Formal Analysis of Quantum Optics

At the beginning of the last century, the theory of quantum optics arose and led to a revolution in physics, since it allowed the interpretation of many unknown phenomena and the development of numerous powerful, cutting edge engineering applications, such as high precision laser technology. The analysis and verification of such applications and systems, however, are very complicated. Moreover, traditional analysis tools, e.g., simulation, numerical methods, computer algebra systems, and paper-and-pencil approaches are not well suited for quantum systems. In the last decade, a new emerging verification technique, called formal methods, became common among engineering domains, and has proven to be effective as an analysis tool. Formal methods consist in the development of mathematical models of the system subject for analysis, and deriving computer-aided mathematical proofs. In this thesis, we propose a framework for the analysis of quantum optics based on formal methods, in particular theorem proving. The framework aims at implementing necessary quantum mechanics and optics concepts and theorems that facilitate the modelling of quantum optical devices and circuits, and then reason about them formally. To this end, the framework consists of three major libraries: 1) Mathematical foundations, which mainly contain the theory of complex-valued-function linear spaces, 2) Quantum mechanics, which develops the general rules of quantum physics, and 3) Quantum Optics, which specializes these rules for light beams and implements all related concepts, e.g., light coherence which is typically emitted by laser sources. On top of these theoretical foundations, we build a library of formal models of a number of optical devices commonly used in quantum circuits, including, beam splitters, light displacers, and light phase shifters. Using the proposed framework, we have been able to formally verify common quantum optical computing circuits, namely the Flip gate, CNOT gate, and Mach-Zehnder interferometer

Automated equivalence checking of quantum information systems

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

Model checking quantum protocols

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

Quantum holonomies in photonic waveguide systems

The thesis at hand deals with the emergence of quantum holonomies in systems of coupled waveguides. Several proposals for their realisation in arrays of laser-written fused-silica waveguides are presented, including experimental results. I develop an operator-theoretic framework for the photon-number independent description of these optical networks. Finally, quantum holonomies will be embedded into schemes for measurement-based quantum computation, with the aim of approximating Jones polynomials.Die vorliegende Arbeit untersucht die Konzipierung von Quantenholonomien in Systemen gekoppelter Wellenleiter. Eine Vielzahl möglicher Realisierungen mittels lasergeschriebener Wellenleiter in Quarzglas wird präsentiert und zugehörige experimentelle Ergebnisse erläutert. Die Entwicklung einer operatortheoretischen Darstellung für die photonenzahlunabhängige Beschreibung dieser optischen Netzwerke wird vorgenommen. Abschließend werden Quantenholonomien für die messinduzierte Quantenberechnung von Jones-Polynomen verwendet

Model checking quantum protocols

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.EThOS - Electronic Theses Online ServiceUniversity of Warwick. Dept. of Computer ScienceEngineering and Physical Sciences Research Council (Great Britain) (EPSRC) (GR/S34090/01, EP/E006833/2, GR/S86037/01)Sixth Framework Programme (European Commission) (SFP)Fundação para a Ciência ea Tecnologia (FCT) (POCI/MAT/55796/2004)Conselho de Reitores das Universidades Portuguesas (CRUP)GBUnited Kingdo

A Verified Software Toolchain for Quantum Programming

Quantum computing is steadily moving from theory into practice, with small-scale quantum computers available for public use. Now quantum programmers are faced with a classical problem: How can they be sure that their code does what they intend it to do? I aim to show that techniques for classical program verification can be adapted to the quantum setting, allowing for the development of high-assurance quantum software, without sacrificing performance or programmability. In support of this thesis, I present several results in the application of formal methods to the domain of quantum programming, aiming to provide a high-assurance software toolchain for quantum programming. I begin by presenting SQIR, a small quantum intermediate representation deeply embedded in the Coq proof assistant, which has been used to implement and prove correct quantum algorithms such as Grover’s search and Shor’s factorization algorithm. Next, I present VOQC, a verified optimizer for quantum circuits that contains state-of-the-art SQIR program optimizations with performance on par with unverified tools. I additionally discuss VQO, a framework for specifying and verifying oracle programs, which can then be optimized with VOQC. Finally, I present exploratory work on providing high assurance for a high-level industry quantum programming language, Q#, in the F* proof assistant