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Formalization of Metatheory of the Quipper Quantum Programming Language in a Linear Logic

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Abstract

We develop a linear logical framework within the Hybrid system and use it to reason about the type system of a quantum lambda calculus. In particular, we consider a practical version of the calculus called Proto-Quipper, which contains the core of Quipper. Quipper is a quantum programming language under active development and recently has gained much popularity among the quantum computing communities. Hybrid is a system that is designed to support the use of higher-order abstract syntax for representing and reasoning about formal systems implemented in the Coq Proof Assistant. In this work, we extend the system with a linear specification logic (SL) in order to reason about the linear type system of Quipper. To this end, we formalize the semantics of Proto-Quipper by encoding the typing and evaluation rules in the SL, and prove type soundness.

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Notes

  1. In general, when we stray from the original presentation, our intention is to simplify formalization, and we only do so when there is a clear equivalence to the original. In this case, we simplify the formalization of the subtyping relation without changing the semantics of types. As we will discuss in the next section, making this kind of change also led to the discovery of a small mistake in the original presentation.

  2. This is straightforward change, but it affects a large portion of the proof development and is left for (very near) future work.

  3. We also impose the restriction that the Coq derivation must be minimal in the same sense as described there. See [13] for details.

  4. A variety of other internal adequacy lemmas are shown in [14].

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Acknowledgements

The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada. We would also like to thank the reviewers for their comments, which helped us to improve the paper. In addition, we would like to thank Julien Ross and Peter Selinger for useful discussions on technical details as well as on approaches and directions for this work. Finally, the second author would also like to extend her gratitude to the University of Ottawa’s Writers Retreats.

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  1. School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada

    Mohamed Yousri Mahmoud & Amy P. Felty

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Correspondence to Amy P. Felty.

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Mahmoud, M.Y., Felty, A.P. Formalization of Metatheory of the Quipper Quantum Programming Language in a Linear Logic. J Autom Reasoning 63, 967–1002 (2019). https://doi.org/10.1007/s10817-019-09527-x

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