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Journal of Computer Chemistry Japan

Development of Quantum Algorithm qUCC-LR for Excited-State Calculation Using Dynamic Polarizability

Tomoya TAKANASHI, Takeshi Yoshikawa, Hiromi Nakai

Published Jan 1, 2021• 1 citation
Publisher

Abstract

In this study, we propose the quantum algorithm based on the unitary coupled cluster linear response theory for excited-state calculations with single and double excitations, denoted as qUCCSD-LR. Instead of the standard eigenvalue-problem-based scheme, the algorithm utilizes the dynamical-polarizability-based scheme, where the pole relative to the frequency of an external electric field corresponds to an excited state. Numerical applications of the qUCCSD-LR method to H2 could reproduce the dynamical polarizabilities, excitation energies and oscillator strengths obtained by the standard CCSD-LR method. Furthermore, potential energy curves for the double bond rotation in the ground and excited states of C2H4 were accurately calculated by the proposed method.

DOI
10.2477/jccj.2022-0007

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In this study, we propose the quantum algorithm based on the unitary coupled cluster linear response theory for excited-state calculations with single and double excitations, denoted as qUCCSD-LR. Instead of the standard eigenvalue-problem-based scheme, the algorithm utilizes the dynamical-polarizability-based scheme, where the pole relative to the frequency of an external electric field corresponds to an excited state. Numerical applications of the qUCCSD-LR method to H2 could reproduce the dynamical polarizabilities, excitation energies and oscillator strengths obtained by the standard CCSD-LR method. Furthermore, potential energy curves for the double bond rotation in the ground and excited states of C2H4 were accurately calculated by the proposed method.
This work introduces a divide-and-conquer (DC) quantum linear-response framework for scalable excited-state simulations, in which excitation energies and oscillator strengths are extracted from the poles of the frequency-dependent dynamical polarizability. This feature naturally enables a fragmentation-based formulation, while retaining the ability to describe nonlocal excitations beyond predefined localization regions. The method, termed DC-qUCCSD-LR, builds upon the established self-consistent quantum linear-response (qLR) theory combined with the variational unitary coupled-cluster ansatz with single and double excitations. For linear hydrogen chains, n H 2, the DC-qUCCSD-LR method reproduces full configuration interaction (FCI) excitation energies while significantly reducing quantum resource requirements, achieving favorable scaling of O ( n 1.5 ) for gate counts and O ( n 2.2 ) for measurements with respect to the molecular size, n . These results demonstrate that the polarizability-based qUCC-LR framework and its DC extension provide an accurate and scalable foundation for quantum excited-state simulations.
The development of a quadratic unitary coupled-cluster singles and doubles (qUCCSD) based self-consistent polarization propagator method is reported. We present a simple strategy for truncating the commutator expansion of the unitary version of coupled-cluster transformed Hamiltonian H̄. The qUCCSD method for the electronic ground state includes up to double commutators for the amplitude equations and up to cubic commutators for the energy expression. The qUCCSD excited-state eigenvalue equations include up to double commutators for the singles-singles block of H̄, single commutators for the singles-doubles and doubles-singles blocks, and the bare Hamiltonian for the doubles-doubles block. Benchmark qUCCSD calculations of the ground-state properties and excitation energies for representative molecules demonstrate significant improvement of the accuracy and robustness over the previous UCC3 scheme derived using Møller-Plesset perturbation theory.
An efficient implementation of the quadratic unitary coupled-cluster singles and doubles (qUCCSD) scheme for calculations of electronic ground and excited states using an unrestricted molecular spin–orbital formulation and an efficient tensor contraction library is reported. The accuracy of the qUCCSD scheme and the efficiency of the present implementation are demonstrated using extensive benchmark calculations of excitation energies and an application to S 0 → S 1 vertical excitation energies for cis - and trans -4 a,4 b- dihydrotriphenylene. The qUCCSD scheme has been shown to provide improved excitation energies compared with the UCC3 scheme formulated based on perturbation theory. A UCC truncation scheme that can provide excitation energies correct through the fourth order is also presented to further improve the accuracy of the qUCCSD scheme.
Quantum computing presents a promising avenue for solving complex problems, particularly in quantum chemistry, where it could accelerate the computation of molecular properties and excited states. This work focuses on computing excitation energies with hybrid quantum-classical algorithms for near-term quantum devices, combining the quantum linear response (qLR) method with a polarizable embedding (PE) environment. We employ the self-consistent operator manifold of quantum linear response (q-sc-LR) on top of a unitary coupled cluster (UCC) wave function in combination with a Davidson solver. The latter removes the need to construct the entire electronic Hessian, improving computational efficiency when going toward larger molecules. We introduce a new superposition-state-based technique to compute Hessian-vector products and show that this approach is more resilient toward noise than our earlier gradient-based approach. We demonstrate the performance of the PE-UCCSD model on systems such as butadiene and para-nitroaniline in water and find that PE-UCCSD delivers comparable accuracy to classical PE-CCSD methods on such simple closed-shell systems. We also explore the challenges posed by hardware noise and propose simple error mitigation techniques to maintain accurate results on noisy quantum computers.
We explore Davidson methods for obtaining excitation energies and other linear response properties within the recently developed quantum self-consistent linear response (q-sc-LR) method. Davidson-type methods allow for obtaining only a few selected excitation energies without explicitly constructing the electronic Hessian since they only require the ability to perform Hessian-vector multiplications. We apply the Davidson method to calculate the excitation energies of hydrogen chains (up to H 10 ) and analyze aspects of statistical noise for computing excitation energies on quantum simulators. Additionally, we apply Davidson methods for computing linear response properties such as static polarizabilities for H 2, LiH, H 2 O, OH –, and NH 3, and show that unitary coupled cluster outperforms classical projected coupled cluster for molecular systems with strong correlation. Finally, we formulate the Davidson method for damped (complex) linear response, with application to the nitrogen K-edge X-ray absorption of ammonia, and the C 6 coefficients of H 2, LiH, H 2 O, OH –, and NH 3 .
A unitary coupled cluster (UCC) form for the wave function in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled cluster excitations, practically, the accuracy of this approach depends on the number and type of terms are included in the wave function parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixed Fermionic excitation rank. Not all quantum computing architectures can cope with this growth. To address both problems, we introduce a qubit coupled cluster (QCC) method that starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers for the variational energy minimization. Also, we provide an exact factorization of a unitary rotation of more than two qubits to a product of two-qubit unitary rotations. Thus, the QCC method with the factorization technique can be limited to only two-qubit entanglement gates and allows for very efficient use of quantum resources in terms of the number of coupled cluster operators. The method performance is illustrated by calculating ground-state potential energy curves of H 2 and LiH molecules with chemical accuracy, ≤1 kcal/mol, and a symmetric water dissociation curve.