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arXiv:2604.05577

Resource Implications of Different Encodings for Quantum Computational Fluid Dynamics

Hans A. Kösel, Roland Ewert, Jan W. Delfs

quant-phSubmitted Apr 7, 2026• Announced Apr 8, 2026• 0 citations
arXivHTMLPDF

Abstract

For quantum algorithms for problems in which the task is to compute an entire field of values, like e.g. computational fluid dynamics (CFD), it is often proposed amplitude encoding w.r.t. multiple qubits; however, the efforts implied by it for initialization and read-out are not addressed. This work is devoted specifically to this issue: It reviews different encoding schemes in quantum computing, discussing their computational costs for initialization and read-out as well as resulting aspects for their usage via minimal examples. The considerations in previous literature on the required computational resources for amplitude encoding w.r.t. multiple qubits are extended in the presented quantification by explicitly deducing the circuit depth that results for the decomposed initialization procedure of V. V. Shende et al. [1, 2] and deriving an upper bound for the necessary number of executions of a quantum algorithm to extract the encoded values with a specific accuracy. For these two results, an empirical verification via the means provided by IBM's quantum computing simulation framework $\textit{Qiskit}$ [3] is given. In the framework of the study on the required number of runs to achieve a desired accuracy, it is however found that the derived upper bound, scaling like $ {\tilde{n}^2} ~ {\ln( {\tilde{n}} )} $ with the number of encoded values $ {\tilde{n}} $, is too conservative to be used for precise estimations. Therefore, a corresponding study of the required runs for the reference distribution of equal probabilities for all basis states is done in particular, which suggests $ {\tilde{n}} ~ { \ln( {\tilde{n}} ) } $ as an empirical scaling law. Since the view regarding CFD applications is taken here, it is presented in particular that the insights from this work lead to a new encoding approach, which is proposed specifically for a quantum algorithm for the lattice Boltzmann method.

