With the previous notions of bisimulation presented in the literature, to check if two quantum processes are bisimilar, we have to instantiate their free quantum variables with arbitrary quantum states, and verify the bisimilarity of the resulting configurations. This makes checking bisimilarity infeasible from an algorithmic point of view, because quantum states constitute a continuum. In this article, we introduce a symbolic operational semantics for quantum processes directly at the quantum operation level, which allows us to describe the bisimulation between quantum processes without resorting to quantum states. We show that the symbolic bisimulation defined here is equivalent to the open bisimulation for quantum processes in previous work, when strong bisimulations are considered. An algorithm for checking symbolic ground bisimilarity is presented. We also give a modal characterisation for quantum bisimilarity based on an extension of Hennessy-Milner logic to quantum processes.
With the previous notions of bisimulation presented in the literature, to check if two quantum processes are bisimilar, we have to instantiate their free quantum variables with arbitrary quantum states, and verify the bisimilarity of the resulting configurations. This makes checking bisimilarity infeasible from an algorithmic point of view, because quantum states constitute a continuum. In this article, we introduce a symbolic operational semantics for quantum processes directly at the quantum operation level, which allows us to describe the bisimulation between quantum processes without resorting to quantum states. We show that the symbolic bisimulation defined here is equivalent to the open bisimulation for quantum processes in previous work, when strong bisimulations are considered. An algorithm for checking symbolic ground bisimilarity is presented. We also give a modal characterisation for quantum bisimilarity based on an extension of Hennessy-Milner logic to quantum processes.
Bisimulation is a fundamental concept in the classical concurrency theory for comparing the behaviour of nondeterministic processes. It admits elegant characterisations from various perspectives such as fixed point theory, modal logics, game theory, coalgebras etc. In this paper, we review some key ideas used in the formulations and characterisations of reasonable notions of bisimulations for both probabilistic and quantum processes. To some extent the transition from probabilistic to quantum concurrency theory is smooth and natural. However, new ideas need also to be introduced. We have not yet reached the stage of formally verifying quantum communication protocols and quantum algorithms using bisimulations implemented by automatic tools. We discuss some recent efforts in this direction.
Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the other hand, quantum protocol designers may commit more faults than classical protocol designers since human intuition is poorly adapted to the quantum world. To offer formal techniques for modeling and verification of quantum protocols, several quantum extensions of process algebra have been proposed. An important issue in quantum process algebra is to discover a quantum generalization of bisimulation preserved by various process constructs, in particular, parallel composition, where one of the major differences between classical and quantum systems, namely quantum entanglement, is present. Quite a few versions of bisimulation have been defined for quantum processes in the literature, but in the best case they are only proved to be preserved by parallel composition of purely quantum processes where no classical communication is involved. Many quantum cryptographic protocols, however, employ the LOCC (Local Operations and Classical Communication) scheme, where classical communication must be explicitly specified. So, a notion of bisimulation preserved by parallel composition in the circumstance of both classical and quantum communication is crucial for process algebra approach to verification of quantum cryptographic protocols. In this article we introduce novel notions of strong bisimulation and weak bisimulation for quantum processes, and prove that they are congruent with respect to various process algebra combinators including parallel composition even when both classical and quantum communication are present. We also establish some basic algebraic laws for these bisimulations. In particular, we show the uniqueness of the solutions to recursive equations of quantum processes, which proves useful in verifying complex quantum protocols. To capture the idea that a quantum process approximately implements its specification, and provide techniques and tools for approximate reasoning, a quantified version of strong bisimulation, which defines for each pair of quantum processes a bisimulation-based distance characterizing the extent to which they are strongly bisimilar, is also introduced.
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.
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.
Abstract One important application of quantum process algebras is to formally verify quantum communication protocols. With a suitable notion of behavioural equivalence and a decision method, one can determine if an implementation of a protocol is consistent with its specification. Ground bisimulation is a convenient behavioural equivalence for quantum processes because of its associated coinduction proof technique. We exploit this technique to design and implement two on-the-fly algorithms for the strong and weak versions of ground bisimulation to check if two given processes in quantum CCS are equivalent. We then develop a tool that can verify interesting quantum protocols such as the BB84 quantum key distribution scheme.
The development of Quantum Communication Protocols sparked the interest in quantum extensions of process calculi. However, existing semantic models, typically based on probability distributions and probabilistic transition systems, often fail to adequately capture the nuances of Quantum Mechanics. They can spuriously distinguish between systems that quantum theory deems identical, and allow for physically unfeasible non-deterministic capabilities. Furthermore, their reliance on concrete quantum states makes automated verification computationally intractable due to the infinite, continuous nature of the state space. To overcome these challenges, this thesis proposes to model quantum protocols in a novel manner, by leveraging both the mathematical language of Quantum Theory and the verification techniques of Computer Science. The main objective is to develop a verification framework that is both physically sound and amenable to symbolic, tractable analysis. This is achieved through the development of two novel models of quantum concurrent systems. The first major contribution is a new semantic model that respects quantum indistinguishability by construction. Quantum theory teaches us that probabilistic mixtures of quantum states should not be modelled as probability distributions. We thus introduce Density Distributions, where processes are weighted by density operators rather than scalar probabilities. We define an operational semantics based on these distributions and introduce Quantum Labelled Bisimilarity, the first behavioural equivalence to be a congruence under physically admissible scheduling policies. We show that our equivalence coincides with a purely observational, context-based, saturated bisimilarity, thus proving its adequacy in terms of discriminating power. The second core contribution is an elementary framework for the symbolic verification of quantum protocols. To abstract away from concrete quantum inputs, we develop a mathematical theory of Conditional Quantum Effects, a novel generalization of quantum effects capable of representing conditional probabilities arising from quantum measurements. Building on this foundation, we define Effect-weighted Labelled Transition Systems (eLTSs), an input-independent model for concurrent quantum systems. We propose a "Heisenberg-style" symbolic semantics that maps any quantum protocol to a single eLTS. We prove that symbolic bisimilarity on these eLTSs is fully abstract with respect to ground probabilistic bisimilarity: two symbolic systems are equivalent if and only if they yield equivalent pLTSs under all possible quantum inputs.