Unified adiabatic and diabatic excited-state description via the ensemble-variational quantum eigensolver

Abstract: 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.

• The paper introduces a generalized ensemble-VQE method that unifies adiabatic and diabatic state descriptions for systems with more than two coupled states.
• It employs a parameterized orthogonal transformation with a GUCCSD ansatz to achieve accurate FCI-matching results in nonadiabatic regions.
• The approach simplifies multi-state quantum simulations by generating quasi-diabatic states optimized for gradient and nonadiabatic coupling evaluations.

Unified Adiabatic and Diabatic Excited-State Descriptions via Ensemble-VQE

Introduction and Motivation

The simultaneous description of adiabatic and diabatic electronic states is central to understanding nonadiabatic dynamics, including the photophysics of conical intersections and avoided crossings, which are ubiquitous in both natural and synthetic photochemical processes. Classical electronic-structure methods often struggle with the non-trivial coupling between adiabatic states, especially near regions of strong nonadiabatic coupling (NAC) where the Born–Oppenheimer approximation fails. Diabatic states, which minimize or nullify NACs, are sought after for robust quantum dynamical simulations but are challenging to construct beyond two-state systems and without cumbersome adiabatic-to-diabatic transformations.

Quantum algorithms, particularly variational quantum eigensolver (VQE) and its ensemble/state-averaged extensions, are promising for efficiently representing low-lying eigenstates of many-body Hamiltonians in the NISQ era. The ensemble-VQE (also termed state-averaged VQE) treats multiple states on an equal footing and has shown an inherent capability to provide quasi-diabatic states “for free” in two-state models. The present work systematically extends ensemble-VQE to construct unified adiabatic and diabatic representations for systems involving three (or more) coupled states and details the quantum circuit, algebraic optimization, and theoretical foundations for multi-state excited-state quantum simulations.

Ensemble-VQE Extension and Methodological Advances

The authors generalize the ensemble-VQE framework to compute more than two coupled excited states by embedding a parameterized orthogonal transformation—directly implementable on quantum circuits—between an initial chemically-informed model space (built from Slater determinants or configuration state functions) and the target eigenstate subspace of the Hamiltonian. The generalized approach entails:

• Qubit Mapping and State Preparation: Electronic structure encoded to qubits via Jordan–Wigner mapping. Chemically-motivated initial states are used as the ensemble basis. For the H$_4^+$ ion, the model comprises three singly-occupied doublet Slater determinants targeting the first three eigenstates.
• Expressible Unitary Ansatz: Generalized unitary coupled-cluster single-double (GUCCSD), Trotterized to sufficient depth (two repetitions in practice here), ensures access to the entire relevant FCI manifold, preserving orthonormality and total $M_S$.
• Adiabatic State Extraction: After block-diagonalizing the Hamiltonian in the minimal ensemble-energy subspace using ensemble-VQE, a parameterized real orthogonal transformation (Euler-angles-based) is variationally optimized to obtain true eigenstates (adiabatic), either in a post-processing (classical) or a pre-processing (retro-variational) mode, the latter being directly implementable on a quantum device.
• Diagonalization Objective Flavors: Several mathematically sound objective functions for optimizing the orthogonal rotation are discussed—Frobenius norm minimization, extremization of ensemble-weighted energies, and use of partial derivatives linked to off-diagonal Hamiltonian matrix elements. Additionally, ensemble extensions of the Rayleigh–Ritz variational principle are leveraged with flexible weighting of state contributions.

