  Encoding qubits in multiple degrees of freedom (DOFs) of a quantum system
allows less-decoherence quantum information processing with much less quantum
resources. We present a compact and scalable quantum circuit to determinately
implement a hyper-parallel controlled-controlled-phase-flip
(hyper-$\rm{C^2PF}$) gate on a three-photon system in both the polarization and
spatial DOFs. In contrast with the one with many qubits encoding in one DOF
only, our hyper-$\rm{C^2PF}$ gate operating two independent $\rm{C^2PF}$ gates
on a three-photon system with less decoherence, and reduces the quantum
resources required in quantum information processing by a half. Additional
photons, necessary for many approaches, are not required in the present scheme.
Our calculation shows that this hyper-$\rm{C^2PF}$ gate is feasible in
experiment.
