Ground-state reachability for variational quantum eigensolvers: a Rydberg-atom case study

Andreas Kruckenhauser; Juhi Singh; Rick van Bijnen; Robert Zeier

REVIEW 3 major objections 6 minor 105 references

Ground-state reachability for variational quantum eigensolvers: a Rydberg-atom case study

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

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Pith's one-line read A variational quantum eigensolver on a Rydberg-atom ring with global controls can reach a target ground state only when the initial state and the target state share an invariant subspace of the resource Hamiltonians; the paper shows this…

desk verdict A rigorous necessary-condition analysis of VQE reachability for Rydberg global controls, with solid negative predictions; the positive 'reachability' labels outrun what the symmetry argument alone can certify. read the letter →

arxiv 2506.22387 v1 pith:JBTFQQAM submitted 2025-06-27 quant-ph

Juhi Singh , Andreas Kruckenhauser , Rick van Bijnen , Robert Zeier This is my paper · ORCID

classification quant-ph

keywords variationalquantumeigensolverground-statereachabilityinvariantsubspacessymmetryanalysisRydbergatomsIsingmodelHeisenbergsimulability

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 asks a practical question: before running a variational quantum eigensolver (VQE) on a Rydberg-atom simulator with global controls, can one tell whether the optimizer has any chance of reaching the target ground state? It argues that the answer is controlled by the symmetries of the resource Hamiltonians: a VQE can reach the target only when the initial state and the target ground state have nonzero support in the same invariant subspace of those resources. Applying this test to ring geometries of three to ten qubits, it predicts that the Heisenberg ground state is unreachable for 3, 5, 6, 7, and 9 qubits and reachable for 4, 8, and 10, while the Ising ground state is reachable for all sizes considered; VQE simulations match these predictions. The paper also shows that Hamiltonian simulability does not imply ground-state reachability, and that choosing an initial state inside the target's invariant subspace can restore reachability.

What carries the argument

The central object is the decomposition of the $2^N$-dimensional Hilbert space into invariant subspaces of the resource Hamiltonians. The paper computes this decomposition through the commutant (the set of matrices commuting with all resource Hamiltonians) and then projects states onto isotypic and irreducible blocks using projection operators. The mechanism doing the work is support conservation: the variational dynamics is generated by the resources, so the amount of the wavefunction in each block cannot change, and a target ground state with zero overlap on the initial state's blocks is unreachable. This turns a symmetry calculation into a reachability certificate.

What would settle it

Compute the invariant-subspace decomposition for an eleven-qubit Rydberg ring with global controls and the Heisenberg target Hamiltonian, and run VQE from the trivial initial state; the symmetry test predicts failure when the initial and exact target ground states have disjoint support, so a single successful convergence to the ground-state energy would refute the claim that the dynamics cannot leave an invariant subspace.

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Extended reading notes

Core claim

The paper establishes a necessary condition for ground-state reachability in an analog Rydberg VQE. With resource set $G_R = \{H_d, H_\Omega, H_\Delta\}$ under global controls, every unitary produced by the variational circuit preserves the invariant subspaces of $G_R$, so any state reached from the trivial initial state stays inside the union of subspaces where that initial state has support. Consequently, if the target ground state has zero support on those subspaces, no amount of optimization can reach it. The paper computes the commutant and the isotypic and irreducible decompositions of $G_R$ for three to ten qubits, projects the initial state and the exactly known target ground states, and finds that the symmetry verdict matches VQE simulations: the Heisenberg ground state is blocked for $N = 3, 5, 6, 7, 9$ and allowed for $N = 4, 8, 10$, whereas the Ising ground state is allowed for all $N = 3, \ldots, 10$. It further shows that adding the target Hamiltonian to the resources does not remove the obstruction, because the extra center elements act outside the subspace containing both states, and that an initial state chosen inside the target's invariant subspace restores reachability, as demonstrated for six qubits.

Load-bearing premise

The load-bearing premise is that one already knows the target ground state, or at least which invariant subspace contains it; as the paper itself notes in its conclusion, a practical VQE lacks exactly that spectral information, so the criterion currently works as a diagnostic for small systems rather than a predictive tool for large ones.

Editorial extensions

If this is right

If the paper's symmetry criterion is right, then on the Rydberg ring with global controls a VQE aimed at the Heisenberg ground state from the trivial initial state is bound to fail for $N = 3, 5, 6, 7, 9$.
For $N = 4, 8, 10$, the control structure does not block the Heisenberg ground state, so any observed failure there must come from optimization, noise, or other practical limitations.
The Ising ground state is not symmetry-blocked for any considered size, so the device is in-principle capable of preparing it for $N = 3$ through $10$.
Extending the resource set by the target Hamiltonian itself does not lift the obstruction, since the additional symmetries have no effect on the relevant invariant subspace.
A simple workaround is available: starting from an initial state inside the target ground state's invariant subspace removes the symmetry obstruction, as the six-qubit Heisenberg example shows.

