Quantum Strategies Are Better Than Classical in Almost Any XOR Game

Ambainis, Andris; Bačkurs, Artūrs; Balodis, Kaspars; Kravčenko, Dmitrijs; Ozols, Raitis; Smotrovs, Juris; Virza, Madars

doi:10.1007/978-3-642-31594-7_3

Cited by 11 publications

(27 citation statements)

References 30 publications

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“…where we just used that γ 2 (·) is dual to γ * 2 (·) and · ℓ n ∞ ⊗πℓ n ∞ is dual to · ℓ n 1 ⊗ǫℓ n In words, Corollary 4.3 tells us the following: the ratio w(C * )/w(Q * ) is asymptotically at least 16/15, hence stays lower bounded away from 1. This result complements earlier findings on the mean width of Q and C. Indeed, it was proved in [3] that, as n → ∞,…”

Section: 1supporting

confidence: 91%

“…The results established in [3] are actually about random Bernoulli Bell functionals instead of random Gaussian ones, but the estimate on their quantum value and the upper bound on their classical value remain true in the Gaussian case. It was therefore already known that the ratio w(Q)/w(C) is asymptotically at least 1/ √ ln 2, hence it stays lower bounded away from 1.…”

Section: 1mentioning

confidence: 99%

“…One of the first steps in this direction was given in [3], where the authors study the dual question, that is: how likely is it for a random (in a certain sense) Bell inequality to attain a strictly higher value on quantum correlations than on classical correlations? Later, in [11], some of the authors of this note initiated the study of random correlations in the particular case where these correlations arise as the product of two rectangular normalized Gaussian matrices, a setting motivated by a well known result of Tsirelson that we explain below.…”

Section: Introductionmentioning

confidence: 99%

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Random Quantum Correlations are Generically Non-classical

González-Guillén

Lancien

Palazuelos

et al. 2017

Ann. Henri Poincaré

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Abstract. It is now a well-known fact that the correlations arising from local dichotomic measurements on an entangled quantum state may exhibit intrinsically nonclassical features. In this paper we delve into a comprehensive study of random instances of such bipartite correlations. The main question we are interested in is: given a quantum correlation, taken at random, how likely is it that it is truly non-explainable by a classical model? We show that, under very general assumptions on the considered distribution, a random correlation which lies on the border of the quantum set is with high probability outside the classical set. What is more, we are able to provide the Bell inequality certifying this fact. On the technical side, our results follow from (i) estimating precisely the "quantum norm" of a random matrix, and (ii) lower bounding sharply enough its "classical norm", hence proving a gap between the two. Along the way, we need a non-trivial upper bound on the ∞→1 norm of a random orthogonal matrix, which might be of independent interest.

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Section: 1supporting

confidence: 91%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Random Quantum Correlations are Generically Non-classical

González-Guillén

Lancien

Palazuelos

et al. 2017

Ann. Henri Poincaré

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“…Two such formulae are illustrated in Figure 3. There is a quantum algorithm which allows any such formula to be evaluated in slightly more than O(N 1/2 ) operations [5], while it is Figure 3: Two boolean formulae on 4 bits. For x 1 = 1, x 2 = x 3 = x 4 = 0, for example, the first formula evaluates to 1 and the second to 0.…”

Section: Quantum Walksmentioning

confidence: 99%

Quantum algorithms: an overview

2016

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Quantum computers are designed to outperform standard computers by running quantum algorithms. Areas in which quantum algorithms can be applied include cryptography, search and optimisation, simulation of quantum systems and solving large systems of linear equations. Here we briefly survey some known quantum algorithms, with an emphasis on a broad overview of their applications rather than their technical details. We include a discussion of recent developments and near-term applications of quantum algorithms. INTRODUCTIONA quantum computer is a machine designed to use quantum mechanics to do things which cannot be done by any machine based only on the laws of classical physics. Eventual applications of quantum computing range from breaking cryptographic systems to the design of new medicines. These applications are based on quantum algorithms-algorithms that run on a quantum computer and achieve a speedup, or other efficiency improvement, over any possible classical algorithm. Although large-scale general-purpose quantum computers do not yet exist, the theory of quantum algorithms has been an active area of study for over 20 years. Here we aim to give a broad overview of quantum algorithmics, focusing on algorithms with clear applications and rigorous performance bounds, and including recent progress in the field.Contrary to a rather widespread popular belief that quantum computers have few applications, the field of quantum algorithms has developed into an area of study large enough that a brief survey such as this cannot hope to be remotely comprehensive. Indeed, at the time of writing the 'Quantum Algorithm Zoo' website cites 262 papers on quantum algorithms. 1 There are now a number of excellent surveys about quantum algorithms, 2-5 and we defer to these for details of the algorithms we cover here, and many more. In particular, we omit all discussion of how the quantum algorithms mentioned work. We will also not cover the important topics of how to actually build a quantum computer 6 (in theory or in practice) and quantum error-correction, 7 nor quantum communication complexity 8 or quantum Shannon theory. 9

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“…for every x, y, where (Ω, P) is a probability space and for every ω ∈ Ω we have A x (ω) ∈ {−1, 1} for every x (resp. B y (ω) ∈ {−1, 1} for every y); and quantum correlation matrices 1 Formally, Bell talked about a local hidden variable model (LHVM). 2 Note that P is not a probability distribution itself.…”

Section: Introductionmentioning

confidence: 99%

Random Constructions in Bell Inequalities: A Survey

Palazuelos

2018

Found Phys

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Initially motivated by their relevance in foundations of quantum mechanics and more recently by their applications in different contexts of quantum information science, violations of Bell inequalities have been extensively studied during the last years. In particular, an important effort has been made in order to quantify such Bell violations. Probabilistic techniques have been heavily used in this context with two different purposes. First, to quantify how common the phenomenon of Bell violations is; and secondly, to find large Bell violations in order to better understand the possibilities and limitations of this phenomenon. However, the strong mathematical content of these results has discouraged some of the potentially interested readers. The aim of the present work is to review some of the recent results in this direction by focusing on the main ideas and removing most of the technical details, to make the previous study more accessible to a wide audience.

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Quantum Strategies Are Better Than Classical in Almost Any XOR Game

Cited by 11 publications

References 30 publications

Random Quantum Correlations are Generically Non-classical

Random Quantum Correlations are Generically Non-classical

Quantum algorithms: an overview

Random Constructions in Bell Inequalities: A Survey

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