Linear-algebraic lambda-calculus

With a view towards models of quantum computation and/or the interpretation of linear logic, we define a functional language where all functions are linear operators by construction. A small step operational semantic (and hence an interpreter/simulator) is provided for this language in the form of a term rewrite system. The linear-algebraic lambda-calculus hereby constructed is linear in a different (yet related) sense to that, say, of the linear lambda-calculus. These various notions of linearity are discussed in the context of quantum programming languages. KEYWORDS: quantum lambda-calculus, linear lambda-calculus, $\lambda$-calculus, quantum logics.

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

[2]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[3]  Terry Rudolph,et al.  A 2 rebit gate universal for quantum computing , 2002 .

[4]  P. Arrighi,et al.  On quantum operations as quantum states , 2003, quant-ph/0307024.

[5]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[6]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

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

[8]  Laurent Regnier,et al.  The differential lambda-calculus , 2003, Theor. Comput. Sci..

[9]  Andr'e van Tonder,et al.  Quantum Computation, Categorical Semantics and Linear Logic , 2003, ArXiv.

[10]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[11]  Andrew Chi-Chih Yao,et al.  Quantum Circuit Complexity , 1993, FOCS.

[12]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[13]  Jean-Jacques Lévy,et al.  Confluence properties of weak and strong calculi of explicit substitutions , 1996, JACM.

[14]  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.

[15]  Hans Zantema,et al.  Rewrite Systems for Integer Arithmetic , 1995, RTA.

[16]  Leonard M. Adleman,et al.  Quantum Computability , 1997, SIAM J. Comput..

[17]  F. Verstraete,et al.  On quantum channels , 2002, quant-ph/0202124.

[18]  Jan Willem Klop,et al.  Term Rewriting Systems: From Church-Rosser to Knuth-Bendix and Beyond , 1990, ICALP.

[19]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[20]  V. Roychowdhury,et al.  On Universal and Fault-Tolerant Quantum Computing , 1999, quant-ph/9906054.

[21]  Phil Watson,et al.  An Efficient Representation of Arithmetic for Term Rewriting , 1991, RTA.

[22]  Philip Maymin.  Extending the Lambda Calculus to Express Randomized and Quantumized Algorithms , 1996 .

[23]  A. Berthiaume.  Quantum computation , 1998 .

[24]  Pierre Lescanne.  From Lambda-sigma to Lambda-upsilon a Journey Through Calculi of Explicit Substitutions. , 1994 .

[25]  Martín Abadi,et al.  Explicit substitutions , 1989, POPL '90.

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