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Quantum Algorithms for Computational Fluid Dynamics

This paper provides a comprehensive review of quantum algorithms for solving computational fluid dynamics partial differential equations, analyzing both fully quantum and hybrid approaches, exploring tensor-network representations for efficient encoding, and assessing the problem-dependent limitations and challenges toward achieving scalable quantum advantage.

Original authors: Mario Guillaume Cecile, Nis-Luca van Hülst, Tomohiro Hashizume, Pia Siegl, Abhishek Setty, José Diogo da Costa Jesus, Paul Over, Sergio Bengoechea, Muhammad Umer, Spyros Tserkis, Eleftherios Mastoraki

Published 2026-10-01

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mario Guillaume Cecile, Nis-Luca van Hülst, Tomohiro Hashizume, Pia Siegl, Abhishek Setty, José Diogo da Costa Jesus, Paul Over, Sergio Bengoechea, Muhammad Umer, Spyros Tserkis, Eleftherios Mastorakis, Tristan Kraft, Francisco Cárdenas-López, Leonardo Scandurra, Thomas Rung, Felix Motzoi, Belda Yesil, Barbara Kraus, Martin Kiffner, Dimitris G. Angelakis, Eugene de Villiers, Dieter Jaksch

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

Fluids are the silent architects of our world, shaping everything from the lift of an airplane wing to the swirling currents of the atmosphere. To understand these flows, scientists rely on a powerful tool called computational fluid dynamics, which breaks down the continuous motion of liquids and gases into a massive grid of tiny points. At each point, the computer calculates how velocity, temperature, and pressure interact, solving a complex web of equations to predict how the fluid will behave. While this method has revolutionized engineering, it hits a hard wall when the flow becomes truly chaotic. In turbulent conditions, where energy cascades from large eddies down to microscopic swirls, the number of points required to capture every detail grows so explosively that even the world's most powerful supercomputers struggle to keep up. For decades, researchers have hoped that quantum computers, with their ability to process information in fundamentally different ways, could break this barrier and simulate these complex flows with unprecedented speed.

A new review by a large international team of researchers brings this hope down to earth, offering a clear-eyed map of where quantum computing stands in the race to solve fluid dynamics. The authors do not promise a magic wand that will instantly replace current supercomputers. Instead, they meticulously examine the various ways scientists are trying to translate the physics of flowing fluids into the language of quantum bits. They find that while the theoretical potential for speed is real, the path forward is paved with significant hurdles. The most promising results do not come from trying to force a quantum computer to act exactly like a classical one, but rather from hybrid approaches that combine the strengths of both. These methods use quantum circuits to handle specific, difficult parts of the calculation while relying on classical computers to guide the process, a strategy that appears best suited for the noisy, imperfect quantum machines available today.

The review begins by looking at "fully quantum" approaches, which aim to solve the entire problem on a quantum processor. One such method involves encoding the fluid's state into the amplitudes of a quantum wave, a technique that could theoretically represent a massive grid of data using very few qubits. However, the authors point out that this compression comes with a heavy price. To get a useful answer out of the machine, one must be able to prepare the initial data and read the final result efficiently. If the process of loading the data or extracting the answer takes as long as the calculation itself, the quantum advantage disappears. Furthermore, many of these methods rely on the mathematical "condition" of the problem being favorable; if the fluid equations are too sensitive or unstable, the quantum algorithm can fail to converge or require so many repetitions that it becomes slower than a standard computer. While these fully quantum algorithms offer a glimpse of exponential speedups in theory, the practical reality is that they currently require error-free machines that do not yet exist.

Recognizing these limitations, the researchers turn their attention to hybrid methods, which are better suited for the current generation of quantum devices. In these approaches, a quantum computer acts as a specialized engine within a larger classical workflow. One popular technique involves using a quantum circuit as a flexible function approximator, trained to satisfy the physical laws of fluid motion without explicitly solving the equations step-by-step. Another approach encodes the fluid field directly into the quantum state and uses a classical optimizer to tweak the circuit until the solution fits the physics. The review highlights that these methods are particularly good at handling the nonlinear interactions that make fluids so difficult to model, such as the way a fast-moving stream crashes into a slow-moving one. However, the authors caution that these hybrid algorithms are still in their infancy. They require careful tuning to avoid getting stuck in local solutions, and the cost of measuring the quantum state to guide the optimization can quickly add up, potentially negating any speed gains.