Quasi-Diabatic State Construction and Optimality

Recognizing that the adiabatic-to-diabatic transformation is not uniquely defined and often not globally achievable, especially for more than two states, the present framework incorporates a direct quantum-compatible strategy for generating quasi-diabatic states:

• Diabatic Model States: The initial ensemble states are constructed in a “diabatic” MO basis using an orthogonal Procrustes alignment to a reference geometry, maximizing overlaps and ensuring smoothness with respect to nuclear geometry changes.
• Optimal Projections: After ensemble-variational minimization, the overlap matrix between model and target states is subjected to singular value decomposition (SVD). Optimal quasi-diabatic states are obtained by further orthogonal rotation, minimizing the Frobenius norm distance from identity in the overlap, i.e., finding the least-transformed projections.
• Numerical Descriptors: Two key numerical descriptors are introduced—$d$ (minimally achievable distance to the identity overlap in finite dimensions) and $r$ (residual, fully removable via an optimal rotation)—quantifying diabaticity.
• Constrained Optimization: Direct constraints (via Lagrange multipliers or SLSQP) ensure optimal spin and diabaticity properties within the ensemble-VQE parametrization, with the algorithm favoring solutions closest to original model states ("lazy" paths) but finalized for true optimality.

Numerical Validation and Circuit Implementation

Validation is performed on the prototypical H$_4^+$ molecular ion (3 electrons in 4 orbitals, STO-3G), a minimal testbed for three near-degenerate adiabatic states relevant to ultrafast nonadiabatic processes.

• Full Configuration Interaction Baseline: Reference FCI calculations benchmark the quantum algorithmic results, providing exact adiabatic and (symmetry-adapted) diabatic potential energy curves.
• Quantum Circuit Realization: Explicit quantum circuits for the $3 \times 3$ orthogonal transformations required to rotate between basis states are constructed efficiently using sequences of single- and two-qubit gates parameterized by Euler angles.
• State and Energy Tracks: The ensemble-VQE successfully accesses the minimal-energy subspace, and subsequent basis rotation extracts the adiabatic states. The energies and state overlaps fully match FCI benchmarks after rotation; numerical stability and physical smoothness are demonstrated along nuclear deformation paths.
• Diabatic Representation: Without further intervention, the initial block-diagonalization offers quasi-diabatic states that are close to optimal but may lack smoothness in multi-state (three or more) cases. Incorporation of the explicit $r$-minimization rotation ensures the diagonal energies become smooth functions of nuclear geometry, and overlap matrices approach symmetry and near-unity diagonal values, demonstrating effective diabatization.

Implications and Future Perspectives

This unified formalism demonstrates that ensemble-VQE, when equipped with modular parameterized basis transformations, enables consistent and simultaneous access to both adiabatic and diabatic excited-state manifolds on quantum devices for arbitrary numbers of coupled states. This removes the need for explicit NAC-driven adiabatic-to-diabatic transformation procedures and provides robust “diabaticization for free” within quantum-resource-efficient variational frameworks.

The direct implications are:

• Nonadiabatic Quantum Simulation Readiness: The approach directly supports quantum simulation of mixed quantum-classical and fully quantum nonadiabatic dynamics, as needed in photochemistry and ultrafast molecular sciences.
• Analytical Gradient and NAC Accessibility: The construction allows for gradient and NAC evaluation within both adiabatic and (optimized) diabatic representations, a necessity for quantum molecular dynamics and trajectory-surface hopping methodologies.
• Manifest Gauge Invariance and Flexibility: The explicit parameterization of gauge degrees of freedom ensures numerical robustness, seamless extensibility to arbitrary ensemble sizes, and compatibility with state-averaged, weighted, or adaptive ensemble objectives.
• Quantum Circuit Efficiency: The work provides algorithmic and circuit designs suitable for state-of-the-art and future quantum hardware, balancing expressibility and depth required for multi-state problems.

Potential future developments include: systematic optimization of ansatz expressibility for larger state manifolds, further studies of the properties of the rotation/minimization landscapes, and scaling and hardware experiments with larger electronic and nuclear configuration spaces.

Conclusion

The paper establishes a unified, quantum-computable framework for treating adiabatic and quasi-diabatic excited states via generalized ensemble-VQE, with rigorous mathematical, algorithmic, and circuit-level solutions for multi-state systems. The methodology supports robust, scalable, and physically transparent excited-state quantum dynamics, substantially advancing the applicability of quantum algorithms in molecular science, especially for systems with complex nonadiabatic couplings.