Document Zbl 1486.18023

Categorifying the ZX-calculus. (English) Zbl 1486.18023

Coecke, Bob (ed.) et al., Proceedings of the 14th international conference on quantum physics and logic, QPL’17, Nijmegen, The Netherlands, July 3–7, 2017. Waterloo: Open Publishing Association (OPA). Electron. Proc. Theor. Comput. Sci. (EPTCS) 266, 294-314 (2018).

Summary: We build a symmetric monoidal and compact closed bicategory by combining spans and cospans inside a topos. This can be used as a framework in which to study open networks and diagrammatic languages. We illustrate this framework with Coecke and Duncan’s zx-calculus by constructing a bicategory with the natural numbers for 0-cells, the zx-calculus diagrams for 1-cells, and rewrite rules for 2-cells.
For the entire collection see [Zbl 1434.03012].

Cited in 1 Document

MSC:

18M05	Monoidal categories, symmetric monoidal categories
18B25	Topoi
81P15	Quantum measurement theory, state operations, state preparations
81P10	Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)

Cite Review PDF

Full Text: arXiv Link

References:

[1]	S. Abramsky & B. Coecke (2004): A categorical semantics of quantum protocols. In: Logic in Computer Science. Proceedings of the 19th Annual IEEE Symposium, pp. 415-425, doi:10.1109/LICS.2004.1319636. · doi:10.1109/LICS.2004.1319636
[2]	Bob B. Coecke & B. Edwards (2011): Toy quantum categories. Electron. Notes Theor. Comput. Sci. 270(1), doi:10.1016/j.entcs.2011.01.004. Available at https://arxiv.org/abs/0808.1037. · Zbl 1348.81049 · doi:10.1016/j.entcs.2011.01.004
[3]	M. Backens (2016): Completeness and the ZX-calculus, doi:10.1109/LICS.2004.1319636. Available at https://arxiv.org/abs/1602.08954. · doi:10.1109/LICS.2004.1319636
[4]	J. Baez, B. Coya & F. Rebro (2017): Props in network theory. Available at https://arxiv.org/abs/ 1707.08321. · Zbl 1400.18004
[5]	J. Baez & B. Fong (2015): A compositional framework for passive linear networks. Available at https: //arxiv.org/abs/1504.05625.
[6]	J. Baez, B. Fong & B. Pollard (2016): A compositional framework for Markov processes. J. Math. Phys. 57, doi:10.1063/1.4941578. · Zbl 1336.60147 · doi:10.1063/1.4941578
[7]	K. Bar, A. Kissinger & J. Vicary (2016): Globular: an online proof assistant for higher-dimensional rewrit-ing. Available at http://globular.science. · Zbl 1387.68206
[8]	D. Cicala (2016): Spans of cospans. Available at https://arxiv.org/abs/1611.07886. · Zbl 1412.18008
[9]	D. Cicala & K. Courser (2017): Spans of cospans in a topos. Available at https://arxiv.org/abs/1707. 02098. · Zbl 1395.18002
[10]	B. Coecke & R. Duncan (2008): Interacting quantum observables. In: Automata, languages and program-ming. Part II, Lecture Notes in Comput. Sci. 5126, Springer, Berlin, pp. 298-310, doi:10.1007/978-3-540-70583-3 25. · Zbl 1155.81316 · doi:10.1007/978-3-540-70583-3_25
[11]	B. Coecke & R. Duncan (2011): Interacting quantum observables: categorical algebra and diagrammatics. New J. Phys. 13, doi:10.1088/1367-2630/13/4/043016. · Zbl 1448.81025 · doi:10.1088/1367-2630/13/4/043016
[12]	B. Coecke & B. Edwards (2012): Spekkens’s toy theory as a category of processes. In: Mathemat-ical foundations of information flow, Proc. Sympos. Appl. Math. 71, Amer. Math. Soc., pp. 61-68, doi:10.1090/psapm/071/602. · doi:10.1090/psapm/071/602
[13]	B. Coecke, B. Edwards & R. Spekkens (2011): Phase groups and the origin of non-locality for qubits. Electron. Notes Theor. Comput. Sci. 270(2), doi:10.1016/j.entcs.2011.01.021. · Zbl 1348.81050 · doi:10.1016/j.entcs.2011.01.021
[14]	B. Coecke & D. Pavlovic (2008): Quantum measurements without sums. In: Mathematics of quantum computation and quantum technology, Chapman & Hall CRC Appl. Math. Nonlinear Sci. Ser., Chapman & Hall/CRC, Boca Raton, FL, pp. 559-596, doi:10.1201/9781584889007.ch16. · Zbl 1139.81307 · doi:10.1201/9781584889007.ch16
[15]	B. Coecke, D. Pavlovic & J. Vicary (2013): A new description of orthogonal bases. Math. Structures Comput. Sci. 23(3), doi:10.1017/S0960129512000047. · Zbl 1276.46016 · doi:10.1017/S0960129512000047
[16]	B. Coecke & S. Perdrix (2012): Environment and classical channels in categorical quantum mechanics. Log. Methods Comput. Sci. 8(4), doi:10.2168/LMCS-8(4:14)2012. · Zbl 1259.81019 · doi:10.2168/LMCS-8(4:14)2012
[17]	A. Corradini, U. Montanari, F. Rossi, H. Ehrig, R. Heckel & M. Lowe (1997): Algebraic approaches to graph transformation. Basic concepts and double pushout approach. In: Handbook of graph grammars and comput-ing by graph transformation, Vol. 1, World Sci. Publ., River Edge, NJ, doi:10.1142/9789812384720 0003. · doi:10.1142/9789812384720_0003
