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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state |ψ_1⟩ ⊗ … ⊗ |ψ_n⟩ where each qubit is in one of two orthogonal states |ψ⟩ or |ψ^⟂⟩, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of 1/2 + Θ(1/√n). Under the promise that at least 2/3 of the input qubits are in the majority state, the fidelity increases to 1 - Θ(1/n) and approaches 1 as n increases.
We also consider the more general problem of computing any symmetric and equivariant Boolean function f: {0,1}ⁿ → {0,1} in an unknown quantum basis, and show that a generalization of our quantum majority vote algorithm is optimal for this task. The optimal parameters for the generalized algorithm and its worst-case fidelity can be determined by a simple linear program of size O(n). The time complexity of the algorithm is O(n⁴ log n) where n is the number of input qubits.

Harry Buhrman, Noah Linden, Laura Mančinska, Ashley Montanaro, and Maris Ozols. Quantum Majority Vote. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, p. 29:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{buhrman_et_al:LIPIcs.ITCS.2023.29, author = {Buhrman, Harry and Linden, Noah and Man\v{c}inska, Laura and Montanaro, Ashley and Ozols, Maris}, title = {{Quantum Majority Vote}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {29:1--29:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.29}, URN = {urn:nbn:de:0030-drops-175321}, doi = {10.4230/LIPIcs.ITCS.2023.29}, annote = {Keywords: quantum algorithms, quantum majority vote, Schur-Weyl duality} }

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Complete Volume

**Published in:** LIPIcs, Volume 135, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)

LIPIcs, Volume 135, TQC'19, Complete Volume

14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{vandam_et_al:LIPIcs.TQC.2019, title = {{LIPIcs, Volume 135, TQC'19, Complete Volume}}, booktitle = {14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-112-2}, ISSN = {1868-8969}, year = {2019}, volume = {135}, editor = {van Dam, Wim and Man\v{c}inska, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2019}, URN = {urn:nbn:de:0030-drops-105052}, doi = {10.4230/LIPIcs.TQC.2019}, annote = {Keywords: Theory of computation, Quantum computation theory, Quantum complexity theory, Quantum communication complexity} }

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Front Matter

**Published in:** LIPIcs, Volume 135, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)

Front Matter, Table of Contents, Preface, Conference Organization

14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 0:i-0:xiii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{vandam_et_al:LIPIcs.TQC.2019.0, author = {van Dam, Wim and Man\v{c}inska, Laura}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)}, pages = {0:i--0:xiii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-112-2}, ISSN = {1868-8969}, year = {2019}, volume = {135}, editor = {van Dam, Wim and Man\v{c}inska, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2019.0}, URN = {urn:nbn:de:0030-drops-103920}, doi = {10.4230/LIPIcs.TQC.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We introduce a nonlocal game that captures and extends the notion of graph isomorphism. This game can be won in the classical case if and only if the two input graphs are isomorphic. Thus, by considering quantum strategies we are able to define the notion of quantum isomorphism. We also consider the case of more general non-signalling strategies, and show that such a strategy exists if and only if the graphs are fractionally isomorphic. We prove several necessary conditions for quantum isomorphism, including cospectrality, and provide a construction for producing pairs of non-isomorphic graphs that are quantum isomorphic.
We then show that both classical and quantum isomorphism can be reformulated as feasibility programs over the completely positive and completely positive semidefinite cones respectively. This leads us to considering relaxations of (quantum) isomorphism arrived at by relaxing the cone to either the doubly nonnegative (DNN) or positive semidefinite (PSD) cones. We show that DNN-isomorphism is equivalent to the previous defined notion of graph equivalence, a polynomial-time decidable relation that is related to coherent algebras. We also show that PSD-isomorphism implies several types of cospectrality, and that it is equivalent to cospectrality for connected 1-walk-regular graphs. Finally, we show that all of the above mentioned relations form a strict hierarchy of weaker and weaker relations, with non-singalling/fractional isomorphism being the weakest. The techniques used are an interesting mix of algebra, combinatorics, and quantum information.

Laura Mancinska, David E. Roberson, Robert Samal, Simone Severini, and Antonios Varvitsiotis. Relaxations of Graph Isomorphism. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 76:1-76:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mancinska_et_al:LIPIcs.ICALP.2017.76, author = {Mancinska, Laura and Roberson, David E. and Samal, Robert and Severini, Simone and Varvitsiotis, Antonios}, title = {{Relaxations of Graph Isomorphism}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {76:1--76:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.76}, URN = {urn:nbn:de:0030-drops-74697}, doi = {10.4230/LIPIcs.ICALP.2017.76}, annote = {Keywords: graph isomorphism, quantum information, semidefinite programming} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.

