BURA Collection:http://bura.brunel.ac.uk/handle/2438/2352024-03-16T03:20:29Z2024-03-16T03:20:29ZClassical simulation of restricted cluster state quantum circuitsGarn, Michaelhttp://bura.brunel.ac.uk/handle/2438/278102023-12-06T03:01:02Z2023-01-01T00:00:00ZTitle: Classical simulation of restricted cluster state quantum circuits
Authors: Garn, Michael
Abstract: A fundamental open problem in quantum computing is to understand when quantum systems
can or cannot be efficiently classically simulated. In this thesis, we study when cluster state
quantum circuits with alternative input states (parameterised by a radius r) and measurements
in the Z basis and XY-plane, can or cannot be efficiently classically simulated.
In the first part of this thesis, we study when such a system can be efficiently classically
simulated. The main technical tool we consider is a generalised notion of separability in
terms of cylindrical operators. We first show that if a CZ gate acts on cylindrical operators
with radius r, then the output can be given a separable decomposition provided the radius
of the cylindrical operators in the decomposition grows by a constant λ > 0. By combining
this with a modified version of a previous algorithm, we find that this enables an efficient
classical simulation algorithm that can sample from the output distribution to within additive
error. We then use a coarse-graining approach to increase the range of input states that can be
efficiently classically simulated. We then compute the equivalent of λ for arbitrary diagonal
two-qubit gates. Lastly, we use alternative notions of entanglement to show that there are
state spaces that can improve the region of input states that can be classically simulated.
In the second part, we examine potential obstacles that may arise when attempting to
efficiently classically simulate an increased range of quantum input states. Using a percolation
based approach, we show that if the input states are permitted with sufficient radius, then BQP
can be supported and efficient classical simulation is unlikely. Furthermore, using conjectures
about the polynomial hierarchy, we show that there is a threshold for which cluster states
with alternative inputs cannot be efficiently classically simulated with multiplicative error.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2023-01-01T00:00:00ZTaking shape: The data science of elastic shape analysis with practical applicationsSalili-James, Ariannahttp://bura.brunel.ac.uk/handle/2438/262822023-05-15T15:07:03Z2023-01-01T00:00:00ZTitle: Taking shape: The data science of elastic shape analysis with practical applications
Authors: Salili-James, Arianna
Abstract: A mathematical curve can represent many different objects, both physical and abstract,
from the outline curve of an artefact in an image to the weight of growing animal to
the set of frequencies used in a sound. Regardless of these variations, the curves can
almost always vary non-linearly. One way to study shapes and their potential variations
is elastic shape analysis, a rich theory of which has developed over the past twenty years.
However, methods of elastic shape analysis are seldom utilized in practical applications
on real-world data, especially outside of the mathematical shape analysis community.
Our aim in this thesis is to explore some practical applications of elastic shape analysis.
To do this, we work with various types of shape data, the majority of which are based on
image datasets. As our focus is on two-dimensional curves, it is important to be able to
robustly extract contours from images, before we can apply elastic shape analysis tools.
In order to analyse the shapes in a dataset, we turn to methods of machine learning, to
investigate the applications of elastic shape analysis in classification.
In this thesis, we introduce an anthology of projects, in order to emphasise and under-
stand the potential of elastic shape analysis in practical applications. There are four main
projects in this thesis: (i) Classification of objects using outlines and the comparisons
between methods of elastic shape analysis, geometric morphometrics, and human experts,
with a focus on ancient Greek vases, (ii) Mussel species identification and a demonstra-
tion that shape may not be enough in some applications, (iii) A novel tool to monitor
the development of k ̄ak ̄ap ̄o chicks, and (iv) Classifying individual kiwi based on acoustic
data from their calls.