Reading between the lines

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

The paper's method requires the target ground state as input; an extension it leaves open is to predict the symmetry sector from the target Hamiltonian alone, which would turn the diagnostic into a large-scale predictive tool.
The same invariant-subspace test should transfer to other analog simulators and control sets: any global-control device with a nontrivial commutant will have forbidden ground states that could be identified before running VQE.
The six-qubit rescue suggests a practical workflow when VQE fails: compute the projector onto the target's invariant subspace and restart from one of its eigenvectors; but this assumes spectral information that would usually be unavailable, so it is a small-system remedy rather than a general solution.
The adiabatic connection implies that the symmetry test also predicts when a proposed adiabatic path between parent and target Hamiltonians must cross a sector not containing the desired ground state; checking gap closings per sector would be a concrete test of that link.

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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

Here's my take. The necessary-condition analysis is correct and the negative predictions are the real result: for the Rydberg ring with global controls, the Heisenberg ground state is unreachable from the trivial product state for N=3,5,6,7,9. That's rigorous, given the model. The companion observation that simulability of the target Hamiltonian does not imply ground-state reachability (three-qubit Heisenberg is simulable but unreachable; four-qubit is not fully simulable but reachable) is clean and worth stealing for a lecture. What's genuinely new is the application of known Lie-algebra/commutant tools to this specific Rydberg VQE setting, and the N-dependent obstruction pattern. The computational description is careful, with exact-field Magma computations; the VQE numerics for N<=10 are consistent with the negative predictions. The paper also states in Sec. III C that containment in the same irreducible subspace is necessary but not sufficient, and gives the SU(2) x SU(2) example. The soft spots are real but not fatal. The abstract and Table II use 'reachability' for the positive cases (Heisenberg 4,8,10; Ising 3-10) where the symmetry analysis only shows non-obstruction; the actual evidence for those entries is the VQE runs, not a sufficiency proof. So the phrase 'predicting whether ... could successfully reach' overstates what the framework alone certifies. The Conclusion does admit the need to know the target ground state a priori, which limits practical large-scale use. Minor: no code or data posted, and the extension beyond seven qubits depends on a plausible but unproven Lie-center assumption (Appendix C). These are all fixable. I'd send this to a serious referee. The core necessary condition is solid, the negative cases are useful, and the message that symmetry obstructions matter for VQE on analog simulators is worth publishing. I'd ask the authors to recalibrate the positive-claim language and consider adding code or a sufficiency discussion.

Referee Report

3 major / 6 minor

Summary. The paper studies whether a variational quantum eigensolver (VQE) running on a Rydberg-atom analog simulator with global controls can reach the ground state of a target Hamiltonian. The authors consider the resource Hamiltonians G_R = {H_d, H_Ω, H_Δ} and analyze the invariant subspaces of the Lie algebra they generate, obtained through the commutant and isotypic/irreducible projectors. Their central result is a necessary condition: the initial state and the target ground state must have support in the same invariant subspace of G_R; otherwise VQE is provably unable to reach the target. Applying this condition to Heisenberg and Ising target Hamiltonians on rings of 3–10 qubits, they find that the Heisenberg ground state is unreachable for N = 3, 5, 6, 7, 9 and potentially reachable for N = 4, 8, 10, while the Ising ground state is in the same subspace as the initial state for all considered N. These symmetry-based predictions are compared with numerical VQE simulations, which agree. The paper also contrasts simulability with reachability, shows how changing the initial state can bypass symmetry obstructions in the six-qubit Heisenberg case, and draws connections to adiabatic state preparation. Appendices provide Lie-algebra and symmetry computations with Magma.

