Quantum approximation II. Sobolev embeddings

@article{Heinrich2003QuantumAI,
  title={Quantum approximation II. Sobolev embeddings},
  author={Stefan Heinrich},
  journal={J. Complex.},
  year={2003},
  volume={20},
  pages={27-45}
}
  • S. Heinrich
  • Published 6 May 2003
  • Computer Science, Mathematics
  • J. Complex.

Optimal query error of quantum approximation on some Sobolev classes

The 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.

Sobolev Approximation in the Quantum Computation Model

  • Ye PeixinYu Xiuhua
  • Computer Science
    2011 Fourth International Conference on Intelligent Computation Technology and Automation
  • 2011
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(Wrp([0,1]d)) to anisotropic Sobolev

QUANTUM COMPLEXITY OF SOBOLEV IMBEDDINGS

Complexity for the approximation of Sobolev imbeddings in the quantum computation model

A new and elegant reduction approach is derived that quantum algorithms bring speed-up over the classical deterministic and randomized ones and this conjecture was confirmed in the situation s = 0.

On the quantum and randomized approximation of linear functionals on function spaces

  • B. Kacewicz
  • Mathematics, Computer Science
    Quantum Inf. Process.
  • 2011
Lower bounds are provided on the power of quantum, randomized and deterministic algorithms for the exemplary problems, and some cases sharpness of the obtained results is compared.

Convergence rate of quantum algorithm for multivariate approximation

It turns out that for the Sobolev class B B(W<inf>p</inf><sup>r</Sup> ([0, 1]<sup>d</sup>)) (r ∈ ℕ<sup*d</ Sup>), when p < q, the quantum algorithms can bring speedup over classical deterministic and randomized algorithms.

The quantum query complexity of elliptic PDE

On the quantum complexity of computing the median of continuous distribution

It is shown that the ε-complexity up to a logarithmic factor is of order ε−1/(r+ρ+1).

Quantum Approximation Error on Some Sobolev Classes

  • Peixin Ye
  • Computer Science, Physics
    Third International Conference on Natural Computation (ICNC 2007)
  • 2007
The 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.

References

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Developing quantum algorithms for computing the mean of sequences that satisfy a p-summability condition and for integration of functions from Lebesgue spaces Lp(0, 1]d, and proving lower bounds showing that the proposed algorithms are, in many cases, optimal within the setting of quantum computing.

Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

The worst case, randomized, and quantum settings are considered and it is proved that strong tractability and tractability in the class $\lall$ are equivalent and this holds under the same assumption as for the class £lall in the worst case setting.

On a problem in quantum summation

The finite element method for elliptic problems

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From the Publisher: This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional

Random Approximation in Numerical Analysis

Discretization of the problem of diameters

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