Nonlinear terms present a fundamental challenge for quantum computational fluid dynamics, as their implementation on inherently linear quantum hardware typically requires resource-intensive workarounds that limit scalability to large-scale simulations. We present a hybrid quantum-classical tensor network algorithm that addresses this bottleneck by combining variational time-stepping with quantum tensor programming to efficiently compile operators and time-dependent fields into quantum circuits. Within a probabilistic framework, we replace prior state-based nonlinear implementations with tensor-based block encodings, stabilizing success probabilities that otherwise decay exponentially with system size. Benchmarking on turbulent flow fields demonstrates that the algorithm maintains high success probabilities and moderate measurement overhead across increasing Reynolds numbers and grid resolutions. Compared to fully classical tensor network solvers, our hybrid approach yields substantial reductions in both memory footprint and computational cost, establishing a scalable pathway toward practical quantum advantage in scale-resolving CFD simulations.
Nonlinear terms present a fundamental challenge for quantum computational fluid dynamics, as their implementation on inherently linear quantum hardware typically requires resource-intensive workarounds that limit scalability to large-scale simulations. We present a hybrid quantum-classical tensor network algorithm that addresses this bottleneck by combining variational time-stepping with quantum tensor programming to efficiently compile operators and time-dependent fields into quantum circuits. Within a probabilistic framework, we replace prior state-based nonlinear implementations with tensor-based block encodings, stabilizing success probabilities that otherwise decay exponentially with system size. Benchmarking on turbulent flow fields demonstrates that the algorithm maintains high success probabilities and moderate measurement overhead across increasing Reynolds numbers and grid resolutions. Compared to fully classical tensor network solvers, our hybrid approach yields substantial reductions in both memory footprint and computational cost, establishing a scalable pathway toward practical quantum advantage in scale-resolving CFD simulations.
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.
Quantum computing (QC) represents an emerging frontier with significant potential for transformative advancements in Computational Fluid Dynamics (CFD). This potential has been explicitly recognized by the CFD Vision 2030 report, emphasizing the strategic importance of exploring QC for future breakthroughs. Among quantum algorithms relevant to CFD, the Variational Quantum Linear Solver (VQLS) is particularly promising due to its suitability for solving linear and Hamiltonian-based problems, characteristic of fundamental CFD equations such as the pressure-Poisson equation. However, the practical implementation of VQLS for large-scale CFD problems faces critical scalability challenges, most notably the exponential increase in the number of quantum circuit evaluations and the barren plateau phenomenon, which severely hampers trainability and algorithm convergence. In this study, two novel strategies are introduced to address these challenges. First, we utilize a block encoding technique based on efficient oracles, thereby significantly reducing quantum circuit complexity and qubit overhead while accessing system matrix. Second, we incorporate a quantum-inspired evolutionary optimization (QIEO) approach, demonstrating enhanced training efficiency and effectively mitigating the barren plateau issue. These improvements are integrated within a Hybrid Quantum-Classical Method (HQCM) linear solver framework and demonstrated through solving a 2D steady-state heat equation, representative of the pressure-Poisson equations common in incompressible CFD simulations. The proposed methodologies yield promising results, showcasing improved circuit efficiency, enhanced stability during optimization, and accurate numerical solutions. This work lays a foundational step toward realizing practical, scalable quantum-assisted CFD solvers and sets clear pathways for future research aimed at further algorithmic and architectural enhancements.
