Quantum integration in Sobolev classes

@article{Heinrich2003QuantumII,
  title={Quantum integration in Sobolev classes},
  author={Stefan Heinrich},
  journal={J. Complex.},
  year={2003},
  volume={19},
  pages={19-42}
}
  • S. Heinrich
  • Published 23 December 2001
  • Computer Science
  • J. Complex.

Quantum integration error for some sobolev classes

It is proved that for these classes of functions the optimal convergence rates of quantum algorithms are essential smaller than those of classical deterministic and randomized algorithms.

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 Integration Error on Some Classes of Multivariate Functions

The results show that for these two type of classes the quantum algorithms give significant speed up over classical deterministic and randomized algorithms.

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.

Lower bound for quantum integration error on anisotropic Sobolev classes

The results show that for anisotropic Hölder-Nikolskii and Sobolev classes the quantum algorithms give significant speed up over classical deterministic and randomized algorithms.

Quantum complexity of parametric integration

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

The integration of the Hölder-Nikolskii classes in the randomized and quantum computation model is investigated and it is seen that quantum computing can reach an exponential speed up over deterministic classical computation and a quadratic speedup over randomized classical computation.

Quantum approximation I. Embeddings of finite-dimensional Lp spaces

  • S. Heinrich
  • Mathematics, Computer Science
    J. Complex.
  • 2004

Quantum approximation II. Sobolev embeddings

  • S. Heinrich
  • Computer Science, Mathematics
    J. Complex.
  • 2004

Quantum Summation with an Application to Integration

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.

References

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

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