QUANTUM APPROXIMATION ON ANISOTROPIC SOBOLEV AND HÖLDER-NIKOLSKII CLASSES

We estimate the quantum query error of approximation to functions from the anisotropic Sobolev class ℬ ( W p r ( [ 0 ,   1 ] d ) ) and the Hölder-Nikolskii class ℬ ( H p r ( [ 0 ,   1 ] d ) ) in theLq ([0, 1] d ) norm for all 1 ≤p, q≤ ∞. It turns out that for the class ℬ ( W p r ( [ 0 ,   1 ] d ) )...

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Bibliographic Details
Published inTaiwanese journal of mathematics Vol. 16; no. 1; pp. 71 - 88
Main Author Ye, Peixin
Format Journal Article
LanguageEnglish
Published Mathematical Society of the Republic of China 01.02.2012
Subjects
Approximation
Banach space
Coordinate systems
Cubes
Determinism
Embeddings
Mathematical functions
Quantum computers
Sobolev spaces
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Summary:We estimate the quantum query error of approximation to functions from the anisotropic Sobolev class ℬ ( W p r ( [ 0 ,   1 ] d ) ) and the Hölder-Nikolskii class ℬ ( H p r ( [ 0 ,   1 ] d ) ) in theLq ([0, 1] d ) norm for all 1 ≤p, q≤ ∞. It turns out that for the class ℬ ( W p r ( [ 0 ,   1 ] d ) ) (r∈ ℕ d ), whenp<q, the quantum algorithms can essentially improve the rate of convergence of classical deterministic and randomized algorithms; while for the class ℬ ( H p r ( [ 0 ,   1 ] d ) ) and ℬ ( W p r ( [ 0 ,   1 ] d ) )   ( r ∈ ℝ + d ) , whenp≥q, the optimal convergence rate is the same for all three settings. 2010Mathematics Subject Classification: 41A63, 65D15, 65Y20. Key words and phrases: Quantum approximation, Anisotropic classes, Minimal query error.
ISSN:1027-5487
2224-6851
DOI:10.11650/twjm/1500406528