Significance. If taken as a framework for necessary conditions, the paper is a solid and useful contribution: the invariant-subspace criterion is mathematically airtight, and the negative predictions for Heisenberg N = 3, 5, 6, 7, 9 are rigorous given the model. The computational approach via the commutant scales better than full Lie-algebra closure and is clearly explained, with reproducible mathematical tools (Magma computations). The explicit acknowledgment that containment in the same irreducible subspace is not sufficient is honest and important. The main weakness is that the positive 'reachability' classifications in Table II are not derived from the symmetry analysis; they are only consistent with VQE simulations, so the abstract's claim of 'predicting whether a given quantum architecture could successfully reach the ground state' overstates what the method delivers. The paper also correctly notes in the Conclusion that the method requires prior knowledge of the target ground state, which limits its practical predictive power for large systems.

major comments (3)

[Sec. III C and Table II; Abstract] The paper explicitly states in Sec. III C that the condition that the initial and target states lie in the same irreducible invariant subspace is not sufficient for reachability, giving the SU(2) ⊗ SU(2) versus SU(4) example. Nevertheless, Table II and the abstract present same-subspace support as 'could reach' or 'could successfully reach' for the Ising target (all N) and for Heisenberg N = 4, 8, 10. These positive entries are not consequences of the symmetry analysis; they are, at best, statements that the symmetry does not forbid reaching the target, corroborated by VQE runs. Please reword the positive claims as 'not ruled out by symmetry' and restrict the predictive claim of the framework to necessary (obstruction) conditions, or supply an additional sufficiency/controllability argument within the irreducible subspace.
[Sec. VII (Conclusion) and Abstract] The method requires knowing the target ground state(s) or at least their invariant-subspace support, which in this paper is obtained by exact diagonalization. In a genuine VQE task the ground state is the quantity being sought, so the framework as presented cannot be used as an a priori predictor for large systems. The authors acknowledge this in the Conclusion, but the abstract's claim that the approach 'demonstrates the reliability of our approach in predicting whether a given quantum architecture could successfully reach the ground state' is too strong. Please qualify the abstract and title-level claims, e.g., by describing the result as necessary-condition diagnostics for small systems with empirical corroboration.
[Appendix C] The extrapolation beyond seven qubits is explicitly predicated on the observation that extending the resource Hamiltonians by the target Hamiltonian adds at most one center element. The authors state, 'our analysis beyond seven qubits is predicated by general correctness of this observation,' but no proof is provided for N = 8, 9, 10. While the central necessary-condition results for G_R alone do not depend on this assumption, the Appendix C conclusion that adding the target Hamiltonian 'does not help' for N = 8, 9, 10 does. Please either prove the observation for these sizes (the center dimension can be computed via the commutant, as the authors do) or explicitly label that conclusion as conjectural.

minor comments (6)

[Appendix B] The word 'isoyptic' appears twice in the text around the meataxe description; it should be 'isotypic'.
[Sec. VI] In the sentence 'Note that this assumption is not statisfied in all possible cases', 'statisfied' should be 'satisfied'.
[Sec. VI] The phrase 'If the the system starts at τ = 0' contains a duplicated 'the'.
[Fig. 3 caption] The caption contains the typo 'inital' for 'initial'.
[Appendix B] In the final sentence of the appendix, the enumeration 'C8 ⊞ C6 ⊞ C6 ⊞ C6 ⊞ C6' is described as 'three invariant subspaces', but it lists four components; this should be harmonized with Table IV, which groups them as C8 ⊞ (C6 ⊞ C6) ⊞ (C6 ⊞ C6).
[Sec. II B] The number of layers M used in the VQE simulations for Fig. 6 is not specified; only iteration counts are given. Please state M (and the optimizer details) for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant-subspace necessary condition is derived from the stated Hamiltonians and checked against independent VQE runs, not fit to them.

full rationale

The central reachability test (Sec. III C) is a self-contained necessary condition: any infinitesimal action of the resource Hamiltonians preserves each invariant subspace, so if the initial and target states have support in different invariant subspaces, the target cannot be reached. This is proved in the text directly from the definition of an invariant subspace, and the load-bearing step does not reduce to a self-citation. The support data in Tables II and IV are computed from the resource Hamiltonians and from target ground states obtained by exact diagonalization; the paper explicitly acknowledges this input in the Conclusion: "we have assumed that we have access to the target ground state (which VQE is supposed to find)". That is an honest limitation for practical VQE, not a fitted parameter renamed as a prediction. The VQE simulations in Figs. 3 and 6 are independent numerical experiments with random parameter initializations, and no parameter is tuned to force agreement with the symmetry conclusions. The paper also explicitly states that containment in the same irreducible subspace is necessary but not sufficient, using the SU(2) x SU(2) example in Sec. III C, so the positive "reachability" entries in Table II are compatibility statements rather than sufficiency proofs. The abstract's wording that the approach predicts whether the architecture "could successfully reach" the ground state is somewhat stronger than the proved necessary condition, but this is a sufficiency/overstatement concern, not circularity. Self-citations to the authors' earlier Lie-algebra and symmetry papers are legitimate reuse of published computational tools; they are not the justification for the central necessary condition, and no uniqueness theorem or ansatz is imported from them to forbid alternatives. No self-definitional, fitted-input, or ansatz-smuggling step was found.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard representation theory, the modeling assumption that VQE explores the full Lie group generated by the resource Hamiltonians, and the stated availability of the target ground state. The paper also assumes an extrapolated one-center-element observation beyond seven qubits for a secondary claim.

