Quantum games as quantum types

In this thesis, we present a new model for higher-order quantum programming languages. The proposed model is an adaptation of the probabilistic game semantics developed by Danos and Harmer [DH02]: we expand it with quantum strategies which enable one to represent quantum states and quantum operations. Some of the basic properties of these strategies are established and then used to construct denotational semantics for three quantum programming languages. The first of these languages is a formalisation of the measurement calculus proposed by Danos et al. [DKP07]. The other two are new: they are higher-order quantum programming languages. Previous attempts to define a denotational semantics for higher-order quantum programming languages have failed. We identify some of the key reasons for this and base the design of our higher-order languages on these observations.;Quantum strategies are a new mathematical model which describes the interaction between classical and quantum data using system-environment dialogues. The interactions between the different parts of a quantum system are described using the rich structure generated by composition of strategies. This approach has enough generality to be put in relation with other work in quantum computing. Quantum strategies could thus be useful for other purposes than the study of quantum programming languages.;The game semantics proposed in this thesis is the first denotational semantics for a lambda-calculus equipped with quantum types and with extra operations which allow one to program quantum algorithms. The results presented validate the two different approaches used in the design of these two new higher-order languages: a first one where quantum states are used through references and a second one where they are introduced as constants in the language. The quantum strategies presented in this thesis allow one to understand the constraints that must be imposed on quantum type systems with higher-order types. The most significant constraint is the fact that abstraction over part of the tensor product of many unknown quantum states must not be allowed.