  We present a product formula to approximate the exponential of a
skew-Hermitian operator that is a sum of generators of a Lie algebra. The
number of terms in the product depends on the structure factors. When the
generators have large norm with respect to the dimension of the Lie algebra, or
when the norm of the effective operator resulting from nested commutators is
less than the product of the norms, the number of terms in the product is
significantly less than that obtained from well-known results. We apply our
results to construct product formulas useful for the quantum simulation of some
continuous-variable and bosonic physical systems, including systems whose
potential is not quadratic. For many of these systems, we show that the number
of terms in the product can be sublinear or subpolynomial in the dimension of
the relevant local Hilbert spaces, where such a dimension is usually determined
by an energy scale of the problem. Our results emphasize the power of quantum
computers for the simulation of various quantum systems.
