Abstract
We introduce a Geometry of Interaction model for higher-order quantum computation, and prove its adequacy for a fully fledged quantum programming language in which entanglement, duplication, and recursion are all available.
This model is an instance of a new framework which captures not only quantum but also classical and probabilistic computation. Its main feature is the ability to model commutative effects in a parallel setting. Our model comes with a multi-token machine, a proof net system, and a -style language. Being based on a multi-token machine equipped with a memory, it has a concrete nature which makes it well suited for building low-level operational descriptions of higher-order languages.
- K. de Leeuw, E. F. Moore, et al. Computability by probabilistic machine. In Automata Studies. Princeton U. Press, 1955.Google Scholar
- M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 10th edition, 2011. Google Scholar
Digital Library
- G. L. Miller. Riemann’s hypothesis and tests for primality. In STOC, pages 234–239, 1975. Google Scholar
Digital Library
- M. O. Rabin. Probabilistic algorithm for testing primality. Journal of Number Theory, 12(1):128 – 138, 1980.Google Scholar
Cross Ref
- P. W. Shor. Algorithms for quantum computation: Discrete logarithms and factoring. In SFCS, pages 124–134, 1994. Google Scholar
Digital Library
- J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik. Simulation of electronic structure hamiltonians using quantum computers. Molecular Physics, 109(5):735–750, 2011.Google Scholar
Cross Ref
- A. W. Harrow, A. Hassidim, and S. Lloyd. Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 103:150502, Oct 2009.Google Scholar
Cross Ref
- D. Kozen. Semantics of probabilistic programs. Journal of Computer System Sciences, 22:328–350, 1981.Google Scholar
Cross Ref
- G. Plotkin. Probabilistic powerdomains. In CAAP, 1982.Google Scholar
- S. J. Gay. Quantum programming languages: survey and bibliography. Mathematical Structures in Computer Science, 16(4):581–600, 2006. Google Scholar
Digital Library
- P. Selinger and B. Valiron. A lambda calculus for quantum computation with classical control. Math. Struct. in Comp. Sc., 16(3):527–552, 2006. Google Scholar
Digital Library
- A. S. Green, P. LeFanu Lumsdaine, et al. Quipper: A scalable quantum programming language. In PLDI, pages 333–342, 2013. Google Scholar
Digital Library
- M. Pagani, P. Selinger, and B. Valiron. Applying quantitative semantics to higher-order quantum computing. In POPL, pages 647–658, 2014. Google Scholar
Digital Library
- J.-Y. Girard. Linear logic. Theor. Comput. Sci., 50:1–102, 1987. Google Scholar
Digital Library
- S. Abramsky, R. Jagadeesan, and P. Malacaria. Full abstraction for PCF. Inf. Comput., 163(2):409–470, 2000. Google Scholar
Digital Library
- J. M. E. Hyland and C.-H. Luke Ong. On full abstraction for PCF: I, II, and III. Inf. Comput., 163(2):285–408, 2000. Google Scholar
Digital Library
- J.-Y. Girard. Geometry of interaction I: Interpretation of system F. Logic Colloquium 88, 1989.Google Scholar
- Y. Lafont. Logiques, catégories et machines. PhD thesis, Université Paris 7, 1988.Google Scholar
- P.-A. Melliès. Categorical semantics of linear logic. Panoramas et Synthèses, 12, 2009.Google Scholar
- P. Selinger. Towards a quantum programming language. Math. Struct. in Comp. Sc., 14(4):527–586, 2004. Google Scholar
Digital Library
- I. Hasuo and N. Hoshino. Semantics of higher-order quantum computation via geometry of interaction. In LICS, pages 237–246, 2011. Google Scholar
Digital Library
- Y. Delbecque. Quantum games as quantum types. PhD thesis, McGill University, 2009.Google Scholar
- Y. Delbecque and P. Panangaden. Game semantics for quantum stores. Electr. Notes Theor. Comput. Sci., 218:153–170, 2008. Google Scholar
