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Documents authored by Berta, Mario

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.82
Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension

Authors: Hyejung H. Jee, Carlo Sparaciari, Omar Fawzi, and Mario Berta

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
In a recent landmark result [Ji et al., arXiv:2001.04383 (2020)], it was shown that approximating the value of a two-player game is undecidable when the players are allowed to share quantum states of unbounded dimension. In this paper, we study the computational complexity of two-player games when the dimension of the quantum systems is bounded by T. More specifically, we give a semidefinite program of size exp(𝒪(T^{12}(log²(AT)+log(Q)log(AT))/ε²)) to compute additive ε-approximations on the value of two-player free games with T× T-dimensional quantum entanglement, where A and Q denote the number of answers and questions of the game, respectively. For fixed dimension T, this scales polynomially in Q and quasi-polynomially in A, thereby improving on previously known approximation algorithms for which worst-case run-time guarantees are at best exponential in Q and A. For the proof, we make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints that we derive via quantum entropy inequalities.
Cite as

Hyejung H. Jee, Carlo Sparaciari, Omar Fawzi, and Mario Berta. Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 82:1-82:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jee_et_al:LIPIcs.ICALP.2021.82,
  author =	{Jee, Hyejung H. and Sparaciari, Carlo and Fawzi, Omar and Berta, Mario},
  title =	{{Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{82:1--82:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.82},
  URN =		{urn:nbn:de:0030-drops-141514},
  doi =		{10.4230/LIPIcs.ICALP.2021.82},
  annote =	{Keywords: non-local game, semidefinite programming, quantum correlation, approximation algorithm, Lasserre hierarchy, de Finetti theorem}
}
Document
DOI: 10.4230/LIPIcs.TQC.2015.73
Semidefinite Programs for Randomness Extractors

Authors: Mario Berta, Omar Fawzi, and Volkher B. Scholz

Published in: LIPIcs, Volume 44, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)

Abstract
Randomness extractors are an important building block for classical and quantum cryptography. However, for many applications it is crucial that the extractors are quantum-proof, i.e., that they work even in the presence of quantum adversaries. In general, quantum-proof extractors are poorly understood and we would like to argue that in the same way as Bell inequalities (multiprover games) and communication complexity, the setting of randomness extractors provides a operationally useful framework for studying the power and limitations of a quantum memory compared to a classical one. We start by recalling how to phrase the extractor property as a quadratic program with linear constraints. We then construct a semidefinite programming (SDP) relaxation for this program that is tight for some extractor constructions. Moreover, we show that this SDP relaxation is even sufficient to certify quantum-proof extractors. This gives a unifying approach to understand the stability properties of extractors against quantum adversaries. Finally, we analyze the limitations of this SDP relaxation.
Cite as

Mario Berta, Omar Fawzi, and Volkher B. Scholz. Semidefinite Programs for Randomness Extractors. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 73-91, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{berta_et_al:LIPIcs.TQC.2015.73,
  author =	{Berta, Mario and Fawzi, Omar and Scholz, Volkher B.},
  title =	{{Semidefinite Programs for Randomness Extractors}},
  booktitle =	{10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)},
  pages =	{73--91},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-96-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{44},
  editor =	{Beigi, Salman and K\"{o}nig, Robert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2015.73},
  URN =		{urn:nbn:de:0030-drops-55507},
  doi =		{10.4230/LIPIcs.TQC.2015.73},
  annote =	{Keywords: Randomness Extractors, Quantum adversaries, Semidefinite programs}
}
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