Matrix Phase-Space Representations for Gaussian Boson Sampling

We introduce coherent matrix phase-space distributions. These use conservation laws and symmetries to improve the accuracy and speed of quantum phase-space representations. As an example, this is applied to the validation of low-loss Gaussian boson sampling (GBS) quantum computational advantage experiments, where classical generation of the random photon-number counts is exponentially hard. Large improvements in sampling errors are demonstrated compared to previous methods. Matrix phase-space representations also provide a large numerical speedup due to their (at worst) quadratic scaling, compared to other methods for validating total count probabilities of large-scale, low-loss GBS networks.