free parameters (3)

Target Hamiltonian coupling point (Heisenberg) = h = J (J = 1 in simulations)
The Heisenberg case study is evaluated only at h = J; the target ground state's symmetry sector, and hence reachability, could differ for other ratios.
Target Hamiltonian coupling point (Ising) = Omega = Delta = J (J = 1 in simulations)
The Ising case study is evaluated only at Omega = Delta = J; other field ratios could change the ground-state sector.
Ring-geometry drift coefficients = C6/|R_n-R_m|^6 with N-gon ratios, radius unspecified
The resource drift Hamiltonian is defined by the ring geometry but the explicit coefficients are never written; the Lie algebra and invariant subspaces depend on these ratios.

assumptions (5)

standard math The resource Hamiltonians generate a compact Lie group acting unitarily on the Hilbert space, so all representations are completely reducible and the invariant subspace decomposition is valid.
Invoked in Sec. III C to justify decomposing the Hilbert space into irreducible/isotypic invariant subspaces under the dynamics generated by G_R.
domain assumption The VQE circuit consists of piecewise-constant time evolutions generated by the resource Hamiltonians with arbitrary durations and amplitudes, so the reachable unitaries are exactly the connected Lie group generated by G_R.
Sec. II B defines the VQE circuit; classical optimization over Omega_j, Delta_j, delta_t_j can in principle realize any product of exponentials, generating the full group. Any restriction on controls beyond this would further limit reachability.
domain assumption The target ground state is known (or its symmetry sector is known) in order to apply the symmetry test; the paper assumes access to exact ground states for the case study.
Sec. III C and the Conclusion state that the analysis requires the target ground state, which is not available in a practical VQE run; this is a stated limitation, not an error.
domain assumption For the adiabatic connection, the resource Lie algebra g'_R is assumed to contain both iH0 and iH_T; the paper explicitly notes this is not satisfied in all cases.
Sec. VI states this simplifying assumption before the three examples.
ad hoc to paper The observation that extending the resource Hamiltonians by the target Hamiltonian adds at most one center element is assumed to hold beyond seven qubits.
Appendix C: 'our analysis beyond seven qubits is predicated by general correctness of this observation.' This is a stated extrapolation.

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Cite this review

Pith. "Pith review of Ground-state reachability for variational quantum eigensolvers: a Rydberg-atom case study." pith.science (2026). https://pith.science/paper/JBTFQQAM

@misc{pith2026250622387,
  author       = {Pith},
  title        = {Pith review of: Ground-state reachability for variational quantum eigensolvers: a Rydberg-atom case study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JBTFQQAM}},
  note         = {Machine review of arXiv:2506.22387}
}

read the original abstract

As quantum computing progresses, variational quantum eigensolvers (VQE) for ground-state preparation have become an attractive option in leveraging current quantum hardware. However, a major challenge in implementing VQE is understanding whether a given quantum system can even reach the target ground state. In this work, we study reachability conditions for VQE by analyzing their inherent symmetries. We consider a Rydberg-atom quantum simulator with global controls and evaluate its ability to reach ground states for Ising and Heisenberg target Hamiltonians. Symmetry-based conclusions for a smaller number of qubits are corroborated by VQE simulations, demonstrating the reliability of our approach in predicting whether a given quantum architecture could successfully reach the ground state. Our framework also suggests approaches to overcome symmetry restrictions by adding additional quantum resources or choosing different initial states, offering practical guidance for implementing VQE in quantum simulation architectures. Finally, we illustrate connections to adiabatic state preparation.

Figures

Figures reproduced from arXiv: 2506.22387 by the authors.

**Figure 2.** FIG. 2. VQE circuit executed on the analog Rydberg-atom [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. Block structure and invariant subspaces: Symme [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5. Flowchart for the symmetry analysis. (a) First, the commutant [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. VQE for the Heisenberg and Ising ground state showing the expectation value of the energy compared to the eigenen [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. Comparison of VQE performance with different ini [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: (a) highlights a crossing in the eigenspectrum of H(τ ) related to the ground-state energy. This results in a zero gap and cannot confirm an adiabatic path to reach the target ground state. We can also validate this using our symmetry analysis: The Lie algebra generate…

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