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arXiv:1910.05168 · International Journal of Quantum Chemistry

Generalized unitary coupled cluster excitations for multireference molecular states optimized by the variational quantum eigensolver

Gabriel Greene‐Diniz, David Muñoz Ramo

Published Nov 27, 2020• 42 citations
arXivHTMLPDF

Abstract

Abstract The variational quantum eigensolver (VQE) algorithm, designed to calculate the energy of molecular ground states on near‐term quantum computers, requires specification of symmetries that describe the system, for example, spin state and number of electrons. This opens the possibility of using VQE to obtain excited states as the lowest‐energy solutions of a given set of symmetries. In this paper, the performances of various unitary coupled cluster (UCC) ansätze applied to VQE calculations on excited states are investigated using quantum circuits designed to represent single reference and multireference wavefunctions to calculate energy curves with respect to variations in the molecular geometry. These ansätze include standard Unitary Coupled Cluster Singles and Doubles (UCCSD), as well as modified versions of Unitary Coupled Cluster Generalized Singles and Doubles (UCCGSD) and k‐UpCCGSD, which are engineered to tackle excited states without undesired spin symmetry crossover to lower states during VQE optimization. These studies are carried out on a range of systems, including H2, H3, H4, NH, OH+, CH2, and , covering examples of spin singlet, doublet, and triplet molecular ground states with single and multireference excited states. In most cases, our calculations are in excellent agreement with results from full‐configuration interaction calculations on classical machines, thus showing that the VQE algorithm is capable of calculating the lowest excited state at a certain symmetry, including multireference closed‐ and open‐shell states, by setting appropriate restrictions on the excitations considered in the cluster operator and appropriate constraints in the qubit register encoding the starting mean field state.

DOI
10.1002/qua.26352

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Abstract The variational quantum eigensolver (VQE) algorithm, designed to calculate the energy of molecular ground states on near‐term quantum computers, requires specification of symmetries that describe the system, for example, spin state and number of electrons. This opens the possibility of using VQE to obtain excited states as the lowest‐energy solutions of a given set of symmetries. In this paper, the performances of various unitary coupled cluster (UCC) ansätze applied to VQE calculations on excited states are investigated using quantum circuits designed to represent single reference and multireference wavefunctions to calculate energy curves with respect to variations in the molecular geometry. These ansätze include standard Unitary Coupled Cluster Singles and Doubles (UCCSD), as well as modified versions of Unitary Coupled Cluster Generalized Singles and Doubles (UCCGSD) and k‐UpCCGSD, which are engineered to tackle excited states without undesired spin symmetry crossover to lower states during VQE optimization. These studies are carried out on a range of systems, including H2, H3, H4, NH, OH+, CH2, and , covering examples of spin singlet, doublet, and triplet molecular ground states with single and multireference excited states. In most cases, our calculations are in excellent agreement with results from full‐configuration interaction calculations on classical machines, thus showing that the VQE algorithm is capable of calculating the lowest excited state at a certain symmetry, including multireference closed‐ and open‐shell states, by setting appropriate restrictions on the excitations considered in the cluster operator and appropriate constraints in the qubit register encoding the starting mean field state.
Abstract Variational quantum eigensolver (VQE) with a unitary coupled cluster (UCC) ansatz has been suggested as a promising method for electronic structure calculations on future quantum computers. However, the complexity of the excitation terms for UCC with the single and double excitations (UCCSD) ansatz is up‐bounded to , where N is the number of electrons and M is the number of spin orbitals. The gate complexity of quantum circuit for the UCCSD ansatz is up‐bounded to using the Jordan–Wigner transformation. These complexities significantly limit the implementation of UCCSD on current Noisy Intermediate‐Scale Quantum (NISQ) devices. Herein, we developed a k‐QUpCCGSD ansatz which is based on the generalized paired double excitation operators and the particle preserving exchange gate. The former reduces the number of the excitation operators, the latter reduces the number of qubit gates for transforming excitation operators to quantum circuit. The gate complexity of the proposed k‐QUpCCGSD ansatz is up‐bounded to , which significantly reduce the complexity of the VQE algorithm on NISQ devices. The performance of the proposed ansatz on VQE is demonstrated by calculating ground‐state dissociation energy curves of the H6, LiH, H2O, and BeH2 molecules with the STO‐3G minimal basis set, and the accuracy is evaluated by comparing to the full configuration interaction (FCI) benchmarks. Moreover, we compare the number of quantum gates, especially the CNOT gates, and accuracies of various ansatzes. The assessments have shown that the accuracy of qubit unitary coupled cluster (QUCC) ansatzes is slightly worse than that of the UCC ones, but the circuit complexity of QUCC is much less than that of UCC. Among the tested QUCC ansatzes, k‐QUpCCGSD achieves higher accuracy with fewer quantum gates than QUCCSD, and k‐QUpCCGSD is a promising ansatz for VQE calculation on NISQ devices.
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.
Classical simulation of molecular electronic structure is often limited by the rapid increase in computational cost associated with describing electron correlation. The Variational Quantum Eigensolver (VQE) is one hybrid quantum-classical approach ideally adapted to the Noisy Intermediate-Scale Quantum (NISQ) regime. This work applies the VQE algorithm to compute ground state energies of a set of molecules:H2,H3+,LiH,HF,OH−,H2O,BH3,CH4\mathrm{H}_{2}, \mathrm{H}_{3}^{+}, \text{LiH}, \text{HF}, \text{OH}^{-}, \mathrm{H}_{2} \mathrm{O}, \text{BH}_{3}, \text{CH}_{4}H2​,H3+​,LiH,HF,OH−,H2​O,BH3​,CH4​, andNH3\text{NH}_{3}NH3​. The approach utilizes the Unitary Coupled Cluster for Single and Double excitations (UCCSD) ansatz along with the Bravyi-Kitaev transformation for fermionic-to-qubit encoding. Quantum simulation outputs are systematically benchmarked against Full Configuration Interaction (FCI), a numerically exact traditional approach, and Unrestricted Hartree-Fock (UHF). The results establish a robust consistency of the VQE-obtained molecular ground state energies against standard FCI and UHF benchmarks. This research provides a critical benchmark, confirming that VQE has the capability for high-precision molecular simulations and playing the role of a pivotal stepping stone toward revolutionizing applications in drug discovery and materials science.
Abstract Variational quantum eigensolver (VQE) for electronic structure calculations is believed to be one major potential application of near term quantum computing. Among all proposed VQE algorithms, the unitary coupled cluster singles and doubles excitations (UCCSD) VQE ansatz has achieved high accuracy and received a lot of research interest. However, the UCCSD VQE based on fermionic excitations needs extra terms for the parity when using Jordan–Wigner transformation. Here we introduce a new VQE ansatz based on the particle preserving exchange gate to achieve qubit excitations. The proposed VQE ansatz has gate complexity up-bounded to O ( n 4 ) for all-to-all connectivity where n is the number of qubits of the Hamiltonian. Numerical results of simple molecular systems such as BeH 2 , H 2 O, N 2 , H 4 and H 6 using the proposed VQE ansatz gives very accurate results within errors about 10 −3 Hartree.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4–8. With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H4-square, H4-chain, and H6-hexagon molecules, while the number of cnot gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient “score” parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using the H4 square and H4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.
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.