Adam Bouland, Laura Mancinska, and Xue Zhang. Complexity Classification of Two-Qubit Commuting Hamiltonians. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 28:1-28:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bouland_et_al:LIPIcs.CCC.2016.28, author = {Bouland, Adam and Mancinska, Laura and Zhang, Xue}, title = {{Complexity Classification of Two-Qubit Commuting Hamiltonians}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {28:1--28:33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.28}, URN = {urn:nbn:de:0030-drops-58469}, doi = {10.4230/LIPIcs.CCC.2016.28}, annote = {Keywords: Quantum Computing, Sampling Problems, Commuting Hamiltonians, IQP, Gate Classification Theorems} }

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**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs G and H. Such vectors exist if and only if theta(G) <= theta(H) where theta represents the Lovász number. We also obtain similar inequalities for the related Schrijver theta^- and Szegedy theta^+ numbers.
These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: alpha^*(G) <= theta^-(G). Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovász number. Beigi introduced a quantity beta as an upper bound on alpha^* and posed the question of whether beta(G) = \lfloor theta(G) \rfloor. We answer this in the affirmative and show that a related quantity is equal to \lceil theta(G) \rceil. We show that a quantity chi_{vect}(G) recently introduced in the context of Tsirelson's conjecture is equal to \lceil theta^+(G) \rceil.

Toby Cubitt, Laura Mancinska, David Roberson, Simone Severini, Dan Stahlke, and Andreas Winter. Bounds on Entanglement Assisted Source-channel Coding Via the Lovász Theta Number and Its Variants. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 48-51, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{cubitt_et_al:LIPIcs.TQC.2014.48, author = {Cubitt, Toby and Mancinska, Laura and Roberson, David and Severini, Simone and Stahlke, Dan and Winter, Andreas}, title = {{Bounds on Entanglement Assisted Source-channel Coding Via the Lov\'{a}sz Theta Number and Its Variants}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {48--51}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.48}, URN = {urn:nbn:de:0030-drops-48054}, doi = {10.4230/LIPIcs.TQC.2014.48}, annote = {Keywords: source-channel coding, zero-error capacity, Lov\'{a}sz theta} }

Document

**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

We introduce a novel technique to give bounds to the entangled value of non-local games. The technique is based on a class of graphs used by Cabello, Severini and Winter in 2010. The upper bound uses the famous Lovàsz theta number and is efficiently computable; the lower one is based on the quantum independence number, which is a quantity used in the study of entanglement-assisted channel capacities and graph homomorphism games.

André Chailloux, Laura Mancinska, Giannicola Scarpa, and Simone Severini. Graph-theoretical Bounds on the Entangled Value of Non-local Games. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 67-75, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chailloux_et_al:LIPIcs.TQC.2014.67, author = {Chailloux, Andr\'{e} and Mancinska, Laura and Scarpa, Giannicola and Severini, Simone}, title = {{Graph-theoretical Bounds on the Entangled Value of Non-local Games}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {67--75}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.67}, URN = {urn:nbn:de:0030-drops-48074}, doi = {10.4230/LIPIcs.TQC.2014.67}, annote = {Keywords: Graph theory, non-locality, entangled games} }

Document

**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

A homomorphism from a graph X to a graph Y is an adjacency preserving
mapping f:V(X) -> V(Y). We consider a nonlocal game in which Alice and
Bob are trying to convince a verifier with certainty that a graph X
admits a homomorphism to Y. This is a generalization of the
well-studied graph coloring game. Via systematic study of quantum
homomorphisms we prove new results for graph coloring. Most
importantly, we show that the Lovász theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that other quantum graph parameters, such as quantum independence number, can differ from their classical counterparts. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum
homomorphisms to construct graphs for which entanglement-assistance
increases their one-shot zero-error capacity.

