Acausal measurement-based quantum computing

In the measurement-based quantum computing, there is a natural "causal cone" among qubits of the resource state, since the measurement angle on a qubit has to depend on previous measurement results in order to correct the effect of byproduct operators. If we respect the no-signaling principle, byproduct operators cannot be avoided. In this paper, we study the possibility of acausal measurement-based quantum computing by using the process matrix framework [O. Oreshkov, F. Costa, and C. Brukner, Nature Communications {\bf3}, 1092 (2012)]. We construct a resource process matrix for acausal measurement-based quantum computing. The resource process matrix is an analog of the resource state of the causal measurement-based quantum computing. We find that the resource process matrix is (up to a normalization factor and trivial ancilla qubits) equivalent to the decorated graph state created from the graph state of the corresponding causal measurement-based quantum computing.