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 $\mathsf{G}$ 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 $m_{\rm eff}$, the protocol reconstructs every coefficient uniformly on a constant interval with error $ε$ and success probability $1-δ$ using $\widetilde{\mathcal{O}}\!\left(ε^{-2}\operatorname{poly}(m_{\rm eff})\log(nδ^{-1})\right)$ 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.