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.
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.
Unitary Coupled Cluster (UCC) theory is a promising variational method for electronic structure calculations, particularly for systems that exhibit strong electronic correlation and for implementation on quantum computers. However, its practical application is limited to small chemical systems with small basis sets due to its steep computational scaling, which results from its nonterminating Baker-Campbell-Hausdorff expansion. Here, 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 potentially greater accuracy. We test our approach on a range of systems, including molecules from the GW100 data set in their equilibrium geometries, a moderately correlated metaphosphate hydrolysis reaction, and the strongly correlated torsion of ethylene. Our results suggest that the interacting method with canonical orbitals is robust and stable for both weakly and moderately correlated systems and accurately reproduces the full UCCSD(4) potential energy curves, including only 15-25% of the virtual orbitals in its active space. In comparison, the composite formulation exhibits greater sensitivity to the choice of 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 the composite and interacting formulations employing canonical orbitals closely track the full UCCSD(4) reference while preserving the qualitative behavior of the parent method, including the breakdown in strongly multireference regimes. This active space framework offers a computationally 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.
The Coupled Cluster (CC) method is used to compute the electronic correlation energy in atoms and molecules and often leads to highly accurate results. However, due to its single-reference nature, standard CC in its projected form fails to describe quantum states characterized by strong electronic correlations and multi-reference projective methods become necessary. On the other hand, quantum algorithms for the solution of many-electron problems have also emerged recently. The quantum unitary variant of CC (UCC) with singles and doubles (q-UCCSD) is a popular wavefunction Ansatz for the variational quantum eigensolver algorithm. The variational nature of this approach can lead to significant advantages compared to its classical equivalent in the projected form, in particular, for the description of strong electronic correlation. However, due to the large number of gate operations required in q-UCCSD, approximations need to be introduced in order to make this approach implementable in a state-of-the-art quantum computer. In this work, we evaluate several variants of the standard q-UCCSD Ansatz in which only a subset of excitations is included. In particular, we investigate the singlet and pair q-UCCD approaches combined with orbital optimization. We show that these approaches can capture the dissociation/distortion profiles of challenging systems, such as H4, H2O, and N2 molecules, as well as the one-dimensional periodic Fermi–Hubbard chain. These results promote the future use of q-UCC methods for the solution of challenging electronic structure problems in quantum chemistry.
The recent quantum information boom has effected a resurgence of interest in the unitary coupled cluster (UCC) theory. Our group’s interest in local energy landscapes of unitary ansätze prompted us to investigate the classical approach of truncating the Taylor series expansion (instead of a perturbative expansion) of the UCC with singles and doubles (UCCSD) energy at the second order. This amounts to an approach where the electron correlation energy is estimated by taking a single Newton–Raphson step from Hartree–Fock toward UCCSD. Such an approach has been explored previously, but the accuracy was not extensively studied. In this paper, we investigate the performance and observe similar pathologies to the linearized coupled cluster with singles and doubles. We introduce the use of derivatives of order three or greater to help partially recover the variational lower bound of true UCCSD, restricting these derivatives to those of the “unmixed” category in order to simplify the model. By testing the approach on several potential energy surfaces and reaction energies, we find this “diagonal” approximation to higher order terms to be effective at reducing sensitivity near singularities for strongly correlated regimes, while not significantly diminishing the accuracy of weakly correlated systems.
We introduce a hybrid quantum-classical algorithm, the localized active space unitary selective coupled cluster singles and doubles (LAS-USCCSD) method. Derived from the localized active space unitary coupled cluster (LAS-UCCSD) method, LAS-USCCSD first performs a classical LASSCF calculation, then selectively identifies the most important parameters (cluster amplitudes used to build the multireference UCC ansatz) for restoring interfragment interaction energy using this reduced set of parameters with the variational quantum eigensolver method. We benchmark LAS-USCCSD against LAS-UCCSD by calculating the total energies of (H 2 ) 2, (H 2 ) 4, and trans -butadiene, and the magnetic coupling constant for a bimetallic compound [Cr 2 (OH) 3 (NH 3 ) 6 ] 3+ . For these systems, we find that LAS-USCCSD reduces the number of required parameters and thus the circuit depth by at least 1 order of magnitude, an aspect which is important for the practical implementation of multireference hybrid quantum-classical algorithms like LAS-UCCSD on near-term quantum computers.
Unitary coupled cluster (UCC), originally developed as a variational alternative to the popular traditional coupled cluster method, has seen a resurgence as a functional form for use on quantum computers. However, the number of excitors present in the Ansatz often presents a barrier to implementation on quantum computers. Given the natural sparsity of wavefunctions obtained from quantum Monte Carlo methods, we consider here a stochastic solution to the UCC problem. Using the coupled cluster Monte Carlo framework, we develop cluster selection schemes that capture the structure of the UCC wavefunction, as well as its Trotterized approximation, and use these to solve the corresponding projected equations. Due to the fast convergence of the equations with order in the cluster expansion, this approach scales polynomially with the size of the system. Unlike traditional UCC implementations, our approach naturally produces a non-variational estimator for the energy in the form of the projected energy. For unitary coupled cluster singles and doubles (UCCSD) in small systems, we find that this agrees well with the expectation value of the energy and, in the case of two electrons, with full configuration interaction results. For the larger N2 system, the two estimators diverge, with the projected energy approaching the coupled cluster result, while the expectation value is close to results from traditional UCCSD.
Quantum chemistry calculations of large, strongly correlated systems are typically limited by the computation cost that scales exponentially with the size of the system. Quantum algorithms, designed specifically for quantum computers, can alleviate this, but the resources required are still too large for today's quantum devices. Here, we present a quantum algorithm that combines a localization of multireference wave functions of chemical systems with quantum phase estimation (QPE) and variational unitary coupled cluster singles and doubles (UCCSD) to compute their ground-state energy. Our algorithm, termed "local active space unitary coupled cluster" (LAS-UCC), scales linearly with the system size for certain geometries, providing a polynomial reduction in the total number of gates compared with QPE, while providing accuracy above that of the variational quantum eigensolver using the UCCSD ansatz and also above that of the classical local active space self-consistent field. The accuracy of LAS-UCC is demonstrated by dissociating (H2)2 into two H2 molecules and by breaking the two double bonds in trans-butadiene, and resource estimates are provided for linear chains of up to 20 H2 molecules.