Almost optimal solution of initial-value problems by randomized and quantum algorithms
Boleslaw Z. Kacewicz · 2006
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.