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arXiv:2009.11417 · Quantum Science and Technology

A state-averaged orbital-optimized hybrid quantum–classical algorithm for a democratic description of ground and excited states

Saad Yalouz, Bruno Senjean, Jakob Günther, Francesco Buda, Thomas E O’Brien, Lucas Visscher

Published Dec 14, 2020• 83 citations
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Abstract

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.

DOI
10.1088/2058-9565/abd334

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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.
We introduce several technical and analytical extensions to our recent state-averaged orbital-optimized variational quantum eigensolver (SA-OO-VQE) algorithm (see Yalouz et al. Quantum Sci. Technol . 2021, 6, 024004). Motivated by the limitations of current quantum computers, the first extension consists of an efficient state-resolution procedure to find the SA-OO-VQE eigenstates, and not just the subspace spanned by them, while remaining in the equi-ensemble framework. This approach avoids expensive intermediate resolutions of the eigenstates by postponing this problem to the very end of the full algorithm. The second extension allows for the estimation of analytical gradients and nonadiabatic couplings, which are crucial in many practical situations ranging from the search of conical intersections to the simulation of quantum dynamics, in, for example, photoisomerization reactions. The accuracy of our new implementations is demonstrated on the formaldimine molecule CH 2 NH (a minimal Schiff base model relevant for the study of photoisomerization in larger biomolecules), for which we also perform a geometry optimization to locate a conical intersection between the ground and first-excited electronic states of the molecule.
The accurate calculation of electronic potential energy surfaces for ground and excited states is crucial for understanding photochemical processes, particularly near conical intersections. While classical methods are limited by scaling and quantum algorithms by hardware, this thesis focuses on the State-Averaged Orbital-Optimized Variational Quantum Eigensolver (SA-OO-VQE). This hybrid quantum-classical algorithm provides a balanced description of multiple electronic states by combining quantum state preparation with classical state-averaged orbital optimization. A key contribution is the implementation and evaluation of the Differential Evolution algorithm within the SA-OO-VQE framework, with a comparative study against classical optimizers like the Broyden-Fletcher-Goldfarb-Shanno (BFGS) and Sequential Least Squares Programming (SLSQP) algorithms. The performance of these optimizers is assessed by calculating ground and first excited state energies for H2_22​, H4_44​, and LiH. The thesis also demonstrates SA-OO-VQE's capability to accurately model potential energy surfaces near conical intersections, using formaldimine as a case study. The results show that orbital optimization is essential for correctly capturing the potential energy surface topology, a task where standard methods with fixed orbitals fail. Our findings indicate that while Differential Evolution presents efficiency challenges, gradient-based methods like BFGS and SLSQP offer superior performance, confirming that the SA-OO-VQE approach is crucial for treating complex electronic structures.
Within the present noisy intermediate-scale quantum-computing era, hybrid quantum-classical-processor algorithms have emerged as promising avenues for tackling electronic-structure eigenproblems. Among them, the so-called ensemble-variational quantum eigensolver has been designed to treat ground and excited states on an equal footing and proven effective in capturing features such as conical intersections and avoided crossings between two electronic states, as we recently demonstrated for formaldimine. We also showed on that example how the underlying ensemble-variational principle was prone to provide a quasi-diabatic representation "for free". To date, this method has been limited to computing only two eigenstates of a Hamiltonian. The aim of the present paper is to show how and under what conditions this can be generalized to models that involve three coupled electronic states or more. Our approach relies on designing a parameterized basis transformation that can directly be implemented on a quantum computer for further post-treatment. This nontrivial step is accompanied by the development of quantum circuits specifically adapted to the several states of interest. An algebraic optimization strategy for the parameters of the basis transformation is formulated to obtain the target eigenstates as well as the optimally diabatic states under various objective flavors of the ensemble-variational principle. Our approach was tested for addressing the first three coupled electronic states of the H4+_4^+4+​ molecular ion as a proof of principle, with three electrons in four spatial orbitals, along various geometries.
In this paper, we investigate the electronic structure of methanimine (CH 2 NH) and water (H 2 O) molecules in an effort to locate conical intersections (CIs) using variational quantum algorithms. Our approach implements and compares a range of hybrid quantum-classical methods, including the variational quantum eigensolver (VQE), variational quantum deflation (VQD), VQE with automatically-adjusted constraints (VQE-AC) and we explore molecular configurations of interest using a state-average (SA) approach. Exact diagonalization is employed as the classical benchmark to evaluate the accuracy of the quantum algorithms. We perform simulations across a range of molecular geometries, basis sets and active spaces to compare each algorithm’s performance and accuracy, and to enhance the detectability of CIs. This work confirms the quantum variational algorithms’ capability of describing CIs in both molecules, as long as appropriate active spaces and geometries of the molecule are chosen. We also compare the accuracy and reliability of VQE-based methods for computing excited states with classical benchmark methods, and we demonstrate good agreement within desired regions.
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.
Conical intersections (CIs) are pivotal in many photochemical processes. Traditional quantum chemistry methods, such as the state-average multiconfigurational methods, face computational hurdles in solving the electronic Schrödinger equation within the active space on classical computers. While quantum computing offers a potential solution, its feasibility in studying CIs, particularly on real quantum hardware, remains largely unexplored. Here, we present the first successful realization of a hybrid quantum-classical state-average complete active space self-consistent field method based on the variational quantum eigensolver (VQE-SA-CASSCF) on a superconducting quantum processor. This approach is applied to investigate CIs in two prototypical systems─ethylene (C 2 H 4 ) and triatomic hydrogen (H 3 ). We illustrate that VQE-SA-CASSCF, coupled with ongoing hardware and algorithmic enhancements, can lead to a correct description of CIs on existing quantum devices. These results lay the groundwork for exploring the potential of quantum computing to study CIs in more complex systems in the future.