accurate modeling, and that kinetic corrections remain significant even with small velocity perturbations. A quantum algorithm can be constructed to solve the shifted incompressible lattice Boltzmann equation by first employing a quantum linear solver to prepare solutions coherently encoded in quantum state amplitudes, then post-processing to extract classical fluid information.

This process relies on solving a set of linear equations, either those described by Eqs. (90) or (93), and efficiently translating the quantum state into measurable parameters representing the fluid system. The algorithm’s success hinges on both the accuracy of the quantum linear solver and the effectiveness of the post-processing steps in retrieving meaningful data.

Complexity Challenges in Quantum Lattice Boltzmann Algorithms

Carleman linearization transforms nonlinear dynamics into a linear, albeit infinite-dimensional, system, a technique recently adapted for quantum algorithms simulating the lattice Boltzmann equation (LBE). This approach, explored by previous work, seeks to use quantum computation for fluid dynamics modeling, but faces inherent limitations in translating theoretical gains into practical advantages. The current work builds on these foundations by focusing on a shifted, linearized version of the incompressible LBE, solved using a quantum linear solver, and detailing a pathway toward potential, though modest, quantum advantage.

Analysis of query and gate complexities, supported by numerical investigations, suggests the algorithm can offer benefits; however, the method’s convergence may be limited to low-Reynolds-number computational fluid dynamics (CFD) simulations. Existing literature often concentrates on generating the final quantum state representing the fluid dynamics, overlooking a critical step: extracting usable classical information from this coherent quantum encoding.

Two prior studies address data extraction, yet they omit a key ratio, detailed in Eq. (34) of Ref. This oversight highlights a significant challenge: even with a successful quantum solution to the linear equations, efficiently retrieving classical data remains a hurdle. The researchers note that a quantum advantage may be restricted to scenarios accepting high error outputs, suggesting a trade-off between accuracy and computational efficiency. Estimating the query complexity, the number of times the quantum linear solver must be called, requires careful consideration of these limitations.

The paper outlines its organization, beginning with an overview of main results and obstacles overcome, followed by background on the LBE and existing quantum algorithms, and then a detailed assessment of computational complexity and potential quantum advantage. “However, we also highlight the challenges that algorithms based on Carleman embedding face when applied to industrially relevant problems,” the authors write, emphasizing the need for further refinement to extend the method’s applicability to more complex, real-world simulations.

Quantum Algorithm for LBE Drag Force Simulation

This finding, presented in section VII of the paper, underscores a critical hurdle in achieving quantum advantage; simply solving the linear equations efficiently is not enough if extracting meaningful data remains computationally expensive. Researchers detail a query complexity, denoted as $q_{Q}$ , for the quantum linear solver that is subject to caveats on the condition number, $κ_{A}$ . The team’s approach centers on an analysis of recent approaches and the development of a new algorithm.

They constructed an end-to-end quantum algorithm, beginning with data encoding and culminating in a data extraction subroutine to determine the drag force through measurements of the coherent output state. This involved developing new block-encodings for discrete Carleman matrices. The algorithm’s performance analysis considers both the accuracy of the simulation and the associated computational costs, including query and gate complexity.

Consequently, even with an efficient quantum solution to the linear equations, achieving a quantum advantage hinges on mitigating this data extraction bottleneck. The analysis suggests that the convergence of the method may be limited to low-Reynolds-number CFD simulations.

Modest Quantum Advantage and Convergence Limitations

A shifted version of the incompressible lattice Boltzmann equation, linearized with a discrete Carleman embedding method, forms the basis of a new quantum algorithm assessed for its potential to model fluid dynamics; however, the analysis reveals limitations to its applicability at higher Reynolds numbers. Extensive numerical investigations, coupled with theoretical analysis of query and gate complexities, suggest a modest quantum advantage is achievable, but not without caveats regarding the condition number.

The work identifies that the Carleman embedding introduces errors dependent on a truncation parameter, $N_{C}$ , and prior studies have often failed to fully account for these or demonstrate consistent convergence. The research team’s assessment of recent quantum linear solver (QLS)-based proposals employing Carleman embedding to simulate fluid dynamics reveals a common algorithmic pipeline. This pipeline centers on analyzing recent approaches and developing a new algorithm, but does not characterize the reformulation as novel or specifically designed to address challenges.

Several challenges arise at different stages of this process, impacting both computational complexity and the potential for quantum advantage; one key issue is the applicability of the Carleman approach itself, as the embedding introduces errors proportional to the core Carleman truncation parameter. The analysis indicates that while a quantum advantage might be expected for two-dimensional systems, and potentially extrapolated to three dimensions, it will be constrained by low $N_{C}$ values, limiting simulations to low Reynolds numbers or accepting higher error outputs.

The paper details that the convergence of the method may be limited, and the researchers conclude that a modest polynomial quantum advantage is the most likely outcome. Throughout the analysis, the Dirac ket notation is used to represent normalized quantum states encoding vectors. The fluid is modeled using the lattice Boltzmann equation, where a vector-valued function, f(r,t), represents the probability density of fluid particles with discrete velocities.

👉 More information
🗞 End-to-End Quantum Algorithm for Nonlinear Fluid Dynamics with Bounded Quantum Advantage
✍️ David Jennings, Kamil Korzekwa, Matteo Lostaglio, Richard Ashworth, Emanuele Marsili and Stephen Rolston
🧠 DOI: http://link.aps.org/doi/10.1103/xysy-q3fp

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