Universal quantum gates for path photonic qubit

The codification of qubits in internal degrees of freedom (DoF) of light as polarization, transverse modes, and path has received a lot of attention lately. However, the qubit codified in path DoF often plays a limited role in the circuits. In this work, we present a linear optical circuit to codify qubits in path on different bases. We also present geometrical representations for path DoF as Poincaré-like sphere and the Bloch sphere for path qubit. In addition, we designed linear optical circuits to construct universal quantum gates for photonic propagation path DoF. The well-known analogy between degrees of freedom of a coherent laser beam and quantum systems is explored to experimentally simulate the preparation of the qubits as well as the implementation of the proposed circuit for the quantum gates. The results show a clear agreement with the predictions of quantum theory.