The Sturm-Liouville eigenvalue problem and NP-complete problems in the quantum setting with queries

Details The Sturm-Liouville eigenvalue problem and NP-complete problems in the quantum setting with queries The Sturm-Liouville eigenvalue problem and NP-complete problems in the quantum setting with queries

2005-04-26

arxiv.org/abs/quant-ph/0504194v1

Physics

Quantum Physics

18 pages

Scientific paper

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 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.

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