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Quantum Information Processing

, Volume 6, Issue 2, pp 101–120 | Cite as

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

  • A. Papageorgiou
  • H. Woźniakowski
Article

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 qubits needed to solve the problems studied in this paper is a low degree polynomial. The implementation of power queries by a polynomial number of elementary quantum gates is an open issue. If this problem is solved positively for the power queries used for the Sturm-Liouville eigenvalue problem then a quantum computer would be a very powerful computation device allowing us to solve NP-complete problems in polynomial time.

Keywords

Complexity quantum algorithms appoximation NP complete problems 

PACS

03.67.Lx 02.60.-x 

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Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of Computer ScienceColumbia UniversityColumbiaUSA
  2. 2.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland

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