Quantum approximation I. Embeddings of finite-dimensional Lp spaces

@article{Heinrich2004QuantumAI,
  title={Quantum approximation I. Embeddings of finite-dimensional Lp spaces},
  author={Stefan Heinrich},
  journal={J. Complex.},
  year={2004},
  volume={20},
  pages={5-26}
}
  • S. Heinrich
  • Published 6 May 2003
  • Mathematics, Computer Science
  • J. Complex.

On the quantum complexity of computing the median of continuous distribution

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

On the Power of Quantum Algorithms for Vector Valued Mean Computation

  • S. Heinrich
  • Mathematics, Computer Science
    Monte Carlo Methods Appl.
  • 2004
TLDR
It turns out that in contrast to the known superiority of quantum algorithms in the scalar case, in high dimensional L M P spaces classical randomized algorithms are essentially as powerful as quantum algorithms.

On the quantum complexity of approximating median of continuous distribution

TLDR
The approximating of the median of absolutely continuous distribution given by a probability density function $f$ is considered and it is shown that the $\epsilon$-complexity up to logarithmic factor is of order $Epsilon^{-1/((r+\rho+1))}$.

Randomized and quantum complexity of nonlinear two-point BVPs

  • Maciej Gocwin
  • Computer Science, Mathematics
    Appl. Math. Comput.
  • 2014

QUANTUM COMPLEXITY OF SOBOLEV IMBEDDINGS

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

On the complexity of a two-point boundary value problem in different settings

TLDR
This work studies the complexity of a two-point boundary value problem, and shows that the use of linear information gives us a speed-up of at least one order of magnitude compared with the standard information.

Convergence rate of quantum algorithm for multivariate approximation

TLDR
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.

References

SHOWING 1-10 OF 24 REFERENCES

Quantum Summation with an Application to Integration

TLDR
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.

On a problem in quantum summation

Information-Based Complexity

This book provides a comprehensive treatment of information-based complexity, the branch of computational complexity that deals with the intrinsic difficulty of the approximate solution of problems

Eigenvalues and S-Numbers

Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms

We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Holder or Sobolev spaces. First we discuss optimal deterministic and

Deterministic and Stochastic Error Bounds in Numerical Analysis

An Introduction to Quantum Computing Algorithms

TLDR
This monograph is a good self-contained introductory resource for newcomers to the field of quantum computing algorithms, as well as a useful self-study guide for the more specialized scientist, mathematician, graduate student, or engineer.

ON CONVERGENCE OF STOCHASTIC PROCESSES

It is clear that for given I,un } and t, the better theorem of this kind would be the one in which (2) is proved for the larger class of functions f. In this paper we shall show that certain known

Quantum integration in Sobolev classes

Random Approximation in Numerical Analysis