Recent advancements in quantum computing provide opportunities to address the challenges of computational resource limitations in computational fluid dynamics (CFD). This work presents a hybrid quantum-classical CFD framework that leverages quantum linear algorithms to practical flow simulations. By transforming nonlinear fluid dynamics problems into linear systems, the framework employs quantum linear algorithms to compute solutions, effectively integrating quantum and classical computing approaches. Furthermore, this framework employs subspace methods to map the original large-scale linear systems to small ones solved by quantum linear algorithms, thereby enabling the solution of large-scale problems using the currently limited quantum resources. We applied the famous Harrow–Hassidim–Lloyd (HHL) algorithm and variational quantum linear solver to simulate complex flows, including unsteady flows around a cylinder, simulation of an aircraft with turbulence model, and combustion flows. The performance and quantum resource consumption of these algorithms were evaluated in practical flow scenarios, demonstrating their effectiveness with an average relative error lower than 0.001%. We conducted quantum resource and convergence analyses to adapt this framework for use with near-term quantum computers. Our framework supports simulations of up to 9.4 × 106 grid cells, marking a significant advancement from previous quantum approaches limited to simple flows. This paper not only offers a way for employing quantum linear algorithms in solving complex flow problems but also provides insights into optimizing quantum algorithms for CFD applications, pushing forward the practical utilization of quantum CFD.
High-fidelity reconstruction of computational fluid dynamics (CFD) fields is a central challenge in reduced-order modelling, flow control, and large-scale scientific simulations, where classical approaches face severe scalability and nonlinearity limitations. In this work, a hybrid quantum–classical framework is proposed for reconstructing CFD solution snapshots using variational quantum algorithms (VQAs) on near-term quantum hardware. The method formulates CFD field reconstruction as a variational quantum state learning problem using amplitude encoding and shallow parameterized quantum circuits. Reconstruction fidelities ranging from 0.8696 to 0.9369 and low L2 errors between 0.2532 and 0.3673 are achieved, demonstrating accurate quantum approximation of high-dimensional fluid fields. The results significantly outperform earlier low-fidelity attempts and confirm that VQA-based surrogate modelling can capture dominant spatial flow structures even with limited circuit depth. This study highlights the potential of hybrid variational quantum frameworks as scalable alternatives to classical CFD post-processing and reduced-order modelling.
From fluid flow and transport to collective dynamics, numerical simulation of nonlinear partial differential equations underpins modern scientific computing. Extending this capability to quantum computers remains a longstanding challenge because nonlinear and non-Hermitian evolution is fundamentally incompatible with conventional Hamiltonian-based quantum simulation. Here we experimentally realize the time evolution of nonlinear fluid dynamics on a quantum processor using a hybrid variational framework for the viscous and inviscid Burgers equations. Our approach directly encodes the nonlinear dynamics into a variational optimization procedure, avoiding the enlarged linear embeddings and truncation overhead associated with Carleman linearization-based quantum algorithms. We further demonstrate convection-dominated dynamics corresponding to Reynolds numbers of order 102. We encode the governing evolution into parametrized quantum circuits and iteratively reconstruct the time-dependent field through quantum-classical optimization. By introducing a zero-noise extrapolation method without additional circuit-folding overhead, we accurately execute deep error-circuits with entangling-gate counts beyond those typical of Hadamard test circuits. We accurately reconstruct the time evolution across multiple timesteps despite hardware noise and finite device coherence. Our results constitute, to our knowledge, the first experimental realization of nonlinear time propagation on a quantum processor, extending quantum simulation beyond predominantly linear settings and establishing a route toward quantum computation for nonlinear continuum dynamics.
As the energy sector transitions towards more sustainable forms of energy production, classical Computational Fluid Dynamics (CFD) increasingly struggles with the bottle- necks of simulating high complexity turbulent flows. While quantum computing (QC) theoretically offers an exponential speedup for large-scale linear systems, current Noisy Intermediate-Scale Quantum (NISQ) hardware lacks the error correction required for deep and complex circuits that mathematically ideal algorithms require. Furthermore, the inherent linearity of quantum mechanics seriously impedes the solving non-linear partial differential equations (PDEs), which are critical for fluid modeling. To bridge thisgap, thisthesisevaluatestheVariationalQuantumLinearSolver(VQLS),a shallow- circuit, hybrid algorithm, applied to the 1D viscous Burgers’ equation while utilizing the Cole-Hopf transformation to analytically linearize the equation. In both simulated and actual hardware, the VQLS algorithm is implemented across increasingly scaled topographies (2, 4, and 8 qubits). This work intends to identify the limitations of implementing quantum fluid dynamics in near-term quantum hardware.