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PHYSICAL REVIEW A 00, 002300 (2011)

Optimal preparation of graph states

Adán Cabello,^1,* Lars Eirik Danielsen,^2,† Antonio J. L ́ópez-Tarrida,^1,‡ and José R. Portillo^3,§

1 Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain

2 Department of Informatics, University of Bergen, P.O. Box 7803, Bergen N-5020, Norway

3 Departamento de Matemática Aplicada I, Universidad de Sevilla, E-41012 Sevilla, Spain

(Received 25 November 2010; published xxxxx)

We show how to prepare any graph state of up to 12 qubits with (a) the minimum number of controlled-Z gates and (b) the minimum preparation depth. We assume only one-qubit and controlled-Z gates. The method exploits the fact that any graph state belongs to an equivalence class under local Clifford operations. We extend up to 12 qubits the classification of graph states according to their entanglement properties, and identify each class using only a reduced set of invariants. For any state, we provide a circuit with both properties (a) and (b), if it does exist, or, if it does not, one circuit with property (a) and one with property (b), including the explicit one-qubit gates needed.

DOI: 10.1103/PhysRevA.00.002300
PACS number(s): 03.67.Mn, 03.65.Ud, 03.65.Ta, 42.50.Xa