The main aim of this doctoral study project is the calculation of “how much” entanglement
and discord is generated in certain kinds of quantum random walks. The main calculations in
this doctoral study will be to investigate how these types of quantum random walks can
generate both entanglement and quantum discord. In this project we are also interested in
quantum random walks on more general graphs, and on closed surfaces such as the torus.
It has been found for example ["Alternate two-dimensional quantum walk with a single-qubit
coin", C. DiFranco, M. Mc Gettrick, T. Machida, Th. Busch, Physical Review A, Vol. 84, Issue
4, 042337 (2011)] that an alternating one dimensional quantum random walks can simulate a
two dimensional quantum random walk. By "simulate" here, we mean
the two processes give identical probability distributions. But, it is expected that the
entanglement in the two processes should be different, since one involves a "1 dimensional
coin"
while the second necessitates a "2 dimensional coin". These entanglement parameters have
not been computed: It is the central purpose of this PhD to determine this quantity. This
should allow us to construct walks that generate larger quantities of entanglement, that may
be used in efficient quantum algorithms.

PLAN:
YEAR 1:
Familiarisation with the field.
Taking some postgraduate modules in maths
Writing some small computer simulation programs
YEAR 2:
Calculating entanglement & discord in 1-dim and 2-dim quantum random walks.
Determination of how entanglement/discord may depend on other parameters in the walk.
YEAR 3:
Publishing & presenting results.
YEAR 4:
Investigating applications to quantum algorithms for searching graphs.