{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article13982","name":"Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language","abstract":"We study the quantum query complexity of two problems.\r\nFirst, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyck_{k,n} problem. We prove a lower bound of \u03a9(c^k \u221an), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative example of star-free languages for which a surprising O\u0303(\u221an) query quantum algorithm was recently constructed by Aaronson et al. [Scott Aaronson et al., 2018]. Their proof does not give rise to a general algorithm. When k is not a constant, Dyck_{k,n} is not context-free. We give an algorithm with O(\u221an(log n)^{0.5k}) quantum queries for Dyck_{k,n} for all k. This is better than the trival upper bound n for k = o({log(n)}\/{log log n}). \r\nSecond, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the \"balanced parentheses\" problem into the grid, we show a lower bound of \u03a9(n^{1.5-\u03b5}) for the directed 2D grid and \u03a9(n^{2-\u03b5}) for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.","keywords":["Quantum query complexity","Quantum algorithms","Dyck language","Grid path"],"author":[{"@type":"Person","name":"Ambainis, Andris","givenName":"Andris","familyName":"Ambainis","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Balodis, Kaspars","givenName":"Kaspars","familyName":"Balodis","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Iraids, J\u0101nis","givenName":"J\u0101nis","familyName":"Iraids","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Khadiev, Kamil","givenName":"Kamil","familyName":"Khadiev","affiliation":"Kazan Federal University, Russia"},{"@type":"Person","name":"K\u013cevickis, Vladislavs","givenName":"Vladislavs","familyName":"K\u013cevickis","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Pr\u016bsis, Kri\u0161j\u0101nis","givenName":"Kri\u0161j\u0101nis","familyName":"Pr\u016bsis","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Shen, Yixin","givenName":"Yixin","familyName":"Shen","affiliation":"Universit\u00e9 de Paris, CNRS, IRIF, F-75006 Paris, France"},{"@type":"Person","name":"Smotrovs, Juris","givenName":"Juris","familyName":"Smotrovs","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Vihrovs, Jevg\u0113nijs","givenName":"Jevg\u0113nijs","familyName":"Vihrovs","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"}],"position":8,"pageStart":"8:1","pageEnd":"8:14","dateCreated":"2020-08-18","datePublished":"2020-08-18","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ambainis, Andris","givenName":"Andris","familyName":"Ambainis","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Balodis, Kaspars","givenName":"Kaspars","familyName":"Balodis","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Iraids, J\u0101nis","givenName":"J\u0101nis","familyName":"Iraids","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Khadiev, Kamil","givenName":"Kamil","familyName":"Khadiev","affiliation":"Kazan Federal University, Russia"},{"@type":"Person","name":"K\u013cevickis, Vladislavs","givenName":"Vladislavs","familyName":"K\u013cevickis","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Pr\u016bsis, Kri\u0161j\u0101nis","givenName":"Kri\u0161j\u0101nis","familyName":"Pr\u016bsis","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Shen, Yixin","givenName":"Yixin","familyName":"Shen","affiliation":"Universit\u00e9 de Paris, CNRS, IRIF, F-75006 Paris, France"},{"@type":"Person","name":"Smotrovs, Juris","givenName":"Juris","familyName":"Smotrovs","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"},{"@type":"Person","name":"Vihrovs, Jevg\u0113nijs","givenName":"Jevg\u0113nijs","familyName":"Vihrovs","affiliation":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia"}],"copyrightYear":"2020","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.MFCS.2020.8","funding":"Supported by QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union\u2019s Horizon 2020 Programme (QuantAlgo project) and ERDF project 1.1.1.5 A\/020 \"Quantum algorithms: from complexity theory to experiment\". The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities.","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["https:\/\/doi.org\/10.1137\/1.9781611975482.107","http:\/\/arxiv.org\/abs\/2007.03402","https:\/\/doi.org\/10.1007\/978-3-642-33090-2_18","http:\/\/arxiv.org\/abs\/1911.05686","http:\/\/arxiv.org\/abs\/1810.03664","http:\/\/dl.acm.org\/citation.cfm?id=2133036.2133080"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6373","volumeNumber":170,"name":"45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)","dateCreated":"2020-08-18","datePublished":"2020-08-18","editor":[{"@type":"Person","name":"Esparza, Javier","givenName":"Javier","familyName":"Esparza","email":"mailto:esparza@in.tum.de","sameAs":"https:\/\/orcid.org\/0000-0001-9862-4919","affiliation":"TU Munich, Germany"},{"@type":"Person","name":"Kr\u00e1l', Daniel","givenName":"Daniel","familyName":"Kr\u00e1l'","email":"mailto:dkral@fi.muni.cz","sameAs":"https:\/\/orcid.org\/0000-0001-8680-0890","affiliation":"Masaryk University, Brno, Czech Republic"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article13982","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6373"}}}