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A lower bound for the Sturm–Liouville eigenvalue problem on a quantum computer
- Source :
-
Journal of Complexity . Oct2006, Vol. 22 Issue 5, p660-675. 16p. - Publication Year :
- 2006
-
Abstract
- Abstract: We study the complexity of approximating the smallest eigenvalue of a univariate Sturm–Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schrödinger equation with a given potential for the ground state energy. The Sturm–Liouville problem depends on a function q, which, in the case of the Schrödinger equation, can be identified with the potential function . Recently Papageorgiou and Woźniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query (“power queries”). They showed that the Sturm–Liouville equation can be solved with power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in . This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer. In this paper we show that the number of queries in Papageorgiou''s and Woźniakowski''s algorithm is asymptotically optimal. In particular we prove a matching lower bound of power queries, therefore showing that power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input. [Copyright &y& Elsevier]
Details
- Language :
- English
- ISSN :
- 0885064X
- Volume :
- 22
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- Journal of Complexity
- Publication Type :
- Academic Journal
- Accession number :
- 22592784
- Full Text :
- https://doi.org/10.1016/j.jco.2006.03.006