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. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These algorithms yield new upper complexity bounds, which differ from known lower bounds by only an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both the randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration. 
Boleslaw Kacewicz