Matrix phase-space representations for quantum symmetries

Larisa Jonke, Stjepan Meljanac

Peter D. Drummond, Alexander S. Dellios, Margaret D. Reid

We introduce a general phase-space representation that includes global quantum symmetries in the basis expansion. This method, called matrix phase-space, projects the basis onto a reduced Hilbert space, which can greatly reduce sampling errors of many-body quantum simulations and unifies several previous phase-space methods. The purpose of this paper is to provide detailed proofs of basic theorems and operator identities. We also treat several different types of symmetries. To illustrate the benefits of matrix phase-space methods, we give a detailed derivation of a recent application to the topical problem of verifying the outputs of Gaussian boson sampling (GBS) quantum computers with photon number resolving detectors. This has exponential complexity, and using parity symmetry reduces sampling errors by very large factors relative to earlier methods.

Anatoli Polkovnikov

P. D. Drummond, Alexander S. Dellios, M. D. Reid

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.

P. Huguenin

The Weyl-Wigner transformation enables one to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined bv the quasi-probability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix. (GE)