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(mod4). The task of constructing generalised SWAP gates based on transpositions of qudit states is argued in terms of the signature of a permutation.
The concern of quantum computation is the computation of\nquantum phenomena as observed in Nature. A prerequisite for the\nattainment of such computation is a set of unitary\ntransformations that describe the operational process within\nthe quantum system. Since operational transformations\ninevitably interact with elements outside of the quantum\nsystem, we therefore have quantum gate evolutions determined\nwith less than absolute precision. Consequently, the theory of\nquantum error-correction has developed to meet this difficulty.\nFurther, it is increasingly evident that much effort is being\nmade into finding efficient quantum circuits in the sense that\nfor a library of realisable quantum gates there is no smaller\ncircuit that achieves the same task with the same library of\ngates. A reason for this concerted effort is primarily due to\nthe principle of decoherence. I give the construction for the\nset of unitary transformations that describe an error model\nthat acts on a d-dimensional quantum system. I also give an\noverview of the theoretical framework associated with such\nunitary transformations and generalise results to cater for\nd-dimensional quantum states. I introduce two quantum gate\nconstructions that generalise the qubit SWAP gate to higher\ndimensions. The first of these constructions is the WilNOT gate\nand the second is an efficient design also based on binomial\nsummations. Both of these constructions yield a quantum qudit\nSWAP gate determined only in the CNOT gate. Furthermore, the\ntask of constructing generalised SWAP gates based on\ntranspositions of qudit states is argued in terms of the\nsignature of a permutation. Based on this argument, we show\nthat circuit architectures completely described by instances of\nthe CNOT gate cannot implement a transposition of a pair of\nqudits over dimensions d = 3 mod 4. Consequently, our quantum\ncircuits are of interest because it is not possible to\nimplement a SWAP of qutrits by a sequence of transpositions of\nqutrits if only CNOT gates are used. I also give bounds on\nnumbers of quantum codes predicated on d-dimensional quantum\nsystems, and generalise the encoding and decoding architectures\nfor qudit codes.\n