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arxiv: 2604.17317 · v2 · pith:HL4XHPREnew · submitted 2026-04-19 · 🪐 quant-ph

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

Christophe Soule , Bruno Senjean , Benjamin Lasorne This is my paper

Pith reviewed 2026-05-21 08:54 UTC · model grok-4.3

classification 🪐 quant-ph
keywords ensemble variational quantum eigensolverexcited electronic statesdiabatic representationquantum computingH4+ molecular ionadiabatic stateselectronic structureNISQ algorithms
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The pith

A parameterized basis transformation run on a quantum computer extends the ensemble variational eigensolver to three or more coupled electronic states while also producing optimally diabatic representations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to generalize an existing quantum-classical algorithm so that it can treat three or more electronic states of a molecule on equal footing. The key step is a basis transformation whose parameters are optimized algebraically and executed directly on the quantum processor. This produces both the usual energy eigenstates and a set of states with improved diabatic character under different versions of the ensemble variational principle. A reader would care because the approach offers a single calculation that captures avoided crossings and conical intersections without separate runs for each state. The method is demonstrated on the first three states of the H4+ ion along several geometries.

Core claim

The authors formulate a parameterized basis transformation that can be implemented with quantum circuits adapted to multiple states. An algebraic optimization strategy for its parameters yields the target eigenstates together with optimally diabatic states under various ensemble-variational objectives. The construction is tested as a proof of principle on the first three coupled electronic states of the H4+ molecular ion with three electrons in four spatial orbitals.

What carries the argument

A parameterized basis transformation implemented on a quantum computer that mixes the states to obtain both the energy eigenstates and the optimally diabatic representations.

If this is right

  • The ensemble variational quantum eigensolver now applies to models involving three or more coupled electronic states.
  • Both adiabatic eigenstates and optimally diabatic states are obtained from the same parameterized transformation.
  • Different flavors of the ensemble variational objective can be chosen to tune the diabatic character.
  • The approach has been verified for the first three states of H4+ across multiple molecular geometries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same transformation technique could be combined with existing quantum error mitigation methods to reach larger active spaces.
  • Photochemical reaction paths might be simulated by following the diabatic states produced in a single run rather than stitching separate adiabatic calculations.
  • The algebraic optimization step suggests a route to parameter-free extensions for even higher numbers of states.

Load-bearing premise

An algebraic optimization of the basis transformation parameters will reliably deliver accurate eigenstates and diabatic states without uncontrolled errors or molecule-specific post-hoc fixes.

What would settle it

Applying the algorithm to the first three states of H4+ and finding that the resulting energies deviate from classical reference values or that the diabatic states exhibit abrupt changes along a geometry path.

Figures

Figures reproduced from arXiv: 2604.17317 by Benjamin Lasorne, Bruno Senjean, Christophe Soule.

Figure 1
Figure 1. Figure 1: (a) Distortion from the tetrahedral Td geometry of H+ 4 to C1 geometries. (b) Corresponding adiabatic potential energies of the first three eigenstates, E0,E1,E2, along ∆z1 (from ∆x2 = 0.1 and ∆y3 = 0.05) obtained from a ’restricted open-shell Hartree–Fock’ (ROHF) FCI/STO-3G calculation with Psi4 [99]. culty as regards our various procedures (not shown herein). A detail account of those will be provided in… view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuits for preparing the state [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Energies of the ensemble-VQE states: (a) [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Diabaticity descriptors r and d. ements of the overlap matrix along the deformation coordinate, as shown in Fig. 4b. Yet, the relatively large values of the di￾agonal terms indicate that the optimization procedure leads, as expected, to target states that remain relatively close to the initial model states (the “guess”). In both cases, the overlap matrices are not symmetric, and for certain pairs of off-di… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Off-diagonal Hamiltonian matrix elements in the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to generalize the ensemble-variational quantum eigensolver (eVQE) from two to three or more coupled electronic states by introducing a parameterized basis transformation that is implemented on a quantum computer via adapted circuits, together with an algebraic optimization strategy over the transformation parameters. This is intended to yield both the target eigenstates and optimally diabatic representations under various ensemble-variational objectives. The approach is presented as a proof-of-principle demonstration on the first three electronic states of H4+ (three electrons, four orbitals) along selected geometries.

