Quantum Information Processing, Vol. 6, No. 2, April 2007 (© 2006)
The Sturm-Liouville Eigenvalue Problem and
NP-Complete Problems in the Quantum Setting
and H. Wo
Received June 11, 2006; Accepted July 18, 2006; Published online November 28, 2006
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 com-
plexity 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 prob-
lem is solved positively for the power queries used for the Sturm-Liouville eigen-
value problem then a quantum computer would be a very powerful computation
device allowing us to solve NP-complete problems in polynomial time.
KEY WORDS: Complexity; quantum algorithms; appoximation; NP complete
PACS: 03.67.Lx; 02.60.-x.
An important question in quantum computing is whether NP-complete
problems can be solved in polynomial time (see Refs. 3, 4 and papers cited
there). We address this question by studying the quantum setting with
queries. This paper is based on results in our previous paper,
we studied the classical and quantum complexity of the Sturm–Liouville
Department of Computer Science, Columbia University, Columbia, NY, USA.
Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw 00-927,
To whom correspondence should be addressed. E-mail: email@example.com
1570-0755/07/0400-0101/0 © 2006 Springer Science+Business Media, Inc.