Circumventing query complexity barriers in learning quantum dynamics via physics-informed kernels

Abstract

Learning continuous quantum dynamical trajectories—essential for understanding non-equilibrium phenomena in quantum chemistry and condensed matter physics—remains prohibitively expensive on quantum computers. Conventional data-driven surrogates treat observables as generic time series, ignoring the governing Schrödinger evolution, and consequently require an infeasible density of samples to resolve highly oscillatory dynamics. We first establish a fundamental information-theoretic lower bound: any incoherent learning protocol that measures independently prepared copies without quantum memory needs Ω(T/ϵ2) oracle queries, revealing a quadratic penalty in 1/ϵ that renders naive dense sampling impossible. To circumvent this barrier, we introduce physics-informed kernel ridge regression (PI-KRR), which encodes the Heisenberg equation as a differentiable constraint. By extracting time derivatives from Hamiltonian commutators at zero additional quantum cost, PI-KRR doubles the information density per simulation shot without violating quantum estimation limits. Furthermore, integrated with classical shadow tomography, our framework reconstructs trajectories for M local observables simultaneously with measurement overhead scaling only as O(log⁡M). We establish robustness guarantees for NISQ devices: isolated measurement outliers are suppressed as 1/m with training size m, and systematic Hamiltonian miscalibrations yield only linearly bounded prediction errors. Numerical experiments on transverse-field Ising models demonstrate that PI-KRR resolves sharp features such as light-cone fronts with drastically fewer simulations than standard kernel methods, achieving up to two orders of magnitude lower mean absolute error. This establishes a practical protocol for compressing complex quantum dynamics into classical predictive models, bridging quantum simulation and machine learning for experimental quantum science.