Significance. If the algebraic optimizer and circuit adaptations prove robust and free of uncontrolled system-dependent tuning, the work would meaningfully extend eVQE applicability to multi-state problems involving avoided crossings and conical intersections while retaining the quasi-diabatic representation benefit. The explicit construction of a quantum-implementable basis transformation is a concrete technical step that could be reused in other hybrid algorithms.

major comments (2)
  1. [Abstract] Abstract and proof-of-principle section: no quantitative results, error bars, convergence data, or circuit-depth/resource estimates are supplied for the H4+ test case; without these the central generalization claim cannot be verified and the load-bearing performance of the algebraic optimizer remains unassessed.
  2. [Optimization Strategy] Algebraic optimization strategy (described after the basis-transformation definition): the procedure is asserted to locate the desired stationary point for both eigenstates and diabatic frames, yet the manuscript does not demonstrate that the optimizer avoids multiple roots or requires manual root selection; if such selection is needed, the claim of a reliable, parameter-free generalization is weakened.
minor comments (2)
  1. [Methods] Clarify how the ensemble-variational objective functionals are modified when moving from two to three or more states and whether any additional constraints are introduced.
  2. [Quantum Circuits] Provide at least one explicit circuit diagram or gate count for the adapted multi-state ansatz to allow reproducibility assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the paper to strengthen the presentation of the H4+ results and the optimization procedure.

read point-by-point responses

  1. Referee: [Abstract] Abstract and proof-of-principle section: no quantitative results, error bars, convergence data, or circuit-depth/resource estimates are supplied for the H4+ test case; without these the central generalization claim cannot be verified and the load-bearing performance of the algebraic optimizer remains unassessed.

    Authors: We agree that the current proof-of-principle demonstration would be strengthened by the inclusion of quantitative metrics. In the revised manuscript we will add error bars obtained from repeated optimizations, convergence data for the algebraic optimizer across the sampled geometries, and explicit circuit-depth and gate-count estimates for the adapted basis-transformation circuits. These additions will allow a clearer assessment of the method's performance on the three-state H4+ model. revision: yes

  2. Referee: [Optimization Strategy] Algebraic optimization strategy (described after the basis-transformation definition): the procedure is asserted to locate the desired stationary point for both eigenstates and diabatic frames, yet the manuscript does not demonstrate that the optimizer avoids multiple roots or requires manual root selection; if such selection is needed, the claim of a reliable, parameter-free generalization is weakened.

    Authors: The algebraic optimizer solves the stationarity conditions of the chosen ensemble-variational objective directly. For the H4+ test cases examined, the procedure consistently yields the physically relevant solutions corresponding to the target eigenstates and diabatic frames. In the revision we will include a brief numerical check (or supplementary table) confirming that, for the geometries and objective flavors considered, no manual root selection is required. Should isolated cases arise where multiple stationary points exist, we will explicitly state the selection criterion used. revision: partial

Circularity Check

1 steps flagged

Minor self-citation to prior two-state eVQE work; new basis transformation and algebraic optimizer add independent content for three-state generalization.

specific steps
  1. self citation load bearing [Abstract]
    "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."

