Edge local complementation for logical cluster states

A method is presented for the implementation of edge local complementation (ELC) 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 for constructing a one-dimensional logical cluster state based on the five-qubit quantum error-correcting code, using a sequence of ELCs. A single physical CZ operation, together with local operations, is sufficient to create a logical CZ operation between two logical qubits. This approach in concatenation may allow one to create a hierarchical quantum network for quantum information tasks.