Is the Trotterized UCCSD Ansatz Chemically Well-Defined?

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.

Orbital optimized unitary coupled cluster theory for quantum computer

“…As mentioned above, the difference between UCCSD 1 and OO-UCCD 1 essentially comes from the order dependency of the doubles. Indeed, Grimsley et al reported that the typical doubles-operator-order dependency results in an energy difference of an order of less than a microhartree for UCCSD 1 near the equilibrium geometry for LiH, whereas the magnitude of the order-dependency increases with an increase in the LiH bond length [81]. Our results are consistent with theirs.…”

“…The double excitation operatorsT n = t rs pqĉ † pĉ † qĉ rĉs are generated so that the index strings psqr are in lexicographical order, where t rs pq is a doubles amplitude. Although Grimsley et al have illustrated that the effect of the operator ordering could be significant, their results suggest that the impact of the ordering is not important in the systems used in this study [81].…”

“…Although a single-Trotter-step UCC ansatz appears to be a crude approximation, Barkoutsos [18]. Note that the Trotter error depends on the ordering of the operators [81,82]. This order dependency makes UCCSD n inequivalent to OO-UCCD n even when n = 1.…”

Benchmarking Adaptive Variational Quantum Eigensolvers

“…The COBYLA (Powell, 1994) algorithm was used as a gradient-free optimizer, while gradient-based optimizations were carried out with the L-BFGS algorithm (Nocedal, 1980;Liu and Nocedal, 1989), with all parameters being initialized at 0 at each optimization cycle for both optimizers. Other approaches have been reported in the literature, such as random initialization (Grimsley et al, 2020), or as in the original implementation of ADAPT-VQE (Grimsley et al, 2019) where the new parameter is initialized at 0, while the previous parameters are initialized from the optimal values obtained in the previous ADAPT iteration. XACC offers both optimizers via its interface with NLOpt (Johnson).…”

Efficient Step-Merged Quantum Imaginary Time Evolution Algorithm for Quantum Chemistry

“…It has been discussed recently that the Pauli operator ordering in the Trotterized circuits of the VQE-UCCSD approach can introduce significant errors in energy evaluations beyond chemical accuracy. 36 As a Trotterized form is also adopted in the smQITE method, similar operator ordering effects could exist. Nevertheless, we will demonstrate that decent numerical results from smQITE calculations can already be obtained without exploiting optimum Pauli operator orderings.…”