Towards an optimal swap gate

We present a novel approach that generalizes the well- known quantum SWAP gate to higher dimensions and construct a regular quantum gate composed entirely in terms of the generalized CNOT gate that cyclically permutes the states of qudits for prime. We also investigate the case for other than prime. A key feature of the construction design relates to the periodicity evaluation for a family of linear recurrences which we achieve by exploiting generating functions and their factorization over the complex reals.