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 H$_4^+$ molecular ion as a proof of principle, with three electrons in four spatial orbitals, along various geometries.
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+ molecular ion as a proof of principle, with three electrons in four spatial orbitals, along various geometries.
In the present work, we examine how the recent quantum-computing algorithm known as the state-average orbital-optimized variational quantum eigensolver (SA-OO-VQE), viewed within the context of quantum chemistry as a type of multiconfiguration self-consistent field (MCSCF) electronic-structure approach, exhibits a propensity to produce an ab initio quasi-diabatic representation “for free” if considered as a least-transformed block-diagonalization procedure, as alluded to in our previous work (Yalouz, S. et al. J. Chem. Theory Comput. 2022, 18, 776−794) and thoroughly assessed herein. To this end, we introduce intrinsic and residual descriptors of diabaticity and re-explore the definition and linear-algebra properties, as well as their consequences on the vibronic nonadiabatic couplings, of an optimal diabatic representation within this context and how much one may deviate from it. Such considerations are illustrated numerically in the prototypical case of formaldimine, which presents a well-known conical intersection between its ground and first-excited singlet electronic states.
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.
Calculating excited states in chemistry is crucial to providing insight into photoinduced molecular behavior beyond the ground state, enabling innovations in spectroscopy, material sciences, and drug design. While several approaches have been developed to compute excited-state properties, finding the best ratio between the computational cost and accuracy remains challenging. The advent of quantum computers brings new perspectives with the development of quantum algorithms that promise an advantage over classical ones. Most of these new algorithms are inspired by previous classical ones but with different pros and cons. In this work, we focus on the choice of the weights within the ensemble variational quantum eigensolver (VQE) based on the generalization of the variational principle for many-body excited states. We compare the performance of the equi-ensemble and weighted ensemble VQE on two types of problems that cover different correlation regimes: (1) the many-body electronic structure problem of formaldimine using the generalized unitary coupled cluster ansatz and (2) the one-body noninteracting Kohn–Sham problem of hydrogen chains using the R Y CNOT hardware-efficient ansatz. While the equi-ensemble objective is to perform a block-diagonalization only, the weighted ensemble has the additional complexity to reach the targeted eigenstates and eigenvalues. In this work, we show that such a difference leads to several significant consequences, such as the increase in circuit depth together with optimization issues. We conclude that one should always favor the use of equi-weights, even if it requires an additional postprocessing step to extract the eigenvalues of the problem.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) hardware to estimate eigenvalues of many-body Hamiltonians. Unlike fully quantum approaches such as quantum phase estimation (QPE), VQE trades deep coherent circuits for repeated state preparation, measurement, and classical optimization, making it more compatible with limited qubit counts and finite coherence times. Recent developments have focused on reducing quantum resource requirements while retaining chemically meaningful wavefunction structure. In this paper, we examine recent progress in VQE methods for molecular electronic structure with an emphasis on three themes: (i) strategies for circuit and ansatz complexity reduction, including adaptive and selectively screened approaches, (ii) chemically motivated workflows that combine VQE with orbital optimization, fragmentation, and localized active-space ideas to better address strong correlation, and (iii) extensions of VQE to excited-state calculations. Throughout, we emphasize the tradeoffs among parameter count, gate depth, symmetry preservation, measurement overhead, and classical preprocessing, and discuss where these approaches may become most useful for chemically challenging active spaces. We also highlight benchmarking considerations for assessing both accuracy and resource requirements, and conclude with a perspective on regimes in which VQE may offer the greatest long-term value, particularly multireference active spaces and low-lying excited-state manifolds.
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.