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Trotter-Suzuki decompositions are frequently used in the quantum simulationof quantum chemistry. They transform the evolution operator into a formimplementable on a quantum device, while incurring an error---the Trottererror. The Trotter error can be made arbitrarily small by increasing theTrotter number. However, this increases the length of the quantum circuitsrequired, which may be impractical. It is therefore desirable to find methodsof reducing the Trotter error through alternate means. The Trotter error isdependent on the order in which individual term unitaries are applied. Due tothe factorial growth in the number of possible orderings with respect to thenumber of terms, finding an optimal strategy for ordering Trotter sequences isdifficult. In this paper, we propose three ordering strategies, and assesstheir impact on the Trotter error incurred. Initially, we exhaustively examinethe possible orderings for molecular hydrogen in a STO-3G basis. We demonstratehow the optimal ordering scheme depends on the compatibility graph of theHamiltonian, and show how it varies with increasing bond length. We then use 44molecular Hamiltonians to evaluate two strategies based on coloring theirincompatibility graphs, while considering the properties of the obtainedcolorings. We find that the Trotter error for most systems involving heavyatoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relativeto this, the difference between ordering schemes can be substantial, beingapproximately on the order of millihartrees. The coloring-based orderingschemes are reasonably promising, however further work is required. Finally, weconsider ordering strategies based on the norm of the Trotter error operator,including an iterative method for generating the new error operator terms addedupon insertion of a term into an ordered Hamiltonian.

Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure