A Method of Determining Excited-States for Quantum Computation

M. Kostuk; P. Jouzdani; S. Bringuier

REVIEW 4 major objections 4 minor 25 references

A Method of Determining Excited-States for Quantum Computation

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Projecting out the ground state turns excited-state energies into ground-state problems.

desk verdict A useful iterative projection heuristic for excited-state VQE, honestly caveated in the supplement, but with an uncontrolled truncation that limits it to near-Cz Hamiltonians; H2 works, LiH partially fails. read the letter →

arxiv 1908.05238 v1 pith:J3VU72NU submitted 2019-08-12 quant-ph

P. Jouzdani , S. Bringuier , M. Kostuk This is my paper · ORCID

classification quant-ph PACS 03.67.Ac31.15.-p

keywords excitedstateseffectiveHamiltoniancovarianceassertionvariationalquantumeigensolverPaulistringoperatorsperturbed-Czclasschemistry

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 proposes a way to compute excited-state energies of a molecular Hamiltonian using the same hybrid quantum-classical machinery that already computes ground-state energies, such as the variational quantum eigensolver. The idea is to construct, from a known ground state, a projected Hamiltonian whose lowest eigenvalue is the first excited-state energy of the original problem, and then iterate. Because the exact projection would contain exponentially many Pauli-string terms, the paper introduces a covariance assertion: keep only the string operators already present in the original Hamiltonian, renormalizing their coefficients using ground-state expectation values. The method is shown to reproduce the full hydrogen-molecule spectrum, to obtain the first excited state of LiH with high fidelity, and to become exact in the thermodynamic limit for a perturbed classical (Cz) class of Hamiltonians.

What carries the argument

The load-bearing mechanism is the covariance assertion: the assumption that the ground-state projection can be represented using only the Pauli string operators that already appear in the Hamiltonian, with renormalized coefficients $\lambda_j' = \lambda_j - E_g f_j$, where $f_j$ is the ground-state expectation value of the $j$-th string operator. This keeps the number of terms constant across iterations, avoiding the exponential growth that a general projection operator would incur. Its justification is group-theoretic: if the Hamiltonian is a weighted sum over all elements of a finite Abelian group of commuting string operators and the ground state stabilizes the group's generators, then the projection operator has the product form $\hat{P} = \frac{1}{2^{N_g}}\prod_g (\hat{I} - \hat{h}_g)$, making the covariance assertion exact.

What would settle it

Take a molecule beyond H2 or LiH (such as water, or LiH at a non-equilibrium bond length), compute the exact projected Hamiltonian $\hat{H}^{(1)} = (\hat{I}-\hat{P})\hat{H}(\hat{I}-\hat{P})$ by full diagonalization, expand it in Pauli string operators, and compare the ground-state energy of this exact projected Hamiltonian with that of the covariance-asserted effective Hamiltonian $\sum_j(\lambda_j - E_g f_j)\hat{h}_j$; if the difference exceeds chemical accuracy while the discarded operator weights are non-negligible, the covariance assertion is falsified for that system.

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Extended reading notes

Core claim

The central discovery is the identity $\hat{H}^{(1)} = (\hat{I} - \hat{P}^{(0)})\hat{H}^{(0)}(\hat{I} - \hat{P}^{(0)}) = \hat{H}^{(0)} - E_g^{(0)}\hat{P}^{(0)}$, where $\hat{P}^{(0)}$ projects onto the ground state of $\hat{H}^{(0)}$. In a basis where $\hat{H}^{(0)}$ is diagonal, this projected Hamiltonian has eigenvalue zero for the ground state, and its remaining spectrum is exactly the spectrum of $\hat{H}^{(0)}$ above the ground state. The paper's contribution is to make this projection practical by the covariance assertion: the projection operator is replaced by a stabilizer projection of a set of commuting generators, so that the effective Hamiltonian at each iteration is $\hat{H}^{(i+1)} = \sum_j (\lambda_j^{(i)} - E_g^{(i)} f_j^{(i)})\,\hat{h}_j$, with $f_j^{(i)} = \langle\varphi_{\mathrm{int}}| (\hat{U}^{(i)})^{\dagger} \hat{h}_j \hat{U}^{(i)} |\varphi_{\mathrm{int}}\rangle$ directly measurable on a quantum device. Iterating this update yields successive low-lying excited states, and the paper demonstrates the procedure on H$_2$ and partially on LiH.

Load-bearing premise

The entire scheme rests on the covariance assertion: that after the ground state is removed, the remaining Hamiltonian is still faithfully described by the same set of Pauli string operators, with only their coefficients changed.

