Report on manuscript AC7208 ``Higher order methods...''

This paper addresses a practically important problem in dealing with
quantum computation and quantum simulation. Though rather technical,
the results should be of immediate interest to people working in the
field. Previous results are available in the literature which relate
to this approximation method. However, the authors are able to make
some decisive steps forward. Within clearly stated criteria they are
able to expose efficient methods which should have considerable impact
on numerical simulations as well as on investigations on computational
complexity. The comparison with symplectic methods known from
classical Hamilton systems is illuminating. Unfortunately, there are
only few numerical examples given, but they are quite convincing and
show the ideas at work. Introduction and the discussions are very well
written and should attract attention also from a broader readership;
in fact, I enjoyed reading the paper. I recommend publication.