  A method is presented for the implementation of edge local complementation in
graph states, based on the application of two Hadamard operations and a single
controlled-phase (CZ) gate. As an application, we demonstrate an efficient
scheme to construct a one-dimensional logical cluster state based on the
five-qubit quantum error-correcting code, using a sequence of edge local
complementations. A single physical CZ operation, together with local
operations, is sufficient to create a logical CZ operation between two logical
qubits. The same construction can be used to generate any encoded graph state.
This approach in concatenation may allow one to create a hierarchical quantum
network for quantum information tasks.
