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 summation with an application to integration
Journal of Complexity
Path Integration on a Quantum Computer
Quantum Information Processing
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
Adaptive Itô-Taylor algorithm can optimally approximate the Itô integrals of singular functions
Journal of Computational and Applied Mathematics
Numerical analysis on a quantum computer
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
Hi-index | 0.00 |
We study the problem, initiated by Kacewicz [Randomized and quantum algorithms yield a speed-up for initial-value problems, J. Complexity 20 (2004) 821-834; see also http://arXiv.org/abs/quant-ph/0311148], of finding randomized and quantum complexity of initial-value problems. We showed in Kacewicz (2004) 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 Holder class of right-hand side functions with r continuous bounded partial derivatives, with rth derivative being a Holder function with exponent @r, the @?-complexity is shown to be O(1/@?)^1^/^(^r^+^@r^+^1^/^3^) in the randomized setting, and O(1/@?)^1^/^(^r^+^@r^+^1^/^2^) on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from Kacewicz (2004). 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 @Q(1/@?)^1^/^(^r^+^@r^+^1^/^2^) in the randomized setting, and @Q(1/@?)^1^/^(^r^+^@r^+^1^) in the quantum setting (again up to logarithmic factors). Hence, this problem is essentially as hard as the integration problem.