Laura Mancinska and David Roberson. Graph Homomorphisms for Quantum Players. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 212-216, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{mancinska_et_al:LIPIcs.TQC.2014.212, author = {Mancinska, Laura and Roberson, David}, title = {{Graph Homomorphisms for Quantum Players}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {212--216}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.212}, URN = {urn:nbn:de:0030-drops-48179}, doi = {10.4230/LIPIcs.TQC.2014.212}, annote = {Keywords: graph homomorphism, nonlocal game, Lov\'{a}sz theta, quantum chromatic number, entanglement} }

Document

**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state |ψ_1⟩ ⊗ … ⊗ |ψ_n⟩ where each qubit is in one of two orthogonal states |ψ⟩ or |ψ^⟂⟩, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of 1/2 + Θ(1/√n). Under the promise that at least 2/3 of the input qubits are in the majority state, the fidelity increases to 1 - Θ(1/n) and approaches 1 as n increases.
We also consider the more general problem of computing any symmetric and equivariant Boolean function f: {0,1}ⁿ → {0,1} in an unknown quantum basis, and show that a generalization of our quantum majority vote algorithm is optimal for this task. The optimal parameters for the generalized algorithm and its worst-case fidelity can be determined by a simple linear program of size O(n). The time complexity of the algorithm is O(n⁴ log n) where n is the number of input qubits.

Harry Buhrman, Noah Linden, Laura Mančinska, Ashley Montanaro, and Maris Ozols. Quantum Majority Vote. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, p. 29:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{buhrman_et_al:LIPIcs.ITCS.2023.29, author = {Buhrman, Harry and Linden, Noah and Man\v{c}inska, Laura and Montanaro, Ashley and Ozols, Maris}, title = {{Quantum Majority Vote}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {29:1--29:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.29}, URN = {urn:nbn:de:0030-drops-175321}, doi = {10.4230/LIPIcs.ITCS.2023.29}, annote = {Keywords: quantum algorithms, quantum majority vote, Schur-Weyl duality} }

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Complete Volume

**Published in:** LIPIcs, Volume 135, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)

LIPIcs, Volume 135, TQC'19, Complete Volume

14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{vandam_et_al:LIPIcs.TQC.2019, title = {{LIPIcs, Volume 135, TQC'19, Complete Volume}}, booktitle = {14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-112-2}, ISSN = {1868-8969}, year = {2019}, volume = {135}, editor = {van Dam, Wim and Man\v{c}inska, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2019}, URN = {urn:nbn:de:0030-drops-105052}, doi = {10.4230/LIPIcs.TQC.2019}, annote = {Keywords: Theory of computation, Quantum computation theory, Quantum complexity theory, Quantum communication complexity} }

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Front Matter

**Published in:** LIPIcs, Volume 135, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)

Front Matter, Table of Contents, Preface, Conference Organization

14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 0:i-0:xiii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{vandam_et_al:LIPIcs.TQC.2019.0, author = {van Dam, Wim and Man\v{c}inska, Laura}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)}, pages = {0:i--0:xiii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-112-2}, ISSN = {1868-8969}, year = {2019}, volume = {135}, editor = {van Dam, Wim and Man\v{c}inska, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2019.0}, URN = {urn:nbn:de:0030-drops-103920}, doi = {10.4230/LIPIcs.TQC.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We introduce a nonlocal game that captures and extends the notion of graph isomorphism. This game can be won in the classical case if and only if the two input graphs are isomorphic. Thus, by considering quantum strategies we are able to define the notion of quantum isomorphism. We also consider the case of more general non-signalling strategies, and show that such a strategy exists if and only if the graphs are fractionally isomorphic. We prove several necessary conditions for quantum isomorphism, including cospectrality, and provide a construction for producing pairs of non-isomorphic graphs that are quantum isomorphic.
We then show that both classical and quantum isomorphism can be reformulated as feasibility programs over the completely positive and completely positive semidefinite cones respectively. This leads us to considering relaxations of (quantum) isomorphism arrived at by relaxing the cone to either the doubly nonnegative (DNN) or positive semidefinite (PSD) cones. We show that DNN-isomorphism is equivalent to the previous defined notion of graph equivalence, a polynomial-time decidable relation that is related to coherent algebras. We also show that PSD-isomorphism implies several types of cospectrality, and that it is equivalent to cospectrality for connected 1-walk-regular graphs. Finally, we show that all of the above mentioned relations form a strict hierarchy of weaker and weaker relations, with non-singalling/fractional isomorphism being the weakest. The techniques used are an interesting mix of algebra, combinatorics, and quantum information.

Laura Mancinska, David E. Roberson, Robert Samal, Simone Severini, and Antonios Varvitsiotis. Relaxations of Graph Isomorphism. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 76:1-76:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mancinska_et_al:LIPIcs.ICALP.2017.76, author = {Mancinska, Laura and Roberson, David E. and Samal, Robert and Severini, Simone and Varvitsiotis, Antonios}, title = {{Relaxations of Graph Isomorphism}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {76:1--76:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.76}, URN = {urn:nbn:de:0030-drops-74697}, doi = {10.4230/LIPIcs.ICALP.2017.76}, annote = {Keywords: graph isomorphism, quantum information, semidefinite programming} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.

