Coherent matrix projectors for quantum simulations

We introduce a novel type of phase-space: a coherent matrix projector (CPM) distribution. This extends the idea of a density matrix to incorporate phase-space variables. The purpose of this is to separate global eigenvalues from local fluctuations, greatly improving efficiency and precision. The method is applied to quantum computer validation. We treat quantum supremacy experiments on Gaussian boson sampling (GBS), where generating photon-number counts requires an exponentially hard (#P) computation.

The well-known positive-P method is efficient for precision validation of current, lossy GBS experiments, via random sampling. We show that in future low loss experiments, this older method has very large sampling errors for high-order correlations, which can reach 250-th order. The cause of this is subtle. With conservation laws, a positive-P method that treats the whole Hilbert space has a skew distribution with large tails. This leads to growing sampling errors in the limit of exponentially large Hilbert space dimension.

This sampling problem is efficiently solved with coherent matrix projectors, which include global symmetries to implement conservation laws at the level of the basis set. For the case of squeezed state experiments, the inclusion of a parity symmetry gives very large improvements. In lossless GBS cases of 10,000 modes, larger than any planned experiment, the photon number distribution sampling variance in CPM phase-space is reduced by millions or more compared to previous methods.