Unified adiabatic and diabatic excited-state description via the ensemble-variational quantum eigensolver
Christophe Soule, Bruno Senjean, Benjamin Lasorne
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 molecular ion as a proof of principle, with three electrons in four spatial orbitals, along various geometries.
AI Impact Assessments
(3 models)Scientific Impact Assessment
Core Contribution
This paper extends the ensemble-variational quantum eigensolver (ensemble-VQE) framework from two electronic states to three (and in principle more) coupled states, with simultaneous capability to produce both adiabatic eigenstates and optimally quasi-diabatic representations. The key innovations are: (1) a parameterized SO(3) basis transformation decomposed into three elementary rotations that can be implemented directly on quantum circuits, (2) quantum circuit designs for preparing rotated initial states among three Slater determinants, (3) multiple algebraic optimization strategies for the rotation parameters (Frobenius norm, partial derivative, and ensemble-variational approaches), and (4) a rigorous strategy for achieving optimal quasi-diabatic states by minimizing a removable diabaticity descriptor *r*. The proof-of-concept is demonstrated on the H₄⁺ molecular ion (3 electrons in 4 spatial orbitals) along C₁ symmetry-breaking coordinates.
Methodological Rigor
The theoretical framework is carefully constructed. The parameterization of the similarity transformation via Rodrigues' formula and the Euler-angle-like decomposition into three rotations is mathematically sound. Three distinct objective functions for the eigenstate resolution are derived and compared, with analytical conditions connecting partial derivatives to off-diagonal Hamiltonian elements (Eqs. 25-28). The diabaticity framework, building on prior work, properly distinguishes between the irreducible descriptor *d* (reflecting MO-level unavoidable variations) and the removable descriptor *r* (reflecting CI-level mixing within the target subspace).
However, there are notable methodological concerns. The work uses a classical state-vector emulator rather than actual quantum hardware, sidestepping noise effects entirely. The system studied is extremely small (3 electrons, 4 spatial orbitals, STO-3G basis), making it unclear how the approach scales. The need for a twofold Trotterized GUCCSD ansatz to achieve expressibility, acknowledged but not analyzed in depth, raises questions about circuit depth requirements for larger systems. The spin-constraint handling via inequality constraints (Eq. 6) rather than equality constraints, chosen for numerical stability, is pragmatic but somewhat ad hoc. The paper also acknowledges that the "lazy optimizer" heuristic explaining near-diabatic convergence in the two-state case does not generalize well to three states, necessitating the explicit optimization procedure.
Potential Impact
The paper addresses an important interface between quantum computing and photochemistry/nonadiabatic dynamics. The ability to produce quasi-diabatic representations directly from quantum algorithms—without the notoriously difficult adiabatic-to-diabatic transformation via NAC integration—could be significant for:
1. Nuclear wavepacket dynamics: Quasi-diabatic representations are essential for MCTDH and related methods, where singular NACs in the adiabatic basis cause severe numerical difficulties.
2. Trajectory surface hopping: While TSH typically works in the adiabatic representation, having both representations available provides flexibility.
3. Quantum-classical nonadiabatic dynamics: The approach opens pathways to hybrid quantum-classical molecular dynamics where the electronic structure is solved on quantum computers.
The generalization from two to three states is nontrivial and practically important, as many photochemical processes involve three or more coupled states (e.g., through higher-lying conical intersections). The mathematical framework for the rotation parameterization is generalizable to N states, though circuit design complexity would grow.
Timeliness & Relevance
The work is well-positioned at the intersection of two active fields: NISQ-era quantum algorithms and computational photochemistry. The demand for excited-state methods on quantum computers is growing, and the diabatic representation capability distinguishes this work from most competing approaches. The paper cites and builds upon a rapidly growing literature on excited-state quantum algorithms (VQE variants, quantum subspace methods, etc.), and the 2025-2026 references indicate the field's currency.
However, the practical utility remains limited by current quantum hardware capabilities. The eight-qubit circuits shown are far from the scale needed for chemically interesting photochemical problems. The real impact may be more theoretical/methodological at this stage—establishing that the mathematical and algorithmic framework exists and works in principle.
Strengths
Limitations
Overall Assessment
This is a methodologically sound extension of the ensemble-VQE framework to three-state problems with a novel quasi-diabatization capability. The theoretical contributions are solid, but the practical impact is currently limited by the minimal scale of demonstration and absence of noise studies. The work is incremental relative to the group's previous two-state results but addresses a genuinely nontrivial generalization. Its long-term significance depends on whether the approach can scale to chemically meaningful systems on future quantum hardware.
Generated Apr 21, 2026
Comparison History (57)
Paper 1 extends the ensemble-variational quantum eigensolver to three or more coupled electronic states, addressing a fundamental challenge in quantum chemistry (diabatic/adiabatic representations, conical intersections) on quantum computers. This has broad implications for computational chemistry and materials science. Paper 2 presents DART-Q, a practical framework for real-time QLDPC decoding under operational constraints—important for fault-tolerant quantum computing engineering but more narrowly focused on decoder scheduling/systems design. Paper 1's methodological contribution to excited-state quantum chemistry is more novel and has broader scientific reach across multiple disciplines.
