Logical network implementation for cluster states and graph codes

In a previous paper a straight forward construction method for quantum error correcting codes, based on graphs, has been presented. These graph codes are directly related to cluster states which have been introduced by Briegel and Raussendorf. We show that the preparation of a cluster state as well as the coding operation for a graph code, can be implemented by a logical network by only using one type of two-qubit gate, e.g. CNOT or controlled Phase, and one type of one-qubit gate, e.g. the Hadamard transform. Concerning the qubit case each vertex corresponds to an Hadamard gate and each edge corresponds to a controlled not gate.