Unitary cluster expansions of the electronic wavefunction have recently gained much interest because of their use in conjunction with quantum algorithms. In this contribution, we investigate some aspects of an ansatz, using generalized two-body excitation operators, which have been considered in some recent studies on quantum algorithms for quantum chemistry. Our numerical results show that, in particular, two-body operators with effective particle-hole excitation level of one in connection with the usual particle-hole double excitation operators lead to a very accurate, yet compact representation of the wavefunction. Generalized two-body operators with effective excitation rank zero have a considerably less pronounced effect. We compare with standard and unitary coupled-cluster expansions and show that the above mentioned approach matches or even surpasses the accuracy of expansions with three-body particle-hole excitations, in particular at the onset of strong correlation. A downside of the approach is that it is rather difficult to rigorously converge it to its variational minimum.
Unitary cluster expansions of the electronic wavefunction have recently gained much interest because of their use in conjunction with quantum algorithms. In this contribution, we investigate some aspects of an ansatz, using generalized two-body excitation operators, which have been considered in some recent studies on quantum algorithms for quantum chemistry. Our numerical results show that, in particular, two-body operators with effective particle-hole excitation level of one in connection with the usual particle-hole double excitation operators lead to a very accurate, yet compact representation of the wavefunction. Generalized two-body operators with effective excitation rank zero have a considerably less pronounced effect. We compare with standard and unitary coupled-cluster expansions and show that the above mentioned approach matches or even surpasses the accuracy of expansions with three-body particle-hole excitations, in particular at the onset of strong correlation. A downside of the approach is that it is rather difficult to rigorously converge it to its variational minimum.
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.
We introduce a unitary coupled-cluster (UCC) ansatz termed k -UpCCGSD that is based on a family of sparse generalized doubles operators, which provides an affordable and systematically improvable unitary coupled-cluster wave function suitable for implementation on a near-term quantum computer. k -UpCCGSD employs k products of the exponential of pair coupled-cluster double excitation operators (pCCD), together with generalized single excitation operators. We compare its performance in both efficiency of implementation and accuracy with that of the generalized UCC ansatz employing the full generalized single and double excitation operators (UCCGSD), as well as with the standard ansatz employing only single and double excitations (UCCSD). k -UpCCGSD is found to show the best scaling for quantum computing applications, requiring a circuit depth of, compared with for UCCGSD, and for UCCSD, where N is the number of spin orbitals and η is the number of electrons. We analyzed the accuracy of these three ansätze by making classical benchmark calculations on the ground state and the first excited state of H 4 (STO-3G, 6-31G), H 2 O (STO-3G), and N 2 (STO-3G), making additional comparisons to conventional coupled cluster methods. The results for ground states show that k -UpCCGSD offers a good trade-off between accuracy and cost, achieving chemical accuracy for lower cost of implementation on quantum computers than both UCCGSD and UCCSD. UCCGSD is also found to be more accurate than UCCSD but at a greater cost for implementation. Excited states are calculated with an orthogonally constrained variational quantum eigensolver approach. This is seen to generally yield less accurate energies than for the corresponding ground states. We demonstrate that using a specialized multideterminantal reference state constructed from classical linear response calculations allows these excited state energetics to be improved.
Adapting a veritable many-body method to a system of non-interacting particles (NIP), while being trivial from a physical point of view, can be of interest with regard to methodological aspects. In this article we study the NIP versions of the bi-orthogonal (BCC) and unitary (UCC) coupled-cluster methods both for the ground state and generalized excitations. The essential simplification here is that the cluster operator is confined to single (or p-h) excitations. For the ground state, specifically the CC amplitude equations, the NIP versions illustrate the enormous complexity of the UCC approach as compared to the BCC concept. In the treatment of excitations, on the other hand, the UCC outperforms the BCC method, as the UCC secular matrix is block-diagonal with respect to the excitation classes, e.g., 1h (one-hole), 2h-1p (two-hole-one particle), … excitations of N−1 particles. Moreover, the UCC basis states are essentially equivalent with the states used in the ADC/ISR (algebraic-diagrammatic construction/intermediate state representation) approach, whose NIP version has been discussed in a recent paper [J. Schirmer, arXiv:2309.15721].
In this paper, we propose a novel many-body approach for determining the amplitudes of generalized operators in a projection-based formalism. To implicitly account for the effects of higher-order excitations, we begin with the well-established double-exponential coupled-cluster (CC) ansatz, parametrized by both one- and two-body excitation operators, complemented by a set of vacuum-annihilating two-body generalized operators with effective excitation rank of one. A systematic formalism is developed that effectively bypasses the constraints due to the vacuum-annihilation property of the generalized operators toward a set of closed-form residual equations for their optimization. Such a strategy requires the removal of the underlying redundancy in high-rank excited determinants, generated due to the presence of the generalized operators in the ansatz, by projecting them onto an internally contracted lower-dimensional manifold. This many-body formalism is integrated with the near-term projective quantum eigensolver (PQE) framework that leverages the conventional CC-like residual minimization to iteratively decouple the excited manifold from the reference. With the application of several molecular systems within PQE architecture, we demonstrate that the developed methodology enables us to achieve similar accuracy to the disentangled unitary coupled cluster with singles, doubles, and triples ansatz while utilizing an order of magnitude fewer quantum resources. Furthermore, when simulated under stochastic Gaussian noise or depolarizing hardware noise, our method shows significantly improved noise resilience compared to the other members of PQE family and the state-of-the-art variational quantum eigensolver.
In this paper, we have developed a unitary variant of a double exponential coupled cluster theory, which is capable of handling molecular strong correlation with arbitrary electronic complexity. With the Hartree-Fock determinant taken as the reference, we introduce a sequential product of parameterized unitary Ansätze. While the first unitary, containing the excitation operators, acts directly on the reference determinant, the second unitary, containing a set of rank-two, vacuum-annihilating scattering operators, has nontrivial action only on certain entangled states. We demonstrate the theoretical bottleneck of such an implementation in a classical computer, whereas the same is implemented in the hybrid quantum-classical variational quantum eigensolver framework with a reasonably shallow quantum circuit without any additional approximation. We have further introduced a number of variants of the proposed Ansatz with different degrees of sophistication by judiciously approximating the scattering operators. With a number of applications on strongly correlated molecules, we have shown that all our schemes can perform uniformly well throughout the molecular potential energy surface without significant additional implementation cost over the conventional unitary coupled cluster approach with single and double excitations.
A prominent goal in quantum chemistry is to solve the molecular electronic structure problem for the ground state energy with high accuracy. While classical quantum chemistry is a relatively mature field, the accurate and scalable prediction of strongly correlated states found, e.g., in bond breaking and polynuclear transition metal compounds remains an open problem. Within the context of a variational quantum eigensolver, we propose a new family of ansatzes which provides a more physically appropriate description of strongly correlated electrons than unitary coupled cluster with singles and doubles excitations, with vastly reduced quantum resource requirements. Specifically, we present a set of local approximations to the unitary cluster Jastrow wavefunction motivated by Hubbard physics. The resulting ansatz removes the need for SWAP gates and can be tailored to arbitrary qubit topologies (e.g., square, hex, heavy-hex). Our ansatz is well-suited to take advantage of continuous sets of quantum gates recently realized on superconducting devices with tunable couplers, while retaining a unique level of physical transparency and interpretability. As the capabilities of quantum computing devices continue to progress, we expect that the local cluster Jastrow ansatz to be a natural choice to encode both statically and dynamically correlated electronic wavefunctions.