# Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem

@article{Papageorgiou2005ClassicalAQ, title={Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem}, author={Anargyros Papageorgiou and Henryk Wozniakowski}, journal={Quantum Information Processing}, year={2005}, volume={4}, pages={87-127} }

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(ɛ) of function evaluations or queries that are necessary to compute an ɛ-approximation of the smallest eigenvalue. We prove that n(ɛ)=Θ(ɛ−1/2) in the (deterministic) worst case setting, and n(ɛ)=Θ(ɛ−2/5) in the randomized setting…

## 21 Citations

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It is shown how a number of NP-complete as well as NP-hard problems can be reduced to the Sturm-Liouville eigenvalue problem in the quantum setting with queries, and that the number of power queries as well thenumber of qubits needed to solve the problems studied in this paper is a low degree polynomial.

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It is proved that for path integration the authors have an exponential improvement for the qubit complexity over the quantum setting with deterministic queries, which limits the power of quantum computation for continuous problems.

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