Implicit methods are attractive for hybrid quantum-classical CFD solvers as the flow equations are combined into a single coupled matrix that is solved on the quantum device, leaving only the CFD discretisation and matrix assembly on the classical device. In this paper, an implicit hybrid solver is investigated using emulated HHL circuits. The hybrid solutions are compared with classical solutions including full eigen-system decompositions. A thorough analysis is made of how the number of qubits in the HHL eigenvalue inversion circuit affect the CFD solver's convergence rates. Loss of precision in the minimum and maximum eigenvalues have different effects and are understood by relating the corresponding eigenvectors to error waves in the CFD solver. An iterative feed-forward mechanism is identified that allows loss of precision in the HHL circuit to amplify the associated error waves. These results will be relevant to early fault tolerant CFD applications where every (logical) qubit will count. The importance of good classical estimators for the minimum and maximum eigenvalues is also relevant to the calculation of condition number for Quantum Singular Value Transformation approaches to matrix inversion.
Implicit methods are attractive for hybrid quantum-classical CFD solvers as the flow equations are combined into a single coupled matrix that is solved on the quantum device, leaving only the CFD discretisation and matrix assembly on the classical device. In this paper, an implicit hybrid solver is investigated using emulated HHL circuits. The hybrid solutions are compared with classical solutions including full eigen-system decompositions. A thorough analysis is made of how the number of qubits in the HHL eigenvalue inversion circuit affect the CFD solver's convergence rates. Loss of precision in the minimum and maximum eigenvalues have different effects and are understood by relating the corresponding eigenvectors to error waves in the CFD solver. An iterative feed-forward mechanism is identified that allows loss of precision in the HHL circuit to amplify the associated error waves. These results will be relevant to early fault tolerant CFD applications where every (logical) qubit will count. The importance of good classical estimators for the minimum and maximum eigenvalues is also relevant to the calculation of condition number for Quantum Singular Value Transformation approaches to matrix inversion.
There has been significant progress in the development of quantum algorithms for solving linear systems of equations with a growing body of applications to Computational Fluid Dynamics (CFD) and CFD-like problems. This work extends previous work by developing a non-linear hybrid quantum-classical CFD solver and using it to generate fully converged solutions. The hybrid solver uses the SIMPLE CFD algorithm, which is common in many industrial CFD codes, and applies it to the 2-dimensional lid driven cavity test case. A theme of this work is the classical processing time needed to prepare the quantum circuit with a focus on the decomposition of the CFD matrix into a linear combination of unitaries (LCU). CFD meshes with up to 65x65 nodes are considered with the largest producing a LCU containing 32,767 Pauli strings. A new method for rapidly re-computing the coefficients in a LCU is proposed, although this reduces, rather than eliminates, the classical scaling issues. The quantum linear equation solver uses the Harrow, Hassidim, Lloyd (HHL) algorithm via a state-vector emulator. Test matrices are sampled from the classical CFD solver to investigate the solution accuracy that can be achieved with HHL. For the smallest 5x5 and 9x9 CFD meshes, full non-linear hybrid CFD calculations are performed. The impacts of approximating the LCU and the varying the number of ancilla rotations in the eigenvalue inversion circuit are studied. Preliminary timing results indicate that the classical computer preparation time needed for a hybrid solver is just as important to the achievement of quantum advantage in CFD as the time on the quantum computer. The reported HHL solutions and LCU decompositions provide a benchmark for future research. The CFD test matrices used in this study are available upon request.
Quantum linear solvers are well studied as standalone quantum algorithms; however, in hybrid classical-quantum routines, their practical value must be evaluated at the level of the full non-linear application. A central issue is whether the approximation error of the quantum linear solver remains controlled once embedded in a full iterative workflow. We study this question in the context of a hybrid computational fluid dynamics (CFD) scheme. Through numerical simulations, we analyze how an approximate quantum linear solver affects the convergence of the overall CFD iteration. We show that convergence can be preserved for a non-exact quantum solver with only a moderate overhead in iteration count, provided that the high-frequency components of the linear system are resolved with sufficient accuracy. In addition, we develop an approximate qubitization-based solver (Cheb-LCU) that can reduce quantum resource requirements relative to a Quantum Singular Value Transformation (QSVT)-based solver while inducing only a small loss in convergence performance. This claim is demonstrated through explicit implementation and compilation of the quantum algorithms and by examining their impact on the convergence of the full CFD scheme. We find that our approximate approach reduces the required number of single-qubit rotations by over an order of magnitude relative to the QSVT-based solver, while requiring only a modest increase in CFD iteration count.
We present a classical enhancement to improve the accuracy of the Hybrid variant (Hybrid HHL) of the quantum algorithm for solving linear systems of equations proposed by Harrow, Hassidim, and Lloyd (HHL). We achieve this by using higher precision quantum estimates of the eigenvalues relevant to the linear system, and a new classical step to guide the eigenvalue inversion part of Hybrid HHL. We show that eigenvalue estimates with just two extra bits of precision result in tighter error bounds for our Enhanced Hybrid HHL compared to HHL. Our enhancement reduces the error of Hybrid HHL by an average of 57 percent on an ideal quantum processor for a representative sample of 2x2 systems. On IBM Torino and IonQ Aria-1 hardware, we see that the error of Enhanced Hybrid HHL is on average 13 percent and 20 percent (respectively) less than that of HHL for the same set of systems.
To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated computational fluid dynamic algorithms coupled with powerful supercomputers can not find approximate solutions for several design challenges in both adequate time and scale-resolving accuracy. One difficulty arises from solving high dimensional, strongly nonlinear partial differential equations, such as the Navier-Stokes equations, which capture the underlying physics. For nearly all classical algorithms, methods closely related to Newton's method are used to approximate a solution to the problem. Approximately solving the large-scale linear systems of equations occurring in this iterative scheme is generally a main contributor to the total computational complexity. In this paper a new quantum linear system solver supporting Newton's classical method to solve nonlinear partial differential equations is introduced. We present a new variant of the HHL algorithm, requiring less apriori information regarding the eigenvalues of the corresponding matrix. We apply this quantum linear system solver in a hybrid quantum-classical fashion to solve nonlinear partial differential equations. Moreover, a resource estimation for advanced use-cases of practical relevance is provided. Our results demonstrate how quantum computation may improve existing classical methodologies for solving nonlinear partial differential equations. This approach provides another promising application of quantum computers and presents a possible way forward for handling nonlinearities on inherently linear quantum systems.
Numerical solutions for fluid flow problems are challenging and have been focus of Computational Fluid Dynamics (CFD) research for past several decades. The advent of quantum computing promises exponential speedup in comparison to existing classical methods and alleviate computational constraints posed by CFD problems. Although solutions for most problems of interest in fluid dynamics using quantum computing are distant, recent advances in algorithms, software and hardware provide a path towards realizing this goal. Quantum linear solver algorithms (QLSA) such as Harrow–Hassidim–Lloyd (HHL) and Variational Quantum Linear Solver (VQLS) have been successfully implemented to solve for canonical problems such as Hele-Shaw flow. However, these algorithms still suffer to scale and address problems with ill-conditioned Jacobians. In the current paper, we alleviate these restrictions with a new quantum solver based on Singular Value Decomposition (SVD) and simulate flow past a 2D cylinder. The fidelity of the SVD based quantum solver in predicting the flow past 2D cylinder is computed along with an assessment of errors. Classical and quantum solutions for the flow are compared for different resolutions. Finally, we discuss variation in the solutions based on number of shots used.
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.