In this talk, we will discuss two important tasks in quantum information science: Hamiltonian simulation and Hamiltonian learning. The accuracy of quantum dynamics simulation (Hamiltonian 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, which may not be tight for specific initial states and observables. In the first part, we will discuss a few ways to weaken the strong operator norm dependence and provide a case study on the semiclassical Schrödinger equation. In the second part, we will introduce the task of learning a many-body Hamiltonian. We will then highlight the first algorithm to achieve the Heisenberg limit for efficiently learning an interacting N-qubit local Hamiltonian.
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