Information-based complexity
The algebraic eigenvalue problem
The algebraic eigenvalue problem
Adapted wavelet analysis from theory to software
Adapted wavelet analysis from theory to software
SIAM Journal on Computing
Applied numerical linear algebra
Applied numerical linear algebra
The quantum query complexity of approximating the median and related statistics
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Quantum complexity of integration
Journal of Complexity
Quantum computation and quantum information
Quantum computation and quantum information
Quantum summation with an application to integration
Journal of Complexity
Quantum integration in Sobolev classes
Journal of Complexity
Quantum Lower Bounds by Polynomials
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Classical and Quantum Complexity of the Sturm--Liouville Eigenvalue Problem
Quantum Information Processing
The Quantum Setting with Randomized Queries for Continuous Problems
Quantum Information Processing
A lower bound for the Sturm-Liouville eigenvalue problem on a quantum computer
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
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We study the complexity of approximating the smallest eigenvalue of -@D+q with Dirichlet boundary conditions on the d-dimensional unit cube. Here @D is the Laplacian, and the function q is non-negative 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 @?. 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 algorithms is not exponential in d, regardless of the type of queries they use. Power queries enjoy a clear advantage over bit queries and lead to an optimal complexity algorithm.