A central theme of the paper is the role of tensor networks, a mathematical framework originally developed for quantum physics that has found a surprising second life in classical fluid dynamics. Tensor networks allow scientists to compress the massive amount of data in a fluid simulation by identifying patterns and correlations, effectively discarding redundant information while keeping the essential physics intact. The authors propose a novel bridge between these two worlds: using tensor networks as an intermediate language to translate classical fluid data into quantum circuits. By representing the fluid's operators and states as these compressed networks, researchers can systematically build the quantum circuits needed to solve the problem, rather than designing them by hand for every new scenario. This "tensor-programmable" approach offers a structured way to move from classical simulations to quantum ones, ensuring that the quantum circuits remain shallow and manageable.

The researchers tested these ideas on several benchmark problems, including the flow of air over a wing and the decay of swirling vortices. In simulations, they found that the number of parameters needed for the quantum circuits to achieve a high-fidelity solution grew much more slowly than the number of parameters needed for the best classical compression methods. This suggests that as the problems get larger and more complex, the quantum approach might eventually pull ahead. However, the review is careful to note that these are still simulations and small-scale experiments. When the team looked at the actual hardware requirements for a real-world industrial problem, the numbers were sobering. A fully fault-tolerant quantum computer capable of solving a standard engineering problem would currently require millions of physical qubits and months of runtime, far beyond what is available today.

The paper concludes that the road to quantum advantage in fluid dynamics is not a straight line but a series of trade-offs. The most likely path forward involves a gradual transition where quantum computers handle specific, structured parts of the calculation while classical computers manage the rest. The researchers emphasize that success will depend on finding problems where the fluid flow has enough inherent structure to be compressed efficiently, and where the output required is a specific quantity of interest rather than a complete reconstruction of the entire flow field. While the dream of simulating a hurricane on a quantum chip remains distant, the work outlined in this review provides a clear, realistic roadmap for how scientists might eventually get there, turning the chaotic beauty of fluid motion into a solvable puzzle for the next generation of computing.

Technical Summary: Quantum Algorithms for Computational Fluid Dynamics

Problem Statement

Computational Fluid Dynamics (CFD) is a cornerstone of modern engineering, yet it faces severe computational bottlenecks when resolving complex, multiscale turbulent flows. Fully resolved Direct Numerical Simulation (DNS) requires degrees of freedom scaling as $N \sim \text{Re}^{9/4}$ in three dimensions, where $\text{Re}$ is the Reynolds number. This scaling arises from the need to resolve Kolmogorov scales governing energy dissipation. Consequently, high-fidelity simulations of realistic industrial configurations are often prohibitive even for national-scale supercomputers, forcing reliance on approximations like Reynolds-Averaged Navier–Stokes (RANS) or Large-Eddy Simulation (LES).

The paper addresses the potential of quantum computing to accelerate CFD by solving the underlying partial differential equations (PDEs). However, it emphasizes that a viable quantum approach must navigate the transition from Noisy Intermediate-Scale Quantum (NISQ) devices to early fault-tolerant architectures. The core challenge is not merely the theoretical speedup of linear algebra, but the end-to-end efficiency of state preparation, operator encoding, time evolution, and the extraction of physically relevant observables without reconstructing the full classical flow field.

Methodology

The review categorizes quantum approaches into three primary frameworks, analyzing their mathematical formulations, algorithmic structures, and limitations:

1. Fully Quantum, Fault-Tolerant Paradigms

These approaches aim for exponential or polynomial speedups but generally require fault-tolerant hardware.

Quantum Linear System Algorithms (QLSAs): Based on the Harrow–Hassidim–Lloyd (HHL) algorithm and extended by Quantum Singular Value Transformation (QSVT). These solve systems $Ax=b$ by encoding the solution in a quantum state. The complexity depends heavily on the condition number $\kappa(A)$ , which often worsens with mesh refinement ( $\kappa \sim O(N^{2/d})$ ). Nonlinear PDEs are handled via Carleman linearization, which lifts the system to a higher-dimensional linear space, though this introduces a curse of dimensionality ( $O(N^K)$ ).
Hamiltonian Simulation: Encodes time evolution as $e^{-iHt}$ . This is suitable for unitary, conservative systems. For dissipative systems (e.g., advection-diffusion), the non-unitary nature of the generator requires embedding into a larger Hermitian system using techniques like Linear Combination of Unitaries (LCU) or block encoding. Recent work (e.g., Over et al.) has optimized the success probability for these embeddings to avoid exponential overhead.
Quantum Lattice Boltzmann Methods (QLBMs): Adapt the mesoscopic streaming-collision structure of classical LBM. Streaming is naturally unitary, but collision steps (often nonlinear and dissipative) require approximations, linearizations, or learned surrogate circuits to remain compatible with quantum evolution.

