Low Depth Unitary Coupled Cluster Algorithm for Large Chemical Systems

TL;DR

Low Depth Unitary Coupled Cluster (qUCC) introduces a scalable strategy for quantum chemistry on quantum processors by combining a small set of large-angle UCC factors treated exactly with a quadratic Taylor expansion for the remaining factors. The method uses MP2-based grouping to identify which UCC factors require exact treatment, and solves for small-angle parameters via a linear system, reducing quantum circuit depth while preserving accuracy. Benchmarking on linear hydrogen chains and BeH2 demonstrates systematic convergence toward full UCCSD with only a fraction of total factors, with the hardest regime occurring at the weak-to-strong coupling crossover. This approach offers a practical route for large, strongly correlated systems on near- to fault-tolerant quantum hardware by shifting much of the computation to classical processing and memory.

Abstract

The unitary coupled cluster (UCC) algorithm is one of the most promising implementations of the variational quantum eigensolver for quantum computers. However, for large systems, the number of UCC factors leads to deep quantum circuits, which are prohibitive for execution on quantum hardware. To address this, circuit depth can be reduced at the cost of more measurements with a Taylor series expansion of UCC factors with small angles, while treating the large-angle factors exactly. We implement this approach to quadratic order (qUCC) for systems with strong correlations and systems where conventional methods like coupled cluster (CC) with low excitation levels fail, but UCC and qUCC perform well. We study hydrogen chains and the BeH2 molecule that allow us to change the degree of strong correlation due to geometrical distortions. We show, via a dramatic increase in number of factors able to handle exactly, a systematic convergence of these results as more exact UCC factors are included in the calculations -- the hardest to converge regime is in the crossover from weak to strong coupling. In all cases the total number of UCC factors needed to be treated exactly is much less than the total number of UCC factors available (typically about one-third to one-half of the total number of factors).

Figures (4)

• Figure 1: Flowchart outlining the procedure of the qUCC algorithm.
• Figure 2: Left: total energy for the H$_6$ linear chain as a function of bond length. Right: the error from FCI, as $E-E_\text{FCI}$. Here, 30 exact factors are used for qUCCSD. We observe that qUCCSD performs similarly to both UCCSD and CCSD for the equilibrium bond length while outperforming CCSD in the presence of strong correlations. The poorest agreement is observed in the crossover region around 2 Å. Note that while UCC is always variational, qUCC need not be due to the truncation of the quadratic expansion. For smaller $L$ cases, such as shown here, it does go non variational in the intermediate coupling regime.
• Figure 3: Energy difference from FCI for the 6- (left), 8- (right), and 10- (bottom) atom hydrogen chains. We illustrate how the accuracy improves with more exact factors, eventually converging to the full UCC result. Convergence is shown for H$_6$ and H$_8$, where we use a small fraction of the total doubles factors, and a small number of singles (58 and 126 factors, respectively). We cannot achieve full convergence For H$_{10}$, with the largest number of factors listed (which is about one-fourth of the total number available), we do not achieve a perfect convergence to the UCC result. We cannot increase the number of exact factors due to the computational constrains, however, for this case, even in the intermediate regime, one observes fast convergence to the UCC result as the number of the exact factors is increasing.
• Figure 4: Left: Total energy for the BeH$_2$ system as the beryllium atom is inserted. Right: The error from FCI, as $E-E_\text{FCI}$, for several different methods. Here, 40 exact factors are used for qUCCSD and we see that already, it performs similar to both UCCSD and CCSD, and outperforms CCSD in the region of strong correlations. Bottom: Energy difference as we add more factors to achieve convergence. We find that with 56 factors (40 doubles and 16 singles), qUCCSD recovers almost the full UCCSD with the order of one-fourth of the UCC factors. In all panels, the gray region denotes where the single reference state produces the first excited state.