  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 Hamiltonian 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 Hamiltonian
is a density matrix, can be efficiently classically simulated under specific
structural conditions. Our main technical contribution is a randomized
algorithm for approximating Hermitian matrix exponentials. The proof leverages
a low-rank, symmetric approximation via the Nystr\"om method. Our results
suggest that under strong sampling assumptions there exist classical
poly-logarithmic time simulations of quantum computations.
