The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries

@article{Papageorgiou2007TheSE,
  title={The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries},
  author={Anargyros Papageorgiou and Henryk Wozniakowski},
  journal={Quantum Information Processing},
  year={2007},
  volume={6},
  pages={101-120}
}
We show how a number of NP-complete as well as NP-hard problems can be reduced to the Sturm-Liouville eigenvalue problem in the quantum setting with queries. We consider power queries which are derived from the propagator of a system evolving with a Hamiltonian obtained from the discretization of the Sturm-Liouville operator. We use results of our earlier paper concering the complexity of the Sturm-Liouville eigenvalue problem. We show that the number of power queries as well the number of… 

Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem

A formula is derived that relates the Sturm–Liouville eigenvalue problem to a weighted integration problem, which allows us to solve them with a polylog number of power queries and the lower bound on the number of quantum queries is proven.

Quantum Algorithms and Complexity for Numerical Problems

This thesis designs an adiabatic quantum algorithm for the counting problem, and derives the optimal order of convergence, given e and the cost of the resulting algorithm, which is close to the best lower bound on query complexity known for the classical PAC learning model.

Continuous Quantum Computation

New algorithms and quantum speedups were obtained for a number of important problems such as path integration, eigenvalues of Hermitian operators, Feynman-Kac path Integration, high-dimensional approximation, and the Sturm-Liouville eigenvalue problem.

A brief history of information-based complexity

This paper discusses precursors to Information-based complexity and reports on the beginning of optimal iteration theory in the early 60s which was published in Traub's 1964 monograph.

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.

Tractability of Multivariate Problems

The main purpose of this book is to study weighted spaces and to obtain conditions on the weights that are necessary and sufficient to achieve various notions of tractability, depending on how to measure the lack of exponential dependence.

A summary on two new algorithms for Grover's unsorted database search problem

This summary discusses two new algorithms for Grover’s unsorted database search problem that claimed to have reached exponential speedup over Grover's original algorithm in the quantum setting with “power queries” and to use “dubit queries" on a duality computer.

Henryk Woźniakowski and the Complexity of Continuous Problems

Henryk Woźniakowski was for many years one of the leaders of the Solidarnośc movement at the University of Warsaw and was elected chairman of the Department of Mathematics, Computer Science and Mechanics in 1981.

References

SHOWING 1-10 OF 27 REFERENCES

Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem

A formula is derived that relates the Sturm–Liouville eigenvalue problem to a weighted integration problem, which allows us to solve them with a polylog number of power queries and the lower bound on the number of quantum queries is proven.

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.

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.

Quantum complexity theory

This paper gives the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis, and proves that bits of precision suffice to support a step computation.

Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation.

The method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schrödinger equation, on a discrete grid, and efficiently construct an approximation of the same eigenvector on a fine grid.

Strengths and Weaknesses of Quantum Computing

It is proved that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$.

Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors

A new polynomial time quantum algorithm is described that uses the quantum fast Fourier transform to find eigenvalues and eigenvectors of a local Hamiltonian and that can be applied in cases for which all known classical algorithms require exponential time.

Quantum approximation I. Embeddings of finite-dimensional Lp spaces

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

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