We provide a recursive method for constructing product formula approximations

to exponentials of commutators, giving the first approximations that are

accurate to arbitrarily high order. Using these formulas, we show how to

approximate unitary exponentials of (possibly nested) commutators using

exponentials of the elementary operators, and we upper bound the number of

elementary exponentials needed to implement the desired operation within a

given error tolerance. By presenting an algorithm for quantum search using

evolution according to a commutator, we show that the scaling of the number of

exponentials in our product formulas with the evolution time is nearly optimal.

Finally, we discuss applications of our product formulas to quantum control and

to implementing anticommutators, providing new methods for simulating many-body

interaction Hamiltonians.