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End-to-End Quantum Algorithm for Nonlinear Fluid Dynamics with Bounded Quantum Advantage

David Jennings1, Kamil Korzekwa1,*, Matteo Lostaglio1, Richard Ashworth2, Emanuele Marsili2, and Stephen Rolston2

  • *Contact author: kkorzekwa@psiquantum.com
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PRX Quantum 7, 033060 – Published 18 September, 2026

DOI: https://doi.org/10.1103/xysy-q3fp

Abstract

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 a detailed analysis, including all potential sources of algorithmic complexity and 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 D at Kolmogorov microscale resolution with O(Re34(1+D2)×qM), where qM is a multiplicative overhead for data extraction with qM=O(Re38) for the drag force. This upper bounds the scaling improvement over classical algorithms by O(Re3D8). However, our numerical investigations suggest a lower speedup, with a scaling estimate of O(Re1.936×qM) for D=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.

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Accurate simulation of fluid flows is critical across science and engineering, from predicting aerodynamic performance to understanding turbulent transport. However, high-fidelity computational fluid dynamics (CFD) remains one of the most resource-intensive tasks in classical computing. This has motivated interest in whether quantum computers could offer computational advantages in this field and bring transformative changes. Several recent approaches rely on Carleman linearization, a method that embeds nonlinear fluid equations into much higher-dimensional linear systems amenable to quantum computers. Despite the appeal of this approach, significant challenges remain unresolved, including convergence of the linearization procedure, run-time scaling for outputting the solution, and difficulties in extracting physically meaningful quantities from the quantum output. In this work, we introduce a new quantum algorithm for the lattice Boltzmann equation (LBE) that is designed explicitly to overcome these issues. By employing a shifted incompressible formulation of LBE together with a discrete Carleman embedding, we develop a representation of the dynamics that is compatible with fault-tolerant quantum computation while mitigating prior bottlenecks. We perform a comprehensive analysis of every component of the algorithm, down to the elementary-gate level, and quantify all sources of computational cost. This is complemented by extensive numerical studies assessing the validity and limitations of the underlying Carleman strategy. Our findings indicate that a modest but nontrivial quantum advantage is achievable for selected observables in specific parameter regimes. More broadly, this work provides a realistic and rigorous foundation for the emerging field of quantum-enabled CFD, clarifying both its potential and its fundamental limitations.

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PRX Quantum 7, 033060– Published 18 September, 2026
  • Received 19 December 2025
  • Accepted 17 July 2026
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