Physics – Quantum Physics
Scientific paper
2004-05-04
Journal of Complexity 21 (2005), 740-756
Physics
Quantum Physics
17 pages, extended version (new section added), to appear in the Journal of Complexity
Scientific paper
We deal with the problem, initiated in [8], of finding randomized and quantum complexity of initial-value problems. We showed in [8] that a speed-up in both settings over the worst-case deterministic complexity is possible. In the present paper we prove, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved. In the H\"older class of right-hand side functions with r continuous bounded partial derivatives, with r-th derivative being a H\"older function with exponent \rho, the \epsilon-complexity is shown to be O((1/\epsilon)^{1/(r+\rho+1/3)}) in the randomized setting, and O((1/\epsilon)^{1/(r+\rho+1/2)}) on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from [8]. The gap still remaining between upper and lower bounds on the complexity is further discussed for a special problem. We consider scalar autonomous problems, with the aim of computing the solution at the end point of the interval of integration. For this problem, we fill up the gap by establishing (essentially) matching upper and lower complexity bounds. We show that the complexity in this case is of order (1/\epsilon)^{1/(r+\rho+1/2)} in the randomized setting, and (1/\epsilon)^{1/(r+\rho+1)} in the quantum setting (again up to logarithmic factors).
No associations
LandOfFree
Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-346469