Editorial extensions

If this is right

Excited-state energies of molecules whose Hamiltonians lie near the perturbed-Cz class can be obtained with the same quantum resources as a single ground-state VQE computation, since the coefficients $f_j$ are already measured during the ground-state optimization.
The iterative procedure produces not only excited-state energies but also the corresponding state vectors, with high fidelity when the covariance assertion holds, such as above 95% fidelity for H2 under noise and roughly 97% for the LiH first excited state.
For the perturbed-Cz class, the error in the predicted first excited-state energy and the state infidelity both go to zero as the number of qubits grows, so the method becomes exact in the thermodynamic limit.
If the covariance assertion fails at some iteration, the method signals its own breakdown: the ground-state energy of the next effective Hamiltonian violates the ordering constraint $E_g \le E_1 \le E_2 \le \cdots$.
The method can also produce orthogonal degenerate ground states when the target subspace is degenerate, which is useful for computing degenerate spectra.

Reading between the lines

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

For generic molecular Hamiltonians, the covariance assertion is unlikely to hold beyond low-lying states, as the LiH example already shows at the second excited state; a quantitative measure of the discarded part of the projection operator would be a useful diagnostic.
The method effectively treats the molecule as a member of the fixed-point class of a renormalization-group flow, which suggests that combining it with symmetry-adapted ansatze that enforce stabilizer structure could push the covariance assertion closer to exactness.
Because the $f_j$ coefficients are standard VQE expectation values, the approach has almost no additional quantum overhead; the main cost is classical, and the authors' heuristic for choosing the energy shift could be systematized into a classical optimization.
A direct comparison with quantum subspace expansion on the same H2 and LiH data sets would clarify whether the covariance assertion yields competitive accuracy with a simpler classical post-processing step.

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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

Dear colleague,

Quick read: the paper is a real algorithmic proposal, not a repackaged trick. The deflation step (I-P)H(I-P) is textbook, but the covariance assertion — keep the operator basis fixed and update coefficients as λ_j - E_g f_j — is a concrete, testable heuristic. For Hamiltonians that are weighted sums over an Abelian Pauli group with a stabilizer ground state, it is exact. The paper is honest that H2 and LiH are only perturbed-Cz instances of that class.

What it does well: the H2 binding-curve calculation is clean and reproducible, and the added noise model (corrupting the exact state vector) gives some evidence of robustness, at least for small W. The LiH test is also honest: 97% fidelity for the first excited state, good second energy but zero overlap with the exact second state, and failure of the third iteration to satisfy E_g^(3) >= E_g^(2). That is exactly the signature of an uncontrolled truncation error. The supplement also openly admits the energy-shift problem: without a shift, the VQE minimization of the projected Hamiltonian can return the old ground state with eigenvalue zero, and the shift is nontrivial once covariance is imposed.

Soft spots, in proportion: the abstract overstates the result. The ground state of the effective Hamiltonian coincides with an excited state only when the covariance assertion holds, and the paper gives no error bound for the truncation outside the exact Abelian group case. The numerical demonstration is exact diagonalization, not VQE on hardware or a simulator, so the claim about 'existing hybrid-quantum classical techniques' is not actually tested. The noise model is a crude proxy for optimizer noise and ansatz error. The LiH λI sweep in S4 is ad hoc — they tune a constant to satisfy ordering constraints, which works for the first two iterations but not the third. None of these flaws are fatal for the paper as a heuristic, but they should be stated in the main text, not only in the supplement.

The citation pattern looks fine: QSE, VQE, and related excited-state work are all covered. No self-citation issues.

Bottom line: this is a useful, honestly-grounded heuristic for a limited class of Hamiltonians, with a clean H2 demonstration and an instructive LiH failure. It deserves peer review, with referees asked to probe the covariance assertion for a practical error estimate and to push the authors to address the energy-shift issue. I would not cite it in my own work yet, but I'd recommend a careful referee for the archive version.

Referee Report

4 major / 4 minor

Summary. The paper proposes an iterative method for computing excited states of a Hamiltonian by projecting out the already-found ground state and constructing an effective Hamiltonian whose ground state supposedly coincides with the next excited state of the original problem. To avoid an exponential growth in the number of Pauli terms, the authors introduce a "covariance assertion": only the string operators already present in the current Hamiltonian are retained in the projected Hamiltonian, with coefficients updated as lambda'_j = lambda_j - E_g f_j. The method is demonstrated numerically for H2 and LiH, and a class of Hamiltonians (perturbed Cz-type) is discussed in the supplemental material. The central claim is that this gives a resource-efficient way to extend ground-state VQE to excited states.

Significance. If the covariance assertion could be justified with controlled error, the method would be a simple and attractive extension of ground-state VQE, requiring only expectation values that are already measured during the ground-state calculation. The paper gives a clean derivation for the exactly solvable Abelian-stabilizer class and produces clean H2 binding curves. It is also to the authors' credit that the LiH failure and the unresolved energy-shift problem are disclosed explicitly. However, as it stands, the central claim is not established for general or even realistic Hamiltonians: the main numerical example uses exact diagonalization rather than a hybrid quantum-classical procedure, and the covariance assertion is applied without any quantitative error bound.

major comments (4)

