143 research outputs found
Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing
This work is about diagrammatic languages, how they can be represented, and
what they in turn can be used to represent. More specifically, it focuses on
representations and applications of string diagrams. String diagrams are used
to represent a collection of processes, depicted as "boxes" with multiple
(typed) inputs and outputs, depicted as "wires". If we allow plugging input and
output wires together, we can intuitively represent complex compositions of
processes, formalised as morphisms in a monoidal category.
[...] The first major contribution of this dissertation is the introduction
of a discretised version of a string diagram called a string graph. String
graphs form a partial adhesive category, so they can be manipulated using
double-pushout graph rewriting. Furthermore, we show how string graphs modulo a
rewrite system can be used to construct free symmetric traced and compact
closed categories on a monoidal signature.
The second contribution is in the application of graphical languages to
quantum information theory. We use a mixture of diagrammatic and algebraic
techniques to prove a new classification result for strongly complementary
observables. [...] We also introduce a graphical language for multipartite
entanglement and illustrate a simple graphical axiom that distinguishes the two
maximally-entangled tripartite qubit states: GHZ and W. [...]
The third contribution is a description of two software tools developed in
part by the author to implement much of the theoretical content described here.
The first tool is Quantomatic, a desktop application for building string graphs
and graphical theories, as well as performing automated graph rewriting
visually. The second is QuantoCoSy, which performs fully automated,
model-driven theory creation using a procedure called conjecture synthesis.Comment: PhD Thesis. Passed examination. Minor corrections made and one
theorem added at the end of Chapter 5. 182 pages, ~300 figures. See full text
for unabridged abstrac
Transformation of Attributed Structures with Cloning (Long Version)
Copying, or cloning, is a basic operation used in the specification of many
applications in computer science. However, when dealing with complex
structures, like graphs, cloning is not a straightforward operation since a
copy of a single vertex may involve (implicitly)copying many edges. Therefore,
most graph transformation approaches forbid the possibility of cloning. We
tackle this problem by providing a framework for graph transformations with
cloning. We use attributed graphs and allow rules to change attributes. These
two features (cloning/changing attributes) together give rise to a powerful
formal specification approach. In order to handle different kinds of graphs and
attributes, we first define the notion of attributed structures in an abstract
way. Then we generalise the sesqui-pushout approach of graph transformation in
the proposed general framework and give appropriate conditions under which
attributed structures can be transformed. Finally, we instantiate our general
framework with different examples, showing that many structures can be handled
and that the proposed framework allows one to specify complex operations in a
natural way
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