Theory and applications of quantum process calculus

Abstract

Formal methods is an area in theoretical computer science that provides the theories and tools for describing and verifying the correctness of computing systems. Usually, such systems comprise of concurrent and communicating components. The success of this field led to the development of quantum formal methods by transferring the ideas of formal methods to quantum systems. In particular, formal methods provides a systematic methodology for verification of systems. Quantum process calculus is a specialised field in quantum formal methods that helps to describe and analyse the behaviour of systems that combine quantum and classical elements. We focus on the theory and applications of quantum process calculus in particular to use Communicating Quantum Processes (CQP), a quantum process calculus, to model and analyse quantum information processing (QIP) systems. Previous work on CQP defined labelled transition relations for CQP in order to describe external interactions and also established the theory of behavioural equivalence in CQP based on probabilistic branching bisimilarity. This theory formalizes the idea of observational indistinguishability in order to prove or verify the correctness of a system, and an important property of the equivalence is the congruence property. We use the theory to analyse two versions of a quantum error correcting code system. We use the equational theory of CQP from the previous work and define an additional three new axioms in order to analyse quantum protocols comprising quantum secret-sharing, quantum error correction, remote-CNOT and superdense coding. We have expanded the framework of modelling in CQP from providing an abstract view of the quantum system to describe a realistic QIP system such as linear optical quantum computing (LOQC) and its associated experimental processes. By extending the theory of behavioural equivalence of CQP, we have formally verified two models of an LOQC CNOT gate using CQP. The two models use different measurement semantics in order to work at different levels of abstraction. This flexibility of the process calculus approach allows descriptions from detailed hardware implementations up to more abstract specifications. The orbital angular momentum (OAM) property of light allows us to perform experiments in studying higher dimensional quantum systems and their applications to quantum technologies. In relation to this work, we have extended CQP to model higher dimensional quantum protocols