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For quantum algorithms for problems in which the task is to compute an entire field of values, like e.g. computational fluid dynamics (CFD), it is often proposed amplitude encoding w.r.t. multiple qubits; however, the efforts implied by it for initialization and read-out are not addressed. This work is devoted specifically to this issue: It reviews different encoding schemes in quantum computing, discussing their computational costs for initialization and read-out as well as resulting aspects for their usage via minimal examples. The considerations in previous literature on the required computational resources for amplitude encoding w.r.t. multiple qubits are extended in the presented quantification by explicitly deducing the circuit depth that results for the decomposed initialization procedure of V. V. Shende et al. [1, 2] and deriving an upper bound for the necessary number of executions of a quantum algorithm to extract the encoded values with a specific accuracy. For these two results, an empirical verification via the means provided by IBM's quantum computing simulation framework Qiskit\textit{Qiskit}Qiskit [3] is given. In the framework of the study on the required number of runs to achieve a desired accuracy, it is however found that the derived upper bound, scaling like n~2 ln⁡(n~){\tilde{n}^2} ~ {\ln( {\tilde{n}} )}n~2 ln(n~) with the number of encoded values n~{\tilde{n}}n~, is too conservative to be used for precise estimations. Therefore, a corresponding study of the required runs for the reference distribution of equal probabilities for all basis states is done in particular, which suggests n~ ln⁡(n~){\tilde{n}} ~ { \ln( {\tilde{n}} ) }n~ ln(n~) as an empirical scaling law. Since the view regarding CFD applications is taken here, it is presented in particular that the insights from this work lead to a new encoding approach, which is proposed specifically for a quantum algorithm for the lattice Boltzmann method.
Classical computational fluid dynamics (CFD) methods encounter significant computational bottlenecks in large systems and at high Reynolds number flows due to the rapid increase in degrees of freedom, even on flagship supercomputers. As quantum computers emerge, the Quantum Lattice Boltzmann Method (QLBM) has been proposed as a quantum-native alternative for solving fluid dynamics problems. However, until recently, nearly all studies have relied on classical quantum simulators rather than real quantum hardware. This study presents a benchmark of a one-dimensional QLBM implementation executed on three different IBM Quantum hardware. We design and test a quantum circuit for the advection-diffusion equation using amplitude encoding, a Bhatnagar-Gross-Krook (BGK)-type collision operator, and a streaming step with periodic boundary conditions. Our numerical experiments evaluate transpiled two-qubit gate count, total circuit depth, runtime, and accuracy metrics including maximum absolute error (L∞L_\inftyL∞​ norm) and root-mean-square error (RMSE). We find that the present collision and streaming operators dominate the overall cost and become the primary scalability bottleneck on current hardware. To mitigate the prohibitive cost of full quantum state tomography on hardware-based benchmarking exercises, we introduce a sampler-based measurement strategy that reconstructs macroscopic density fields directly from computational-basis measurements. This approach substantially reduces measurement overhead while preserving the expected advection-diffusion dynamics on hardware. These results provide a quantitative assessment of QLBM performance in the noisy intermediate-scale quantum (NISQ) regime and identify the principal algorithmic and experimental limitations that must be addressed for QLBM to become viable for fluid dynamics.
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.
Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. The existing proposals for fault-tolerant quantum algorithms for CFD have almost exclusively been based on the Carleman embedding method, used to encode nonlinearities on a quantum computer. Here, we first identify several key challenges that must be addressed in order to establish quantum advantage using this method, including Carleman convergence, time-stepping requirements, condition-number scaling, and data extraction. Building on this analysis, we develop a novel algorithm for the incompressible lattice Boltzmann equation that addresses them, and provide its detailed analysis, including all potential sources of algorithmic complexity, as well as gate count estimates. We find that for an end-to-end problem, a modest quantum advantage may be preserved for selected observables in the high-error-tolerance regime. We establish a lower bound for the Reynolds number scaling of our quantum algorithm in dimension DDD at Kolmogorov microscale resolution with O(Re34(1+D2)×qM)O(\mathrm{Re}^{\frac{3}{4}(1+\frac{D}{2})} \times q_M)O(Re43​(1+2D​)×qM​), where qMq_MqM​ is a multiplicative overhead for data extraction with qM=O(Re38)q_M = O(\mathrm{Re}^{\frac{3}{8}})qM​=O(Re83​) for the drag force. This upper bounds the scaling improvement over classical algorithms by O(Re3D8)O(\mathrm{Re}^{\frac{3D}{8}})O(Re83D​). However, our numerical investigations suggest a lower speedup, with a scaling estimate of O(Re1.936×qM)O(\mathrm{Re}^{1.936} \times q_M)O(Re1.936×qM​) for D=2D=2D=2. Our results give robust evidence that small, but nontrivial, quantum advantages can be achieved in the context of CFD, and motivate the need for additional rigorous end-to-end quantum algorithm development.
The computational power to resolve large complex aerospace flow problems has increased over the past decades but is still yet not able to simulate these problems sufficiently accurate to a scale resolved level. At the same time, the interest in quantum computing has recently increased rapidly due to the massive improvements of the quantum computer hardware technology. Many fields of research explore the possible advantages that quantum computing can offer in their respective fields. For fluid flow problems, the exponentially increasing storage capacity of flow properties by quantum amplitude encoding offers an approach to efficiently deal with highly resolved flow fields. Within the DLR-QCI project ToQuaFlics, we adjust and extend the quantum lattice Boltzmann method (QLBM) from the literature for single time steps and present our extension to multiple time steps. The structure of our algorithm is shown in Fig. 1. Our extended QLBM can perform all time steps of a simulation without having to extract the quantum state or parts of it at any time. Therefore, with our algorithm there is no need to measure any qubit during or at the end of the simulation. This saves the computationally expensive read out and reinitialization after every time step and enables an efficient time stepping scheme. We verify our algorithm on 1D and 2D test cases simulating a linear advection-diffusion process with periodic boundary conditions. The algorithm is simulated using the IBM simulator package Qiskit with the shot method. For verification reasons, we measure the entire flow field at the end of the simulation to demonstrate the agreement with the reference solution given by the classic lattice Boltzmann solution. The results show excellent agreement and arbitrary convergence towards the reference. We further discuss the inherent amplitude decay of a QLBM time step and how that effects our solution in terms of signal loss of the quantum state. An outlook to include a body with a surface interaction model and to capture nonlinearities is given and briefly sketched.
Quantum computing has emerged as a powerful potential accelerator for computational fluid dynamics (CFD), but whether this promise can be realized in practice depends on how fluid information is encoded on quantum hardware. This review provides an architecture-agnostic assessment of encoding strategies for quantum-enhanced fluid simulation, focusing on the trade-offs they impose on state preparation, measurement, boundary treatment, nonlinear dynamics, and temporal evolution. We examine the principal encoding paradigms used in the literature and relate them to representative quantum algorithms for fluid simulation. Through these examples, we show that encoding choices fundamentally shape both the algorithm itself and also the practical feasibility of quantum CFD. For example, highly compact encodings can offer attractive asymptotic advantages but might introduce severe bottlenecks in readout, state preparation, and nonlinear processing, whereas less compact representations may simplify interactions and improve compatibility with analog and near-term hardware. No single encoding is universally optimal, rather the most suitable choice depends strongly on the structure of the fluid problem, the computational objective and the constraints of the target quantum platform. We therefore argue that encoding should be treated as a primary design variable in quantum CFD and revisited iteratively throughout the design pipeline, as different algorithmic components interact and influence one another.
Abstract Solving computational fluid dynamics (CFD) problems requires the inversion of a linear system of equations, which can be done using a quantum algorithm for matrix inversion (Gilyén et al 2019 Proc. 51st Annual ACM SIGACT Symp. on Theory of Computing 193–204). However, the number of shots required to measure the output of the system can be prohibitive and remove any advantage obtained by quantum computing. In this work we propose a scheme for measuring the output of a quantum singular value transform (QSVT) matrix inversion algorithm specifically for the CFD use case. We use a quantum signal processing based amplitude estimation algorithm (Rall and Fuller 2023 Quantum 7 937) and show how it can be combined with the QSVT matrix inversion algorithm. We perform a detailed resource estimation of the amount of computational resources required for a single iteration of amplitude estimation, and compare the costs of amplitude estimation with the cost of not doing amplitude estimation and measuring the whole wavefunction. We also propose a measurement scheme to reduce the number of amplitudes measured in the CFD example by focussing on large amplitudes only. We simulate the whole CFD loop, finding that thus measuring only a small number of the total amplitudes in the output vector still results in an acceptable level of overall error.