Quantum approximation I. Embeddings of finite-dimensional Lp spaces

  title={Quantum approximation I. Embeddings of finite-dimensional Lp spaces},
  author={Stefan Heinrich},
  journal={J. Complex.},
  • S. Heinrich
  • Published 6 May 2003
  • Mathematics, Computer Science
  • J. Complex.

Quantum approximation II. Sobolev embeddings

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

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


Numerical Analysis on a Quantum Computer

Having matching upper and lower complexity bounds for the quantum setting, this work is in a position to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.

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).

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.

On the Power of Quantum Algorithms for Vector Valued Mean Computation

  • S. Heinrich
  • Mathematics, Computer Science
    Monte Carlo Methods Appl.
  • 2004
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.

The quantum query complexity of elliptic PDE

Randomized and quantum algorithms yield a speed-up for initial-value problems

  • B. Kacewicz
  • Computer Science, Mathematics
    J. Complex.
  • 2004



Quantum approximation II. Sobolev embeddings

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

Quantum integration in Sobolev classes

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

Path Integration on a Quantum Computer

A lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved, and it is proved that path integration on a quantum computer is tractable.

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.

From Monte Carlo to quantum computation

Quantum lower bounds by polynomials

This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.

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.


  • P. Shor
  • Computer Science, Physics
  • 2000
These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms. So far, we have only discovered a few techniques which

Sharp error bounds on quantum Boolean summation in various settings