Simulations of quantum walks on beam splitter arrays modeled as higher-order rotations

We present a comprehensive matrix representation of a beam splitter array, incorporating multiple input and output channels. We propose treating each beam splitter as rotational matrices of a 2n-th-dimensional space. With these operators, the matrix that describes the entire square array and, consequently, the final probability distribution of an input photon state can be calculated. Furthermore, square and non-square arrays are explored using the same approach, encompassing certain interferometer arrays, the quantum quincunx, and the discrete quantum Zeno effect. In addition, we show the versatility of the proposed operator by using it to map the action of field operators in the description of multiphoton inputs.