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This research introduces a robust, near-optimal quantum algorithm designed to solve time-dependent nonlinear partial differential equations (PDEs) governing fluid flow and other physical systems. By embedding these complex problems into a truncated linear space via quantum homotopy analysis, the method improves upon existing approaches in simulation time and gate complexity while adapting to specific nonlinearity profiles. The work demonstrates significant potential for hybrid quantum-classical workflows on both near-term and fault-tolerant devices, marking a step toward practical quantum advantage in scientific computing.
Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems
arXiv:2512.21033v2 Announce Type: replace Abstract: Quantum algorithms to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, nonlinear PDEs. We embed the PDEs in a truncated, high-dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods with a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of the time-marching system size, simulation time, accuracy parameters, and post-selection parameters. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts, and query complexity. A physically motivated measure of nonlinearity is connected to a parameter similar to the flow Reynolds number $Re_{\textrm{H}}$, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of Burgers, Fisher--KPP, damped Kuramoto--Sivashinsky, and real Ginzburg--Landau/Allen--Cahn equations. Together, these examples encompass cubic nonlinearity and saturation, quadratic transport, and fourth-order dissipative stiffness. This work shows the potential of hybrid quantum algorithms for simulating nonlinear problems on near-term and fault-tolerant devices.
Abstract:Quantum algorithms to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, nonlinear PDEs. We embed the PDEs in a truncated, high-dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods with a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of the time-marching system size, simulation time, accuracy parameters, and post-selection parameters. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts, and query complexity. A physically motivated measure of nonlinearity is connected to a parameter similar to the flow Reynolds number $Re_{\textrm{H}}$, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of Burgers, Fisher--KPP, damped Kuramoto--Sivashinsky, and real Ginzburg--Landau/Allen--Cahn equations. Together, these examples encompass cubic nonlinearity and saturation, quadratic transport, and fourth-order dissipative stiffness. This work shows the potential of hybrid quantum algorithms for simulating nonlinear problems on near-term and fault-tolerant devices.
Quantum Physics (quant-ph); Computational Engineering, Finance, and Science (cs.CE); Applied Physics (physics.app-ph); Computational Physics (physics.comp-ph); Fluid Dynamics (physics.flu-dyn)
arXiv:2512.21033 [quant-ph]
(or arXiv:2512.21033v2 [quant-ph] for this version)
Full text under the publisher's open licence · 226 words
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