Continuous-variable (CV) quantum information processing is a promising candidate for large-scale fault-tolerant quantum computation. However, analysis of CV quantum process relies mostly on direct computation of the evolution of operators in the Heisenberg picture, and the features of CV space has yet to be thoroughly investigated in an intuitive manner. One key ingredient for further exploration of CV quantum computing is the construction of a computational model that brings visual intuition and new tools for the analysis of quantum protocols, especially non-Gaussian processes. In this paper, we delve into a graphical computational model, inspired by a similar model for qubit-based systems called the ZX calculus, which enables the representation of arbitrary CV quantum process as a simple directed graph. We demonstrate the utility of our model as a graphical tool to comprehend CV processes intuitively by showing how equivalences between two distinct quantum processes can be proven as a sequence of diagrammatic transformations for various quantum protocols. We also examine possible applications of our model, such as measurement-based quantum computing, characterization of Gaussian and non-Gaussian processes, and circuit optimization.
Quantum-Inspired Variational Calculus (QIVC) presents a novel approach to variational calculus that integrates quantum mechanics principles to enhance the accuracy and efficiency of optimization algorithms. The core mechanism involves the implementation of a variational optimization framework utilizing quantum-inspired techniques to improve the exploration of the search space. This research addresses the limitations of traditional optimization methods by introducing a quantum-based enhancement, potentially yielding superior results in complex optimization problems. This paper details the theoretical foundation, implementation, and preliminary results of QIVC, highlighting its potential impact on various fields where optimization is crucial.
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.