Is the Trotterized UCCSD Ansatz chemically well-defined?

TL;DR

This paper investigates how the order of operators in Trotterized UCCSD ansatz affects energy calculations in VQE, revealing significant variations and emphasizing the importance of defining operator order for chemical accuracy and reproducibility.

Contribution

It demonstrates that operator ordering in Trotterized UCCSD significantly impacts energy results, highlighting the need for a well-defined ordering to ensure reproducibility and chemical relevance.

Findings

Operator ordering causes significant energy variation.

Energy differences can reach hundreds of kcal/mol.

A strategy for selecting a consistent operator order is proposed.

Abstract

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 non-uniqueness 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. 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.