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International Conference on Intelligent Computation Technology and Automation

Sobolev Approximation in the Quantum Computation Model

Peixin Ye, Yuan Xiuhua

Published Mar 1, 2011• 0 citations
Publisher

Abstract

Using a new and elegant reduction approach we derive a lower bound of quantum complexity for the approximation of imbeddings from anisotropic Sobolev classes B(Wpr([0,1]d)) to anisotropic Sobolev space (Wps([0,1]d) for all 1 ≥ p, q ≤ ∞ . When p≥q we show this bound is optimal by deriving the matching upper bound. In this case the quantum algorithms are not significantly better than the classical deterministic or randomized algorithms.

DOI
10.1109/icicta.2011.69

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Using a new and elegant reduction approach we derive a lower bound of quantum complexity for the approximation of imbeddings from anisotropic Sobolev classes B(Wpr([0,1]d)) to anisotropic Sobolev space (Wps([0,1]d) for all 1 ≥ p, q ≤ ∞ . When p≥q we show this bound is optimal by deriving the matching upper bound. In this case the quantum algorithms are not significantly better than the classical deterministic or randomized algorithms.
Using a new and elegant reduction approach we derive a lower bound of quantum complexity for the approximation of imbeddings from anisotropic Sobolev classes B(W p r([0, 1]d)) to anisotropic Sobolev space W p s([0, 1]d) for all 1 ≤ p, q ≤ ∞. When p ≥ q this bound is optimal. In this case the quantum algorithms are not significantly better than the classical deterministic or randomized algorithms. When p ≥ q we conjecture that quantum algorithms bring speed-up over the classical deterministic and randomized ones. This conjecture was confirmed in the situation s = 0.
We study the approximation of functions from anisotropic and generalized Sobolev classes under Lq([0, 1]d) norm in the quantum model of computation. Based on the quantum algorithm for approximation of finite imbedding from LNPto LiNQ, we develop a quantum algorithm for approximating the imbedding from anisotropic Sobolev classes B(Wpr(|0, 1|d)) to Lq(|0, 1|d) space for all 1 les q,p les infin and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup of roughly squaring the rate of classical deterministic and randomized settings.
We study the approximation of the integration of multivariate functions in the quantum model of computation. Using a new reduction approach we obtain a lower bound of the n-th minimal query error on anisotropic Sobolev classes B(Wpr([0, 1]d)), (r isin R+d). Then combining our previous results we determine the optimal bound of nth minimal query error for anisotropic Holder-Nikolskii class B(Hinfinr([0, 1]d)), (r isin R+d) and Sobolev class B(Wpr([0, 1]d)) (r isin Nd). The results show the quantum algorithms give speed up over classical algorithms.
We obtain the optimal order of high-dimensional integration complexity in the quantum computation model in anisotropic Sobolev classes W∞r([0,1]d)W_{\infty}^{\bf r}([0,1]^d)W∞r​([0,1]d) and Hölder Nikolskii classes H∞r([0,1]d)H_{\infty}^{\bf r}([0,1]^d)H∞r​([0,1]d). It is proved that for these classes of functions there is a speed-up of quantum algorithms over deterministic classical algorithms due to factor n−1n^{-1}n−1 and over randomized classical methods due to factor n−1/2n^{-1/2}n−1/2. Moreover, we give an estimation for optimal query complexity in the class H∞Λ(D)H_{\infty}^{\Lambda}(D)H∞Λ​(D) whose smoothness index is the boundary of some complete set in Z+d\mathbb{Z}_+^dZ+d​.
We estimate the quantum query error of approximation to functions from the anisotropic Sobolev class B(Wpr([0,1]d)){\cal B}(W_p^{\bf r}([0,1]^d))B(Wpr​([0,1]d)) and the Hölder-Nikolskii class B(Hpr([0,1]d)){\cal B}(H_p^{\bf r}([0,1]^d))B(Hpr​([0,1]d)) in the Lq([0,1]d)L_q([0,1]^d)Lq​([0,1]d) norm for all 1≤p,q≤∞1 \leq p,q \leq \infty1≤p,q≤∞. It turns out that for the class B(Wpr([0,1]d)){\cal B}(W_p^{\bf r}([0,1]^d))B(Wpr​([0,1]d)) (r∈Nd)({\bf r}\in {\mathbb{N}^d})(r∈Nd), when p<qp \lt qp<q, the quantum algorithms can essentially improve the rate of convergence of classical deterministic and randomized algorithms; while for the class B(Hpr([0,1]d)){\cal B}(H_p^{\bf r}([0,1]^d))B(Hpr​([0,1]d)) and B(Wpr([0,1]d)){\cal B}(W_p^{\bf r}([0,1]^d))B(Wpr​([0,1]d)) (r∈R+d)({\bf r}\in {\mathbb{R}_+^d})(r∈R+d​), when p≥qp \geq qp≥q, the optimal convergence rate is the same for all three settings.