  We provide a new approach for compiling quantum simulation circuits that
appear in Trotter, qDRIFT and multi-product formulas to Clifford and
non-Clifford operations that can reduce the number of non-Clifford operations
by a factor of up to $4$. In fact, the total number of gates reduce in many
cases. We show that it is possible to implement an exponentiated sum of
commuting Paulis with at most $m$ (controlled)-rotation gates, where $m$ is the
number of distinct non-zero eigenvalues (ignoring sign). Thus we can collect
mutually commuting Hamiltonian terms into groups that satisfy one of several
symmetries identified in this work which allow an inexpensive simulation of the
entire group of terms. We further show that the cost can in some cases be
reduced by partially allocating Hamiltonian terms to several groups and provide
a polynomial time classical algorithm that can greedily allocate the terms to
appropriate groupings. We further specifically discuss these optimizations for
the case of fermionic dynamics and provide extensive numerical simulations for
qDRIFT of our grouping strategy to 6 and 4-qubit Heisenberg models, $LiH$,
$H_2$ and observe a factor of 1.8-3.2 reduction in the number of non-Clifford
gates. This suggests Trotter-based simulation of chemistry in second
quantization may be even more practical than previously believed.