Paper 2 addresses a foundational bottleneck in variational quantum algorithms (trainability and optimization landscapes) by moving beyond gate-level abstractions to pulse-level control. This fundamentally alters the optimization landscape, offering broader applicability across quantum machine learning and variational algorithms compared to Paper 1, which provides a highly specialized, domain-specific extension of VQE for molecular excited states.
Paper 1 likely has higher impact due to timeliness and applicability: extending ensemble-variational VQE to >2 coupled excited states directly addresses a key limitation in NISQ-era quantum chemistry, with clear relevance to photochemistry (conical intersections/diabatization) and broader electronic-structure workflows. It includes implementable circuit constructions and an optimization strategy, suggesting methodological depth and near-term experimental/use potential. Paper 2 offers elegant exact SU(2) wave solutions, but such ansatz-based exact families may have narrower immediate applicability and uncertain downstream use compared with a scalable quantum-algorithm advance.
Paper 1 addresses a timely and practically important problem in quantum computing—extending the ensemble-variational quantum eigensolver to three or more coupled electronic states with both adiabatic and diabatic descriptions. This has broad real-world applications in quantum chemistry (photochemistry, nonadiabatic dynamics) and is directly relevant to the NISQ era. Paper 2 makes rigorous contributions to quantum hypothesis testing theory with optimal error exponents, but its impact is more niche within quantum information theory. Paper 1's combination of algorithmic innovation, quantum circuit design, and chemical applications gives it broader cross-disciplinary impact.
Paper 2 explores fundamental quantum information dynamics, uncovering a novel phase with volume-law entanglement and rapid purification without scrambling. This conceptual breakthrough in measurement-induced phase transitions offers broad implications for quantum state preparation and many-body physics. In contrast, Paper 1 presents an incremental, though useful, algorithmic extension of the ensemble-VQE for quantum chemistry, which is currently limited to small proof-of-principle models. Therefore, Paper 2 demonstrates higher methodological innovation and broader potential impact across quantum science.
Paper 2 addresses a fundamental problem in quantum physics—approaching the theoretical limit of clock precision—with broad implications for metrology, quantum foundations, and practical timekeeping applications. Its results (achieving the fundamental precision-resolution bound with physically realistic systems and a robust quench protocol) represent a significant advance with cross-disciplinary relevance. Paper 1, while technically solid in extending ensemble-variational quantum eigensolvers to three or more states, represents a more incremental methodological advance within the specialized niche of quantum chemistry on near-term quantum computers, with narrower impact scope.
Paper 1 addresses a fundamental question in quantum physics—achieving the theoretical limit of clock precision—with a concrete blueprint using physically realistic systems. Its novelty (achieving the precision-resolution trade-off bound with time-independent interactions and a robust quench protocol) has broad implications for metrology, quantum foundations, and practical timekeeping. Paper 2, while a solid methodological advance in quantum computing for excited states, is more incremental (extending from 2 to 3+ states) and narrower in scope, targeting a specific computational chemistry application within the NISQ era that may have limited longevity.
Paper 2 presents a practical advancement for near-term quantum computing by generalizing the ensemble-VQE to three or more excited states. This has direct, high-impact applications in quantum chemistry and materials science for simulating complex molecular dynamics and photochemical reactions. While Paper 1 offers deep theoretical insights into quantum entanglement and phase transitions, Paper 2's focus on implementable quantum circuits for NISQ devices provides a more immediate and tangible real-world impact across applied quantum technologies.
Paper 1 offers a highly timely and practical advancement for the NISQ era by generalizing the ensemble-variational quantum eigensolver to handle three or more excited states. This directly addresses a major bottleneck in quantum chemistry simulations on near-term quantum hardware, unlocking potential real-world applications in modeling complex photochemical processes and conical intersections. While Paper 2 provides valuable fundamental insights into cavity QED and quantum Rabi models, Paper 1 has a broader and more immediate impact trajectory across the rapidly growing fields of quantum algorithms and computational chemistry.
Paper 2 introduces a fundamentally novel concept—quantum synchronizing automata—bridging classical automata theory with quantum mechanics in a creative way. It addresses a conceptual tension (irreversibility vs. unitarity) with an elegant solution using auxiliary qubits, opening new research directions in quantum computation, quantum information theory, and entanglement generation. Paper 1, while technically solid, represents an incremental extension (from 2 to 3+ states) of an existing method (ensemble-VQE) within the relatively narrow domain of quantum chemistry on NISQ devices. Paper 2's broader conceptual novelty gives it higher cross-disciplinary impact potential.
