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arXiv:2110.14448 · npj Computational Materials

Excited state calculations using variational quantum eigensolver with spin-restricted ansätze and automatically-adjusted constraints

Shigeki Gocho, Hajime Nakamura, Shu Kanno, Qi Gao, Takao Kobayashi, Taichi Inagaki, Miho Hatanaka

Published Jan 21, 2023• 35 citations
arXivHTMLPDF

Abstract

Abstract The ground and excited state calculations at key geometries, such as the Frank–Condon (FC) and the conical intersection (CI) geometries, are essential for understanding photophysical properties. To compute these geometries on noisy intermediate-scale quantum devices, we proposed a strategy that combined a chemistry-inspired spin-restricted ansatz and a new excited state calculation method called the variational quantum eigensolver under automatically-adjusted constraints (VQE/AC). Unlike the conventional excited state calculation method, called the variational quantum deflation, the VQE/AC does not require the pre-determination of constraint weights and has the potential to describe smooth potential energy surfaces. To validate this strategy, we performed the excited state calculations at the FC and CI geometries of ethylene and phenol blue at the complete active space self-consistent field (CASSCF) level of theory, and found that the energy errors were at most 2 kcal mol −1 even on the ibm_kawasaki device.

DOI
10.1038/s41524-023-00965-1

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Abstract The ground and excited state calculations at key geometries, such as the Frank–Condon (FC) and the conical intersection (CI) geometries, are essential for understanding photophysical properties. To compute these geometries on noisy intermediate-scale quantum devices, we proposed a strategy that combined a chemistry-inspired spin-restricted ansatz and a new excited state calculation method called the variational quantum eigensolver under automatically-adjusted constraints (VQE/AC). Unlike the conventional excited state calculation method, called the variational quantum deflation, the VQE/AC does not require the pre-determination of constraint weights and has the potential to describe smooth potential energy surfaces. To validate this strategy, we performed the excited state calculations at the FC and CI geometries of ethylene and phenol blue at the complete active space self-consistent field (CASSCF) level of theory, and found that the energy errors were at most 2 kcal mol −1 even on the ibm_kawasaki device.
Abstract In the noisy intermediate-scale quantum (NISQ) era, solving the electronic structure problem from chemistry is considered as the ‘killer application’ for near-term quantum devices. In spite of the success of variational hybrid quantum/classical algorithms in providing accurate energy profiles for small molecules, careful considerations are still required for the description of complicated features of potential energy surfaces. Because the current quantum resources are very limited, it is common to focus on a restricted part of the Hilbert space (determined by the set of active orbitals). While physically motivated, this approximation can severely impact the description of these complicated features. A perfect example is that of conical intersections (i.e. a singular point of degeneracy between electronic states), which are of primary importance to understand many prominent reactions. Designing active spaces so that the improved accuracy from a quantum computer is not rendered useless is key to finding useful applications of these promising devices within the field of chemistry. To answer this issue, we introduce a NISQ-friendly method called ‘state-averaged orbital-optimized variational quantum eigensolver’ which combines two algorithms: (1) a state-averaged orbital-optimizer, and (2) a state-averaged VQE. To demonstrate the success of the method, we classically simulate it on a minimal Schiff base model (namely the formaldimine molecule CH 2 NH) relevant also for the photoisomerization in rhodopsin—a crucial step in the process of vision mediated by the presence of a conical intersection. We show that merging both algorithms fulfil the necessary condition to describe the molecule’s conical intersection, i.e. the ability to treat degenerate (or quasi-degenerate) states on the same footing.
Calculations of molecular spectral properties, like photodissociation rates and absorption bands, rely on knowledge of the excited state energies of the molecule of interest. Protocols based on the variational quantum eigensolver (VQE) are promising candidates to calculate such energies on emerging noisy intermediate scale quantum (NISQ) computers. The successful implementation of these protocols on NISQ computers, relies on ansatze that can accurately approximate the molecular states and that can be implemented by shallow quantum circuits. In this paper, we introduce the ``excited iterative qubit-excitation-based variational quantum eigensolver'' (e-IQEB-VQE) protocol to calculate molecular excited state energies. The e-IQEB-VQE constructs efficient problem-tailored ansatze by iteratively appending evolutions of qubit excitation operators. The e-IQEB-VQE is an adaptation of the IQEB-VQE protocol, designed to be less dependent on the choice of an initial reference state. We perform classical numerical simulations for LiH and BeH2_22​ to benchmark the performance of the IQEB-VQE. We demonstrate that the e-IQEB-VQE can construct highly accurate ansatze that require at least an order of magnitude fewer CNOTCNOTCNOTs than standard fixed UCC ansatze, such as the UCCSD and the GUCCSD.
