Hyper-parallel Toffoli gate on three-photon system with two degrees of freedom assisted by single-sided optical microcavities.

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-C²PF) gate in a three-photon system in both the polarization and spatial DOFs. In contrast with the one with many qubits encoding on one DOF only, our hyper-C²PF gate operating two independent C²PF 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-C²PF gate is feasible in experiment.