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Quantum systems are modelled as different mathematical structures, depending on their nature and complexity. This chapter considers one of the simplest (discrete-time) models of quantum systems, namely quantum automata. It introduces a way of describing linear-time (dynamic) properties of quantum systems and presents several algorithms for checking certain linear-time properties of quantum automata, for example, invariants and reachability.
Model checking is an algorithmic technique for verification of computing and communication hardware and software. This book extends the technique of model checking for quantum systems. As preliminaries, this chapter introduces basics of model checking for both classical non-probabilistic and probabilistic systems.
This chapter develops model-checking techniques for a much larger class of quantum systems modelled as quantum Markov chains or more generally, quantum Markov decision processes. The differences between quantum automata and quantum Markov systems require us to develop algorithms for the latter that are fundamentally different from those for the former.
This chapter is intended to introduce some basic notions of quantum theory needed in the subsequent chapters for the reader who is not familiar with them. Quantum mechanics is a fundamental physics subject that studies the phenomena at the atomic and subatomic scales. This chapter introduces the required mathematical tools and presents the postulates mainly through their mathematical formalisms. The physics interpretation of these is only very briefly discussed.
This chapter is devoted to studying a class of even more complex quantum systems modelled as so-called super-operator-valued Markov chains (SVMCs). This new model is particularly useful in modelling the high-level structure of quantum programs and quantum communication protocols. Several algorithms for checking SVMCs are presented in this chapter.
This is the concluding chapter of the book. It briefly discusses several possible directions for the further development, including the problem of state space explosion in model checking quantum systems, possible applications in verification and analysis of quantum circuits, quantum cryptographic protocols, and more generally, quantum programs.
A chief executive officer (CEO) acting as the firm's transformational leader is typically viewed as instrumental to corporate entrepreneurship in established firms, but how exactly does a higher level of corporate entrepreneurship come about, given a transformational CEO's actions? We suggest that organizational ambidexterity can function as a core mediating mechanism between transformational CEOs and the observed level of corporate entrepreneurship and that the effectiveness of this mediating process varies as a function of critical contingencies related to characteristics of the top management team (TMT), the environment and the organization's design. Our empirical evidence, based on a sample of 145 Chinese private sector firms, and using three primary sources of data (145 CEOs, 506 TMT members, and 1,981 middle managers), provides support for a moderated mediation process. We find that the mediating pathway from transformational leadership to corporate entrepreneurship through organizational ambidexterity is not significant when boundary conditions are ignored. However, when environmental dynamism, TMT collectivism, and structural differentiation are included as moderators, CEO transformational leadership does affect corporate entrepreneurship via the creation and effective functioning of organizational ambidexterity.
The chapter focuses on network flow problems, which form a very important part of practical applications. Routing, distribution, and scheduling problems often belong to this category of formulations, while a large number of other optimization problems encountered in diverse areas of applications may contain elements of network flow problems.
Quadratic multidimensional functions play a very important role in the understanding of general nonlinear functions. Convexity of quadratic functions is linked in a natural way from its geometrical definition all the way to the properties of its matrix eigenspectrum. Indeed, to second order expansion, and close to the expansion point, any nonlinear function can be approximated by a quadratic – thus providing a crucial link and understanding of the local behaviour and convexity properties of general functions.