[18]	V. Danos, E. Kashefi & P. Panangaden (2007): The measurement calculus. J. ACM 54(2), doi:10.1145/1219092.1219096. · Zbl 1311.81076 · doi:10.1145/1219092.1219096
[19]	L. Dixon, R. Duncan & A. Kissinger: Quantomatic. Available at https://sites.google.com/site/ quantomatic/.
[20]	R. Duncan & S. Perdrix (2009): Graph states and the necessity of Euler decomposition. In: Mathematical theory and computational practice, Lecture Notes in Comput. Sci. 5635, Springer, Berlin, pp. 167-177, doi:10.1007/978-3-642-03073-4 18. · Zbl 1268.81046 · doi:10.1007/978-3-642-03073-4_18
[21]	R. Duncan & S. Perdrix (2010): Rewriting measurement-based quantum computations with generalised flow. Automata, Languages and Programming, doi:10.1007/978-3-642-14162-1 24. · Zbl 1288.68069 · doi:10.1007/978-3-642-14162-1_24
[22]	J. Evans, R. Duncan, A. Lang & P. Panangaden (2009): Classifying all mutually unbiased bases in Rel. Available at https://arxiv.org/abs/0909.4453.
[23]	B. Fong (2016): The Algebra of Open and Interconnected Systems. Available at https://arxiv.org/abs/ arXiv:1609.05382.
[24]	A. Habel, J. Muller & D. Plump (2001): Double-pushout graph transformation revisited. Math. Structures Comput. Sci. 11(5), doi:10.1017/S0960129501003425. · Zbl 0987.18005 · doi:10.1017/S0960129501003425
[25]	John J. Baez & B. Pollard (2017): A compositional framework for reaction networks. Rev. Math. Phys. 29(9), doi:10.1142/S0129055X17500283. · Zbl 1383.68053 · doi:10.1142/S0129055X17500283
[26]	A. Joyal & R. Street (1991): The geometry of tensor calculus. I. Adv. Math. 88(1), doi:10.1016/0001-8708(91)90003-P. · Zbl 0738.18005 · doi:10.1016/0001-8708(91)90003-P
[27]	A. Kissinger (2012): Pictures of processes: automated graph rewriting for monoidal categories and applica-tions to quantum computing. Ph.D. Thesis, University of Oxford. Available at https://arxiv.org/abs/ 1203.0202.
[28]	A. Kissinger & V. Zamdzhiev (2015): Quantomatic: a proof assistant for diagrammatic reasoning. In: Automated deduction-CADE 25, Lecture Notes in Comput. Sci. 9195, Springer, Cham, pp. 326-336, doi:10.1007/978-3-319-21401-6 22. · Zbl 1465.68288 · doi:10.1007/978-3-319-21401-6_22
[29]	Lucas L. Dixon, R. Duncan & A. Kissinger (2010): Electron. Proc. Theor. Comput. Sci. 26, doi:10.4204/EPTCS.26.16. · doi:10.4204/EPTCS.26.16
[30]	S. MacLane & I. Moerdijk (2012): Sheaves in geometry and logic: A first introduction to topos theory. Springer Science & Business Media.
[31]	A. Merry (2014): Reasoning with !-Graphs. CoRR abs/1403.7828. Available at http://arxiv.org/abs/ 1403.7828.
[32]	M. Nielsen & I. Chuang (2010): Quantum computation and quantum information. Cambridge University Press, Cambridge, doi:10.1017/CBO9780511976667. · Zbl 1288.81001 · doi:10.1017/CBO9780511976667
[33]	D. Pavlovic (2009): Quantum and classical structures in nondeterministic computation. In: Quantum inter-action, Lecture Notes in Comput. Sci. 5494, Springer, Berlin, pp. 143-157, doi:10.1007/978-3-642-00834-4 13. · Zbl 1229.68039 · doi:10.1007/978-3-642-00834-4_13
[34]	R. Penrose (1971): Applications of negative dimensional tensors. In: Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, pp. 221-244. Available at http:// homepages.math.uic.edu/ kauffman/Penrose.pdf. · Zbl 0216.43502
[35]	B. Pollard (2016): Open Markov Processes: A Compositional Perspective on Non-Equilibrium Steady States in Biology. Entropy 18(4), doi:10.3390/e18040140. · doi:10.3390/e18040140
[36]	P. Pstragowski (2014): On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis. Available at https://arxiv.org/abs/1411.6691.
[37]	V. Sassone & P. Sobocinski (2005): A congruence for Petri nets. Electronic Notes in Theoretical Computer Science 127(2), doi:10.1016/j.entcs.2005.02.008. · doi:10.1016/j.entcs.2005.02.008
[38]	P. Selinger (2011): A survey of graphical languages for monoidal categories. In: New structures for physics, Lecture Notes in Phys. 813, Springer, Heidelberg, pp. 289-355, doi:10.1007/978-3-642-12821-9 4. · Zbl 1217.18002 · doi:10.1007/978-3-642-12821-9_4
[39]	M. Shulman (2010): Constructing symmetric monoidal bicategories. Available at http://arxiv.org/ abs/1004.0993.
[40]	M. Stay (2016): Compact closed bicategories. Theory Appl. Categ. 31. Available at https://arxiv.org/ abs/1301.1053.

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.