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Model checking quantum protocols

Nickolas K. Papanikolaou

Published Sep 9, 2026• 18 citations
Publisher

Abstract

This thesis describes model checking techniques for protocols arising in quantum information theory and quantum cryptography. We discuss the theory and implementation of a practical model checker, QMC, for quantum protocols. In our framework, we assume that the quantum operations performed in a protocol are restricted to those within the stabilizer formalism; while this particular set of operations is not universal for quantum computation, it allows us to develop models of several useful protocols as well as of systems involving both classical and quantum information processing. We detail the syntax, semantics and type system of QMC’s modelling language, the logic QCTL which is used for verification, and the verification algorithms that have been implemented in the tool. We demonstrate our techniques with applications to a number of case studies.

DOI
10.82444/warw.33484729

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This thesis describes model checking techniques for protocols arising in quantum information theory and quantum cryptography. We discuss the theory and implementation of a practical model checker, QMC, for quantum protocols. In our framework, we assume that the quantum operations performed in a protocol are restricted to those within the stabilizer formalism; while this particular set of operations is not universal for quantum computation, it allows us to develop models of several useful protocols as well as of systems involving both classical and quantum information processing. We detail the syntax, semantics and type system of QMC’s modelling language, the logic QCTL which is used for verification, and the verification algorithms that have been implemented in the tool. We demonstrate our techniques with applications to a number of case studies.
We describe model-checking techniques for protocols arising in quantum information theory and quantum cryptography. We discuss the theory and implementation of a practical model checker, QMC, for quantum protocols. In our framework, we assume that the quantum operations performed in a protocol are restricted to those within the stabilizer formalism; while this particular set of operations is not universal for quantum computation, it allows us to develop models of several useful protocols as well as of systems involving both classical and quantum information processing. We discuss the modeling language of QMC, the logic used for verification, and the verification algorithms that have been implemented in the tool. We demonstrate our techniques with applications to a number of case studies, including quantum teleportation and the BB84 quantum coin-flipping protocol. 11.1 Introduction Model-checking is a method of computer-aided verification used widely in both academia and industry for the analysis of software and communication systems. The method essentially consists of three phases: (1) modeling the system of interest at a suitable level of abstraction, usually by describing it in a formal specification language; (2) specifying the properties that the system must satisfy – these properties are typically expressed with reference to the system model, using a logic designed specially for this purpose ( e.g. , linear or branching time temporal logic); and (3) supplying the model and the properties to a model-checking tool, which automatically constructs the state space of the model and checks whether or not the given properties are satisfied (and if not, usually providing a counterexample).
We introduce a model-checking tool intended specially for the analysis of quantum information protocols. The tool incorporates an efficient representation of a certain class of quantum circuits, namely those expressible in the so-called stabiliser formalism. Models of protocols are described using a simple, imperative style simulation language which includes commands for the unitary operators in the Clifford group as well as classical integer and boolean variables. Formulas for verification are expressed using a subset of quantum computational tree logic (QCTL). The model-checking procedure treats quantum measurements as the source of non-determinism, leading to multiple protocol runs, one for each outcome. Verification is performed for each run.
Quantum technologies have progressed beyond the laboratory setting and are beginning to make an impact on industrial development. The construction of practical, general purpose quantum computers has been challenging, to say the least. But quantum cryptographic and communication devices have been available in the commercial marketplace for a few years. Quantum networks have been built in various cities around the world, and plans are afoot to launch a dedicated satellite for quantum communication. Such new technologies demand rigorous analysis and verification before they can be trusted in safety and security-critical applications. In this thesis we investigate the theory and practice of equivalence checking of quantum information systems. We present a tool, Quantum Equivalence Checker (QEC), which uses a concurrent language for describing quantum systems, and performs verification by checking equivalence between specification and implementation. For our process algebraic language CCSq, we define an operational semantics and a superoperator semantics. While in general, simulation of quantum systems using current computing technology is infeasible, we restrict ourselves to the stabilizer formalism, in which there are efficient simulation algorithms and representation of quantum states. By using the stabilizer representation of quantum states we introduce various algorithms for testing equality of stabilizer states. In this thesis, we consider concurrent quantum protocols that behave functionally in the sense of computing a deterministic input-output relation for all interleavings of a concurrent system. Crucially, these input-output relations can be abstracted by superoperators, enabling us to take advantage of linearity. This allows us to analyse the behaviour of protocols with arbitrary input, by simulating their operation on a finite basis set consisting of stabilizer states. We present algorithms for the checking of functionality and equivalence of quantum protocols. Despite the limitations of the stabilizer formalism and also the range of protocols that can be analysed using equivalence checking, QEC is applied to specify and verify a variety of interesting and practical quantum protocols from quantum communication and quantum cryptography to quantum error correction and quantum fault tolerant computation, where for each protocol different sequential and concurrent model are defined in CCSq. We also explain the implementation details of the QEC tool and report on the experimental results produced by using it on the verification of a number of case studies.
Quantum communication is a rapidly growing area of research and development. Quantum cryptography has already been implemented for secure communication, and commercial solutions are available. The application of formal methods to classical computing and communication systems has been very successful, and is widely used by industry. We expect similar benefits for the verification of quantum systems. Communicating Quantum Processes (CQP) is a process calculus based on the π-calculus with the inclusion of primitives for quantum information. Process calculi provide an algebraic approach to system specification and behavioural analysis. The Quantum Model Checker (QMC) is a tool for the automated verification of system correctness. Through an exhaustive search of the possible executions, QMC can check that correctness properties expressed using temporal logic formulae are satisfied. In this paper we describe our approach to the verification of quantum systems using a combination of process calculus and model checking. We also define a formal translation from CQP to the modelling language used by QMC and prove that this preserves the semantics of all supported CQP processes. © 2012 Old City Publishing, Inc.
Our objective in this thesis is to perform formal specification and automated validation of the class of protocols associated with quantum cryptography. Current analyses and proofs regarding quantum cryptographic protocols and their security are information-theoretic and also require in--depth understanding of the underlying physics. The alternative approach presented here involves the use of computer modelling languages and verification software. In particular, the logical model checker SPIN and the probabilistic model checker PRISM are used to analyse quantum key distribution, entanglement and quantum teleportation. The value of our approach lies not only in the fact that it is conceptually simpler than existing proofs, but in that it allows us to disambiguate protocol definitions and to assess their properties in various circumstances. By varying the values of certain parameters, different attacks on the protocols can be attempted and implementation-specific attributes can be analysed. The quantum key distribution protocol BB84 has been proved to be unconditionally secure against all types of eavesdropping. Although a general proof of this result is extremely hard to generate automatically, it is possible to develop specific protocol attacks and validate the protocol by computer. Assuming two types of eavesdropping attack (intercept-resend and random-substitute), we have used the PRISM tool to compute exactly the probability of successful eavesdropping, and found it to diminish as the number of transmitted qubits is increased. This result is linked to Mayers' security criteria for BB84. Also, we briefly describe a new quantum protocol definition language, named qSpec. The recent CQP process algebra, due to Nagarajan and Gay, is reviewed and examples of its application are provided. Alternative formalisms and validation tools are discussed, including QPAlg, PROBMELA and probUSM, the logic of knowledge and time for quantum systems due to Van der Meyden, and the QCSim simulator.