  We study the efficiency of algorithms simulating a system evolving with
Hamiltonian $H=\sum_{j=1}^m H_j$. We consider high order splitting methods that
play a key role in quantum Hamiltonian simulation. We obtain upper bounds on
the number of exponentials required to approximate $e^{-iHt}$ with error $\e$.
Moreover, we derive the order of the splitting method that optimizes the cost
of the resulting algorithm. We show significant speedups relative to previously
known results.
