Quantum Theory from Principles, Quantum Software from Diagrams

TL;DR

This thesis explores foundational principles of quantum theory and applies diagrammatic reasoning, specifically the ZX-calculus, to optimize quantum computing processes and develop new models and algorithms.

Contribution

It reconstructs quantum theory from first principles and introduces novel diagrammatic methods for quantum circuit optimization and measurement-based quantum computing.

Findings

Reconstructed quantum theory from properties of sequential measurement.

Developed a new model for measurement-based quantum computing.

Created algorithms for circuit simplification and T-count reduction.

Abstract

This thesis consists of two parts. The first part is about how quantum theory can be recovered from first principles, while the second part is about the application of diagrammatic reasoning, specifically the ZX-calculus, to practical problems in quantum computing. The main results of the first part include a reconstruction of quantum theory from principles related to properties of sequential measurement and a reconstruction based on properties of pure maps and the mathematics of effectus theory. It also includes a detailed study of JBW-algebras, a type of infinite-dimensional Jordan algebra motivated by von Neumann algebras. In the second part we find a new model for measurement-based quantum computing, study how measurement patterns in the one-way model can be simplified and find a new algorithm for extracting a unitary circuit from such patterns. We use these results to develop a circuit optimisation strategy that leads to a new normal form for Clifford circuits and reductions in the T-count of Clifford+T circuits.