Quantum approximation I. Embeddings of finite-dimensional Lp spaces
Journal of Complexity
Sharp error bounds on quantum Boolean summation in various settings
Journal of Complexity
Classical and Quantum Complexity of the Sturm--Liouville Eigenvalue Problem
Quantum Information Processing
On the Complexity of Searching for a Maximum of a Function on a Quantum Computer
Quantum Information Processing
Improved bounds on the randomized and quantum complexity of initial-value problems
Journal of Complexity
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 quantum query complexity of elliptic PDE
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
Improved bounds on the randomized and quantum complexity of initial-value problems
Journal of Complexity
Adiabatic quantum counting by geometric phase estimation
Quantum Information Processing
Numerical analysis on a quantum computer
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
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We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j-k with k1. For the Wiener measure occurring in many applications we have k=2. We want to compute an ϵ-approximation to path integrals whose integrands are at least Lipschitz. We prove:• Path integration on a quantum computer is tractable.• Path integration on a quantum computer can be solved roughly ϵ-1 times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance.• The number of quantum queries needed to solve path integration is roughly the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most 4.46 ϵ-1. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved.• The number of qubits is polynomial in ϵ-1. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands.PACS: 03.67.Lx; 31.15Kb; 31.15.-p; 02.70.-c