Quantum programming languages permit a hardware independent, high-level description of quantum algorithms. In particular, the quantum λ-calculus is a higher-order programming language with quantum primitives, mixing quantum data and classical control. Giving satisfactory denotational semantics to the quantum λcalculus 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 λ-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 λ-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 model 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 λ-calculus. We do this both from a sequential and a parallel interpretation, the latter representing faithfully the causal independence between sub-computations.

Quantum programming languages permit a hardware independent, high-level description of quantum algorithms. In particular, the quantum λ-calculus is a higher-order language with quantum primitives, mixing quantum data and classical control. Giving satisfactory denotational semantics to the quantum λ-calculus is a challenging problem that has attracted significant interest. In the past few years, both static (the quantum relational model) and dynamic (quantum game semantics) denotational models were given, with matching computational adequacy results. However, no model was known to be fully abstract. Our first contribution is a full abstraction result for the games model of the quantum λ-calculus. Full abstraction holds with respect to an observational quotient of strategies, obtained by summing valuations of all states matching a given observable. Our proof method for full abstraction extends a technique recently introduced to prove full abstraction for probabilistic coherence spaces with respect to probabilistic PCF. Our second contribution is an interpretation-preserving functor from quantum games to the quantum relational model, extending a long line of work on connecting static and dynamic denotational models. From this, it follows that the quantum relational model is fully abstract as well. Altogether, this gives a complete denotational landscape for the semantics of the quantum λ-calculus, with static and dynamic models related by a clean functorial correspondence, and both fully abstract. CCS Concepts: • Theory of computation → Denotational semantics; • Computer systems organization → Quantum computing.

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