Three-dimensional compact photonic circuits for quantum state tomography of multipath qudits in any finite dimension

We propose compact three-dimensional photonic circuit designs that guarantee a remarkable reduction in the complexity of circuits to perform quantum state tomography of N-dimensional path qudits. We obtain solutions such that the number of beam splitters obeys a polynomial function of degree three with the quantum system dimension, whereas, in current proposals, this quantity grows with a polynomial function of degree four. In addition, the optical depth reduces from quadratic to linear over the dimension of the Hilbert space.