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.

变分量子本征求解器（variational quantum eigensolver, VQE）已成为最具前景的近期量子算法之一，可用于模拟分子电子结构等多体系统。幺正耦合簇（unitary coupled cluster, UCC）理论作为VQE算法中极具吸引力的试探波函数，近年来在相关学术文献中重新受到关注。然而，与经典原始UCC理论不同，在量子计算机上实现该理论需采用有限阶铃木-特罗特分解（Suzuki-Trotter decomposition），以拆分大量泡利算符（Pauli operators）之和的指数形式。尽管已有文献指出UCC方法的特罗特化形式中，算符的不同排序存在非唯一性，但不同排序在化学精度尺度下是否会产生影响这一问题尚未得到解决。在本研究快报中，我们探究了算符排序对特罗特化UCCSD试探波函数，以及Lee等人近期提出的更为紧凑的k-UpCCGSD试探波函数的影响[J. Chem. Theory Comput., 2019, 15, 311. arXiv, 2019, quant-ph:1909.09114. https://arxiv.org/abs/1909.09114]。我们发现，采用不同算符排序的特罗特化方案，其计算得到的能量存在显著的、依赖于体系的差异。这种能量差异处于化学精度尺度范围内，有时可达数百千卡每摩尔（kcal/mol）。本研究快报明确指出，不仅需要定义试探波函数中包含的算符，还需明确算符的排列顺序。这对于遵循量子化学中的"model chemistry"概念而言是必要的，同时也对科学可重复性具有普遍的重要意义。最后，我们提出了一种实用策略，可从组合爆炸的可能性中筛选出一套定义明确且高效的算符排序方案。