Reducing the Cost of Unitary Coupled Cluster via Active Space Partitioning

TL;DR

This work presents an active-space Unitary Coupled Cluster method, UCCSD(4)/MP2, to reduce the cost of accurately modeling electron correlation. By partitioning the cluster operator into internal (active) and external (inactive) parts and treating the external space with MP2, the authors compare composite and interacting coupling schemes using canonical and frozen natural orbitals. Across small molecules, the GW100 set, phosphate hydrolysis, and ethylene torsion, the interacting scheme with canonical orbitals most faithfully reproduces full UCCSD(4) behavior while requiring only 15–25% of the virtual space; natural orbitals can improve composite results but destabilize the interacting approach under strong correlation. The framework offers a tractable path for classical and quantum-resource-constrained calculations, guiding active-space choices and coupling strategies for scalable UCC implementations.

Abstract

Unitary Coupled Cluster (UCC) theory is a promising variational method for electronic structure calculations, especially for strongly correlated systems and quantum computers. However, its practical application is limited by the steep scaling of its non-terminating Baker-Campbell-Hausdorff expansion. We introduce an active space UCCSD(4)/MP2 approach that leverages a fourth-order many-body perturbation theory truncation of UCCSD within a selected active space, while treating external excitations at the MP2 level. We explore two variants: a composite method that sums separate internal and external contributions and an interacting method that couples the amplitudes for greater accuracy. We test our approach on the GW100 dataset, a metaphosphate hydrolysis reaction, and the strongly correlated torsion of ethylene. Our results suggest that the interacting method with canonical orbitals is robust for weakly and moderately correlated systems and accurately reproduces the full UCCSD(4) potential energy curves using only 15-25% of the virtual orbitals in its active space. In comparison, the composite formulation exhibits greater sensitivity to the orbital basis and active space size, leading to less systematic behavior across the benchmark set. For ethylene torsion, a system dominated by strong static correlation, both composite and interacting formulations employing canonical orbitals closely track the full UCCSD(4) reference but do not alleviate the unphysical features inherited from the underlying single-reference UCCSD(4) description. This active space framework offers a tractable approach for modeling correlated molecules and reactions on classical computers and provides a viable path for scaling UCC calculations for resource-constrained quantum hardware.

Figures (5)

• Figure 1: Absolute errors in total energies (in milliHartrees) relative to the UCCSD(4) reference for a range of molecules. Each bar represents a different approximation: CCSD (blue); composite UCCSD (c-UCCSD(4)/MP2) with canonical orbitals (CO) (dark teal); interacting UCCSD (i-UCCSD(4)/MP2) with canonical orbitals (red); composite UCCSD (c-UCCSD(4)/MP2) with natural orbitals (NO) (cyan); and interacting UCCSD (i-UCCSD(4)/MP2) with natural orbitals (magenta). Only 60% of the virtual orbitals are retained in the active space.
• Figure 2: Percentage error in correlation energies with respect to the CCSD(T) reference for (a) medium‐sized and (b) large molecules from the GW100 dataset. Each violin envelope shows the distribution of errors over all molecules in the set; the thick horizontal bar marks the median and the whiskers span the full range, while the overlaid points correspond to individual molecules. "CCSD" is the full canonical CCSD result. All other methods employ the active–space UCCSD(4)/MP2 method. "CO" and "NO" indicate that the calculations were performed using canonical and natural orbitals, respectively.
• Figure 3: Reaction Energy Profile for the PO$_3^-$ + H$_2$O reaction, computed using the cc-pVDZ basis set. Electronic energies are reported relative to the reactant energy. The CCSD(T) curve (green, dashed line) serves as the high-level reference, while the blue trace (solid line) and yellow squares show the corresponding CCSD and UCCSD(4) results. All remaining curves were obtained with the active-space UCCSD(4)/MP2 methods, where the active space contains all occupied valence orbitals and 11 virtual orbitals. "i-" denotes the interacting UCCSD(4)/MP2 method, "c-" the composite UCCSD(4)/MP2 method, and "CO" and "NO" refer to canonical orbitals and natural orbitals, respectively.
• Figure 4: Dependence of the barrier height for the forward (left panel) and backward (right panel) PO$_3^-$ + H$_2$O reaction on the number of virtual orbitals included in the active space. All calculations were performed using the cc-pVDZ basis set. The active space for these calculations contained all occupied valence orbitals. The full-space CCSD, UCCSD(4), and CCSD(T) values are provided as horizontal lines for reference. The "i-" and "c-" prefixes denote the interacting and composite UCCSD(4)/MP2 methods, respectively, while "CO" and "NO" refer to the use of canonical and natural orbitals.
• Figure 5: Potential energy curve for the torsion of the ethylene molecule. Electronic energies are reported in kcal/mol relative to the planar structure (0$^\circ$ torsion angle). The plot compares results from several electronic structure methods. The CCSD(T) curve (orange squares) and the CIPSI curve (pink diamonds) serve as high-level references. The corresponding results for HF, CCSD, and UCCSD(4) are also shown. The remaining curves were obtained with active-space UCCSD(4)/MP2 methods, where the active space contains all occupied valence orbitals and 22 virtual orbitals. "i" denotes the interacting method, "c-" denotes the composite method, and "CO" and "NO" refer to the use of canonical orbitals and natural orbitals, respectively.