Paper 1 introduces a broadly applicable, scalable Hessian-vector product kernel for arbitrary linear-map compositions in tensor networks, enabling practical second-order optimization without forming Hessians. This is methodologically innovative and likely to impact many areas using tensor networks (quantum simulation, ML, condensed matter), with immediate performance gains shown via trust-region circuit compression. Paper 2 extends an existing EVQE framework from two to multiple excited states with tailored circuits, valuable for quantum chemistry but demonstrated on a small proof-of-principle system and more constrained to NISQ electronic-structure workflows. Overall, Paper 1 has wider cross-field reach and nearer-term utility.
Paper 1 provides fundamental theoretical limits and optimal measurement strategies for an experimental technique already seeing active use in diverse fields like microscopy and bio-sensing. Its immediate applicability to existing setups and broad real-world utility gives it higher potential impact compared to Paper 2, which offers an algorithmic extension for noisy intermediate-scale quantum (NISQ) computers whose practical large-scale applications remain constrained by hardware.
Paper 2 advances a fundamental quantum chemistry method (ensemble-variational quantum eigensolver) by generalizing it from two to three or more electronic states, addressing conical intersections and diabatic/adiabatic representations—core challenges in computational chemistry. This has broad implications for photochemistry, molecular dynamics, and materials science. Paper 1, while practically useful, addresses a narrower meta-learning optimization problem for quantum kernel circuit selection with incremental contributions. Paper 2's methodological depth, novel quantum circuit design, and relevance to both quantum computing and chemistry give it higher potential impact.
Paper 1 addresses quantum sensing with undetected photons—a rapidly growing experimental field—by providing the first comprehensive multiparameter quantum estimation framework with clear practical guidance for optimizing experiments. It fills an important theoretical gap for a technique with broad applications (spectroscopy, microscopy, bio-sensing) and provides actionable results (optimal measurement strategies, multipass scaling). Paper 2 extends the ensemble-variational quantum eigensolver to three+ states, which is a meaningful but incremental advance in quantum computing for chemistry, limited to small proof-of-principle systems within the NISQ era's inherent constraints.
Paper 1 is likely higher impact: it advances a core quantum chemistry capability by generalizing ensemble-VQE from two to three+ coupled excited states, enabling unified adiabatic/diabatic descriptions relevant to photochemistry (conical intersections, nonadiabatic dynamics). This is a substantive methodological extension with clear scientific applications and potential cross-field influence (quantum algorithms + electronic structure theory). Paper 2 is useful for NISQ quantum ML workflow automation, but its contribution is more incremental (model selection via classical meta-features) and may be narrower and more dependent on specific benchmarks/candidate sets.
Paper 1 addresses a fundamental bottleneck in Quantum Machine Learning by significantly reducing the sampling overhead using amplitude estimation. This provides a clear algorithmic advantage (quadratic improvement in sampling error) that can broadly impact QNN training and inference across many applications. Paper 2, while methodologically rigorous, offers a more incremental advance (scaling from two to three or more states) within the specific niche of quantum chemistry and electronic-structure problems.
Paper 2 likely has higher impact: it targets a core bottleneck in QNNs (sampling cost) with a broadly applicable algorithmic primitive (amplitude estimation) offering asymptotic improvement from O(1/√N) to O(1/N). This could affect many quantum ML/QNN architectures and near-/mid-term workflows, with cross-field relevance (quantum algorithms, ML, hardware-efficient inference). Paper 1 is technically novel and rigorous for quantum chemistry excited states, but its impact is narrower (specialized electronic-structure regime, small proof-of-principle system) and more contingent on scalable fault-tolerant resources.
Paper 2 addresses a critical bottleneck in the highly active field of quantum computing for quantum chemistry: simulating multiple excited states on NISQ devices. Its extension of the EVQE to three or more states directly impacts molecular simulations, materials science, and photochemistry. While Paper 1 provides an interesting mathematical bridge between quantum control and classical mechanics, Paper 2 aligns with a broader, heavily invested technological frontier with transformative potential across multiple scientific disciplines.
Paper 1 targets a key bottleneck in quantum networking: interfacing broadband, short telecom photons with narrowband quantum memories via integrated resonant frequency conversion and extreme bandwidth compression. If realized, this directly enables scalable quantum repeaters/interconnects with broad cross-field impact (integrated photonics, nonlinear optics, quantum communication). Paper 2 extends an existing NISQ algorithm to more excited states with solid methodological development, but its near-term impact is narrower (quantum chemistry simulation on limited hardware) and more incremental relative to the fundamental systems-level need addressed by Paper 1.
Paper 1 presents a novel algorithmic advancement in quantum computing for electronic structure calculations, extending the ensemble-variational quantum eigensolver to three or more coupled electronic states with diabatic representations. This addresses a fundamental challenge in quantum chemistry with clear methodological innovation and theoretical depth. Paper 2, while useful for software engineering practices, is an empirical bug study with more incremental, field-specific impact limited to quantum software development practices. Paper 1's contributions to quantum algorithms and computational chemistry have broader and longer-lasting scientific impact across physics, chemistry, and quantum computing.