Excited States from ADAPT-VQE convergence path in Many-Body Problems: application to nuclear pairing problem and $H_4$ molecule dissociation

Denis Lacroix; Jing Zhang

REVIEW 3 major objections 6 minor 2 cited by

Excited States from ADAPT-VQE convergence path in Many-Body Problems: application to nuclear pairing problem and $H_4$ molecule dissociation

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that the intermediate states generated during ADAPT-VQE's descent to the ground state span a subspace whose diagonalization gives accurate low-lying excited states, with only a small overhead beyond the ground-state…

desk verdict Useful, honest demonstration that ADAPT-VQE path states can seed a QSD subspace for low-lying states in the tested sectors, but the abstract overclaims; the key premise is unproven and H4 shows symmetry-sector failures without a state-averaged add-on. read the letter →

arxiv 2506.22275 v1 pith:UN35GXJV submitted 2025-06-27 quant-ph nucl-th

Jing Zhang , Denis Lacroix This is my paper

classification quant-phnucl-th MSC 81P68 PACS 03.67.Ac03.67.Lx

keywords ADAPT-VQEquantumsubspacediagonalizationexcitedstatesnuclearpairingneutron-protonH4dissociationmany-bodysystemslow-lyingspectra

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's claim is that the intermediate states produced while ADAPT-VQE, an adaptive variational quantum eigensolver that builds its ansatz step by step from an operator pool, descends toward the ground state are not just a computational by-product: their span is a compact trial space for low-lying excited states. Solving a generalized eigenvalue problem for the Hamiltonian in this span reproduces accurate ground- and excited-state energies for like-particle and neutron-proton pairing and for H4 dissociation, with a small additional quantum cost beyond the ground-state calculation. This matters because excited-state spectra are normally a separate and more expensive task, while here they come almost for free from the convergence history. The paper also shows that the same subspace diagonalization improves the ground-state estimate obtained from single-parameter optimization, and that symmetry sectors missed by the descent can be recovered by additional starting states or a state-averaged variant.

What carries the argument

The machinery is a quantum subspace diagonalization built from the path states: one forms the matrices $H_{lk}=\langle l|H|k\rangle$ and $O_{lk}=\langle l|k\rangle$ and solves $H c = E O c$. The matrix elements are measured with Hadamard-test circuits that evaluate $\langle l|P_\beta|k\rangle$ for the Pauli string $P_\beta$ in the Hamiltonian decomposition, using the unitary that maps state $|k\rangle$ to $|l\rangle$ along the ADAPT-VQE path. The paper uses the qubit-excitation and Qubit operator pools, compares full versus single-parameter optimization, thresholds the overlap eigenvalues to remove nearly dependent states, and invokes a state-averaged cost function when different symmetry sectors coexist.

What would settle it

Take a small Hamiltonian that is exactly diagonalizable and whose first excited state lies in a symmetry sector orthogonal to the initial ADAPT-VQE state. If after a large number of iterations the generalized eigenvalue problem still yields no eigenvalue within numerical tolerance of that excited state, the generic claim that the convergence path contains the low-lying spectrum fails; the paper itself exhibits this behavior for H4 before the state-averaged strategy is added.

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Extended reading notes

Core claim

The central discovery is that the ADAPT-VQE convergence path carries low-lying spectroscopy. During iterations the energy fluctuation $\sigma_n^2$ decreases and the occupation of low-energy eigenstates increases while high-energy components are depopulated, so the states $\{|k\rangle\}$ along the path form a subspace with substantial overlap on low-lying eigenstates. The paper diagonalizes the Hamiltonian in this subspace through the generalized eigenvalue equation $H_{lk} c_k = E\, O_{lk} c_k$ and shows numerically that the ten lowest states of a like-particle pairing problem are reproduced with about thirty path states out of a 252-dimensional seniority-zero space. In the full neutron-proton pairing case, the method works within each seniority sector, and different symmetry sectors are obtained by restarting from a different initial state. For H4 dissociation, the lowest Ag states are captured directly, while states of other symmetries require restarting from states of those sectors, and the B2u-B3u level crossing requires the state-averaged strategy; after that, QSD again recovers the missing states.