2. Hybrid Quantum-Classical Algorithms

Designed for NISQ and early fault-tolerant devices, these methods replace global matrix inversion with variational optimization.

Quantum Physics-Informed Neural Networks (QPINNs): Use parameterized quantum circuits as function approximators where spatial coordinates are input features. The circuit minimizes a loss function composed of PDE residuals and boundary conditions.
Amplitude-Encoded Variational PDE Solvers: Encode the discretized solution field directly into the amplitudes of a quantum state. The cost function evaluates residuals or energy. Nonlinear terms are handled via Quantum Nonlinear Processing Units (QNPU), which use multiple copies of the state and probabilistic post-selection. Recent advances include adaptive variational frameworks and regularization strategies (e.g., quantum sparsity) to mitigate barren plateaus and improve trainability.

3. Tensor-Programmable Variational Quantum Algorithms (TP-VQAs)

This framework bridges classical tensor network (TN) methods and quantum circuits.

Tensor Network Representation: CFD fields and operators are represented as Matrix Product States (MPS) and Matrix Product Operators (MPO). This exploits the low-rank structure often found in turbulent flows.
Compilation: The MPO representation of differential operators is systematically compiled into quantum circuits (block encoding) without manual gate design.
Nonlinearity Handling: A key contribution is the "diagonal block encoding" strategy for nonlinear terms. Instead of state-based Hadamard products (which suffer from decreasing success probability with system size), this method reconstructs the field classically as a diagonal MPO and compiles it, maintaining an $O(1)$ success probability for investigated turbulent regimes.

Key Contributions and Results

The paper synthesizes recent experimental and theoretical progress, providing a unified perspective on the viability of quantum CFD:

Resource Analysis: The authors detail the resource requirements for specific benchmarks. For example, QLSA implementations for the heat equation on IBM hardware require circuit depths in the range of $10^5$ to $10^6$ two-qubit gates for small grids, with post-selection probabilities of 4–8%.
Nonlinearity Scaling: Experimental demonstrations on the Burgers' equation and Nonlinear Schrödinger Equation (NLSE) show that while variational approaches can achieve high fidelity (>99%), the evaluation of nonlinear cost functions remains the primary hardware bottleneck. The "diagonal block encoding" approach significantly improves the success probability of nonlinear operations compared to state-based methods.
Tensor Network Efficiency: Numerical studies indicate that turbulent flow fields often admit compact MPS representations. The number of variational parameters in a quantum circuit can scale more favorably ( $N^{0.52}$ ) than classical MPS parameters ( $N^{0.65}$ ) for fixed accuracy, suggesting a potential advantage in the "sweet spot" where classical TN contractions become expensive but quantum circuits remain shallow.
Hardware Trade-offs: The review highlights that current NISQ devices struggle with the depth required for nonlinear cost evaluation. However, early fault-tolerant devices (with $\sim 10^4$ physical qubits) could enable deeper, more expressive ansätze and stable evaluation of nonlinear terms, potentially offering a performance metric ( $Q_{5E7}$ ) improvement over current NISQ prototypes.

Significance and Claims

The paper asserts that quantum advantage in CFD is highly problem-dependent and cannot be claimed based on asymptotic complexity alone. The significance of the work lies in its realistic assessment of the "end-to-end" workflow:

Conditioning and Structure: The potential advantage is governed by the condition number of the discretized operators, the representational complexity (bond dimension) of the flow fields, and the efficiency of state preparation.
The Role of Tensor Networks: TNs are presented not just as quantum-inspired solvers but as a critical intermediate layer. They provide a structured pathway to translate classical discretizations into quantum circuits, enabling compact representations that are compatible with both classical compression and quantum hardware.
Hybrid Viability: Variational methods (VQAs and TP-VQAs) are identified as the most plausible route for near-term and early fault-tolerant applications. They avoid the explicit condition-number dependence of QLSAs and the deep coherent evolution requirements of Hamiltonian simulation.
Future Outlook: The authors conclude that while current demonstrations are too small to outperform optimized classical solvers (with current performance gaps estimated at orders of magnitude), the field is moving toward reproducible scaling studies. The path to a decisive quantum advantage requires addressing complex geometries, unstructured grids, and the development of efficient methods for extracting coarse-grained observables without full state reconstruction.

The paper does not claim that quantum CFD is ready for industrial deployment. Instead, it outlines a roadmap where the transition from NISQ to early fault-tolerant hardware, combined with tensor-programmable frameworks, could eventually enable simulations of high-Reynolds turbulence and complex transport processes that are currently intractable.