Graphical Calculi and their Conjecture Synthesis

TL;DR

This paper advances graphical calculi for quantum computing by developing algebraic inference frameworks, introducing new calculi like RING and ZQ, and linking these to algebraic fields, aiding theorem synthesis and optimization.

Contribution

It introduces inference and verification frameworks for graphical calculi, and presents two new calculi, RING and ZQ, enhancing quantum diagram reasoning and conjecture synthesis.

Findings

Developed a generalisation step for conjecture synthesis.

Introduced the RING calculus for ring-based qubit diagrams.

Created the ZQ calculus for arbitrary qubit rotations.

Abstract

Categorical Quantum Mechanics, and graphical calculi in particular, has proven to be an intuitive and powerful way to reason about quantum computing. This work continues the exploration of graphical calculi, inside and outside of the quantum computing setting, by investigating the algebraic structures with which we label diagrams. The initial aim for this was Conjecture Synthesis; the algorithmic process of creating theorems. To this process we introduce a generalisation step, which itself requires the ability to infer and then verify parameterised families of theorems. This thesis introduces such inference and verification frameworks, in doing so forging novel links between graphical calculi and fields such as Algebraic Geometry and Galois Theory. These frameworks inspired further research into the design of graphical calculi, and we introduce two important new calculi here. First is the calculus RING, which is initial among ring-based qubit graphical calculi, and in turn inspired the introduction and classification of phase homomorphism pairs also presented here. The second is the calculus ZQ, an edge-decorated calculus which naturally expresses arbitrary qubit rotations, eliminating the need for non-linear rules such as (EU) of ZX. It is expected that these results will be of use to those creating optimisation schemes and intermediate representations for quantum computing, to those creating new graphical calculi, and for those performing conjecture synthesis.