  We propose efficient algorithms for classically simulating Gaussian unitaries
and measurements applied to non-Gaussian initial states. The constructions are
based on decomposing the non-Gaussian states into linear combinations of
Gaussian states. We use an extension of the covariance matrix formalism to
efficiently track relative phases in the superpositions of Gaussian states. We
get an exact simulation that scales quadratically with the number of Gaussian
states required to represent the initial state and an approximate simulation
algorithm that scales linearly with the degree. We define measures of
non-Gaussianty quantifying this simulation cost, which we call the Gaussian
rank and the Gaussian extent. From the perspective of quantum resource
theories, we investigate the properties of this type of non-Gaussianity measure
and compute optimal decomposition for states relevant to continuous-variable
quantum computing.
