Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing

by   Aleks Kissinger, et al.

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.




Promonads and String Diagrams for Effectful Categories

Premonoidal and Freyd categories are both generalized by non-cartesian F...

Synthesising Graphical Theories

In recent years, diagrammatic languages have been shown to be a powerful...

Graphical Conjunctive Queries

The Calculus of Conjunctive Queries (CCQ) has foundational status in dat...

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

Symmetric monoidal theories (SMTs) generalise algebraic theories in a wa...

String Diagram Rewrite Theory III: Confluence with and without Frobenius

In this paper we address the problem of proving confluence for string di...

The Cost of Compositionality: A High-Performance Implementation of String Diagram Composition

String diagrams are an increasingly popular algebraic language for the a...

Completeness and expressiveness for gs-monoidal categories

Formalised in the study of symmetric monoidal categories, string diagram...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.