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

Towards the Design Automation of Quantum Circuits

Quantum mechanics based computing systems are expected to have high capabilities and are considered good candidates to replace classical cryptography and supercomputing systems. Among many implementations, quantum optics systems provide a promising platform to implement universal quantum computers, since they link quantum computation and quantum communication in the same framework. Recently, several quantum gates, circuits, and protocols have been experimentally realized using optics. Despite the fact that big advances in building the physical quantum computers were achieved, there are no currently available industrial computer aided tools that can perform the modeling, analysis, and verification of optical quantum computing systems. In this thesis, we tackle the idea of design automation for quantum circuits, where we use a sound language, higher order logic, to model and reason about quantum circuits formally. In particular, we propose a framework for the hierarchical modeling and automated verification of quantum computing circuits. The modeling approach captures quantum models built hierarchically from quantum gates, which models are readily available in a library. The analysis and verification of composed circuits is done seamlessly based on dedicated mathematical foundations formalized in the theorem prover. Specifically, the tensor product and linear projection are used to extract the quantum circuit outputs. Subsequently, a rich library of quantum gates which includes 1-qubit, 2-qubit, and 3-qubit gates is formalized. In order to automate the analysis process, we developed a decision procedure to eliminate the need of user guidance throughout the formal proofs. To demonstrate the effectiveness of the proposed framework, we conduct the formal analysis of a benchmark of quantum circuits including the Shor's integer factorization algorithm, the Grover's oracle, and the quantum full adder

Computer theorem proving in math

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003)

Formal Analysis of Geometrical Optics using Theorem Proving

Geometrical optics is a classical theory of Physics which describes the light propagation in the form of rays and beams. One of its main advantages is efficient and scalable formalism for the modeling and analysis of a variety of optical systems which are used in ubiquitous applications including telecommunication, medicine and biomedical devices. Traditionally, the modeling and analysis of optical systems have been carried out by paper-and-pencil based proofs and numerical algorithms. However, these techniques cannot provide perfectly accurate results due to the risk of human error and inherent incompleteness of numerical algorithms. In this thesis, we propose a higher-order logic theorem proving based framework to analyze optical systems. The main advantages of this framework are the expressiveness of higher-order logic and the soundness of theorem proving systems which provide unrivaled analysis accuracy. In particular, this thesis provides the higher-order logic formalization of geometrical optics including the notion of light rays, beams and optical systems. This allows us to develop a comprehensive analysis support for optical resonators, optical imaging and Quasi-optical systems. This thesis also facilitates the verification of some of the most interesting optical system properties like stability, chaotic map generation, beam transformation and mode analysis. We use this infrastructure to build a library of commonly used optical components such as lenses, mirrors and optical cavities. In order to demonstrate the effectiveness of our proposed approach, we conduct the formal analysis of some real-world optical systems, e.g., an ophthalmic device for eye, a Fabry-P\'{e}rot resonator, an optical phase-conjugated ring resonator and a receiver module of the APEX telescope. All the above mentioned work is carried out in the HOL Light theorem prover

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 26th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The total of 60 regular papers presented in these volumes was carefully reviewed and selected from 155 submissions. The papers are organized in topical sections as follows: Part I: Program verification; SAT and SMT; Timed and Dynamical Systems; Verifying Concurrent Systems; Probabilistic Systems; Model Checking and Reachability; and Timed and Probabilistic Systems. Part II: Bisimulation; Verification and Efficiency; Logic and Proof; Tools and Case Studies; Games and Automata; and SV-COMP 2020