Quantum algorithms and complexity for certain continuous and related discrete problems

  title={Quantum algorithms and complexity for certain continuous and related discrete problems},
  author={Henryk Wozniakowski and Marek Kwas},
The thesis contains an analysis of two computational problems. The first problem is discrete quantum Boolean summation. This problem is a building block of quantum algorithms for many continuous problems, such as integration, approximation, differential equations and path integration. The second problem is continuous multivariate Feynman-Kac path integration, which is a special case of path integration. The quantum Boolean summation problem can be solved by the quantum summation (QS) algorithm… 

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Quantum Algorithms and Complexity for Continuous Problems

High-dimensional integration, path Integration, Feynman path integration, the smallest eigenvalue of a differential equation, approximation, partial differential equations, ordinary differential equations and gradient estimation are reported on.

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A brief introduction to quantum algorithms is given and a wide range of applications including high dimensional integration, path integration, Hamiltonian simulation, and ground state energy estimation are reported on.

D ec 2 00 7 Quantum Algorithms and Complexity for Continuous Problems ∗

Glossary I. Abstract I



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.

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.

A new algorithm and worst case complexity for Feynman-Kac path integration

A new algorithm is presented and an explicit bound on its cost to compute an e-approximation to the Feynman–Kac path integral is established, which is equal to the cost of the new algorithm and is given in terms of the complexity of a certain function approximation problem.

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

Complexity of multivariate Feynman-Kac path integration in randomized and quantum settings

  • M. Kwas
  • Computer Science, Mathematics
  • 2004
It is shown that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the number of function evaluations and/or quantum queries required to compute an e-approximation has a bound independent of d and depending polynomially on 1/e.

On the power of quantum computation

  • Daniel R. Simon
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
This work presents here a problem of distinguishing between two fairly natural classes of function, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class.