Tim Moebus

Hamiltonian learning protocols are quickly establishing themselves as valuable tools to benchmark and verify quantum computers and simulators. However, virtually no rigorous protocols exist to learn time-dependent Hamiltonians and Lindbladians, despite their widespread applications. In this work, we address this gap and show how to learn the time-dependent evolution of a locally interacting $n$-qubit system arranged on a graph $\mathsf{G}$ of effective dimension $D$ by resorting only to the preparation of product Pauli eigenstates, evolution by the time-dependent generator for given times and measurements in product Pauli bases. We assume that the time-dependent parameters are well-approximated by functions in a known space of dimension $m$ and for which we can efficiently perform stable interpolation, say by polynomial functions. Our protocol outputs an expansion in that basis that approximates the parameters up to $\epsilon$ in an interval. The protocol only requires $\widetilde{\cO}\big(\epsilon^{-2}\,\poly{m}\,\log(n\delta^{-1})\big)$ samples and $\poly{n,m}$ preprocessing and postprocessing to learn the parameters with probability of success $1-\delta$, making it highly scalable. Importantly, the scaling in the dimension $m$ is polynomial, whereas naive extensions of previous methods yield a dependency that is exponential in $m$. Like previous protocols for the time-independent case, ours is mostly based on estimating time derivatives of expectation values of various observables through interpolation techniques. We then obtain well-conditioned linear equations that allow us to evaluate the value of the time-dependent function for a local generator. However, whereas in the time-independent case it sufficed to only consider derivatives at time $t=0$, here we need to evaluate them at finite times while still being able to relate the derivatives to parameters of the evolution. Thus, besides dealing with technical intricacies related to the time-dependent case, our main innovation is to show how to combine Lieb-Robinson bounds, process shadows and semidefinite programs to estimate the parameters of the evolution efficiently at constant times. Along the way, we extend state-of-the-art Lieb-Robinson bounds on general graphs to the time-dependent, dissipative setting, a result of independent interest. In addition, we show how our technique can be used to verify the outputs of time-dependent dynamics for polynomial times from access to short-time dynamics for cases of interest like linear adiabatic schedules. As such, our protocol is a valuable tool to verify various state preparation procedures on quantum computers and simulators, such as adiabatic preparation, or to characterize time-dependent Markovian noise.