Nowadays quantum SWAP gate has become an integral part of quantum computing, so investigation of methods of its realization seems to be an important practical problem for various quantum-optical and information applications. In the present paper we propose a scheme for performing a SWAP logic operation in discrete variables in the framework of quantum non-demolition interaction between an atomic ensemble and a multimode light with orbital angular momentum. We discuss in detail the procedure for revealing two-qubit closed subsystems on a set of atomic and field states for different values of the driving field orbital momentum. We also demonstrate the possibility of implementing a parallel multi-two-qubit quantum SWAP gate.