A programmable quantum device that has a large number of qubits without fault-tolerance has emerged recently. Variational quantum eigensolver (VQE) is one of the most promising ways to utilize the computational power of such devices to solve problems in condensed matter physics and quantum chemistry. As the size of the current quantum devices is still not large for rivaling classical computers at solving practical problems, Fujii et al. proposed a method called ``Deep VQE'', which can provide the ground state of a given quantum system with the smaller number of qubits by combining the VQE and the technique of coarse graining [K. Fujii, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, arXiv:2007.10917]. In this paper, we extend the original proposal of Deep VQE to obtain the excited states and apply it to quantum chemistry calculation of a periodic material, which is one of the most impactful applications of the VQE. We first propose a modified scheme to construct quantum states for coarse graining in Deep VQE to obtain the excited states. We also present a method to avoid a problem of meaningless eigenvalues in the original Deep VQE without restricting variational quantum states. Finally, we classically simulate our modified Deep VQE for quantum chemistry calculation of a periodic hydrogen chain as a typical periodic material. Our method reproduces the ground-state energy and the first-excited-state energy with the errors up to O(1)O(1)%O(1) despite the decrease in the number of qubits required for the calculation by two or four compared with the naive VQE. Our result will serve as a beacon for tackling quantum chemistry problems with classically-intractable sizes by smaller quantum devices in the near future.
We introduce a spin zzz-component (SzS_{z}Sz​) conserving symmetry-preserving ansatz and a shallow quantum phase estimation (QPE) routine of spin xxx (SxS_xSx​), and combine them into a spin-filtering variational quantum deflation (sfVQD) scheme for noisy intermediate-scale quantum (NISQ) computing era excited state calculations. The scheme encodes the spin information into a small ancilla register through controlled rotations under exp(iθS^x)\mathrm{exp} (iθ\hat{S}_{x})exp(iθS^x​) with only modest circuit overhead. The encoded information is then utilized to suppress spin contamination by screening, avoiding costly explicit evaluation on the total spin ⟨S^2⟩\langle\hat{S}^{2}\rangle⟨S^2⟩. Because the screening module operates independently of the variational ansatz, it can also be employed with other excited-state calculation schemes based on variational quantum eigensolvers. As a demonstration, we apply sfVQD to LiH and BeH2_22​ with varying geometries to show markedly improved separation of singlet and triplet manifolds over conventional VQD without QPE-derived screening. These results suggest that ancilla-assisted symmetry screening provides a modular and NISQ-compatible route to securing excited state calculations of physically meaningful properties. We discuss how our scheme may naturally be extended to computing other conserved quantities.
Variational quantum eigensolver (VQE) is a promising method for ground-state calculations on current noisy intermediate-scale quantum computers. However, the research progress of excited-state calculations on quantum computers is relatively slow. In order to extend the framework of VQE for excited-state calculations, we propose a new algorithm, orthogonal state reduction variational eigensolver (OSRVE), to determine the energies of excited states. Theoretical derivations prove that the optimized state in the OSRVE method can ensure the energy minimum and orthogonality constraint simultaneously, and OSRVE is also applicable for the degenerate state. The performance of OSRVE is demonstrated by numerical calculations of the H 4 and H 2 O molecules. Compared with other excited-state calculation algorithms, OSRVE has obvious advantages in calculating lower-order excited states. This work extends the VQE algorithm to excited-state calculations, and OSRVE can be implemented on near-term quantum computers.
The variational quantum eigensolver (VQE) is one of the most promising algorithms for low-lying eigenstates calculation on Noisy Intermediate-Scale Quantum (NISQ) computers. Specifically, VQE has achieved great success for ground state calculations of a Hamiltonian. However, excited state calculations arising in quantum chemistry and condensed matter often requires solving more challenging problems than the ground state as these states are generally further away from a mean-field description, and involve less straightforward optimization to avoid the variational collapse to the ground state. Maintaining orthogonality between low-lying eigenstates is a key algorithmic hurdle. In this work, we analyze three VQE models that embed orthogonality constraints through specially designed cost functions, avoiding the need for external enforcement of orthogonality between states. Notably, these formulations possess the desirable property that any local minimum is also a global minimum, helping address optimization difficulties. We conduct rigorous landscape analyses of the models' stationary points and local minimizers, theoretically guaranteeing their favorable properties and providing analytical tools applicable to broader VQE methods. A comprehensive comparison between the three models is also provided, considering their quantum resource requirements and classical optimization complexity.