Abstract
We develop a new framework of algebraic theories with linear parameters, and use it to analyze the equational reasoning principles of quantum computing and quantum programming languages. We use the framework as follows:
we present a new elementary algebraic theory of quantum computation, built from unitary gates and measurement;
we provide a completeness theorem or the elementary algebraic theory by relating it with a model from operator algebra;
we extract an equational theory for a quantum programming language from the algebraic theory;
we compare quantum computation with other local notions of computation by investigating variations on the algebraic theory.
- S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In Proc. LICS 2004, 2004. Google Scholar
Digital Library
- J. Aczél. On mean values. Bull. Am. Math. Soc., 54:392--400, 1948.Google Scholar
Cross Ref
- R. Adams. QPEL: Quantum program and effect language. In Proc. QPL 2014, 2014. To appear.Google Scholar
Cross Ref
- D. Ahman and S. Staton. Normalization by evaluation and algebraic effects. In Proc. MFPS XXIX, pages 51--69, 2013. Google Scholar
Digital Library
- T. Altenkirch, J. Chapman, and T. Uustalu. Monads need not be endofunctors. In FOSSACS 2010. Google Scholar
Digital Library
- T. Altenkirch and A. Green. The Quantum IO Monad. In Semantic Techniques in Quantum Computation. CUP, 2009.Google Scholar
Cross Ref
- J. Baez and J. Dolan. From finite sets to Feynman diagrams. In Mathematics Unlimited -- 2001 and Beyond. Springer, 2001.Google Scholar
Cross Ref
- N. Benton and P. Wadler. Linear logic, monads and the lambda calculus. In LICS 1996. Google Scholar
Digital Library
- C. Berger, P.-A. Melliès, and M. Weber. Monads with arities and their associated theories. J. Pure Appl. Algebra, 216, 2012.Google Scholar
- S. L. Bloom and Z. Esik. Iteration Theories: The Equational Logic of Iterative Processes. Springer, 1993. Google Scholar
Digital Library
- I. Cervesato and F. Pfenning. A Linear Logical Framework. Inform. Comput., 179(1):19--75, 2002. Google Scholar
Digital Library
- G. Chiribella, G. M. D'Ariano, and P. Perinotti. Informational derivation of quantum theory. Phys. Rev. A, 84, 2011.Google Scholar
- K. Cho. Semantics for a quantum programming language by operator algebras. In Proc. QPL 2014, 2014. To appear.Google Scholar
Cross Ref
- B. Coecke and R. Duncan. Interacting quantum observables. In Proc. ICALP 2008, pages 298--310, 2008. Google Scholar
Digital Library
- V. Danos, E. Kashefi, and P. Panangaden. The measurement calculus. J. ACM, 54(2), 2007. Google Scholar
Digital Library
- P. A. Fillmore. A User's Guide to Operator Algebras. Wiley-Interscience, 1996.Google Scholar
- M. Fiore. Notes on combinatorial functors. Draft, 2001.Google Scholar
- M. P. Fiore and C.-K. Hur. Second-order equational logic. In CSL'10. Google Scholar
Digital Library
- A. S. Green, P. L. Lumsdaine, N. J. Ross, P. Selinger, and B. Valiron. Quipper: a scalable quantum programming language. In PLDI 2013. Google Scholar
Digital Library
- L. Hardy. Reformulating and reconstructing quantum theory. arXiv:1104.2066, 2011.Google Scholar
- C. Heunen, A. Kissinger, and P. Selinger. Completely positive projections and biproducts. arXiv:1308.4557.Google Scholar
- B. Jacobs. On block structures in quantum computation. In Proc. MFPS XXIX, 2013. Google Scholar
Digital Library
- G. Janelidze and G. Kelly. A note on actions of a monoidal category. Theory Appl. Categ., 9(4), 2001.Google Scholar
- O. Kammar and G. D. Plotkin. Algebraic foundations for effect-dependent optimisations. In POPL 2012, pages 349--360, 2012. Google Scholar
Digital Library
- G. M. Kelly and A. J. Power. Adjunctions whose counits are coequalisers. J. Pure Appl. Algebra, 89:163--179, 1993.Google Scholar
Cross Ref
- F. E. J. Linton. Autonomous equational categories. J. Math. Mech., 15:637--642, 1966.Google Scholar
