Multimode homodyne detection as a tool for cluster state generation and Gaussian quantum computation

This paper theoretically demonstrates a compact method for the production of cluster states, based on recent experiments with multimode cavities, by using a so-called multi-pixel homodyne detection (MPHD) system and classical post-processing. This effectively allows to incorporate the linear optic network in the stage of the measurement. The physical transformations which the modes undergo due to the measurement (i.e., propagation from the cavity to the multipixel homodyne detector and multipixel homodyne detection) can be modeled by symplectic matrices, acting on the vector which regroups the annihilation operators associated to the various input squeezed mode. The method proposed can mimic the action of a desired matrix U_th if there exists a suitable combination of the experimental parameters such that U_MPHD = U_th. It is therefore important to characterize from a theoretical point of view the class of unitary operations which are experimentally implementable by the use of this method. A condition which allows to test the implementability of a given unitary matrix U_th, and to find the experimental parameters to use in order to emulate it is determined. Some examples of cluster state generation by this method are presented. Finally, how this procedure can be generalized to perform Gaussian quantum computation is further analyzed. In particular, the possibility of implementing the elementary operations which compose a Gaussian universal set, including the Fourier transform of a single-mode input state and a two-mode C_Z interaction, is studied.