Hybrid Quantum Algorithms for Summary Statistics of Nonlinear PDEs with Random Initial Data

I develop a hybrid classical–quantum algorithm for summary statistics of nonlinear PDEs with Gaussian random initial data. Perturbative expansion and Wick’s theorem reduce ensemble averages to deterministic momentum and time integrals. Classical inter- polation and quantum amplitude estimation then evaluate these integrals with improved accuracy dependence for sufficiently smooth integrands, without quantum random-access memory. I apply the method to the Einstein–de Sitter Euler–Poisson perturbation theory developed by Goroff, Grinstein, Rey, and Wise. For the one-loop matter power spectrum, I establish the integrand regularity required by the algorithm under specified assumptions on the initial spectrum. Cubic nonlinear Schrödinger evolution provides a comparison involving resonances and diagram cancellations. I account separately for preprocessing and circuit costs.