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Advanced materials research

Information-Based Complexity of Integration in the Randomized and Quantum Computation Model

Wen Gao, Duan Li, Wei Zhou, Peixin Ye

Published Nov 1, 2011• 0 citations
Publisher

Abstract

In this paper, we investigate the integration of the Hölder-Nikolskii classes in the randomized and quantum computation model. We develop randomized and quantum algorithms for integration of functions from this class and analyze their convergence rates. Comparing our result with the convergence rates in the deterministic setting, we see that quantum computing can reach an exponential speedup over deterministic classical computation and a quadratic speedup over randomized classical computation.

DOI
10.4028/www.scientific.net/amr.403-408.367

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In this paper, we investigate the integration of the Hölder-Nikolskii classes in the randomized and quantum computation model. We develop randomized and quantum algorithms for integration of functions from this class and analyze their convergence rates. Comparing our result with the convergence rates in the deterministic setting, we see that quantum computing can reach an exponential speedup over deterministic classical computation and a quadratic speedup over randomized classical computation.
In this paper, we study the numerical integration problem of functions from the generalized Hölder class [Formula: see text], which is determined by a generalized modulus of smoothness [Formula: see text], in the quantum model of computation. We obtain the exact order of complexity of this problem. The results show that quantum algorithms have faster convergence rates than those of deterministic and randomized ones. Quantitatively, the improvement amounts to the factors [Formula: see text] for the deterministic case and [Formula: see text] for the randomized case, where [Formula: see text] is the number of function evaluations involved in the computational process. Furthermore, we apply our general results to the case [Formula: see text] with [Formula: see text] and [Formula: see text] and obtain the corresponding complexity.
We obtain the optimal order of high-dimensional integration complexity in the quantumcomputation model in anisotropic Sobolev classes W∞^r ([0, 1]^d) and Hǒlder Nikolskii classes 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^-1 and over randomized classical methods due to factor n^-1/2. Moreover, we give an estimation for optimal query complexity in the class H∞^∧ (D) whose smoothness index is the boundary of some complete set in Z+^d.
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 give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bounds for the quantum setting, we are in a position to compare them with those for the classical deterministic and randomized setting, previously obtained in information-based complexity theory. This enables us to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.
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.