Learning and certification of local time-dependent quantum dynamics and noise

Hamiltonian learning protocols are becoming valuable tools for benchmarking and verifying quantum computers and simulators, yet few rigorous protocols address explicitly time-dependent dynamics. We give a scalable protocol for learning the time-dependent coefficients of a locally interacting n-qubit system on a graph of effective dimension D. The unknown generator contains geometrically local Hamiltonian terms and weak onsite Pauli-dephasing terms with time-dependent, nonnegative rates. Its coefficient functions and their derivatives admit a certified simultaneous C 1 approximation in a known finite-dimensional function class that supports stable interpolation. The only experimental primitives are preparation of product Pauli eigenstates, evolution for chosen times, and product-Pauli measurements. Under an explicit weak-dissipation condition, the Lieb--Robinson estimates proved here, and a certified effective approximation/interpolation degree m eff, the protocol reconstructs every coefficient uniformly on a constant interval with error ε and success probability 1-δ using ! (ε-2 poly(m eff) (nδ-1)) samples and polynomial classical processing. The method estimates derivatives of finite-time channel overlaps by stable interpolation, uses process shadows to learn the required local transfer matrices, and constructs sparse diagonally dominant systems through localized semidefinite programs.