- O. Malherbe, P. J. Scott, and P. Selinger. Presheaf models of quantum computation: an outline. In Computation, Logic, Games, and Quantum Foundations. Springer, 2013.Google Scholar
Cross Ref
- P.-A. Melliès. Local stores in string diagrams. In RTA-TLCA 2014.Google Scholar
- P.-A. Melliès. Segal condition meets computational effects. In Proc. LICS 2010, pages 150--159, 2010. Google Scholar
Digital Library
- R. E. Møgelberg and S. Staton. Linear usage of state. Logical Methods Comput. Sci., 10(1), 2014.Google Scholar
- M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. CUP, 2011. Google Scholar
Digital Library
- P. W. O'Hearn. On bunched typing. J. Funct. Program., 2003. Google Scholar
Digital Library
- M. Pagani, P. Selinger, and B. Valiron. Applying quantitative semantics to higher-order quantum computing. In Proc. POPL 2014. Google Scholar
Digital Library
- V. Paulsen. Completely Bounded Maps and Operator Algebras. CUP, 2003.Google Scholar
Cross Ref
- G. Plotkin. Some varieties of equational logic. In Algebra, meaning and computation. Springer, 2006.Google Scholar
Cross Ref
- G. D. Plotkin and J. Power. Notions of computation determine mon- ads. In Proc. FOSSACS'02, 2002. Google Scholar
Digital Library
- G. D. Plotkin and J. Power. Algebraic operations and generic effects. Applied Categorical Structures, 11(1):69--94, 2003.Google Scholar
Cross Ref
- J. Power. Generic models for computational effects. Theor. Comput. Sci., 364(2):254--269, 2006. Google Scholar
Digital Library
- J. Power. Indexed Lawvere theories for local state. In Models, logics and higher-dimensional categories: a tribute to the work of Mihály Makkai, pages 213--230. American Mathematical Society, 2011.Google Scholar
Cross Ref
- J. Power and M. Tanaka. Binding signatures for generic contexts. In Proc. TLCA 2005, volume 308--323, 2005. Google Scholar
Digital Library
- M. Rennela. Towards a quantum domain theory: Order-enrichment and fixpoints in W*-algebras. In Proc. MFPS XXX, 2014.Google Scholar
Digital Library
- P. Selinger. Towards a quantum programming language. Mathematical Structures in Computer Science, 14(4):527--586, 2004. Google Scholar
Digital Library
- P. Selinger. Generators and relations for n-qubit Clifford operators. arXiv:1310.6813, 2013.Google Scholar
- I. Stark. Categorical models for local names. LISP and Symb. Comp., 9(1):77--107, 1996.Google Scholar
Cross Ref
- S. Staton. An algebraic presentation of predicate logic. In FOSSACS 2013. Google Scholar
Digital Library
- S. Staton. Completeness for algebraic theories of local state. In FOSSACS 2010. Google Scholar
Digital Library
- S. Staton. Instances of computational effects. In LICS 2013. Google Scholar
Digital Library
- S. Staton. Two cotensors in one. In Proc. MFPS XXV, 2009.Google Scholar
- S. Staton. Freyd categories are enriched lawvere theories. In Proc. Workshop on Algebra, Coalgebra and Topology, volume 303 of ENTCS, pages 197--206, 2013. Google Scholar
Digital Library
- S. Staton and P. B. Levy. Universal properties for impure programming languages. In POPL 2013, 2013. Google Scholar
Digital Library
- A. van Tonder. A lambda calculus for quantum computation. SIAM J. Comput., 33(5):1109--1135, 2004. Google Scholar
Digital Library
- J. Vicary. Higher semantics of quantum protocols. In LICS 2012. Google Scholar
Digital Library
- J. K. Vizzotto, G. R. Librelotto, and A. Sabry. Reasoning about general quantum programs over mixed states. In SBMF 2009, 2009. Google Scholar
Digital Library
- M. Ying and Y. Feng. An algebraic language for distributed quantum computing. IEEE Trans. Computers, 58(6):728--743, 2009. Google Scholar
Digital Library
- M. Ying, N. Yu, and Y. Feng. Alternation in quantum programming: from superposition of data to superposition of programs. arXiv:1402.5172, 2014.Google Scholar
Index Terms
Algebraic Effects, Linearity, and Quantum Programming Languages
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