Almost optimal solution of initial-value problems by randomized and quantum algorithms

We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. We define for this purpose a sequence of new algorithms, whose error/cost properties improve from step to step. This leads to new upper complexity bounds, which differ from known lower bounds only by an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration. 1 This research was partly supported by AGH grant No. 10.420.03 Department of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, paw. A3/A4, III p., pok. 301, 30-059 Cracow, Poland [email protected], tel. +48(12)617 3996, fax +48(12)617 3165