Load-bearing premise

The load-bearing premise is that the ADAPT-VQE descent enriches low-energy eigenstate components fast enough that the span of the intermediate states is a good trial space for low-lying excited states; for symmetry sectors not represented in the initial state this premise is false unless extra starting states or state-averaging are added.

Editorial extensions

If this is right

Low-lying excited-state energies become a nearly free by-product of a ground-state ADAPT-VQE run, needing only the measurement of H and overlap matrix elements among the path states.
Single-parameter optimization, which is much cheaper than full re-optimization, becomes competitive because the later QSD step improves the ground-state energy by an order of magnitude in the like-particle pairing test.
For Hamiltonians that are block-diagonal by symmetry, the method should be run once per symmetry sector, starting from a state of that sector, to cover all low-lying states.
In the H4 example, state-averaged ADAPT-VQE plus QSD solves the B2u-B3u level-crossing problem and reproduces spectra that a single descent from an Ag seed misses.
The subspace dimension used in the pairing benchmark, about thirty path states, is far below the 252-dimensional seniority-zero full configuration-interaction space for the same problem, indicating the method extracts spectra from a compact trial space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the path states' low-energy overlap grows systematically with iteration count, then the required subspace size may scale with the number of target low-lying states rather than with Hilbert-space dimension; a size-scaling study on larger pairing or molecular models would test this implied scaling.
The overlap matrix's condition number and the threshold epsilon will likely control error propagation; a per-state error bound in terms of the overlap of the exact excited state with the path subspace would turn the mean-relative-error benchmarks into a certificate, which the paper does not provide.
A practical extension is to use the path states of a symmetry-breaking ADAPT-VQE run as generator coordinates for a quantum generator-coordinate-method calculation, connecting the technique to nuclear potential-energy-surface studies without any new hardware requirement.
On noisy hardware the dominant cost will be the O(n_f^2) Hamiltonian matrix elements among path states; using randomized sampling of path states or classically post-processing subsets of them, as the paper notes but does not explore, could reduce that overhead.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid, honest demonstration that the states picked up along an ADAPT-VQE run can be reused as a QSD subspace to get low-lying excited states in the particular systems tested. The nuclear pairing applications and the single-parameter optimization (SPO) variant are the genuinely new pieces. But the abstract overreaches: "small overhead" and "low-lying states from the convergence path" are only true within a symmetry sector, and the H4 results show whole sectors are missed unless you bolt on a state-averaged strategy from Ref [61]. Treat it as a heuristic with demonstrated sector-wise success, not a general method.

What I liked: The paper is transparent about the limitations. Sec. 3.2 is honest that the subspace-richness claim is an intuition; Fig. 2 is schematic. The numerical comparisons against full CI are clean; the MRE metric and forward/backward iteration selection are sensible. SPO is a practical twist: if you don't re-optimize old parameters, the states along the path are related by simple products of exponentials, which makes the overlap and Hamiltonian measurements cheaper and more consistent. The like-particle pairing results are convincing, with n_f ~ 30 states enough to get ten low-lying states to good precision. And the authors explicitly flag the symmetry-sector failures in both Fig. 6 and Fig. 7, then show the state-averaged fix works.

Soft spots, in proportion: The load-bearing premise in Sec. 3.2 is not proven. The Rayleigh-Ritz bound they quote lower-bounds the ground-state fidelity; it says nothing about the overlap of the first excited state with the path subspace. The claim that the descent "depopulates high-energy states" is an interpretation, not a consequence. H4 is the concrete counterexample: with an initial Ag state, the B2u and B3u sectors are entirely absent until you add a state-averaged optimization with K=4 initial states. So the abstract's unqualified statement is wrong as written. The "small overhead" claim is also qualitative; the paper gives no resource estimate beyond the K^2 cost factor for state averaging. Finally, no code or data is shipped, and everything is a noiseless statevector simulation, so practicality on real hardware is untested. These are real constraints but they don't undo the demonstrated results.

