Information-based complexity
The algebraic eigenvalue problem
The algebraic eigenvalue problem
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
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
Complexity and information
Quantum complexity of integration
Journal of Complexity
Quantum computation and quantum information
Quantum computation and quantum information
Numerical Mathematics and Computing
Numerical Mathematics and Computing
Quantum summation with an application to integration
Journal of Complexity
Quantum integration in Sobolev classes
Journal of Complexity
Path Integration on a Quantum Computer
Quantum Information Processing
Quantum Lower Bounds by Polynomials
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Quantum approximation I. Embeddings of finite-dimensional Lp spaces
Journal of Complexity
Quantum approximation II. Sobolev embeddings
Journal of Complexity
Randomized and quantum algorithms yield a speed-up for initial-value problems
Journal of Complexity
On the Complexity of Searching for a Maximum of a Function on a Quantum Computer
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
The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries
Quantum Information Processing
Quantum lower bounds by entropy numbers
Journal of Complexity
On the complexity of the multivariate Sturm--Liouville eigenvalue problem
Journal of Complexity
Adiabatic quantum counting by geometric phase estimation
Quantum Information Processing
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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(驴) of function evaluations or queries that are necessary to compute an 驴-approximation of the smallest eigenvalue. We prove that n(驴)=驴(驴驴1/2) in the (deterministic) worst case setting, and n(驴)=驴(驴驴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((1/2) iM), where M is an n脳 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. In particular, we show how to compute an 驴-approximation with probability (3/4) using n(驴)=驴(驴驴1/3) bit queries. For power queries, we use the phase estimation algorithm as a basic tool and present the algorithm that solves the problem using n(驴)=驴(log 驴驴1) power queries, log 2驴驴1 quantum operations, and (3/2) log 驴驴1 quantum bits. We also prove that the minimal number of qubits needed for this problem (regardless of the kind of queries used) is at least roughly (1/2) log 驴驴1. The lower bound on the number of quantum queries is proven in Bessen (in preparation). We derive a formula that relates the Sturm--Liouville eigenvalue problem to a weighted integration problem. Many computational problems may be recast as this weighted integration problem, which allows us to solve them with a polylog number of power queries. Examples include Grover's search, the approximation of the Boolean mean, NP-complete problems, and many multivariate integration problems. In this paper we only provide the relationship formula. The implications are covered in a forthcoming paper (in preparation).