A Residual-Adaptive Preconditioned ψ-Fractional Quantum Pseudo-Spectral Method: Delay-Memory Differential Equations

We develop a residual-adaptive preconditioned quantum pseudo-spectral method for generalised ψ-Caputo initial-value problems containing a discrete delay, weakly singular hereditary memory, and nonlinear reaction terms. A ψ-fractional Chebyshev basis yields closed-form operational matrices that are exact on the chosen finite spectral space. To make the hereditary term compatible with block encoding, the power-law kernel is approximated by a sum of exponentials and supplemented by an explicit local near-field correction, converting global memory into finitely many local auxiliary modes. A structure-preserving preconditioner controls the condition number, while a residual-adaptive multidomain strategy and damped Newton iteration treat layers and nonlinearities. We prove well-posedness in Mittag–Leffler weighted graph spaces, derive a combined spectral–kernel–residual error estimate, and state the quantum linear-system complexity with explicit block-encoding normalisations and right-hand-side preparation assumptions. Numerical tests show high accuracy for solutions smooth in the ψ-coordinate, improved robustness for singular and layered solutions, substantial condition-number reduction, and lower history cost under sum-of-exponentials compression. To evaluate performance, we compare against L1 product integration and Jacobi collocation, systematically quantifying their respective accuracy, computational cost, and conditioning characteristics.