We study theoretically the teleportation of controlled-phase (CZ) gatethrough measurement-based quantum information processing forcontinuous-variable systems. We examine the degree of entanglement in theoutput modes of the teleported CZ-gate for two classes of resource states: thecanonical cluster states that are constructed via direct implementations oftwo-mode squeezing operations, and the linear-optical version of cluster stateswhich are built from linear-optical networks of beam splitters and phaseshifters. In order to reduce the excess noise arising from finite-squeezedresource states, teleportation through resource states with differentmulti-rail designs will be considered and the enhancement of entanglement inthe teleported CZ-gates will be analyzed. For multi-rail cluster with anarbitrary number of rails, we obtain analytical expressions for theentanglement in the output modes and analyze in detail the results for bothclasses of resource states. At the same time, we also show that for uniformlysqueezed clusters the multi-rail noise reduction can be optimized when theexcess noise is allocated uniformly to the rails. To facilitate the analysis,we develop a trick with manipulations of quadrature operators that can revealrather efficiently the measurement sequence and corrective operations neededfor the measurement-based gate teleportation, which will also be explained indetail.
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