Quantum Information Processing, Vol. 4, No. 2, June 2005 (© 2005)
Classical and Quantum Complexity of the
Sturm–Liouville Eigenvalue Problem
and H. Wo
Received November 18, 2004; accepted February 9, 2005
We study the approximation of the smallest eigenvalue of a Sturm–Liouville
problem in the classical and quantum settings. We consider a univariate Sturm–
Liouville eigenvalue problem with a nonnegative function q from the class
([0,1]) and study the minimal number n(ε) of function evaluations or queries
that are necessary to compute an ε-approximation of the smallest eigenvalue.
We prove that n(ε)= Θ(ε
) in the (deterministic) worst case setting, and
) in the randomized setting. The quantum setting offers a poly-
nomial speedup with bit queries and an exponential speedup with power queries.
Bit queries are similar to the oracle calls used in Grover’s algorithm appropri-
ately extended to real valued functions. Power queries are used for a number
of problems including phase estimation. They are obtained by considering the
propagator of the discretized system at a number of different time moments.
They allow us to use powers of the unitary matrix exp((1/2)iM), where M is
an n× n matrix obtained from the standard discretization of the Sturm–Liou-
ville differential operator. The quantum implementation of power queries by a
number of elementary quantum gates that is polylog in n is an open issue. In
particular, we show how to compute an ε-approximation with probability (3/4)
using n(ε)= Θ(ε
) bit queries. For power queries, we use the phase estima-
tion algorithm as a basic tool and present the algorithm that solves the prob-
lem using n(ε)= Θ(log ε
) power queries, log
quantum operations, and
(3/2) log ε
quantum bits. We also prove that the minimal number of qubits
needed for this problem (regardless of the kind of queries used) is at least
roughly (1/2)log ε
. The lower bound on the number of quantum queries is
proven in Bessen (in preparation). We derive a formula that relates the Sturm–
Liouville eigenvalue problem to a weighted integration problem. Many computa-
tional problems may be recast as this weighted integration problem, which allows
us to solve them with a polylog number of power queries. Examples include
Grover’s search, the approximation of the Boolean mean, NP-complete problems,
and many multivariate integration problems. In this paper we only provide the
relationship formula. The implications are covered in a forthcoming paper (in
Department of Computer Science, Columbia University, New York, USA.
Institute of Applied Mathematics and Mechanics, University of Warsaw, Poland.
1570-0755/05/0600-0087/0 © 2005 Springer Science+Business Media, Inc.