    The generalization claim references the authors' own prior demonstration of quasi-diabatic representation 'for free' as the starting point, but the new algebraic optimization and basis transformation are presented as independent developments rather than a direct reduction to the cited result.

full rationale

The paper extends the authors' earlier two-state ensemble-VQE results (cited in the abstract for formaldimine) to three or more states via a new parameterized basis transformation and algebraic optimization strategy. This extension is formulated explicitly with adapted quantum circuits and tested on H4+, without reducing the central claims to a fit or self-definition by construction. The prior work supports the two-state foundation but is not load-bearing for the generalization step, which introduces new elements that can be evaluated independently. No predictions are shown to be equivalent to inputs by the paper's own equations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract provides limited technical detail; ledger entries are inferred from stated approach and prior context.

free parameters (1)
  • parameters of the basis transformation
    Optimized algebraically to obtain target eigenstates and diabatic states; exact number and fitting procedure not specified in abstract.
axioms (1)
  • domain assumption The ensemble-variational principle extends to three or more states while preserving quasi-diabatic properties.
    Invoked as the foundation for the generalization from two-state case.

pith-pipeline@v0.9.0 · 5786 in / 1236 out tokens · 38629 ms · 2026-05-21T08:54:42.868597+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Workflow of ensemble-VQE. Starting from a numerical one-body basis of real ‘atomic or- bitals’ (AOs){χ µ (r)}Norb µ=1, a set of ‘molecular orbitals’ (MOs) {ϕp(r)}Norb p=1 is constructed as ‘linear combinations of atomic orbitals’ (LCAO). Within an FCI approach (our focus herein), the LCAO coefficients are typically obtained from an initial Hartree–Fock ca...

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  2. [2]

    Defining the qubit-based representations of the electronic Hamiltonian and of the Slater determinants.To implement the electronic Hamiltonian ˆHel within a hybrid quantum-classical approach, the Jordan–Wigner transformation is applied to map the second-quantized Hamiltonian onto the qubit space, where it is usually written as a ‘linear combination of unit...

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    Each of them correspond to a single Slater determinant (encoded as a basic ONV)

    Initialization.The second step of the method consists in preparing three orthonormal initial states on the quantum com- puter, denoted Φ0 A , Φ0 B , and Φ0 C (the model, or “guess”). Each of them correspond to a single Slater determinant (encoded as a basic ONV). This can be generalized to ‘configuration state functions’ (CSFs) of given total spinS. The i...

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  4. [4]

    Here, we used as an ansatz the Trotterized – also called disen- tangled – ‘generalized unitary coupled-cluster single double’ (GUCCSD) [77]

    State preparation.A parameterized unitary operator, ˆU(t), referred to as the ‘ansatz’ is then applied to each of these initial states, leading to multideterminantal states, |ΦA(t)⟩ = ˆU(t) Φ0 A , |ΦB(t)⟩ = ˆU(t) Φ0 B , and |ΦC(t)⟩ = ˆU(t) Φ0 C , that depend on the ensemble-variational parameterst. Here, we used as an ansatz the Trotterized – also called ...

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  5. [5]

    Quantum measurements.To estimate the expectation val- ues of the electronic Hamiltonian ˆHel with respect to the pre- pared states, denotedH AA(t) = ⟨ΦA(t)| ˆHel |ΦA(t)⟩,H BB(t) = ⟨ΦB(t)| ˆHel |ΦB(t)⟩, andH CC(t) = ⟨ΦC(t)| ˆHel |ΦC(t)⟩, respec- tively, statistical sampling is performed through a series of shots on the quantum computer (or deterministicall...

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    Herein, the parameters have been initialised to zero

    Classical optimization.The previous steps are intended to have been performed on a quantum computer for a given set of parameterstprovided by a classical computer. Herein, the parameters have been initialised to zero. Then, in order to de- termine the subspace of minimal ensemble-energy, we invoke the ensemble extension of the Rayleigh–Ritz variational pr...

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    A note on classicalvs.quantum computing. It must be understood that, in the above setting, the ensemble- energy minimization procedure essentially aims at mimicking the result of a(3×3)-block-diagonalization of the many-body Hamiltonian(N det ×N det)-matrix. Indeed, the minimum of the objective function,E ens(t∗), is simply the subtrace of the target bloc...

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    pre-eigenstates

    Taking the spin in to account. Since we are not considering spin-orbit coupling herein, the eigenstates of ˆHel are also eigenstates of ˆS2 and ˆSz. A single Slater determinant is always an eigenstate of ˆSz, but it is not generally an eigenstate of ˆS2. Therefore, an arbitrary linear com- bination of Slater determinants is not necessarily an eigenstate o...

