A new "gold standard": Perturbative triples corrections in unitary coupled cluster theory and prospects for quantum computing.

A major difficulty in quantum simulation is the adequate treatment of a large collection of entangled particles, synonymous with electron correlation in electronic structure theory, with coupled cluster (CC) theory being the leading framework for dealing with this problem. Augmenting computationally affordable low-rank approximations in CC theory with a perturbative account of higher-rank excitations is a tractable and effective way of accounting for the missing electron correlation in those approximations. This is perhaps best exemplified by the "gold standard" CCSD(T) method, which bolsters the baseline CCSD with the effects of triple excitations using considerations from many-body perturbation theory (MBPT). Despite this established success, such a synergy between MBPT and the unitary analog of CC theory (UCC) has not been explored. In this work, we propose a similar approach wherein converged UCCSD amplitudes are leveraged to evaluate energy corrections associated with triple excitations, leading to the UCCSD[T] method. In terms of quantum computing, this correction represents an entirely classical post-processing step that improves the energy estimate by accounting for triple excitation effects without necessitating new quantum algorithm developments or increasing demand for quantum resources. The rationale behind this choice is shown to be rigorous by studying the properties of finite-order UCC energy functionals, and our efforts do not support the addition of the fifth-order contributions as in the (T) correction. We assess the performance of these approaches on a collection of small molecules and demonstrate the benefits of harnessing the inherent synergy between MBPT and UCC theories.

Modeling Singlet Fission on a Quantum Computer.

We demonstrate a practical application of quantum computing by using it to investigate the linear H4 molecule as a simple model for singlet fission. We use the Peeters-Devreese-Soldatov energy functional to calculate the necessary energetics based on the moments of the Hamiltonian estimated on the quantum computer. To reduce the number of required measurements, we use several independent strategies: 1) reduction of the size of the relevant Hilbert space by tapering off qubits; 2) measurement optimization via rotations to eigenbases shared by groups of qubit-wise commuting Pauli strings; and 3) parallel execution of multiple state preparation and measurement operations using all 20 qubits available on the Quantinuum H1-1 quantum hardware. Our results meet the energetic requirements for singlet fission, are in excellent agreement with exact transition energies (for the chosen one-particle basis), and outperform classical methods considered computationally feasible for singlet fission candidates.

Quantum Solvers for Plane-Wave Hamiltonians: Abridging Virtual Spaces Through the Optimization of Pairwise Correlations.

For many-body methods such as MCSCF and CASSCF, in which the number of one-electron orbitals is optimized and independent of the basis set used, there are no problems with using plane-wave basis sets. However, for methods currently used in quantum computing such as select configuration interaction (CI) and coupled cluster (CC) methods, it is necessary to have a virtual space that is able to capture a significant amount of electron-electron correlation in the system. The virtual orbitals in a pseudopotential plane-wave Hartree-Fock calculation, because of Coulomb repulsion, are often scattering states that interact very weakly with the filled orbitals. As a result, very little correlation energy is captured from them. The use of virtual spaces derived from the one-electron operators has also been tried, and while some correlations are captured, the amount is quite low. To overcome these limitations, we have been developing new classes of algorithms to define virtual spaces by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. With these procedures, we have been able to derive virtual spaces, containing only a few orbitals, which are able to capture a significant amount of correlation. The focus in this manuscript is on using these derived basis sets to target full CI (FCI) quality results for H2 on near-term quantum computers. However, the initial results for this approach were promising. We were able to obtain good agreement with FCI/cc-pVTZ results for this system with just 4 virtual orbitals, using both FCI and quantum simulations. The quality of the results using COVOs suggests that it may be possible to use them in other many-body approaches, including coupled cluster and Møller-Plesset perturbation theories, and open up the door to many-body calculations for pseudopotential plane-wave basis set methods.

Benchmarking Adaptive Variational Quantum Eigensolvers.

By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to calculate the electronic ground states and potential energy curves for a few selected diatomic molecules, namely H2, NaH, and KH. Using numerical simulation, we find both methods provide good estimates of the energy and ground state, but only ADAPT-VQE proves to be robust to particularities in optimization methods. Another relevant finding is that gradient-based optimization is overall more economical and delivers superior performance than analogous simulations carried out with gradient-free optimizers. The results also identify small errors in the prepared state fidelity which show an increasing trend with molecular size.

Is the Trotterized UCCSD Ansatz Chemically Well-Defined?

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 nonuniqueness 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. [ J. Chem. Theory Comput. , 2019 , 15 , 311 . arXiv, 2019 , quant-ph:1909.09114. https://arxiv.org/abs/1909.09114 ]. 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.