Digital Library
- U. Dal Lago and M. Zorzi. Wave-style token machines and quantum lambda calculi. In LINEARITY, pages 64–78, 2014.Google Scholar
- U. Dal Lago, C. Faggian, I. Hasuo, and A. Yoshimizu. The geometry of synchronization. In CSL-LICS, pages 35:1–35:10, 2014. Google Scholar
Digital Library
- V. Danos and L. Regnier. Reversible, irreversible and optimal lambdamachines. Theor. Comput. Sci., 227(1-2):79–97, 1999. Google Scholar
Digital Library
- Ian Mackie. The geometry of interaction machine. In POPL, pages 198–208, 1995. Google Scholar
Digital Library
- U. Dal Lago, C. Faggian, B. Valiron, and A. Yoshimizu. Parallelism and synchronization in an infinitary context. In LICS, pages 559–572, 2015. Google Scholar
Digital Library
- U. Dal Lago, C. Faggian, B. Valiron, and A. Yoshimizu. The geometry of parallelism: Classical, probabilistic, and quantum effects (long version). Available at https://arxiv.org/abs/1610.09629, 2016.Google Scholar
- I. Mackie. Applications of the Geometry of Interaction to language implementation. Phd thesis, University of London, 1994.Google Scholar
- D. R. Ghica, A. Smith, and S. Singh. Geometry of synthesis IV. In ICFP, pages 221–233, 2011.Google Scholar
Digital Library
- S. Abramsky, E. Haghverdi, and P. J. Scott. Geometry of interaction and linear combinatory algebras. Math. Struct. in Comp. Sc., 12(5): 625–665, 2002. Google Scholar
Digital Library
- T. Ehrhard, C. Tasson, and M. Pagani. Probabilistic coherence spaces are fully abstract for probabilistic PCF. In POPL, pages 309–320, 2014. Google Scholar
Digital Library
- V. Danos and R. Harmer. Probabilistic game semantics. ACM Trans. Comput. Log., 3(3):359–382, 2002. Google Scholar
Digital Library
- N. Hoshino, K. Muroya, and I. Hasuo. Memoryful geometry of interaction. In CSL-LICS, page 52, 2014. Google Scholar
Digital Library
- S. Staton. Algebraic effects, linearity, and quantum programming languages. In POPL, pages 395–406, 2015. Google Scholar
Digital Library
- B. Coecke, R. Duncan, A. Kissinger, and Q. Wang. Strong complementarity and non-locality in categorical quantum mechanics. In LICS, pages 245–254, 2012. Google Scholar
Digital Library
- D. R. Ghica. Geometry of synthesis: a structured approach to VLSI design. In POPL, pages 363–375, 2007. Google Scholar
Digital Library
- K. Muroya, N. Hoshino, and I. Hasuo. Memoryful geometry of interaction II: recursion and adequacy. In POPL, pages 748–760, 2016. Google Scholar
Digital Library
- M. Bezem and J. W. Klop. Term Rewriting Systems, volume 55 of Cambridge Tracts in Theoretical Computer Science, chapter Abstract Reduction Systems. Cambridge University Press, 2003.Google Scholar
- A. Pitts. Nominal Sets: Names and Symmetry in Computer Science. Cambridge University Press, 2013. Google Scholar
Digital Library
Index Terms
The geometry of parallelism: classical, probabilistic, and quantum effects
Recommendations
The geometry of parallelism: classical, probabilistic, and quantum effects
POPL '17: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming LanguagesWe introduce a Geometry of Interaction model for higher-order quantum computation, and prove its adequacy for a fully fledged quantum programming language in which entanglement, duplication, and recursion are all available.
This model is an instance ...
Semantics of Higher-Order Quantum Computation via Geometry of Interaction
LICS '11: Proceedings of the 2011 IEEE 26th Annual Symposium on Logic in Computer ScienceWhile much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the ...
A Lambda Calculus for Quantum Computation
The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine and continues to be of enormous benefit in ...







Comments