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This thesis studies the categorical formalisation of quantum computing, through the prism of type theory, in a three-tier process. The first stage of our investigation involves the creation of the dagger lambda calculus, a lambda calculus for dagger compact categories. Our second contribution lifts the expressive power of the dagger lambda calculus, to that of a quantum programming language, by adding classical control in the form of complementary classical structures and dualisers. Finally, our third contribution demonstrates how our lambda calculus can be applied to various well known problems in quantum computation: Quantum Key Distribution, the quantum Fourier transform, and the teleportation protocol./

The aim of this thesis is to present an extension to the string graphs of Dixon, Duncan and Kissinger that allows the finite representation of certain infinite families of graphs and graph rewrite rules, and to demonstrate that a logic can be built on this to allow the formalisation of inductive proofs in the string diagrams of compact closed and traced symmetric monoidal categories. String diagrams provide an intuitive method for reasoning about monoidal categories. However, this does not negate the ability for those using them to make mistakes in proofs. To this end, there is a project (Quantomatic) to build a proof assistant for string diagrams, at least for those based on categories with a notion of trace. The development of string graphs has provided a combinatorial formalisation of string diagrams, laying the foundations for this project. The prevalence of commutative Frobenius algebras (CFAs) in quantum information theory, a major application area of these diagrams, has led to the use of variable-arity nodes as a shorthand for normalised networks of Frobenius algebra morphisms, so-called "spider notation". This notation greatly eases reasoning with CFAs, but string graphs are inadequate to properly encode this reasoning. This dissertation extends string graphs to allow for variable-arity nodes to be represented at all, and then introduces !-box notation (and structures to encode it) to represent string graph equations containing repeated subgraphs, where the number of repetitions is abitrary. It then demonstrates how we can reason directly about !-graphs, viewed as (typically infinite) families of string graphs. Of particular note is the presentation of a form of graph-based induction, allowing the formal encoding of proofs that previously could only be represented as a mix of string diagrams and explanatory text., Comment: DPhil (PhD) thesis; University of Oxford; 172 pages

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.While string diagrams are very intuitive, existing methods for defining them rigorously rely on topological notions that do not extend naturally to automated computation. 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. Namely, maximal sets of strongly complementary observables of dimension D must be of size no larger than 2, and are in 1-to-1 correspondence with the Abelian groups of order D. 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. Notably, we illustrate how the algebraic structures induced by these operations correspond to the (partial) arithmetic operations of addition and multiplication on the complex projective line.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.

The aim of this thesis is to present an extension to the string graphs of Dixon, Duncan and Kissinger that allows the finite representation of certain infinite families of graphs and graph rewrite rules, and to demonstrate that a logic can be built on this to allow the formalisation of inductive proofs in the string diagrams of compact closed and traced symmetric monoidal categories. String diagrams provide an intuitive method for reasoning about monoidal categories. However, this does not negate the ability for those using them to make mistakes in proofs. To this end, there is a project (Quantomatic) to build a proof assistant for string diagrams, at least for those based on categories with a notion of trace. The development of string graphs has provided a combinatorial formalisation of string diagrams, laying the foundations for this project. The prevalence of commutative Frobenius algebras (CFAs) in quantum information theory, a major application area of these diagrams, has led to the use of variable-arity nodes as a shorthand for normalised networks of Frobenius algebra morphisms, so-called "spider notation". This notation greatly eases reasoning with CFAs, but string graphs are inadequate to properly encode this reasoning. This dissertation firstly extends string graphs to allow for variable-arity nodes to be represented at all, and then introduces !-box notation – and structures to encode it – to represent string graph equations containing repeated subgraphs, where the number of repetitions is abitrary. This can be used to represent, for example, the "spider law" of CFAs, allowing two spiders to be merged, as well as the much more complex generalised bialgebra law that can arise from two interacting CFAs. This work then demonstrates how we can reason directly about !-graphs, viewed as (typically infinite) families of string graphs. Of particular note is the presentation of a form of graph-based induction, allowing the formal encoding of proofs that previously could only be represented as a mix of string diagrams and explanatory text.

This thesis studies the categorical formalisation of quantum computing, through the prism of type theory, in a three-tier process. The first stage of our investigation involves the creation of the dagger lambda calculus, a lambda calculus for dagger compact categories. Our second contribution lifts the expressive power of the dagger lambda calculus, to that of a quantum programming language, by adding classical control in the form of complementary classical structures and dualisers. Finally, our third contribution demonstrates how our lambda calculus can be applied to various well known problems in quantum computation: Quantum Key Distribution, the quantum Fourier transform, and the teleportation protocol./

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.While string diagrams are very intuitive, existing methods for defining them rigorously rely on topological notions that do not extend naturally to automated computation. 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. Namely, maximal sets of strongly complementary observables of dimension D must be of size no larger than 2, and are in 1-to-1 correspondence with the Abelian groups of order D. 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. Notably, we illustrate how the algebraic structures induced by these operations correspond to the (partial) arithmetic operations of addition and multiplication on the complex projective line.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.