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    post-variational

    Diagonalizingviaan orthogonal similarity transformation: toward adiabatic eigensolutions For any matrix ˘H∈S 3(R), there exists a nonunique orthogonal matrix ˘Q∈O(3)(the group of orthogonal(3×3)-matrices) [85] such that ˘H7→ ˘H ′ = ˘Q⊤ ˘H ˘Q,(8) where ˘H ′ is real-diagonal. Such an orthogonal similarity trans- formation map applied to the variationally-op...

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    (11), associated with a rotation matrix ˘Q, in a parameterized form that can be translated into quantum gates

    Parameterization of the similarity transformation In order to implement the retro-variational approach, it is neces- sary to express the transformation in Eq. (11), associated with a rotation matrix ˘Q, in a parameterized form that can be translated into quantum gates. The mathematical procedure to achieve this, inspired by Ref. 86, consists in expressing...

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    From this per- spective, it might appear natural,a priori, to minimize a func- tion constructed from the off-diagonal elements of ˘H ′(θ,φ,ψ)

    Various flavors of the eigenproblem objective Obtaining the eigenstates relies on determining the an- gles(θ,φ,ψ)for which the matrix ˘H ′(t∗,θ,φ,ψ), denoted ˘H ′(θ,φ,ψ)hereafter for simplicity, is diagonal. From this per- spective, it might appear natural,a priori, to minimize a func- tion constructed from the off-diagonal elements of ˘H ′(θ,φ,ψ). Howeve...

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    as diabatic as pos- sible

    Diabatic molecular orbitals and Slater determinants The model states,{ Φ0 I }I∈{A,B,C}, which are Slater determi- nants (or CSFs if relevant), can be made “as diabatic as pos- sible” through a suitable least-changing choice of the underly- ing MO basis with respect toR. This construction relies on the concept of ‘diabatic orbitals’, as first defined by We...

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    optimal diabati- zation

    Least-transformation diabaticity criterion If we were solving the corresponding block-diagonalization problem with a classical-computing approach, our aim would be to determine the first three columns of the many-body overlap (Ndet ×N det)-matrixO(t ∗), which is orthogonal. The model-to- target(3×3)-submatrix has already been denoted ˘O(t∗). Of course, it...

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    as lazily as possible

    Descriptors of diabaticity Our objective is to obtain a particular value oft∗ such that ˘O(t∗) is as close as possible to1. In order to quantify the deviation from this condition, two descriptors were introduced in Ref. 23. Their definitions are recalled below. They are both based on the ‘singular value decomposition’ (SVD) of˘O(t∗), which reads ˘O(t∗) = ...

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    Variationally exact

    Achieving optimal diabaticity A way to achieve optimal diabaticity,i.e., to turn the value of the removable diabatic descriptorr(t ∗)to zero, is to incorporate it directly as a penalty term in the objective function. However, un- like in the case of spin, our numerical experiments indicate that a direct choice ofλ=1 does not yield satisfactory results, es...

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    for free

    After determining the subspace of minimal ensemble-energy In what follows, we chose a C s reference geometry,R 0, for defining the diabatic orbitals [23], such that∆x 2 =0.1 and ∆z1 =−0.1. This small distortion from T d allows us to clearly discriminate the three singly-occupied orbitals (t2-degenerate in Td), hence to fix the choice of the three singly-e...

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    6a, it is clear that the diabatic energies obtained af- ter the minimization of the descriptorrby a change of basis (as discussed at the end of Sec

    After descriptor r minimization From Fig. 6a, it is clear that the diabatic energies obtained af- ter the minimization of the descriptorrby a change of basis (as discussed at the end of Sec. II D) are smooth over the entire ge- ometry profile, as expected. The corresponding values ofrare close to zero (it remains below 2×10 −8 for all values of∆z 1), sugg...

    work page arXiv 2030