1 Quantum Lambda Calculus

We discuss the design of a typed lambda calculus for quantum computation. After a brief discussion of the role of higher-order functions in quantum information theory, we define the quantum lambda calculus and its operational semantics. Safety invariants, such as the no-cloning property, are enforced by a static type system that is based on intuitionistic linear logic. We also describe a type inference algorithm, and a categorical semantics. 1.

[1]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[2]  John McCarthy,et al.  Recursive functions of symbolic expressions and their computation by machine, Part I , 1960, Commun. ACM.

[3]  Robin Milner,et al.  A Theory of Type Polymorphism in Programming , 1978, J. Comput. Syst. Sci..

[4]  C. Pollard,et al.  Center for the Study of Language and Information , 2022 .

[5]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[6]  G. Rota.  Introduction to higher order categorical logic , 1988 .

[7]  R. A. G. Seely,et al.  Linear Logic, -Autonomous Categories and Cofree Coalgebras , 1989 .

[8]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[9]  Nick Benton,et al.  Linear Lambda-Calculus and Categorial Models Revisited , 1992, CSL.

[10]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[11]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[12]  Charles H. Bennett,et al.  Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. , 1992, Physical review letters.

[13]  Samson Abramsky,et al.  Computational Interpretations of Linear Logic , 1993, Theor. Comput. Sci..

[14]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[15]  Nick Benton,et al.  A Term Calculus for Intuitionistic Linear Logic , 1993, TLCA.

[16]  Nick Benton,et al.  A Mixed Linear and Non-Linear Logic: Proofs, Terms and Models (Extended Abstract) , 1994, CSL.

[17]  Philip Wadler,et al.  Linear logic, monads and the lambda calculus , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[18]  E. Knill,et al.  Conventions for quantum pseudocode , 1996, 2211.02559.

[19]  Gerald J. Sussman,et al.  Scheme: A Interpreter for Extended Lambda Calculus , 1998, High. Order Symb. Comput..

[20]  Paul-André Melliès.  Categorical models of linear logic revisited , 2002 .

[21]  Benjamin C. Pierce,et al.  Types and programming languages: the next generation , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[22]  André van Tonder,et al.  A Lambda Calculus for Quantum Computation , 2003, SIAM J. Comput..

[23]  Benoît Valiron.  A functional programming language for quantum computation with classical control , 2004 .

[24]  Peter Selinger,et al.  Towards a quantum programming language , 2004, Mathematical Structures in Computer Science.

[25]  Benoît Valiron,et al.  Quantum typing , 2004, ArXiv.

[26]  Paola Maneggia.  Models of Linear Polymorphism , 2004 .

[27]  Paul Curzon,et al.  On Intuitionistic Linear Logic , 2005 .

[28]  Jonathan Grattage.  A functional quantum programming language , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[29]  Benoît Valiron,et al.  A Lambda Calculus for Quantum Computation with Classical Control , 2005, TLCA.

[30]  Simon L. Peyton Jones,et al.  A history of Haskell: being lazy with class , 2007, HOPL.

[31]  Benoît Valiron,et al.  Semantics for a Higher Order Functional Programming Language for Quantum Computation , 2008 .

[32]  Benoît Valiron,et al.  A linear-non-linear model for a computational call-by-value lambda calculus (extended abstract) , 2008, FoSSaCS.

[33]  Benoît Valiron,et al.  On a Fully Abstract Model for a Quantum Linear Functional Language: (Extended Abstract) , 2008, QPL.

[34]  David Weimer.  Bibliography , 2018, Medical History. Supplement.