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.
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Gilles BartheMPI for Security and Privacy (MPI-SP) and IMDEA Software Institute, Justin HsuUniversity of Wisconsin-Madison, USA, Mingsheng YingUniversity of Technology Sydney, Australia / Institute of Software at Chinese Academy of Sciences, China/ Department of Computer Science and Technology, Tsinghua University., Nengkun YuUniversity of Technology Sydney, Australia, Li ZhouMax Planck Institute for Security and Privacy/Tsinghua UniversityLink to publication DOI Pre-print Media Attached File Attached
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Gilles BartheMPI for Security and Privacy (MPI-SP) and IMDEA Software Institute, Justin HsuUniversity of Wisconsin-Madison, USA, Kevin LiaoMax Planck Institute for Security and PrivacyLink to publication DOI Media Attached