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

@article{Kacewicz2004RandomizedAQ,
  title={Randomized and quantum algorithms yield a speed-up for initial-value problems},
  author={Boleslaw Z. Kacewicz},
  journal={J. Complex.},
  year={2004},
  volume={20},
  pages={821-834}
}
  • B. Kacewicz
  • Published 21 November 2003
  • Computer Science, Mathematics
  • J. Complex.

Improved Upper Bounds on the Randomized and Quantum Complexity of Initial-Value Problems 1

This paper gives up the deterministic optimality of the basic algorithm, defining a new integral algorithm that is better suited for randomization and implementation of a quantum computer, and applies the optimal algorithms for summation of real numbers.

Almost optimal solution of initial-value problems by randomized and quantum algorithms

  • B. Kacewicz
  • Computer Science, Mathematics
    J. Complex.
  • 2006

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.

Improved bounds on the randomized and quantum complexity of initial-value problems

  • B. Kacewicz
  • Computer Science, Mathematics
    J. Complex.
  • 2005

Randomized and quantum algorithms for solving initial-value problems in ordinary differential equations of order k

This paper considers two models of computation, the randomized model and the quantum model, and constructs almost optimal algorithms adjusted to scalar equations of higher order, without passing to systems of first order equations.

Quantum algorithms and complexity for certain continuous and related discrete problems

The thesis shows that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the minimal number of function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of variables.

Ju n 20 05 Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems 1

It is proved, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved, and in the Hölder class of right-hand side functions with r continuous bounded partial derivatives, the ε-complexity is shown to be O ( (1/ε) ) in the randomized setting, and O (1 /ε) on a quantum computer (up to logarithmic factors).

On the Complexity of Searching for a Maximum of a Function on a Quantum Computer

  • Maciej Gocwin
  • Computer Science, Mathematics
    Quantum Inf. Process.
  • 2006
It is shown that quantum computation yields a quadratic speed-up over deterministic and randomized algorithms.

Randomized and Quantum Solution of Initial-Value Problems for Ordinary Differential Equations of Order k

It is shown in this paper that a speed-up dependent on k is not possible in the randomized and quantum settings, and lower complexity bounds are established, showing that the randomizedand quantum complexities remain at the some level as for systems of the first order, no matter how large k is.

Randomized and quantum complexity of nonlinear two-point BVPs

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

References

SHOWING 1-10 OF 16 REFERENCES

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.

A fast quantum mechanical algorithm for database search

In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .

Algorithms for quantum computation: discrete logarithms and factoring

  • P. Shor
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.

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.

Quantum Complexity of Integration

  • E. Novak
  • Computer Science, Mathematics
    J. Complex.
  • 2001
It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the

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

The quantum query complexity of approximating the median and related statistics

The main ingredient in the proof is a polynomial degree lower bound for real multilinear polynomials that ``approximate'' symmetric partial boolean functions, which immediately yields lower bounds for the problems of approximating the kth-smallest element, approximates the mean of a sequence of numbers, and that of approximately counting the number of ones of a boolean function.

On sequential and parallel solution of initial value problems

  • B. Kacewicz
  • Computer Science, Mathematics
    J. Complex.
  • 1990