Measurement Based CV Parametrized Quantum Circuits

Multimode Gaussian states can be represented as complex undirected graphs [1] and such a representation allows for manipulation of the said graphs given measurements on some of the nodes of the graph. The resulting graph due to such measurements on a partition of of the original graph can thus be parametrized by the measurement parametres. Given a multipartite resource state, for example a cluster state, one can effect large unitaries by measuring part of the resource state. Here we use a deterministically generated 2D CV cluster state [2] to generate large Haar Random unitaries. In particular, we show the implementation of a 48 mode Haar-random unitary with a very high fidelity to the theoretical prediction exceeding previous attempts [3], [4]. Half of the modes of a 96 mode 2D cluster state's graph is measured using a homodyne detector while another homodyne detector is used to characterize the resulting graph. The angles for the measured modes are sampled uniformly randomly from a uniform distribution over angles in the range [-0.5π, 0.5π] which is essentially sampling from the Haar measure over U(1). The resulting covariance matrix is then decomposed into an effective symplectic matrix using σ = ¹/₂SS^T, which is further decomposed into a corresponding unitary using first a Bloch-Messiah decomposition and then a rectangular decomposition. The phase and amplitude distributions are shown in the following image also compared to the theoretical distribution [3].