  Trotter-Suzuki decompositions are frequently used in the quantum simulation
of quantum chemistry. They transform the evolution operator into a form
implementable on a quantum device, while incurring an error---the Trotter
error. The Trotter error can be made arbitrarily small by increasing the
Trotter number. However, this increases the length of the quantum circuits
required, which may be impractical. It is therefore desirable to find methods
of reducing the Trotter error through alternate means. The Trotter error is
dependent on the order in which individual term unitaries are applied. Due to
the factorial growth in the number of possible orderings with respect to the
number of terms, finding an optimal strategy for ordering Trotter sequences is
difficult. In this paper, we propose three ordering strategies, and assess
their impact on the Trotter error incurred. Initially, we exhaustively examine
the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate
how the optimal ordering scheme depends on the compatibility graph of the
Hamiltonian, and show how it varies with increasing bond length. We then use 44
molecular Hamiltonians to evaluate two strategies based on coloring their
incompatibility graphs, while considering the properties of the obtained
colorings. We find that the Trotter error for most systems involving heavy
atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative
to this, the difference between ordering schemes can be substantial, being
approximately on the order of millihartrees. The coloring-based ordering
schemes are reasonably promising, however further work is required. Finally, we
consider ordering strategies based on the norm of the Trotter error operator,
including an iterative method for generating the new error operator terms added
upon insertion of a term into an ordered Hamiltonian.