By combining tools from computer vision and machine learning with methods of elastic
shape analysis, we introduce a practical framework for the application of elastic shape
analysis, through a data science lens.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London.2023-01-01T00:00:00ZEfficient classical simulation of a variant of cluster state quantum computationAtallah, Saharhttp://bura.brunel.ac.uk/handle/2438/257592023-01-13T03:01:02Z2022-01-01T00:00:00ZTitle: Efficient classical simulation of a variant of cluster state quantum computation
Authors: Atallah, Sahar
Abstract: Quantum computers are known for their ability to solve some computational problems faster than classical computers. There is a race to build quantum computers because it is believed they might be better than classical; but it remains unknown
whether quantum computers are in fact better than conventional computers. To understand this problem, we develop a new method of classically simulating certain types of quantum system that are previously unknown to be efficiently simulatable
on classical computers.
We adjust a part of cluster state quantum computation to study the computational power and we demonstrate that there is a finite region of pure states j i around the Z-eigenstates for which the setup can be efficiently simulated classically, given
that the measurements are limited to Z and X - Y plane measurements. This classical simulation works by considering alternative local state spaces that we called "cylinders" and different notion of entanglement to normal quantum entanglement.
Then, we work out similar regions for states created using other diagonal gates instead of the CZ. These diagonal gates are represented by V (θ) = |0><0⊗I+|1><1|⊗ Zθ where Zθ = |0><0| + eiθ|1><1|. It turns out that almost all inputs are classically
simulatable when θ is small.
In addition, we nd that classical simulation also works by considering new type of non-quantum state spaces other than cylinders and maintaining non-entangled representation by growing the size of these state spaces. We search over some state spaces to try optimize our classical simulation and it turns out that, among the state spaces that we searched through, the cylinder is the most optimal state space.
And finally, we will look at a coarse graining version of construction which increases the efficiently simulatable region.
Description: This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University2022-01-01T00:00:00ZLocating the source of an acoustic wave equation using likelihood estimates from the kalman filter applied to surface readingsElliott-Sands, Matthew Peter Francishttp://bura.brunel.ac.uk/handle/2438/228042021-06-08T02:01:17Z2020-01-01T00:00:00ZTitle: Locating the source of an acoustic wave equation using likelihood estimates from the kalman filter applied to surface readings
Authors: Elliott-Sands, Matthew Peter Francis
Abstract: Cardiovascular disease (CVD) was the second-largest cause of death in the United
Kingdom in 2014 [1], accounting for 32% of all deaths in 2009 [2]. CVD encompasses
many diseases, one of which is coronary artery disease (CAD), otherwise known as
atherosclerosis. Atherosclerosis is the build-up of fatty material, called plaque, inside
the wall of the artery. Over time, this plaque will grow too large or break o , causing
a blockage resulting in a heart attack. Currently, mortality from CAD has decreased by
72% between 1979 and 2013 [3]. However, predictions show that if the increasing trend
of Body Mass Index (BMI) continues, then mortality from CAD could start increasing
again [4]. There are several di erent methods currently available to the National Health
Service (NHS) to diagnose CAD. However, there are long waiting lists and expensive costs
associated with current diagnosis methods.
Our aim is to look at a non-invasive approach of diagnosing CAD. We have limited
our investigation to simple model problems. Therefore, further work would be required
to consider more complex cases which align with the real-world application.
In this thesis, we consider both 1-dimensional (1D) and 2-dimensional (2D) problems
modelled by an acoustic wave equation with a forcing function which attempts to emulate
a localised disturbance caused by CAD.We use an explicit nite di erence method (FDM)
to approximate the solution in our partial di erential equation (PDE) and discard the
disturbance location used. Having added noise to these approximations in an attempt to
mimic noise from real readings, we record these approximations at speci c locations on
the surface of our domains to imitate data collected from actual sensors. Using this data
in the Kalman lter (KF), where guesses for the disturbance location are made, we can
estimate the approximation of u throughout our domain. Using data generated by the
KF, we compute likelihood estimates for each guess made and obtain the most probable
disturbance location used to generate our sensor readings.
Description: This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University London2020-01-01T00:00:00Z