Is the Trotterized UCCSD Ansatz Chemically Well-Defined?

The variational quantum eigensolver (VQE) has emerged
as one of
the most promising near-term quantum algorithms that can be used to
simulate many-body systems such as molecular electronic structures.
Serving as an attractive ansatz in the VQE algorithm, unitary coupled
cluster (UCC) theory has seen a renewed interest in recent literature.
However, unlike the original classical UCC theory, implementation
on a quantum computer requires a finite-order Suzuki-Trotter decomposition
to separate the exponentials of the large sum of Pauli operators.
While previous literature has recognized the nonuniqueness of different
orderings of the operators in the Trotterized form of UCC methods,
the question of whether or not different orderings matter at the chemical scale has not been addressed. In this Letter, we
explore the effect of operator ordering on the Trotterized UCCSD ansatz,
as well as the much more compact k-UpCCGSD ansatz
recently proposed by Lee et al. [J. Chem. Theory Comput., 2019, 15, 311. arXiv, 2019, quant-ph:1909.09114. https://arxiv.org/abs/1909.09114]. We observe a significant, system-dependent variation
in the energies of Trotterizations with different operator orderings.
The energy variations occur on a chemical scale, sometimes on the
order of hundreds of kcal/mol. This Letter establishes the need to
define not only the operators present in the ansatz but also the order
in which they appear. This is necessary for adhering to the quantum
chemical notion of a “model chemistry”, in addition
to the general importance of scientific reproducibility. As a final
note, we suggest a useful strategy to select out of the combinatorial
number of possibilities, a single well-defined and effective ordering
of the operators.