Full formal descriptions of algorithms making use of quantum principles must take into account both quantum and classical computing components, as well as communications between these components. Moreover, to model concurrent and distributed quantum computations and quantum communication protocols, communications over quantum channels that move qubits physically from one place to another must also be taken into account. Inspired by classical process algebras, which provide a framework for modelling cooperating computations, a process algebraic notation is defined. This notation provides a homogeneous style for formal descriptions of concurrent and distributed computations comprising both quantum and classical parts. Based upon an operational semantics that makes sure that quantum objects, operations and communications operate according to the postulates of quantum mechanics, an equivalence is defined among process states considered as having the same behaviour. This equivalence is a probabilistic branching bisimulation. From this relation, an equivalence on processes is defined. However, it is not a congruence because it is not preserved by parallel composition.
Full formal descriptions of algorithms making use of quantum principles must take into account both quantum and classical computing components, as well as communications between these components. Moreover, to model concurrent and distributed quantum computations and quantum communication protocols, communications over quantum channels that move qubits physically from one place to another must also be taken into account. Inspired by classical process algebras, which provide a framework for modelling cooperating computations, a process algebraic notation is defined. This notation provides a homogeneous style for formal descriptions of concurrent and distributed computations comprising both quantum and classical parts. Based upon an operational semantics that makes sure that quantum objects, operations and communications operate according to the postulates of quantum mechanics, an equivalence is defined among process states considered as having the same behaviour. This equivalence is a probabilistic branching bisimulation. From this relation, an equivalence on processes is defined. However, it is not a congruence because it is not preserved by parallel composition.
Full formal descriptions of algorithms making use of quantum principles must take into account both quantum and classical computing components and assemble them so that they communicate and cooperate.Moreover, to model concurrent and distributed quantum computations, as well as quantum communication protocols, quantum to quantum communications which move qubits physically from one place to another must also be taken into account. Inspired by classical process algebras, which provide a framework for modeling cooperating computations, a process algebraic notation is defined, which provides a homogeneous style to formal descriptions of concurrent and distributed computations comprising both quantum and classical parts.Based upon an operational semantics which makes sure that quantum objects, operations and communications operate according to the postulates of quantum mechanics, a probabilistic branching bisimulation is defined among processes considered as having the same behavior.
Full formal descriptions of algorithms making use of quantum principles must take into account both quantum and classical computing components and assemble them so that they communicate and cooperate. Moreover, to model concurrent and distributed quantum computations, as well as quantum communication protocols, quantum to quantum communications which move qubits physically from one place to another must also be taken into account. Inspired by classical process algebras, which provide a framework for modeling cooperating computations, a process algebraic notation is defined, named QPAlg for Quantum Process Algebra, which provides a homogeneous style to formal descriptions of concurrent and distributed computations comprising both quantum and classical parts. On the quantum side, QPAlg provides quantum variables, operations on quantum variables (unitary operators and measurement observables), as well as new forms of communications involving the quantum world. The operational semantics makes sure that these quantum objects, operations and communications operate according to the postulates of quantum mechanics.
Quantum communication is a rapidly growing area of research and development. While the successful construction of a large-scale quantum computer may be some years away, there are already commercial implementations of secure communication using quantum cryptography. The application of formal methods to classical communication and cryptographic systems has been very successful, and is now widely used in industry by organisations such as Intel, Microsoft and NASA. There is reason to believe that similar benefits can be expected for the verification of quantum systems. In this thesis, we focus on the use of process calculus, specifically Communicating Quantum Processes (CQP), for the analysis of quantum protocols. Congruence relations are an important aspect of process calculus, since they provide the foundation for equational reasoning. Previous work on congruence relations for quantum processes excluded the classical information arising from measurements, and was therefore unable to analyse many of the interesting known quantum communication protocols. Developing a congruence relation for general quantum processes is difficult because of the interaction between measurement, entanglement and parallel composition. We define a labelled transition relation for CQP in order to describe external interactions. Based on this semantics, we define a notion of observational equivalence for CQP processes, namely probabilistic branching bisimilarity. We find that this relation is not preserved by parallel composition, however we are able to gain a deeper understanding of the link between probabilistic branching and measurement. Based on this newfound understanding, we present a novel semantics for quantum processes, combining mixed quantum states with probabilistic branching. With respect to this new semantic model, we define full probabilistic branching bisimilarity and prove that it is a congruence. We use this congruence relation to discuss an axiomatic approach to the verification of quantum processes. The quantum teleportation protocol is used as a primary example throughout, and we prove that it is congruent to a quantum channel. We define a translation from CQP to the Quantum Model Checker (QMC) in order to provide automated verification techniques using CQP specifications. We prove that this translation preserves the semantics of CQP processes, thereby enabling a multifaceted approach to formal verification by enhancing the manual techniques of process calculus with the benefits of model checking.
Modeling and reasoning about concurrent quantum systems is very important both for distributed quantum computing and for quantum protocol verification. As a consequence, a general framework describing formally the communication and concurrency in complex quantum systems is necessary. For this purpose, we propose a model qCCS which is a natural quantum extension of classical value-passing CCS with the input and output of quantum states, and unitary transformations and measurements on quantum systems. The operational semantics of qCCS is given based on probabilistic labeled transition system. This semantics has many different features compared with the proposals in literature in order to describe input and output of quantum systems which are possibly correlated with other components. Based on this operational semantics, we introduce the notions of strong probabilistic bisimilarity and weak probabilistic bisimilarity between quantum processes and discuss some properties of them, such as congruence under various combinators.
We introduce an algebra qCCS of pure quantum processes in which communications by moving quantum states physically are allowed and computations are modeled by super-operators, but no classical data is explicitly involved. An operational semantics of qCCS is presented in terms of (nonprobabilistic) labeled transition systems. Strong bisimulation between processes modeled in qCCS is defined, and its fundamental algebraic properties are established, including uniqueness of the solutions of recursive equations. To model sequential computation in qCCS, a reduction relation between processes is defined. By combining reduction relation and strong bisimulation we introduce the notion of strong reduction-bisimulation, which is a device for observing interaction of computation and communication in quantum systems. Finally, a notion of strong approximate bisimulation (equivalently, strong bisimulation distance) and its reduction counterpart are introduced. It is proved that both approximate bisimilarity and approximate reduction-bisimilarity are preserved by various constructors of quantum processes. This provides us with a formal tool for observing robustness of quantum processes against inaccuracy in the implementation of its elementary gates.
Modeling and reasoning about concurrent quantum systems is very important for both distributed quantum computing and quantum protocol verification. As a consequence, a general framework formally describing communication and concurrency in complex quantum systems is necessary. For this purpose, we propose a model named qCCS. It is a natural quantum extension of classical value-passing CCS which can deal with input and output of quantum states, and unitary transformations and measurements on quantum systems. The operational semantics of qCCS is given in terms of probabilistic labeled transition system. This semantics has many different features compared with the proposals in the available literature in order to describe the input and output of quantum systems which are possibly correlated with other components. Based on this operational semantics, the notions of strong probabilistic bisimulation and weak probabilistic bisimulation between quantum processes are introduced. Furthermore, some properties of these two probabilistic bisimulations, such as congruence under various combinators, are examined.