Simple and Efficient Truncation of Virtual Spaces in Embedded Wave Functions via Concentric Localization.

We present a strategy to generate "concentrically local orbitals" for the purpose of decreasing the computational cost of wave function-in-density functional theory (WF-in-DFT) embedding. The concentric localization is performed for the virtual orbitals by first projecting the virtual space onto atomic orbitals centered on the embedded atoms. Using a one-particle operator, these projected orbitals are then taken as a starting point to iteratively span the virtual space, recursively creating virtual orbital "shells" with consecutively decreasing correlation energy recovery at each iteration. This process can be repeated to convergence, allowing for tunable accuracy. Assessment of the proposed scheme is performed by application to the potential energy diagram of the Menshutkin reaction of chloromethane and ammonia inside a segment of a carbon nanotube and the torsional potential of a simplified version of the retinal chromophore.

Automatic Partition of Orbital Spaces Based on Singular Value Decomposition in the Context of Embedding Theories.

We present a simple approach for orbital space partitioning to be employed in the projection-based embedding theory developed by Goodpaster and co-workers [ Manby et al. J. Chem. Theory Comput. 2012 , 8 , 2564 ]. Once the atoms are assigned to the desired subspaces, the molecular orbitals are projected onto the atomic orbitals centered on active atoms and then singular value decomposed. The right singular vectors are used to rotate the initial molecular orbitals, taking the largest gap in the singular values spectrum to define the most suitable partition of the occupied orbital space. This scheme is free from numerical parameters, contrary to the Mulliken charge threshold or the completeness criterion previously used. The performance of this new prescription is assessed in a test set of several distinct reactions, the deprotonation of decanoic acid, the Diels-Alder reaction of 1,3-butadiene and octadecanonaene, the torsional potential of a retinal derivative, and the critical points along the reaction coordinate of an example of the Menshutkin SN2 reaction inside a carbon nanotube.

Coupled-cluster based basis sets for valence correlation calculations. New primitives, frozen atomic natural orbitals, and basis sets from double to hextuple zeta for atoms H, He, and B-Ne.

A series of new basis sets generated from coupled-cluster theory are presented. Sets of primitive Gaussians for angular momentum shells containing occupied orbitals are optimized and extended until the Hartree-Fock energy improvement per electron falls below 10 µH. This same energy threshold is applied to the energy from the coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) approximation and taken as a guide to the generation of sets of polarization functions, constructed at the level of second-order many-body perturbation theory, once it is shown that practically indistinguishable results are obtained if the more costly CCSD(T) approach is used. A variant of the usual atomic natural orbitals (ANOs) grounded on the frozen natural orbital idea is explored, leading to the frozen ANO (FANO) contraction scheme, whose truncation pattern and convergence capabilities are thoroughly tested. Due to its superior description of the single determinant solution, the FANOs are proposed as the recommended contraction approach. FANO basis sets of sizes ranging from double to hextuple zeta are provided for atoms H through Ne and examined in atomic and molecular calculations.

Coupled-cluster based basis sets for valence correlation calculations.

Novel basis sets are generated that target the description of valence correlation in atoms H through Ar. The new contraction coefficients are obtained according to the Atomic Natural Orbital (ANO) procedure from CCSD(T) (coupled-cluster singles and doubles with perturbative triples correction) density matrices starting from the primitive functions of Dunning et al. [J. Chem. Phys. 90, 1007 (1989); ibid. 98, 1358 (1993); ibid. 100, 2975 (1993)] (correlation consistent polarized valence X-tuple zeta, cc-pVXZ). The exponents of the primitive Gaussian functions are subject to uniform scaling in order to ensure satisfaction of the virial theorem for the corresponding atoms. These new sets, named ANO-VT-XZ (Atomic Natural Orbital Virial Theorem X-tuple Zeta), have the same number of contracted functions as their cc-pVXZ counterparts in each subshell. The performance of these basis sets is assessed by the evaluation of the contraction errors in four distinct computations: correlation energies in atoms, probing the density in different regions of space via ⟨r(n)⟩ (-3 ≤ n ≤ 3) in atoms, correlation energies in diatomic molecules, and the quality of fitting potential energy curves as measured by spectroscopic constants. All energy calculations with ANO-VT-QZ have contraction errors within "chemical accuracy" of 1 kcal/mol, which is not true for cc-pVQZ, suggesting some improvement compared to the correlation consistent series of Dunning and co-workers.