Who this is for: people working on quantum subspace methods, especially in nuclear structure and quantum chemistry on small systems. It's an incremental but useful extension of Ref [61], and the pairing examples give the nuclear community a concrete benchmark. The citation pattern looks appropriate; the authors are honest that the subspace-expansion idea overlaps with Ref [61] and they are positioning this as an application plus the SPO variant.

Recommendation: Send it to peer review. A competent referee should push for a corrected abstract, a quantitative resource comparison, and ideally a diagnostic or criterion for when the path subspace is spectrally complete. With those revisions it would be a solid contribution to the QSD literature.

Referee Report

3 major / 6 minor

Summary. The paper proposes using the intermediate states generated along the ADAPT-VQE convergence path toward the ground state as a subspace for quantum subspace diagonalization (QSD), solving the generalized eigenvalue problem in Eq. (7) to approximate low-lying excited states. The authors argue that the simultaneous decrease of energy and energy variance along the path implies that the subspace is rich in low-energy spectral information. They demonstrate the approach on like-particle pairing, neutron-proton pairing, and H4 dissociation, using noiseless statevector simulations and comparing against exact FCI spectra. They also introduce a single-parameter-optimization variant and augment the H4 calculation with a state-averaged strategy to recover symmetry sectors missed by the ground-state-only path.

Significance. If the central premise held, the method would be an attractive way to obtain excited states almost as a by-product of a ground-state ADAPT-VQE run, with no fitted constants in the energy predictions and with clear benchmarking against exact FCI in small systems. These are real strengths: the numerical demonstrations are internally consistent, the comparison to exact spectra is transparent, and the authors explicitly identify when symmetry sectors are missed. However, the central premise is only illustrated heuristically, not proved, and the H4 example shows that the pure ground-state path can fail entirely for some symmetry sectors. The resource-overhead claim in the abstract is unquantified. As a numerical study of a promising heuristic, the paper is valuable, but the generality and overhead claims need substantial qualification.

major comments (3)

[Sec. 3.2, Fig. 2; Sec. 4.3, Figs. 7-8] The central premise that the ADAPT-VQE path states span the low-lying spectrum is not established. The Rayleigh-Ritz argument quoted in Sec. 3.2 only lower-bounds the ground-state fidelity; it says nothing about the presence of excited-state amplitudes in the intermediate states. A trajectory can approach the ground state while remaining in a symmetry-restricted subspace that is orthogonal to a given low-lying excited eigenspace. The H4 calculation in Fig. 7 is an explicit counterexample: starting from an Ag-symmetric state, the B1g, B2u, and B3u sectors are missed entirely, and the paper must switch to a state-averaged ADAPT-VQE with K=4 initial states (Fig. 8) to recover them. Thus the abstract's unqualified claim that low-lying excited states are obtained from the ground-state convergence path with small overhead is not supported in general. The paper should either provide a criterion for when the path subspace is spectrally complete or explicitly restrict the claim to the demonstrated sectors.
[Sec. 3.3; Sec. 4.3] The claim of 'small overhead in terms of quantum resources' is unquantified. Computing H_lk and O_lk via the Hadamard-test circuits in Fig. 3 requires, for each pair (l,k) and each Pauli string P_beta, an additional circuit evaluation; with n_f+1 path states this is O(n_f^2 * N_Pauli) extra circuits relative to ground-state ADAPT-VQE. In the state-averaged H4 application, the paper itself states at the end of Sec. 4.3 that the subspace cost is K^2 times higher (with K=4), which is not obviously 'small'. No comparison is made with the measurement cost of the ground-state calculation or with other QSD approaches. The resource claim should be either quantitatively supported or removed/qualified.
[Sec. 4.2, Fig. 6; Sec. 3.1, Fig. 1] It is unclear whether the neutron-proton pairing results in Fig. 6 use SPO or FPO. This matters because Fig. 1 shows that, for the full pairing Hamiltonian, SPO does not converge to the ground state within 40 iterations, while FPO does. Since Sec. 4.2 is the main demonstration of the method for neutron-proton pairing, the reader cannot tell whether the successful QSD spectra there rely on full optimization (which weakens the paper's SPO recommendation) or on single optimization (in which case QSD is compensating for non-convergence and that effect deserves discussion). Please state the optimization strategy used in each figure and discuss the implications for the SPO proposal.

