On interchanging the states of a pair of qudits

The qubit SWAP gate has been shown to be an integral component of quantum circuitry design. It permutes the states of two qubits and allows for the storage quantum information, teleportation of atomic or ionic states, and is a fundamental element in the circuit implementation of Shor’s algorithm. We consider the problem of generalising the SWAP gate beyond the qubit setting. We show that quantum circuit architectures completely described by instances of the CNOT gate can not implement a transposition of a pair of qudits for dimensions d ≡ 3 (mod 4). This is of interest to the question of construction a generalised quantum SWAP gate. The task of constructing generalised SWAP gates based on transpositions of qudit states is argued in terms of the signature of a permutation.