On the complexity of the multivariate Sturm-Liouville eigenvalue problem

We study the complexity of approximating the smallest eigenvalue of −∆ + q with Dirichlet boundary conditions on the d-dimensional unit cube. Here ∆ is the Laplacian, and the function q is nonnegative and has continuous first order partial derivatives. We consider deterministic and randomized classical algorithms, as well as quantum algorithms using quantum queries of two types: bit queries and power queries. We seek algorithms that solve the problem with accuracy ε. We exhibit lower and upper bounds for the problem complexity. The upper bounds follow from the cost of particular algorithms. The classical deterministic algorithm is optimal. Optimality is understood modulo constant factors that depend on d. The randomized algorithm uses an optimal number of function evaluations of q when d ≤ 2. The classical algorithms have cost exponential in d since they need to solve an eigenvalue problem involving a matrix with size exponential in d. We show that the cost of quantum algori...

• Type: Journal
• Year: 2007
• Where: JC
• Authors: A. Papageorgiou
• Keywords: Algorithm, Classical Algorithms, JC 2007, Upper Bounds