We introduce a noval quantum phase-space: a matrix coherent state distribution. This extends and improves any quantum phase-space to include symmetry groups. The purpose of this is to distinguish global symmetries from local fluctuations,greatly improving efficiency and precision. The general method is applicable to any nearly symmetric state of Hamiltonian, and has very many potential uses. This unifies the positive-P method with the Carusotto et al Bloch state method.

As an example, we treat quantum advantage experiments on Gaussian boson sampling (GBS), where computing exact random photon counts would be exponentially hard. The positive-P(+P) method is used to validate current, lossy GBS experiments.However, in future ultra-low experiments, this will no longer be enough. The cause is subtle: with nearly conserved parity symmetry, the +P method develops extended phase-space probability tails, increasing sampling errors.

This sampling problem can be efficiently solved with matrix coherent states, which include such group symmetries in the coherent basis set. In the case of low-loss GBS of >1000 modes, as potential future quantum advantage experiments, the sampling variance using matrix coherent states is reduced by millions of times compared to previous competing methods. This gives speed-up factors of billions for validation checks. We give present and future numerical examples.