Colin Michael Wilmott (2009) On Quantum Codes and Networks.
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The concern of quantum computation is the computation of quantum phenomena as observed in Nature. A prerequisite for the attainment of such computation is a set of unitary transformations that describe the operational process within the quantum system. Since operational transformations inevitably interact with elements outside of the quantum system, we therefore have quantum gate evolutions determined with less than absolute precision. Consequently, the theory of quantum error-correction has developed to meet this difficulty. Further, it is increasingly evident that much effort is being made into finding efficient quantum circuits in the sense that for a library of realisable quantum gates there is no smaller circuit that achieves the same task with the same library of gates. A reason for this concerted effort is primarily due to the principle of decoherence. I give the construction for the set of unitary transformations that describe an error model that acts on a d-dimensional quantum system. I also give an overview of the theoretical framework associated with such unitary transformations and generalise results to cater for d-dimensional quantum states. I introduce two quantum gate constructions that generalise the qubit SWAP gate to higher dimensions. The first of these constructions is the WilNOT gate and the second is an efficient design also based on binomial summations. Both of these constructions yield a quantum qudit SWAP gate determined only in the CNOT gate. Furthermore, the task of constructing generalised SWAP gates based on transpositions of qudit states is argued in terms of the signature of a permutation. Based on this argument, we show that circuit architectures completely described by instances of the CNOT gate cannot implement a transposition of a pair of qudits over dimensions d = 3 mod 4. Consequently, our quantum circuits are of interest because it is not possible to implement a SWAP of qutrits by a sequence of transpositions of qutrits if only CNOT gates are used. I also give bounds on numbers of quantum codes predicated on d-dimensional quantum systems, and generalise the encoding and decoding architectures for qudit codes.
This is a Published version This version's date is: 23/02/2009 This item is peer reviewed
https://repository.royalholloway.ac.uk/items/d0c98b8b-5a4c-9db8-3104-bd17ec8bc4d2/1/
Deposited by () on 24-Jun-2010 in Royal Holloway Research Online.Last modified on 15-Dec-2010
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