Document

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We introduce the LO_v-calculus, a graphical language for reasoning about linear optical quantum circuits with so-called vacuum state auxiliary inputs. We present the axiomatics of the language and prove its soundness and completeness: two LO_v-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LO_v-calculus. We give a confluent and terminating rewrite system to rewrite any polarisation-preserving LO_v-circuit into a unique triangular normal form, inspired by the universal decomposition of Reck et al. (1994) for linear optical quantum circuits.

Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron. LO_v-Calculus: A Graphical Language for Linear Optical Quantum Circuits. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 35:1-35:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2022.35, author = {Cl\'{e}ment, Alexandre and Heurtel, Nicolas and Mansfield, Shane and Perdrix, Simon and Valiron, Beno\^{i}t}, title = {{LO\underlinev-Calculus: A Graphical Language for Linear Optical Quantum Circuits}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {35:1--35:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.35}, URN = {urn:nbn:de:0030-drops-168334}, doi = {10.4230/LIPIcs.MFCS.2022.35}, annote = {Keywords: Quantum Computing, Graphical Language, Linear Optical Circuits, Linear Optical Quantum Computing, Completeness} }

Document

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Coherent control of quantum computations can be used to improve some quantum protocols and algorithms. For instance, the complexity of implementing the permutation of some given unitary transformations can be strictly decreased by allowing coherent control, rather than using the standard quantum circuit model. In this paper, we address the problem of optimising the resources of coherently controlled quantum computations. We refine the PBS-calculus, a graphical language for coherent control which is inspired by quantum optics. In order to obtain a more resource-sensitive language, it manipulates abstract gates - that can be interpreted as queries to an oracle - and more importantly, it avoids the representation of useless wires by allowing unsaturated polarising beam splitters. Technically the language forms a coloured PROP. The language is equipped with an equational theory that we show to be sound, complete, and minimal.
Regarding resource optimisation, we introduce an efficient procedure to minimise the number of oracle queries of a given diagram. We also consider the problem of minimising both the number of oracle queries and the number of polarising beam splitters. We show that this optimisation problem is NP-hard in general, but introduce an efficient heuristic that produces optimal diagrams when at most one query to each oracle is required.

Alexandre Clément and Simon Perdrix. Resource Optimisation of Coherently Controlled Quantum Computations with the PBS-Calculus. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 36:1-36:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2022.36, author = {Cl\'{e}ment, Alexandre and Perdrix, Simon}, title = {{Resource Optimisation of Coherently Controlled Quantum Computations with the PBS-Calculus}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {36:1--36:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.36}, URN = {urn:nbn:de:0030-drops-168348}, doi = {10.4230/LIPIcs.MFCS.2022.36}, annote = {Keywords: Quantum computing, Graphical language, Coherent control, Completeness, Resource optimisation, NP-hardness} }

Document

**Published in:** LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams.
Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules.
We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.

Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova. Addition and Differentiation of ZX-Diagrams. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 13:1-13:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jeandel_et_al:LIPIcs.FSCD.2022.13, author = {Jeandel, Emmanuel and Perdrix, Simon and Veshchezerova, Margarita}, title = {{Addition and Differentiation of ZX-Diagrams}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {13:1--13:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-233-4}, ISSN = {1868-8969}, year = {2022}, volume = {228}, editor = {Felty, Amy P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.13}, URN = {urn:nbn:de:0030-drops-162946}, doi = {10.4230/LIPIcs.FSCD.2022.13}, annote = {Keywords: ZX calculus, Addition of ZX diagrams, Diagrammatic differentiation} }

Document

**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Even though coherent control of quantum operations appears to be achievable in practice, it is still not yet well understood. Among theoretical challenges, standard completely positive trace preserving (CPTP) maps are known not to be appropriate to represent coherently controlled quantum channels. We introduce here a graphical language for coherent control of general quantum channels inspired by practical quantum optical setups involving polarising beam splitters (PBS). We consider different situations of coherent control and disambiguate CPTP maps by considering purified channels, an extension of Stinespring’s dilation.
First, we show that in classical control settings, the observational equivalence classes of purified channels correspond to the standard definition of quantum channels (CPTP maps). Then, we propose a refinement of this equivalence class generalising the "half quantum switch" situation, where one is allowed to coherently control which quantum channel is applied; in this case, quantum channel implementations can be distinguished using a so-called transformation matrix. A further refinement characterising observational equivalence with general extended PBS-diagrams as contexts is also obtained. Finally, we propose a refinement that could be used for more general coherent control settings.

