Constructing the nearly deterministic Toffoli polarization gate with the spatial degree of freedom based on weak cross-Kerr nonlinearities

The quantum Toffoli gate can be utilized as a fundamental building block for translating more complicated classical operations into quantum algorithms, so its efficient construction is more conducive to the realization of quantum computation in large-scale than two-qubit universal quantum logic gates. Assisted by the spatial degree of freedom based on weak cross-Kerr nonlinearities, a construction scheme of the nearly deterministic Toffoli gate is presented. After the measurement on the coherent state, the required swap transformations and single-photon transformations are performed by the classical feed-forward to complete the construction task if nonzero phase shifts are displayed. The scheme proposed here is more efficient than those based on the standard model of linear optics and more feasible than those fulfilled by a variety of two-qubit universal quantum logic gates and one-qubit gates. Moreover, with mature measurement methods and simple optical elements and operations, this scheme enhances the feasibility of experiments to construct the Toffoli gate.