minor comments (6)

[Sec. 4.1] There is a typo: 'descrease' should be 'decrease' in the sentence about MRE1.
[Sec. 4.1] The MRE formula 'MRE_Lambda = 1/Lambda Delta-epsilon ...' is ambiguous; a parenthesized denominator, e.g. 1/(Lambda Delta-epsilon), would be clearer, and the choice of Delta-epsilon as the energy scale should be stated explicitly.
[Sec. 3.1, Eq. (3)] The product of exponentials in Eq. (3) and in the definition of U_lk in Sec. 3.3 should specify the multiplication order (left-to-right or right-to-left), since the operators do not commute.
[Figs. 7-8] The caption of Fig. 7 states that about 12 ADAPT-VQE iterations per r value were used, but Fig. 8 does not specify the number of iterations or the K=4 details; please add these parameters for reproducibility.
[Sec. 4.3] The state-averaged weights c_i are all set to 1 without discussion; some justification or a reference to the sensitivity of the results to these weights would strengthen the presentation.
[General] All numerical results are obtained with a noiseless statevector simulator; a brief comment on the expected sensitivity to measurement noise and on the additional measurement overhead would help readers assess NISQ applicability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: excited-state energies are obtained by Hamiltonian diagonalization in an adaptively generated variational subspace and benchmarked against exact FCI, with acknowledged symmetry-sector limitations rather than fitted feedback.

full rationale

The derivation chain is self-contained with respect to the claimed excited-state predictions. The intermediate states |k> are produced by ADAPT-VQE minimizing only the ground-state energy (Sec. 3.1), and the excited states are then obtained by solving the generalized eigenvalue equation H_lk c_k = E O_lk c_k in Eq. (7). No parameter of that diagonalization is fitted to the target excited-state energies. The only threshold, epsilon = 10^-6, controls subspace truncation and is not an input energy. Benchmarks against FCI are independent: like-particle pairing in Figs. 4-5, neutron-proton pairing in Fig. 6, and H4 in Figs. 7-8 use exact diagonalization from Qiskit/PySCF. The paper's heuristic premise in Sec. 3.2 that energy and variance decrease implies the path subspace becomes rich in low-lying spectral information is an unproved rationale, and the H4 results show it fails for symmetry sectors absent from the initial state; the authors explicitly flag this in Sec. 4.2 ('QSD cannot reproduce the spectrum sector of nu != 0') and Sec. 4.3 ('some states are completely missed... different low-lying states belong to different symmetry blocks'). This is a scope/correctness limitation, not circularity: the missed sectors are recovered by adding state-averaged ADAPT-VQE with K=4 initial states, and the resulting energies are still computed by diagonalization and compared with FCI. Self-citations to Ref. [53] justify the choice of the QEB operator pool and the pairing Hamiltonian, but they do not define the excited-state energies being predicted; the cited pool choice is an algorithmic input, not an output of the method. No fitted quantity is renamed as a prediction, and no uniqueness theorem is imported to force the subspace choice.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or unobserved entities are introduced; the contribution is purely algorithmic. The listed free parameters are numerical tolerances, iteration counts, initial-state choices, and pool choices that shape the subspace used in the demonstration.

