Game Semantics for Quantum Programming

Paper and supplementary material: Abstract: Quantum programming languages permit a hardware independent high-level description of quantum algorithms. In particular the quantum lambda-calculus is a higher-order programming language with quantum primitives mixing quantum data and classical control. Giving satisfactory denotational semantics to the quantum lambda-calculus is a challenging problem that has attracted significant interest in the past few years. Several models have been proposed but for those that address the whole quantum lambda-calculus they either do not represent the dynamics of computation or they lack the compositionality one often expects from denotational models. In this paper we give the first compositional and interactive model of the full quantum lambda-calculus based on game semantics. To achieve this we introduce a model of quantum games and strategies combining quantum data with a representation of the dynamics of computation inspired from causal models of concurrent systems. In this category we first give a computationally adequate interpretation of the affine fragment. Then we extend the model with a notion of symmetry allowing us to deal with replication. In this refined setting we interpret and prove adequacy for the full quantum lambda-calculus. We do this both from a sequential and a parallel interpretation the latter representing faithfully the causal independence between sub-computations.,

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