Strategies for measurement-based quantum computation with cluster states transformed by stochastic local operations and classical communication

Abstract

We examine cluster states transformed by stochastic local operations and classical communication, as a resource for deterministic universal computation driven strictly by projective measurements. We identify circumstances under which such states in one dimension constitute resources for random-length single-qubit rotations, in one case quasideterministically ($N\ensuremath{-}U\ensuremath{-}N$ states) and in another probabilistically ($B\ensuremath{-}U\ensuremath{-}B$ states). In contrast to the cluster states, the $N\ensuremath{-}U\ensuremath{-}N$ states exhibit spin correlation functions that decay exponentially with distance, while the $B\ensuremath{-}U\ensuremath{-}B$ states can be arbitrarily locally pure. A two-dimensional square $N\ensuremath{-}U\ensuremath{-}N$ lattice is a universal resource for quasideterministic measurement-based quantum computation. Measurements on cubic $B\ensuremath{-}U\ensuremath{-}B$ states yield two-dimensional cluster states with bond defects, whose connectivity exceeds the percolation threshold for a critical value of the local purity.