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)

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Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems

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