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Classical approximations of quantum Hamiltonian dynamics with the Nyström method

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Simulating the time-evolution of quantum mechanical systems is BQP -hard and expected to be one of the foremost applications of quantum computers. During this talk I will present a method to approximate Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra and propose conditions for the efficient approximation of state vectors evolving under a given Hamiltonian. As an immediate application, I will show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. The main technical contribution of our method is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages the Nyström method to obtain low-rank approximations of the Hamiltonian, a tool commonly used in the statistical machine learning literature.

The talk is based on joint work with Carlo Ciliberto, Massimiliano Pontil, Alessandro Rudi, Simone Severini and Leonard Wossnig

This talk is part of the CQIF Seminar series.

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