Quantum machine learning and quantum-inspired methods applied to computational fluid dynamics: a short review

TL;DR

This short review surveys quantum computing and quantum-inspired approaches to computational fluid dynamics (CFD), focusing on variational quantum algorithms (VQAs), quantum neural networks (QNNs), quantum physics-informed neural networks (QPINNs), and tensor-network (TN) methods. It highlights how VQAs can serve as hybrid solvers for PDEs and how QPINNs and HQPINNs may improve parameter efficiency and accuracy in CFD benchmarks, while tensor networks offer scalable, memory-efficient CFD solvers with substantial reductions in resources. The authors argue that, in the near term, quantum CFD remains out of reach on current hardware, but quantum-inspired TNs already provide practical benefits and hybrid strategies show the most promise. They advocate a two-pronged path: (i) develop hybrid quantum–classical TN approaches that leverage classical solvers and accelerators, and (ii) advance TN-based preprocessing and compression to enable larger-scale quantum computations as fault-tolerant devices become available. Overall, the article underscores that quantum-inspired methods are ready to impact CFD now, while quantum hardware-driven CFD is a longer-term objective.

Abstract

Computational Fluid Dynamics (CFD) is central to science and engineering, but faces severe scalability challenges, especially in high-dimensional, multiscale, and turbulent regimes. Traditional numerical methods often become prohibitively expensive under these conditions. Quantum computing and quantum-inspired methods have been investigated as promising alternatives. This review surveys advances at the intersection of quantum computing, quantum algorithms, machine learning, and tensor network techniques for CFD. We discuss the use of Variational Quantum Algorithms as hybrid quantum-classical solvers for PDEs, emphasizing their ability to incorporate nonlinearities through Quantum Nonlinear Processing Units. We further review Quantum Neural Networks and Quantum Physics-Informed Neural Networks, which extend classical machine learning frameworks to quantum hardware and have shown advantages in parameter efficiency and solution accuracy for certain CFD benchmarks. Beyond quantum computing, we examine tensor network methods, originally developed for quantum many-body systems and now adapted to CFD as efficient high-dimensional compression and solver tools. Reported studies include several orders of magnitude reductions in memory and runtime while preserving accuracy. Together, these approaches highlight quantum and quantum-inspired strategies that may enable more efficient CFD solvers. This review closes with perspectives: quantum CFD remains out of reach in the NISQ era, but quantum-inspired tensor networks already show practical benefits, with hybrid approaches offering the most promising near-term strategy.

Figures (8)

• Figure 1: Schematic of the hybrid loop in a VQA. (0) The ansatz is chosen adn initialized. (1) The parameterized circuit $U(\boldsymbol{\theta})$ is executed and observables are measured. (2) The cost function (and if necessary, gradients) are computed. (3) A classical optimizer updates the parameters, which are (4) fed back into the quantum circuit. These steps are repeated for a prescribed number of iterations or until a convergence criterion is met.
• Figure 2: Variational circuit for the method in Section \ref{['dietter_method']}. Each bold wire $\ket{\mathbf{0}}$ denotes a register (here shown as four registers for clarity). The circuit prepares multiple functions via gates $f^{(j)}$ (distinct dependent variables or replicated copies), which are fed as inputs (IP) to the controlled QNPU block that evaluates the nonlinear cost components and produces outputs (OP). The ancilla measurement yields the cost (and, with parameter-shift, its gradients), which are used by a classical optimizer to update the parameters. These steps are repeated until a convergence criterion is met.
• Figure 3: Example of ansatz for the variational circuit for $N=6$ qubits, following a Matrix Product State ansatz as described in lubasch_variational_2020.
• Figure 4: Example of discretization procedure for multigrid encoding in a one-dimensional problem. Here, $N=4$, with a total of 16 points. The coarsest subgrid is composed of points $\{X_0, X_7\}$. This separation of space is used to define the matrix space utilized to kickstart the consecutive SVDs.
• Figure 5: Example of QNPU for Burgers equation, realizing the terms pertinent to equation \ref{['cost_burgers_qnpu']}. All blue lines indicate that the gate is controlled by the ancilla qubit, on which upon measured approximates \ref{['cost_burgers_qnpu']}. A) Responsible for term $\bra{\mathbf{0}} \tilde{U}^{\dagger}U(\boldsymbol{\theta})\ket{\mathbf{0}}$. B) A circuit $\mathcal{A}$, that shifts bits of the state (as an adder circuit vlatko_quantum_arithmetic_1996), rendering it as an expectation value $\bra{\mathbf{0}}\tilde{U}^\dagger \mathcal{A}U(\boldsymbol{\theta})\ket{\mathbf{0}}$. C) The adder circuit can then be used to calculate derivative.