Abstract
We study the approximation of the imbedding of functions from anisotropic and generalized Sobolev classes into L q ([0, 1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from L N p to L N q , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(W r p ([0, 1]d)) to L q ([0, 1]d) space for all 1 ⩽ q,p ⩽ ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 10501026, 60675010, 10626029 and 60572113) and the China Postdoctoral Science Foundation (Grant No. 20070420708)
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Song, Z., Ye, P. Optimal query error of quantum approximation on some Sobolev classes. Sci. China Ser. A-Math. 51, 1664–1678 (2008). https://doi.org/10.1007/s11425-008-0077-0
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DOI: https://doi.org/10.1007/s11425-008-0077-0