  We study theoretically the teleportation of controlled-phase (CZ) gate
through measurement-based quantum information processing for
continuous-variable systems. We examine the degree of entanglement in the
output modes of the teleported CZ-gate for two classes of resource states: the
canonical cluster states that are constructed via direct implementations of
two-mode squeezing operations, and the linear-optical version of cluster states
which are built from linear-optical networks of beam splitters and phase
shifters. In order to reduce the excess noise arising from finite-squeezed
resource states, teleportation through resource states with different
multi-rail designs will be considered and the enhancement of entanglement in
the teleported CZ-gates will be analyzed. For multi-rail cluster with an
arbitrary number of rails, we obtain analytical expressions for the
entanglement in the output modes and analyze in detail the results for both
classes of resource states. At the same time, we also show that for uniformly
squeezed clusters the multi-rail noise reduction can be optimized when the
excess noise is allocated uniformly to the rails. To facilitate the analysis,
we develop a trick with manipulations of quadrature operators that can reveal
rather efficiently the measurement sequence and corrective operations needed
for the measurement-based gate teleportation, which will also be explained in
detail.
