(Un)decidable Problems about Reachability of Quantum Systems

We study the reachability problem of a quantum system modelled by a quantum automaton. The reachable sets are chosen to be boolean combinations of (closed) subspaces of the state space of the quantum system. Four different reachability properties are considered: eventually reachable, globally reachable, ultimately forever reachable, and infinitely often reachable. The main result of this paper is that all of the four reachability properties are undecidable in general; however, the last three become decidable if the reachable sets are boolean combinations without negation

Automated equivalence checking of concurrent quantum systems

The novel field of quantum computation and quantum information has gathered significant momentum in the last few years. It has the potential to radically impact the future of information technology and in influence the development of modern society. The construction of practical, general purpose quantum computers has been challenging, but quantum cryptographic and communication devices have been available in the commercial marketplace for several years. Quantum networks have been built in various cities around the world and a dedicated satellite has been launched by China to provide secure quantum communication. Such new technologies demand rigorous analysis and verification before they can be trusted in safety- and security- critical applications. Experience with classical hardware and software systems has shown the difficulty of achieving robust and reliable implementations. We present CCSq, a concurrent language for describing quantum systems, and develop verification techniques for checking equivalence between CCSq processes. CCSq has well-defined operational and superoperator semantics for protocols that are functional, in the sense of computing a deterministic input-output relation for all interleavings arising from concurrency in the system. We have implemented QEC (Quantum Equivalence Checker), a tool which takes the specification and implementation of quantum protocols, described in CCSq, and automatically checks their equivalence. For efficiency purposes, we restrict ourselves to Clifford operators in the stabilizer formalism, but we are able to verify protocols over all input states. We have specified and verified a collection of interesting and practical quantum protocols ranging from quantum communication and quantum cryptography to quantum error correction

Equivalence Checking of Quantum Finite-State Machines

In this paper, we introduce the model of quantum Mealy machines and study the equivalence checking and minimisation problems of them. Two efficient algorithms are developed for checking equivalence of two states in the same machine and for checking equivalence of two machines. They are applied in experiments of equivalence checking of quantum circuits. Moreover, it is shown that the minimisation problem is proved to be in \textbf{PSPACE}

Automated equivalence checking of concurrent quantum systems

The novel field of quantum computation and quantum information has gathered significant momentum in the last few years. It has the potential to radically impact the future of information technology and in influence the development of modern society. The construction of practical, general purpose quantum computers has been challenging, but quantum cryptographic and communication devices have been available in the commercial marketplace for several years. Quantum networks have been built in various cities around the world and a dedicated satellite has been launched by China to provide secure quantum communication. Such new technologies demand rigorous analysis and verification before they can be trusted in safety- and security- critical applications. Experience with classical hardware and software systems has shown the difficulty of achieving robust and reliable implementations. We present CCSq, a concurrent language for describing quantum systems, and develop verification techniques for checking equivalence between CCSq processes. CCSq has well-defined operational and superoperator semantics for protocols that are functional, in the sense of computing a deterministic input-output relation for all interleavings arising from concurrency in the system. We have implemented QEC (Quantum Equivalence Checker), a tool which takes the specification and implementation of quantum protocols, described in CCSq, and automatically checks their equivalence. For efficiency purposes, we restrict ourselves to Clifford operators in the stabilizer formalism, but we are able to verify protocols over all input states. We have specified and verified a collection of interesting and practical quantum protocols ranging from quantum communication and quantum cryptography to quantum error correction

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