Cyril Branciard, Alexandre Clément, Mehdi Mhalla, and Simon Perdrix. Coherent Control and Distinguishability of Quantum Channels via PBS-Diagrams. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 22:1-22:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{branciard_et_al:LIPIcs.MFCS.2021.22, author = {Branciard, Cyril and Cl\'{e}ment, Alexandre and Mhalla, Mehdi and Perdrix, Simon}, title = {{Coherent Control and Distinguishability of Quantum Channels via PBS-Diagrams}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {22:1--22:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.22}, URN = {urn:nbn:de:0030-drops-144629}, doi = {10.4230/LIPIcs.MFCS.2021.22}, annote = {Keywords: Quantum Computing, Diagrammatic Language, Quantum Control, Polarising Beam Splitter, Categorical Quantum Mechanics, Quantum Switch} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

We introduce the PBS-calculus to represent and reason on quantum computations involving coherent control of quantum operations. Coherent control, and in particular indefinite causal order, is known to enable multiple computational and communication advantages over classically ordered models like quantum circuits. The PBS-calculus is inspired by quantum optics, in particular the polarising beam splitter (PBS for short). We formalise the syntax and the semantics of the PBS-diagrams, and we equip the language with an equational theory, which is proved to be sound and complete: two diagrams are representing the same quantum evolution if and only if one can be transformed into the other using the rules of the PBS-calculus. Moreover, we show that the equational theory is minimal. Finally, we consider applications like the implementation of controlled permutations and the unrolling of loops.

Alexandre Clément and Simon Perdrix. PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 24:1-24:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2020.24, author = {Cl\'{e}ment, Alexandre and Perdrix, Simon}, title = {{PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.24}, URN = {urn:nbn:de:0030-drops-126921}, doi = {10.4230/LIPIcs.MFCS.2020.24}, annote = {Keywords: Quantum Computing, Diagrammatic Language, Completeness, Quantum Control, Polarising Beam Splitter, Categorical Quantum Mechanics, Quantum Switch} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representation of sub-diagrams via binary matrices. We show soundness and completeness of the SZX-calculus and provide two examples of applications, for graph states and error correcting codes.

Titouan Carette, Dominic Horsman, and Simon Perdrix. SZX-Calculus: Scalable Graphical Quantum Reasoning. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 55:1-55:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carette_et_al:LIPIcs.MFCS.2019.55, author = {Carette, Titouan and Horsman, Dominic and Perdrix, Simon}, title = {{SZX-Calculus: Scalable Graphical Quantum Reasoning}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {55:1--55:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.55}, URN = {urn:nbn:de:0030-drops-109999}, doi = {10.4230/LIPIcs.MFCS.2019.55}, annote = {Keywords: Quantum computing, categorical quantum mechanics, completeness, scalability} }

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics.
We address the question of the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics.
We introduce a new construction, the discard construction, which transforms any dagger-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal.
Using this construction we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However this construction fails for some fringe cases like the Clifford+T quantum mechanics, as the category does not have enough isometries.

Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Completeness of Graphical Languages for Mixed States Quantum Mechanics (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 108:1-108:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carette_et_al:LIPIcs.ICALP.2019.108, author = {Carette, Titouan and Jeandel, Emmanuel and Perdrix, Simon and Vilmart, Renaud}, title = {{Completeness of Graphical Languages for Mixed States Quantum Mechanics}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {108:1--108:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.108}, URN = {urn:nbn:de:0030-drops-106844}, doi = {10.4230/LIPIcs.ICALP.2019.108}, annote = {Keywords: Quantum Computing, Quantum Categorical Mechanics, Category Theory, Mixed States, Completely Positive Maps} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language - i.e. the ability to derive any true equation - is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles.
We introduce a generalised supplementarity - called cyclotomic supplementarity - which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.
We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart, and Quanlong Wang. ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{jeandel_et_al:LIPIcs.MFCS.2017.11, author = {Jeandel, Emmanuel and Perdrix, Simon and Vilmart, Renaud and Wang, Quanlong}, title = {{ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {11:1--11:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.11}, URN = {urn:nbn:de:0030-drops-81173}, doi = {10.4230/LIPIcs.MFCS.2017.11}, annote = {Keywords: Categorical Quantum Mechanincs, ZX-Calculus, Completeness, Cyclotomic Supplmentarity, Clifford+T} }

Document

**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The ZX-calculus is a powerful diagrammatic language for quantum mechanics and quantum information processing. We prove that its pi/4-fragment is not complete, in other words the ZX-calculus is not complete for the so called "Clifford+T quantum mechanics". The completeness of this fragment was one of the main open problems in categorical quantum mechanics, a programme initiated by Abramsky and Coecke. The ZX-calculus was known to be incomplete for quantum mechanics. On the other hand, its pi/2-fragment is known to be complete, i.e. the ZX-calculus is complete for the so called "stabilizer quantum mechanics". Deciding whether its pi/4-fragment is complete is a crucial step in the development of the ZX-calculus since this fragment is approximately universal for quantum mechanics, contrary to the pi/2-fragment.
To establish our incompleteness result, we consider a fairly simple property of quantum states called supplementarity. We show that supplementarity can be derived in the ZX-calculus if and only if the angles involved in this equation are multiples of pi/2. In particular, the impossibility to derive supplementarity for pi/4 implies the incompleteness of the ZX-calculus for Clifford+T quantum mechanics. As a consequence, we propose to add the supplementarity to the set of rules of the ZX-calculus.
We also show that if a ZX-diagram involves antiphase twins, they can be merged when the ZX-calculus is augmented with the supplementarity rule. Merging antiphase twins makes diagrammatic reasoning much easier and provides a purely graphical meaning to the supplementarity rule.

Simon Perdrix and Quanlong Wang. Supplementarity is Necessary for Quantum Diagram Reasoning. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 76:1-76:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{perdrix_et_al:LIPIcs.MFCS.2016.76, author = {Perdrix, Simon and Wang, Quanlong}, title = {{Supplementarity is Necessary for Quantum Diagram Reasoning}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {76:1--76:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.76}, URN = {urn:nbn:de:0030-drops-65062}, doi = {10.4230/LIPIcs.MFCS.2016.76}, annote = {Keywords: quantum diagram reasoning, completeness, ZX-calculus, quantum computing, categorical quantum mechanics} }

Document

**Published in:** LIPIcs, Volume 22, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)

We give graphical characterisation of the access structure to both classical and quantum information encoded onto a multigraph defined for prime dimension q, as well as explicit decoding operations for quantum secret sharing based on graph state protocols. We give a lower bound on $k$ for the existence of a ((k,n))_q scheme and prove, using probabilistic methods, that there exists alpha such that a random multigraph has an accessing parameter k => alpha*n with high probability.

Anne Marin, Damian Markham, and Simon Perdrix. Access Structure in Graphs in High Dimension and Application to Secret Sharing. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 308-324, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{marin_et_al:LIPIcs.TQC.2013.308, author = {Marin, Anne and Markham, Damian and Perdrix, Simon}, title = {{Access Structure in Graphs in High Dimension and Application to Secret Sharing}}, booktitle = {8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)}, pages = {308--324}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-55-2}, ISSN = {1868-8969}, year = {2013}, volume = {22}, editor = {Severini, Simone and Brandao, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.308}, URN = {urn:nbn:de:0030-drops-43306}, doi = {10.4230/LIPIcs.TQC.2013.308}, annote = {Keywords: Quantum Secret Sharing, Graph State, Multigraph, Access structure} }

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