  We show how an entangled cluster state encoded in the polarization of single
photons can be straightforwardly expanded by deterministically entangling
additional qubits encoded in the path degree of freedom of the constituent
photons. This can be achieved using a polarization--path controlled-phase gate.
We experimentally demonstrate a practical and stable realization of this
approach by using a Sagnac interferometer to entangle a path qubit and
polarization qubit on a single photon. We demonstrate precise control over
phase of the path qubit to change the measurement basis and experimentally
demonstrate properties of measurement-based quantum computing using a 2 photon,
3 qubit cluster state.
