  Up to now, all the works about constructing quantum logic gates, an essential
part in quantum computing, are focused on operating on one degree of freedom
(DOF) of quantum systems. Here, we investigate the possibility to achieve a
scalable photonic quantum computing based on two DOFs of quantum systems and
construct a deterministic hyper-controlled-not (hyper-CNOT) gate operating on
both spatial-mode and polarization DOFs of a photon pair simultaneously, by
using the giant optical Faraday rotation induced by a single-electron spin in a
quantum dot inside a one-side optical microcavity as a result of cavity quantum
electrodynamics. With this hyper-CNOT gate and linear optical elements,
two-photon four-qubit cluster entangled states can be prepared and analyzed,
which gives an application to manipulate more information with less resources.
We analyze the experimental feasibility of this hyper-CNOT gate and show that
it can be implemented with current technology.
