A Hybrid Quantum–Classical Spectral Solver for Nonlinear Differential Equations

We investigate hybrid quantum–classical solvers for nonlinear boundary value problems using Chebyshev spectral collocation. Unlike prior methods such as H–DES, which repeatedly recompile circuits and encode the entire spectral basis on the quantum processor, our framework offloads only the residual minimisation to a quantum backend while retaining classical enforcement of boundary conditions. Two paradigms are considered: (i) gate-based residual minimisation on CUDA-Q using variational circuits to evaluate a Cubic Unconstrained Binary Optimisation (CUBO) cost, which naturally arises from the discretisation, and (ii) a Quadratic Unconstrained Binary Optimisation (QUBO) reformulation, which is required for execution on a quantum annealer, executed via a classical–quantum mapping. We further explore a CUBO extension on CUDA-Q and direct residual-to-energy mapping on annealers. Benchmarks confirm that the classical solver reproduces the analytic solution with spectral accuracy; among quantum-enhanced methods, the annealer-based QUBO yields the closest approximation. The gate-based CUBO solver improves upon a legacy variational baseline but exhibits a small interior bias due to limited circuit depth and precision. These findings underscore the complementary roles of annealers and gate-based devices in hybrid scientific computing and demonstrate a feasible workflow for the NISQ era rather than a speedup over classical methods. Recent progress in quantum algorithms for differential equations signals a rapidly maturing field with significant potential for practical quantum advantage.