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Documents authored by Smotrovs, Juris

Document
DOI: 10.4230/LIPIcs.MFCS.2020.8
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language

Authors: Andris Ambainis, Kaspars Balodis, Jānis Iraids, Kamil Khadiev, Vladislavs Kļevickis, Krišjānis Prūsis, Yixin Shen, Juris Smotrovs, and Jevgēnijs Vihrovs

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

Abstract
We study the quantum query complexity of two problems. First, 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 Ω(c^k √n), 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 Õ(√n) 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(√n(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}). Second, 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 Ω(n^{1.5-ε}) for the directed 2D grid and Ω(n^{2-ε}) 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.
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Andris Ambainis, Kaspars Balodis, Jānis Iraids, Kamil Khadiev, Vladislavs Kļevickis, Krišjānis Prūsis, Yixin Shen, Juris Smotrovs, and Jevgēnijs Vihrovs. Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ambainis_et_al:LIPIcs.MFCS.2020.8,
  author =	{Ambainis, Andris and Balodis, Kaspars and Iraids, J\={a}nis and Khadiev, Kamil and K\c{l}evickis, Vladislavs and Pr\={u}sis, Kri\v{s}j\={a}nis and Shen, Yixin and Smotrovs, Juris and Vihrovs, Jevg\={e}nijs},
  title =	{{Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{8:1--8:14},
  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.2020.8},
  URN =		{urn:nbn:de:0030-drops-126774},
  doi =		{10.4230/LIPIcs.MFCS.2020.8},
  annote =	{Keywords: Quantum query complexity, Quantum algorithms, Dyck language, Grid path}
}
Document
DOI: 10.4230/LIPIcs.CCC.2016.25
Polynomials, Quantum Query Complexity, and Grothendieck's Inequality

Authors: Scott Aaronson, Andris Ambainis, Janis Iraids, Martins Kokainis, and Juris Smotrovs

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

Abstract
We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function f is computable by a 1-query quantum algorithm with error bounded by epsilon<1/2 iff f can be approximated by a degree-2 polynomial with error bounded by epsilon'<1/2. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis [Aaronson/Ambainis, STOC 2015]. The proof uses Grothendieck's inequality to relate two matrix norms, with one norm corresponding to polynomial approximations and the other norm corresponding to quantum algorithms. We also show two results for polynomials of higher degree. First, there is a total Boolean function which requires ~Omega(n) quantum queries but can be represented by a block-multilinear polynomial of degree ~O(sqrt(n)). Thus, in the general case (for an arbitrary number of queries), block-multilinear polynomials are not equivalent to quantum algorithms. Second, for any constant degree k, the two notions of approximation by a polynomial (the standard and the block-multilinear) are equivalent. As a consequence, we solve an open problem from [Aaronson/Ambainis, STOC 2015], showing that one can estimate the value of any bounded degree-k polynomial p:{0,1}^n -> [-1,1] with O(n^{1-1/(2k)) queries.
Cite as

Scott Aaronson, Andris Ambainis, Janis Iraids, Martins Kokainis, and Juris Smotrovs. Polynomials, Quantum Query Complexity, and Grothendieck's Inequality. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{aaronson_et_al:LIPIcs.CCC.2016.25,
  author =	{Aaronson, Scott and Ambainis, Andris and Iraids, Janis and Kokainis, Martins and Smotrovs, Juris},
  title =	{{Polynomials, Quantum Query Complexity, and Grothendieck's Inequality}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{25:1--25:19},
  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.2016.25},
  URN =		{urn:nbn:de:0030-drops-58394},
  doi =		{10.4230/LIPIcs.CCC.2016.25},
  annote =	{Keywords: quantum algorithms, Boolean functions, approximation by polynomials, Grothendieck's inequality}
}
Document
DOI: 10.4230/LIPIcs.TQC.2013.263
Exact Quantum Query Complexity of EXACT and THRESHOLD

Authors: Andris Ambainis, Janis Iraids, and Juris Smotrovs

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

Abstract
A quantum algorithm is exact if it always produces the correct answer, on any input. Coming up with exact quantum algorithms that substantially outperform the best classical algorithm has been a quite challenging task. In this paper, we present two new exact quantum algorithms for natural problems: - for the problem EXACT_k^n in which we have to determine whether the sequence of input bits x_1, ..., x_n contains exactly k values x_i=1; - for the problem THRESHOLD_k^n in which we have to determine if at least k of n input bits are equal to 1.
Cite as

Andris Ambainis, Janis Iraids, and Juris Smotrovs. Exact Quantum Query Complexity of EXACT and THRESHOLD. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 263-269, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{ambainis_et_al:LIPIcs.TQC.2013.263,
  author =	{Ambainis, Andris and Iraids, Janis and Smotrovs, Juris},
  title =	{{Exact Quantum Query Complexity of EXACT and THRESHOLD}},
  booktitle =	{8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)},
  pages =	{263--269},
  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.2013.263},
  URN =		{urn:nbn:de:0030-drops-43261},
  doi =		{10.4230/LIPIcs.TQC.2013.263},
  annote =	{Keywords: Quantum query algorithms, Complexity of Boolean functions}
}
Document
DOI: 10.4230/LIPIcs.STACS.2013.446
Optimal quantum query bounds for almost all Boolean functions

Authors: Andris Ambainis, Arturs Backurs, Juris Smotrovs, and Ronald de Wolf

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Abstract
We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis (A. Ambainis, 1999), and shows that van Dam's oracle interrogation (W. van Dam, 1998) is essentially optimal for almost all functions. Our proof uses the fact that the acceptance probability of a T-query algorithm can be written as the sum of squares of degree-T polynomials.
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Andris Ambainis, Arturs Backurs, Juris Smotrovs, and Ronald de Wolf. Optimal quantum query bounds for almost all Boolean functions. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 446-453, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{ambainis_et_al:LIPIcs.STACS.2013.446,
  author =	{Ambainis, Andris and Backurs, Arturs and Smotrovs, Juris and de Wolf, Ronald},
  title =	{{Optimal quantum query bounds for almost all Boolean functions}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{446--453},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.446},
  URN =		{urn:nbn:de:0030-drops-39557},
  doi =		{10.4230/LIPIcs.STACS.2013.446},
  annote =	{Keywords: Quantum computing, query complexity, lower bounds, polynomial method}
}
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