4 Search Results for "Heunen, Chris"


Document
Localisable Monads

Authors: Carmen Constantin, Nuiok Dicaire, and Chris Heunen

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)


Abstract
Monads govern computational side-effects in programming semantics. A collection of monads can be combined together in a local-to-global way to handle several instances of such effects. Indexed monads and graded monads do this in a modular way. Here, instead, we start with a single monad and equip it with a fine-grained structure by using techniques from tensor topology. This provides an intrinsic theory of local computational effects without needing to know how constituent effects interact beforehand. Specifically, any monoidal category decomposes as a sheaf of local categories over a base space. We identify a notion of localisable monads which characterises when a monad decomposes as a sheaf of monads. Equivalently, localisable monads are formal monads in an appropriate presheaf 2-category, whose algebras we characterise. Three extended examples demonstrate how localisable monads can interpret the base space as locations in a computer memory, as sites in a network of interacting agents acting concurrently, and as time in stochastic processes.

Cite as

Carmen Constantin, Nuiok Dicaire, and Chris Heunen. Localisable Monads. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 15:1-15:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{constantin_et_al:LIPIcs.CSL.2022.15,
  author =	{Constantin, Carmen and Dicaire, Nuiok and Heunen, Chris},
  title =	{{Localisable Monads}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.15},
  URN =		{urn:nbn:de:0030-drops-157353},
  doi =		{10.4230/LIPIcs.CSL.2022.15},
  annote =	{Keywords: Monad, Monoidal category, Presheaf, Central idempotent, Graded monad, Indexed monad, Formal monad, Strong monad, Commutative monad}
}
Document
The Logic of Contextuality

Authors: Samson Abramsky and Rui Soares Barbosa

Published in: LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)


Abstract
Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including: - establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; - formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; - investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations; - developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.

Cite as

Samson Abramsky and Rui Soares Barbosa. The Logic of Contextuality. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 5:1-5:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{abramsky_et_al:LIPIcs.CSL.2021.5,
  author =	{Abramsky, Samson and Barbosa, Rui Soares},
  title =	{{The Logic of Contextuality}},
  booktitle =	{29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-175-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{183},
  editor =	{Baier, Christel and Goubault-Larrecq, Jean},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.5},
  URN =		{urn:nbn:de:0030-drops-134394},
  doi =		{10.4230/LIPIcs.CSL.2021.5},
  annote =	{Keywords: partial Boolean algebras, contextuality, exclusivity principles, Kochen-Specker paradoxes, tensor product}
}
Document
Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem

Authors: Huan Li, He Sun, and Luca Zanetti

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)


Abstract
We study spectral approaches for the MAX-2-LIN(k) problem, in which we are given a system of m linear equations of the form x_i - x_j is equivalent to c_{ij} mod k, and required to find an assignment to the n variables {x_i} that maximises the total number of satisfied equations. We consider Hermitian Laplacians related to this problem, and prove a Cheeger inequality that relates the smallest eigenvalue of a Hermitian Laplacian to the maximum number of satisfied equations of a MAX-2-LIN(k) instance I. We develop an O~(kn^2) time algorithm that, for any (1-epsilon)-satisfiable instance, produces an assignment satisfying a (1 - O(k)sqrt{epsilon})-fraction of equations. We also present a subquadratic-time algorithm that, when the graph associated with I is an expander, produces an assignment satisfying a (1- O(k^2)epsilon)-fraction of the equations. Our Cheeger inequality and first algorithm can be seen as generalisations of the Cheeger inequality and algorithm for MAX-CUT developed by Trevisan.

Cite as

Huan Li, He Sun, and Luca Zanetti. Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 71:1-71:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{li_et_al:LIPIcs.ESA.2019.71,
  author =	{Li, Huan and Sun, He and Zanetti, Luca},
  title =	{{Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{71:1--71:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.71},
  URN =		{urn:nbn:de:0030-drops-111926},
  doi =		{10.4230/LIPIcs.ESA.2019.71},
  annote =	{Keywords: Spectral methods, Hermitian Laplacians, the Max-2-Lin problem, Unique Games}
}
Document
Track A: Algorithms, Complexity and Games
Complexity-Theoretic Limitations on Blind Delegated Quantum Computation

Authors: Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu, and Elham Kashefi

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


Abstract
Blind delegation protocols allow a client to delegate a computation to a server so that the server learns nothing about the input to the computation apart from its size. For the specific case of quantum computation we know, from work over the past decade, that blind delegation protocols can achieve information-theoretic security (provided the client and the server exchange some amount of quantum information). In this paper we prove, provided certain complexity-theoretic conjectures are true, that the power of information-theoretically secure blind delegation protocols for quantum computation (ITS-BQC protocols) is in a number of ways constrained. In the first part of our paper we provide some indication that ITS-BQC protocols for delegating polynomial-time quantum computations in which the client and the server interact only classically are unlikely to exist. We first show that having such a protocol in which the client and the server exchange O(n^d) bits of communication, implies that BQP subset MA/O(n^d). We conjecture that this containment is unlikely by proving that there exists an oracle relative to which BQP not subset MA/O(n^d). We then show that if an ITS-BQC protocol exists in which the client and the server interact only classically and which allows the client to delegate quantum sampling problems to the server (such as BosonSampling) then there exist non-uniform circuits of size 2^{n - Omega(n/log(n))}, making polynomially-sized queries to an NP^{NP} oracle, for computing the permanent of an n x n matrix. The second part of our paper concerns ITS-BQC protocols in which the client and the server engage in one round of quantum communication and then exchange polynomially many classical messages. First, we provide a complexity-theoretic upper bound on the types of functions that could be delegated in such a protocol by showing that they must be contained in QCMA/qpoly cap coQCMA/qpoly. Then, we show that having such a protocol for delegating NP-hard functions implies coNP^{NP^{NP}} subseteq NP^{NP^{PromiseQMA}}.

Cite as

Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu, and Elham Kashefi. Complexity-Theoretic Limitations on Blind Delegated Quantum Computation. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 6:1-6:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{aaronson_et_al:LIPIcs.ICALP.2019.6,
  author =	{Aaronson, Scott and Cojocaru, Alexandru and Gheorghiu, Alexandru and Kashefi, Elham},
  title =	{{Complexity-Theoretic Limitations on Blind Delegated Quantum Computation}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{6:1--6:13},
  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.2019.6},
  URN =		{urn:nbn:de:0030-drops-105826},
  doi =		{10.4230/LIPIcs.ICALP.2019.6},
  annote =	{Keywords: Quantum cryptography, Complexity theory, Delegated quantum computation, Computing on encrypted data}
}
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