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

  • Authors:

  • A. Papageorgiou;H. Woźniakowski

  • Affiliations:

  • Department of Computer Science, Columbia University, Columbia, USA;Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland 00-927

  • Venue:
  • Quantum Information Processing
  • Year:
  • 2007

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Abstract

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.