Quantum dynamics simulation and its application to Hamiltonian learning

Recent years have witnessed tremendous progress in developing and analyzing quantum algorithms for quantum dynamics simulation (Hamiltonian simulation). The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on the operator norm of the Hamiltonian. However, the operator norm measures the worst-case scenario, while practical simulation concerns the error with respect to a given initial vector or given observables at hand. In this talk, we will discuss a few ways to weaken the strong operator norm dependence in quantum simulation tasks by taking into account the the initial condition and observables. We then discuss how such analysis can be applied in the setting of Hamiltonian learning. Using a Hamiltonian reshaping technique, we propose a first learning algorithm to achieve the Heisenberg limit for efficiently learning an interacting N-qubit local Hamiltonian.