Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem
| dc.creator | Papageorgiou, A. | |
| dc.creator | Wozniakowski, H. | |
| dc.date | 2005-02-09 | |
| dc.date.accessioned | 2026-07-07T06:12:08Z | |
| dc.date.available | 2026-07-07T06:12:08Z | |
| dc.description | 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 $n(\e)$ of function evaluations or queries that are necessary to compute an $\e$-approximation of the smallest eigenvalue. We prove that $n(\e)=Θ(\e^{-1/2})$ in the (deterministic) worst case setting, and $n(\e)=Θ(\e^{-2/5})$ in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it 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 $\exp(\tfrac12 {\rm i}M)$, where $M$ is an $n\times n$ 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 polylog in $n$ is an open issue. | |
| dc.description | 33 pages | |
| dc.identifier | https://arxiv.org/abs/quant-ph/0502054 | |
| dc.identifier | http://arxiv.org/abs/quant-ph/0502054 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/92699 | |
| dc.subject | Quantum Physics | |
| dc.title | Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem | |
| dc.type | text |