64 research outputs found
Symmetry-Protected Topological Phases for Robust Quantum Computation
In recent years, topological phases of matter have presented exciting new avenues to achieve scalable quantum computation. In this thesis, we investigate a class of quantum many-body spin models known as symmetry-protected topological (SPT) phases for use in quantum information processing and storage. We explore the fault-tolerant properties of SPT phases, and how they can be utilized in the design of a quantum computer. Of central importance in this thesis is the concept of quantum error-correction, which in addition to its importance in fault-tolerant quantum computation, is used to characterise the stability of topological phases at finite temperature. We begin with an introduction to quantum computation, quantum error correction, and topological phases of matter. We then focus on the fundamental question of whether symmetry-protected topological phases of matter can exist in thermal equilibrium; we prove that systems protected by global onsite symmetries cannot be ordered at nonzero temperature. Subsequently, we show that certain three-dimensional models with generalised higher-form symmetries can be thermally SPT ordered, and we relate this order to the ability to perform fault-tolerant measurement-based quantum computation. Following this, we assess feasibility of these phases as quantum memories, motivated by the fact that SPT phases in three dimensions can possess protected topological degrees of freedom on their boundary. We find that certain SPT ordered systems can be self-correcting, allowing quantum information to be stored for arbitrarily long times without requiring active error correction. Finally, we develop a framework to construct new schemes of fault-tolerant measurement-based quantum computation. As a notable example, we develop a cluster-state scheme that simulates the braiding and fusion of surface-code defects, offering novel alternative methods to achieve fault-tolerant universal quantum computation
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
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Topological and geometric inference of data
The overarching problem under consideration is to determine the structure
of the subspace on which a distribution is supported, given
only a finite noisy sample thereof. The special case in
which the subspace is an embedded manifold is given particular
attention owing to its conceptual elegance, and asymptotic bounds are
obtained on the admissible level of noise such that the
manifold can be recovered up to homotopy equivalence.
Attention is turned on how to accomplish this in practice.
Following ideas from topological data analysis, simplicial complexes are used
as discrete analogues of spaces suitable for computation. By utilising
the prior assumption that the data lie on a manifold, topologically
inspired techniques are proposed for refining the simplicial complex
to better approximate this manifold. This is applied to the
problem of nonlinear dimensionality reduction and found to improve accuracy
of reconstructing several synthetic and real-world datasets.
The second chapter focuses on extending this work to the
case where the ambient space is non-Euclidean. The interfaces between
topological data analysis, functional data analysis, and shape analysis
are thoroughly explored. Lipschitz bounds are proved which relate several
metrics on the space of positive semidefinite matrices; they are then
interpreted in the context of topological data analysis. This is
applied to diffusion tensor imaging and phonology.
The final chapter explores the case where the points are
non-uniformly distributed over the embedded subspace. In particular, a method
is proposed to overcome the shortcomings of witness complex construction
when there are large deviations in the density. The theory
of multidimensional persistence is leveraged to provide a succinct setting
in which the structure of the data can be interpreted
as a generalised stratified space.EPSR
Toric Topology
Toric topology emerged in the end of the 1990s on the borders of equivariant
topology, algebraic and symplectic geometry, combinatorics and commutative
algebra. It has quickly grown up into a very active area with many
interdisciplinary links and applications, and continues to attract experts from
different fields.
The key players in toric topology are moment-angle manifolds, a family of
manifolds with torus actions defined in combinatorial terms. Their construction
links to combinatorial geometry and algebraic geometry of toric varieties via
the related notion of a quasitoric manifold. Discovery of remarkable geometric
structures on moment-angle manifolds led to seminal connections with the
classical and modern areas of symplectic, Lagrangian and non-Kaehler complex
geometry. A related categorical construction of moment-angle complexes and
their generalisations, polyhedral products, provides a universal framework for
many fundamental constructions of homotopical topology. The study of polyhedral
products is now evolving into a separate area of homotopy theory, with strong
links to other areas of toric topology. A new perspective on torus action has
also contributed to the development of classical areas of algebraic topology,
such as complex cobordism.
The book contains lots of open problems and is addressed to experts
interested in new ideas linking all the subjects involved, as well as to
graduate students and young researchers ready to enter into a beautiful new
area.Comment: Preliminary version. Contains 9 chapters, 5 appendices, bibliography,
index. 495 pages. Comments and suggestions are very welcom
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective
In this writing we shall address certain beautiful inter-relations between
the construction of 4-dimensional supersymmetric gauge theories and resolution
of algebraic singularities, from the perspective of String Theory. We review in
some detail the requisite background in both the mathematics, such as
orbifolds, symplectic quotients and quiver representations, as well as the
physics, such as gauged linear sigma models, geometrical engineering,
Hanany-Witten setups and D-brane probes.
We investigate aspects of world-volume gauge dynamics using D-brane
resolutions of various Calabi-Yau singularities, notably Gorenstein quotients
and toric singularities. Attention will be paid to the general methodology of
constructing gauge theories for these singular backgrounds, with and without
the presence of the NS-NS B-field, as well as the T-duals to brane setups and
branes wrapping cycles in the mirror geometry. Applications of such diverse and
elegant mathematics as crepant resolution of algebraic singularities,
representation of finite groups and finite graphs, modular invariants of affine
Lie algebras, etc. will naturally arise. Various viewpoints and generalisations
of McKay's Correspondence will also be considered.
The present work is a transcription of excerpts from the first three volumes
of the author's PhD thesis which was written under the direction of Prof. A.
Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of
MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac
vice; it is his sincerest wish that the ensuing pages might be of some small
use to the beginning student.Comment: 513 pages, 71 figs, Edited Excerpts from the first 3 volumes of the
author's PhD Thesi
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On algebraic singularities, finite graphs and D-brane gauge theories: A String theoretic perspective
In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes.
We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered.
The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student
Notes in Pure Mathematics & Mathematical Structures in Physics
These Notes deal with various areas of mathematics, and seek reciprocal
combinations, explore mutual relations, ranging from abstract objects to
problems in physics.Comment: Small improvements and addition
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