Linear-algebraic lambda-calculus: higher-order, encodings, and confluence

We introduce a minimal language combining higher-ordercomputation and linear algebra. This language extends theλ-calculus with the possibility to make arbitrarylinear combinations of terms λ.t+β.u. We describe how to "execute" thislanguage in terms of a few rewrite rules, and justify them throughthe two fundamental requirements that the language be a language oflinear operators, and that it be higher-order. We mention theperspectives of this work in the field of quantum computation,whose circuits we show can be easily encoded in the calculus.Finally we prove the confluence of the calculus, this is our mainresult.

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