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.