Adam Bouland, Laura Mancinska, and Xue Zhang. Complexity Classification of Two-Qubit Commuting Hamiltonians. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 28:1-28:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bouland_et_al:LIPIcs.CCC.2016.28, author = {Bouland, Adam and Mancinska, Laura and Zhang, Xue}, title = {{Complexity Classification of Two-Qubit Commuting Hamiltonians}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {28:1--28:33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.28}, URN = {urn:nbn:de:0030-drops-58469}, doi = {10.4230/LIPIcs.CCC.2016.28}, annote = {Keywords: Quantum Computing, Sampling Problems, Commuting Hamiltonians, IQP, Gate Classification Theorems} }

Document

**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs G and H. Such vectors exist if and only if theta(G) <= theta(H) where theta represents the Lovász number. We also obtain similar inequalities for the related Schrijver theta^- and Szegedy theta^+ numbers.
These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: alpha^*(G) <= theta^-(G). Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovász number. Beigi introduced a quantity beta as an upper bound on alpha^* and posed the question of whether beta(G) = \lfloor theta(G) \rfloor. We answer this in the affirmative and show that a related quantity is equal to \lceil theta(G) \rceil. We show that a quantity chi_{vect}(G) recently introduced in the context of Tsirelson's conjecture is equal to \lceil theta^+(G) \rceil.

Toby Cubitt, Laura Mancinska, David Roberson, Simone Severini, Dan Stahlke, and Andreas Winter. Bounds on Entanglement Assisted Source-channel Coding Via the Lovász Theta Number and Its Variants. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 48-51, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{cubitt_et_al:LIPIcs.TQC.2014.48, author = {Cubitt, Toby and Mancinska, Laura and Roberson, David and Severini, Simone and Stahlke, Dan and Winter, Andreas}, title = {{Bounds on Entanglement Assisted Source-channel Coding Via the Lov\'{a}sz Theta Number and Its Variants}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {48--51}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.48}, URN = {urn:nbn:de:0030-drops-48054}, doi = {10.4230/LIPIcs.TQC.2014.48}, annote = {Keywords: source-channel coding, zero-error capacity, Lov\'{a}sz theta} }

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**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

We introduce a novel technique to give bounds to the entangled value of non-local games. The technique is based on a class of graphs used by Cabello, Severini and Winter in 2010. The upper bound uses the famous Lovàsz theta number and is efficiently computable; the lower one is based on the quantum independence number, which is a quantity used in the study of entanglement-assisted channel capacities and graph homomorphism games.

André Chailloux, Laura Mancinska, Giannicola Scarpa, and Simone Severini. Graph-theoretical Bounds on the Entangled Value of Non-local Games. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 67-75, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chailloux_et_al:LIPIcs.TQC.2014.67, author = {Chailloux, Andr\'{e} and Mancinska, Laura and Scarpa, Giannicola and Severini, Simone}, title = {{Graph-theoretical Bounds on the Entangled Value of Non-local Games}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {67--75}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.67}, URN = {urn:nbn:de:0030-drops-48074}, doi = {10.4230/LIPIcs.TQC.2014.67}, annote = {Keywords: Graph theory, non-locality, entangled games} }

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**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

A homomorphism from a graph X to a graph Y is an adjacency preserving
mapping f:V(X) -> V(Y). We consider a nonlocal game in which Alice and
Bob are trying to convince a verifier with certainty that a graph X
admits a homomorphism to Y. This is a generalization of the
well-studied graph coloring game. Via systematic study of quantum
homomorphisms we prove new results for graph coloring. Most
importantly, we show that the Lovász theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that other quantum graph parameters, such as quantum independence number, can differ from their classical counterparts. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum
homomorphisms to construct graphs for which entanglement-assistance
increases their one-shot zero-error capacity.

Laura Mancinska and David Roberson. Graph Homomorphisms for Quantum Players. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 212-216, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{mancinska_et_al:LIPIcs.TQC.2014.212, author = {Mancinska, Laura and Roberson, David}, title = {{Graph Homomorphisms for Quantum Players}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {212--216}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.212}, URN = {urn:nbn:de:0030-drops-48179}, doi = {10.4230/LIPIcs.TQC.2014.212}, annote = {Keywords: graph homomorphism, nonlocal game, Lov\'{a}sz theta, quantum chromatic number, entanglement} }

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