free parameters (5)

overlap matrix truncation threshold epsilon = 1e-6
Used in Sec. 4.2 to discard overlap-matrix eigenstates below threshold; chosen by hand, affects subspace dimension and which low-lying states survive.
number of ADAPT-VQE iterations n_f = 40 for pairing, ~12 for H4
Subspace size varies per system; chosen by convergence behavior, not fit to exact energies. The paper notes n_f above about 30 is needed for the like-particle pairing spectrum.
state-averaged weights c_i = 1 for all i
In Sec. 4.3 the H4 state-averaged cost function uses equal weights; the paper states other choices are possible (citing Ref. [87]) but does not explore them.
initial reference states for symmetry sectors = |00001111>, |00011110>, and symmetry variants
The method requires starting ADAPT-VQE in each seniority or point-group symmetry sector to recover states in that sector; these are inputs chosen by physics, not fitted to data.
operator pool choice = QEB-pool for pairing; Qubit-pool for state-averaged H4
The choice of pool affects convergence and symmetry preservation; for the state-averaged H4 run the Qubit pool is needed to allow symmetry breaking. This is an algorithmic hyperparameter.

assumptions (5)

standard math Rayleigh-Ritz variational principle and its ensemble (GOK) extension
Used to justify that subspace diagonalization lowers or bounds energies and that state-averaged optimization can target multiple eigenstates (Secs. 3.2 and 4.3).
domain assumption ADAPT-VQE convergence path depopulates high-energy eigenstates
The central heuristic behind using intermediate states as a subspace; illustrated in Figs. 1 and 2 but not proved.
domain assumption Selected operator pools preserve or can break the symmetry of the initial state
Used to explain missing symmetry sectors and to justify the multi-start and state-averaged strategies (Secs. 4.2 and 4.3).
domain assumption Jordan-Wigner encoding correctly represents the pairing Hamiltonian on qubits
Basis for all simulations, inherited from Ref. [53]; no independent verification needed for the method's algorithmic claim.
domain assumption Noiseless statevector simulation reproduces the ideal quantum circuit output
All results use the Qiskit statevector simulator; the paper does not treat hardware noise or measurement shot noise.

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Cite this review

Pith. "Pith review of Excited States from ADAPT-VQE convergence path in Many-Body Problems: application to nuclear pairing problem and $H_4$ molecule dissociation." pith.science (2026). https://pith.science/paper/UN35GXJV

@misc{pith2026250622275,
  author       = {Pith},
  title        = {Pith review of: Excited States from ADAPT-VQE convergence path in Many-Body Problems: application to nuclear pairing problem and $H_4$ molecule dissociation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UN35GXJV}},
  note         = {Machine review of arXiv:2506.22275}
}

abstract

A quantum computing algorithm is proposed to obtain low-lying excited states in many-body interacting systems. The approximate eigenstates are obtained by using a quantum space diagonalization method in a subspace of states selected from the convergence path of the ADAPT-VQE (adaptive derivative-assembled pseudo-Trotter Ansatz variational quantum eigensolver) towards the ground state of the many-body problem. This method is shown to be accurate with only a small overhead in terms of quantum resources required to get the ground state. We also show that the quantum algorithm might be used to facilitate the convergence of the ADAPT-VQE method itself. Successful applications of the technique are made to like-particle pairing as well as neutron-proton pairing. Finally, the $H_4$ molecule's dissociation also illustrates the technique, demonstrating its accuracy and versatility.

Figures

Figures reproduced from arXiv: 2506.22275 by the authors.

**Figure 2.** Schematic illustration of the occupation amplitudes [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Quantum circuits used to obtain (a) the real and (b) the imaginary [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Mean Relative Error obtained from panel (a) of Fig. [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: Illustration of the low-lying spectra obtained for the neutron-proton [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 8.** Figure 8: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Comparison of variational quantum eigensolvers in light nuclei
nucl-th 2025-07 conditional novelty 6.0 of 10

For p-shell nuclei from 6He to 10B, ADAPT-VQE uses fewer total operations than UCC when the many-body space is small (dim(H) < 51), while UCC wins for mid-shell nuclei with dim(H) at least 51.
Exploring fixed points and eigenstates of quantum systems with reinforcement learning
quant-ph 2025-11 conditional novelty 5.0 of 10

A reward-penalty reinforcement-learning loop learns the unitary that maps the computational basis to the eigenbasis of a Hamiltonian, demonstrated on systems of up to six qubits.

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