On Quantum Codes and Networks

by Colin Michael Wilmott

RHUL-MA-2009-11

Abstract:

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.