Quantum Theory from Principles, Quantum Software from Diagrams

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.

PhD Thesis. Part A is 135 pages. Part B is 95 pages

References in corpus (27)

• Surface codes: Towards practical large-scale quantum computation
• Multi-party entanglement in graph states
• Characterizing quantum theory in terms of information-theoretic constraints
• Measurement-based quantum computation beyond the one-way model
• Quantum circuits of T-depth one
• Unsupervised Machine Learning on a Hybrid Quantum Computer
• From three-photon GHZ states to ballistic universal quantum computation
• Novel constructions for the fault-tolerant Toffoli gate
• Generalized Flow and Determinism in Measurement-based Quantum Computation
• Both Toffoli and Controlled-NOT need little help to do universal quantum computation
• Diagonal gates in the Clifford hierarchy
• A Depth-Optimal Canonical Form for Single-qubit Quantum Circuits
• On the qubit routing problem
• A bridge to lower overhead quantum computation
• A universal completion of the ZX-calculus
• Finding Optimal Flows Efficiently
• Terminality implies non-signalling
• Graphical description of the action of Clifford operators on stabilizer states
• Physical-depth architectural requirements for generating universal photonic cluster states
• Global Quantum Circuit Optimization
• Entanglement as an axiomatic foundation for statistical mechanics
• Strongly symmetric spectral convex bodies are Jordan algebra state spaces
• Computation at a distance
• Graphical Fourier Theory and the Cost of Quantum Addition
• Categorical Operational Physics
• Completeness of the Phase-free ZH-calculus
• Effectuses in Categorical Quantum Foundations