Approximating Hamiltonian dynamics with the Nystrom method

Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Elamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Elamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a, randomized algorithm for approxifnating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nystrom method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.