2005 Reports

Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem

Papageorgiou, Anargyros; Wozniakowski, Henryk

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 C 2 ([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−1/2) in the (deterministic) worst case setting, and n(ε) = Θ(ε−2/5) in the randomized setting. The quantum setting offers a polynomial speedup with bit queries and an exponential speedup with power queries. Bit queries are similar to the oracle calls used in Grover’s algorithm appropriately 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(1iM), where M is an 2 n × n matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is pollywog in n is an open issue. 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 C2([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−1/2) in the (deterministic) worst case setting, and n(ε) = Θ(ε−2/5) in the randomized setting. The quantum setting offers a polynomial speedup with bit queries and an exponential speedup with power queries. Bit queries are similar to the oracle calls used in Grover’s algorithm appropriately 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 al- low us to use powers of the unitary matrix exp(1 iM), where M is an n × n 2 matrix obtained from the standard discretization of the Sturm-Liouville differ- ential operator. The quantum implementation of power queries by a number of elementary quantum gates that is pollywog in n is an open issue.