Near-optimal ground state preparation
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Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has exponentially better dependence on the desired precision than phase estimation. When such upper bound is unknown, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. This estimated ground state energy is then combined with the first algorithm to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.
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