We describe in detail the application of four qubit cluster states, built on the simultaneous entanglement of two photons in the degrees of freedom of polarization and linear momentum, for the realization of a complete set of basic one-way quantum computation operations. These consist of arbitrary single qubit rotations, either probabilistic or deterministic, and simple two qubit gates, such as a c-not gate for equatorial qubits and a universal c-phase (CZ) gate acting on arbitrary target qubits. Other basic computation operations, such as the Grovers search and the Deutschs algorithms, have been realized by using these states. In all the cases we obtained a high value of the operation fidelities. These results demonstrate that cluster states of two photons entangled in many degrees of freedom are good candidates for the realization of more complex quantum computation operations based on a larger number of qubits.
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