Open AccessJournal Article
Quantum boolean functions
TL;DR: This paper introduces the study of quantum boolean functions, which are unitary operators f whose square is the identity: f 2 = I, and proves a quantum extension of the hypercontractive inequality of Bonami, Gross and Beckner.
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Abstract: In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f 2 = I. We describe several generalisations of well-known results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich-Levin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In order to obtain one of these generalisations, we prove a quantum extension of the hypercontractive inequality of Bonami, Gross and Beckner.
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Citations
Recovering Low-Rank Matrices From Few Coefficients in Any Basis
TL;DR: It is shown that an unknown matrix of rank can be efficiently reconstructed from only randomly sampled expansion coefficients with respect to any given matrix basis, which quantifies the “degree of incoherence” between the unknown matrix and the basis.
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Recovering low-rank matrices from few coefficients in any basis
TL;DR: In this paper, it was shown that an unknown (n x n) matrix of rank r can be efficiently reconstructed from only O(n r nu log 2 n) randomly sampled expansion coefficients with respect to any given matrix basis.
656
Quantum logarithmic Sobolev inequalities and rapid mixing
TL;DR: In this article, a family of quantum logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced, and an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup is shown.
Some applications of hypercontractive inequalities in quantum information theory
TL;DR: In this article, the authors discuss how hypercontractive inequalities, in various settings, can be used to obtain (fairly) concise proofs of several results in quantum information theory: a recent lower bound of Lancien and Winter on the bias achievable by local measurements which are 4-designs, spectral concentration bounds for k-local Hamiltonians, and a recent result of Pellegrino and Seoane-Sepulveda giving general lower bounds on the classical bias obtainable in multiplayer XOR games.
167
A Survey of Quantum Property Testing
Ashley Montanaro,Ronald de Wolf +1 more
TL;DR: This survey describes recent results obtained for quantum property testing and surveys known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc.
References
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Quantum Computation and Quantum Information
Michael A. Nielsen,Isaac L. Chuang +1 more
- 01 Jan 2000
TL;DR: In this article, the quantum Fourier transform and its application in quantum information theory is discussed, and distance measures for quantum information are defined. And quantum error-correction and entropy and information are discussed.
Matrix analysis: Frontmatter
Roger A. Horn,Charles R. Johnson +1 more
- 01 Jan 1985
TL;DR: This book presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.
21.4K
Quantum computation and quantum information
TL;DR: This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing, with a focus on entanglement.
15.9K
Quantum Computation and Quantum Information
TL;DR: This paper introduces the basic concepts of quantum computation and quantum simulation and presents quantum algorithms that are known to be much faster than the available classic algorithms and provides a statistical framework for the analysis of quantum algorithms and quantum Simulation.
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Quantum detection and estimation theory
Carl W. Helstrom
- 01 Jan 1969
TL;DR: In this article, the optimum procedure for choosing between two hypotheses, and an approximate procedure valid at small signal-to-noise ratios and called threshold detection, are presented, and a quantum counterpart of the Cramer-Rao inequality of conventional statistics sets a lower bound to the mean-square errors of such estimates.
4.6K
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