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2019 | OriginalPaper | Chapter

Concurrent Quantum Strategies

Authors : Pierre Clairambault, Marc de Visme, Glynn Winskel

Published in: Reversible Computation

Publisher: Springer International Publishing

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Abstract

A game-semantics foundation for quantum computation is presented. It draws on two lines of work: for its temporal dynamics, on concurrent games and strategies, based on event structures; for its quantum interactions, on the mathematical foundations of positive operators and completely positive maps. The two lines are married in the definition of quantum concurrent strategy, obtained via an operator generalisation of the conditions on a probabilistic concurrent strategy. The result is a compact-closed (bi)category of quantum games, whose finite configurations carry finite dimensional Hilbert spaces, and quantum strategies, whose finite configurations carry operators.

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Footnotes
1
The use of subdensity rather than density operators, where \(\mathrm{tr}(\rho )=1\), is natural in quantum systems which may stick with a non-trivial probability.
 
2
A Scott-open subset of configurations is upwards-closed w.r.t. inclusion and such that if it contains the union of a directed subset S of configurations then it contains an element of S. A continuous valuation is a function w from the Scott-open subsets of \(\,\!{\mathcal C}^\infty (E)\) to [0, 1] which is ((normalised)\(w(\,\!{\mathcal C}^\infty (E)) = 1\); (strict) \(w(\emptyset ) = 0 \); (monotone)\(U \subseteq V \implies w(U)\le w(V)\); (modular)\(w(U \cup V) + w(U\cap V) = w(U) + w(V)\); and (continuous) \(w(\bigcup _{i\in I} U_i) = \mathrm{sup}_{i\in I} w(U_i)\), for directed unions.
 
3
We eschew the other obvious possibility in which the game also determines the operators because we want strategies to be quantum, not just probabilistic, in line with the quantum lambda-calculus [5] and earlier definition [6].
 
Literature
1.
2.
go back to reference Winskel, G.: Distributed games and strategies. Festschrift for pierre-louis curien on the occasion of his 60th birthday (2014, as yet unpublished) Winskel, G.: Distributed games and strategies. Festschrift for pierre-louis curien on the occasion of his 60th birthday (2014, as yet unpublished)
3.
go back to reference Winskel, G.: Distributed probabilistic and quantum strategies. Electr. Notes Theor. Comput. Sci. 298, 403–425 (2013)MathSciNetCrossRef Winskel, G.: Distributed probabilistic and quantum strategies. Electr. Notes Theor. Comput. Sci. 298, 403–425 (2013)MathSciNetCrossRef
4.
go back to reference Grabbe, J.: An introduction to quantum game theory. arXiv preprint quant-ph/0506219 (2005) Grabbe, J.: An introduction to quantum game theory. arXiv preprint quant-ph/0506219 (2005)
5.
go back to reference Pagani, M., Selinger, P., Valiron, B.: Applying quantitative semantics to higher-order quantum computing. In: POPL 2014 (2014) Pagani, M., Selinger, P., Valiron, B.: Applying quantitative semantics to higher-order quantum computing. In: POPL 2014 (2014)
6.
go back to reference Delbecque, Y., Panagaden, P.: Game semantics for quantum stores. MFPS XXIV, Electr. Notes Theor. Comput. Sci. 218, 153–170 (2008) Delbecque, Y., Panagaden, P.: Game semantics for quantum stores. MFPS XXIV, Electr. Notes Theor. Comput. Sci. 218, 153–170 (2008)
7.
go back to reference Dal Lago, U., Faggian, C., Valiron, B., Yoshimizu, A.: The geometry of parallelism, classical, probabilistic, and quantum effects. In: POPL 2017 (2017) Dal Lago, U., Faggian, C., Valiron, B., Yoshimizu, A.: The geometry of parallelism, classical, probabilistic, and quantum effects. In: POPL 2017 (2017)
9.
go back to reference Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: LICS 2004 (2004) Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: LICS 2004 (2004)
10.
go back to reference Castellan, S., Clairambault, P., Winskel, G.: The parallel intensionally fully abstract games model of PCF. In: LICS 2015 (2015) Castellan, S., Clairambault, P., Winskel, G.: The parallel intensionally fully abstract games model of PCF. In: LICS 2015 (2015)
11.
go back to reference Castellan, S., Clairambault, P., Paquet, H., Winskel, G.: The concurrent game semantics of probabilistic PCF. In: LICS 2018 (2018) Castellan, S., Clairambault, P., Paquet, H., Winskel, G.: The concurrent game semantics of probabilistic PCF. In: LICS 2018 (2018)
12.
go back to reference Selinger, P., Valiron, B.: On a fully abstract model for a quantum linear functional language. Electr. Notes Theor. Comput. Sci. 210, 123–137 (2008)MathSciNetCrossRef Selinger, P., Valiron, B.: On a fully abstract model for a quantum linear functional language. Electr. Notes Theor. Comput. Sci. 210, 123–137 (2008)MathSciNetCrossRef
13.
go back to reference Clairambault, P., de Visme, M., Winskel, G.: Game semantics for quantum programming. In: POPL 2019 (2019) Clairambault, P., de Visme, M., Winskel, G.: Game semantics for quantum programming. In: POPL 2019 (2019)
14.
go back to reference Joyal, A., Street, R., Verity, D.: Traced monoidal categories. In: Mathematical Proceedings of the Cambridge Philosophical Society (1996) Joyal, A., Street, R., Verity, D.: Traced monoidal categories. In: Mathematical Proceedings of the Cambridge Philosophical Society (1996)
15.
go back to reference Selinger, P.: Dagger compact closed categories and completely positive maps: (extended abstract). Electr. Notes Theor. Comput. Sci. 170, 139–163 (2007)CrossRef Selinger, P.: Dagger compact closed categories and completely positive maps: (extended abstract). Electr. Notes Theor. Comput. Sci. 170, 139–163 (2007)CrossRef
17.
go back to reference Alvarez-Manilla, M., Edalat, A., Saheb-Djahromi, N.: An extension result for continuous valuations. J. LMS 61, 629–640 (2000)MathSciNetMATH Alvarez-Manilla, M., Edalat, A., Saheb-Djahromi, N.: An extension result for continuous valuations. J. LMS 61, 629–640 (2000)MathSciNetMATH
18.
go back to reference Leifer, M.S., Spekkens, R.W.: Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Phys. Rev. 88, 052130 (2013)CrossRef Leifer, M.S., Spekkens, R.W.: Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Phys. Rev. 88, 052130 (2013)CrossRef
19.
go back to reference Saunders-Evans, L., Winskel, G.: Event structure spans for nondeterministic dataflow. Electr. Notes Theor. Comput. Sci. 175(3), 109–129 (2007)CrossRef Saunders-Evans, L., Winskel, G.: Event structure spans for nondeterministic dataflow. Electr. Notes Theor. Comput. Sci. 175(3), 109–129 (2007)CrossRef
Metadata
Title
Concurrent Quantum Strategies
Authors
Pierre Clairambault
Marc de Visme
Glynn Winskel
Copyright Year
2019
DOI
https://doi.org/10.1007/978-3-030-21500-2_1

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