[Eqs. (5)-(6) and S1] The covariance assertion is exact only when the Hamiltonian is a weighted sum over all elements of a finite Abelian group and the ground state stabilizes the generators of that group. For H2 (Eq. (8), where X1X2 and Y1Y2 do not commute with the Z terms) and for LiH (explicitly treated as perturbed-Cz in S4), this condition is not met. S1 states that the projection operator is "replaced by the projection to the stabilizing subspace of a set of commuting independent operators without further assumptions." No norm bound or spectral error estimate is given for the discarded Pauli components of the projector, so the equality in Eq. (5) is uncontrolled. This truncation is load-bearing: without it, the updated Hamiltonian need not be of the form (I-P)H(I-P), and the variational guarantee that its ground state is an eigenstate of the previous Hamiltonian is lost.
[S2] The H2 numerical results are obtained by exact diagonalization, as S2 explicitly states, not by a variational ansatz or on quantum hardware. Therefore the abstract's claim that low-lying excited states can be calculated "using existing hybrid-quantum classical techniques" is not tested by the paper's main numerical example. The noise model in S2 perturbs the coefficients of the exact state and is not a simulation of VQE optimization, so it does not close this gap.
[S1] The energy-shift problem is admitted in S1 as "not addressed in the main text." Equation (S2) shows that when the first excited energy E1 is nonnegative, a variational minimization of the projected Hamiltonian returns the ground state with eigenvalue zero rather than the first excited state. The LiH procedure in S4 sweeps the identity coefficient lambda_I over [-10,10] and imposes monotonicity constraints, and S4 reports that the condition E_g^(3) >= E_g^(2) cannot be satisfied. This unresolved obstacle directly affects the iterative use of the method.
[S4 and LiH discussion in main text] The LiH demonstration fails beyond the first excited state: the main text states that in the second round of iteration the excited-state vector has almost zero overlap with the true second excited state, and S4 reports that the third iteration deviates from the exact value. This is evidence that the covariance assertion breaks down for a realistic molecular Hamiltonian after only one successful iteration, which contradicts the broader claim that the method "determines the excited-state vector with high fidelity" and can be used to iteratively extract eigenstates and eigenenergies.

minor comments (4)

[S1 and main text] There are typos: "reresentation" in S1 should be "representation", and "Hamiltotnian" in the main text should be "Hamiltonian".
[Eq. (9) and Eq. (S4)] The normalization 1/2Ng should be written 1/2^{N_g} to avoid ambiguity between 1/(2 N_g) and 2^{-N_g}.
[Fig. S2] The averages in Fig. S2 are over thirty random realizations, but no error bars or standard deviations are shown, making the claimed convergence to the thermodynamic limit difficult to assess.
[S2 noise model] The noise-corrupted states are not renormalized before computing the expectation values f_j; the authors state this is intentional, but the resulting quantity called "fidelity" is then not a standard state overlap and should be discussed as such.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor parameter choice in LiH demonstration; central H2 derivation is self-contained and not circular.

fitted input called prediction [Supplemental Material S4 (LiH), lambda_I sweep; main text LiH paragraph]
"However, once the covariance assertion is applied, we observe that for different values of λI the results of the method are different... Enforcing these constraints during a sweep over λI, we find an optimized value of λI. At the end of the calculation this shift of energy is subtracted."

The identity coefficient λI is a free parameter that changes the effective Hamiltonian, because the update rule (Eq. 6) subtracts E_g f_j from every coefficient, so a shift in λI changes E_g and hence all λ'_j. The paper selects λI by requiring the output spectrum to satisfy E_g≤E1≤E2≤..., i.e., by imposing a qualitative property of the expected output. The reported LiH excited-state energies are therefore not fully parameter-free predictions; a parameter is tuned to make the outputs look like a valid ordered spectrum. The actual numerical values of the excited-state energies are not used as inputs, so this is a mild fitted-input effect rather than a full reduction.

full rationale

The central derivation chain is not circular: Eq. (3) defines the projected Hamiltonian from the previous ground-state projector, and the covariance assertion (Eqs. (5)-(6)) constructs the effective Hamiltonian using only measured expectation values f_j and the ground-state energy E_g; the target excited-state energies are never used as inputs. The H2 benchmark is an external test against direct diagonalization of the original Hamiltonian and therefore constitutes independent support. The paper's S1 limitation (when E1≥0 the VQE minimization returns |G> with eigenvalue zero) and the LiH third-iteration failure are honest statements about the method's validity, not circular reasoning. The only parameter-selection issue is the LiH sweep over the identity coefficient λI, where the paper tunes λI by enforcing the expected ordering E_g≤E1≤E2; this conditions the LiH demonstration on a parameter chosen to make the output look like a valid spectrum, a mild fitted-input effect. No load-bearing self-citations or imported uniqueness theorems appear. Overall, the central claim retains independent content.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the covariance assertion, an unproven truncation of the projection operator, plus a practical requirement that the spectrum be shifted so excited-state energies are negative, which the paper does not resolve. The H2 demonstration avoids the shift question by using exact diagonalization instead of VQE.

free parameters (2)

Energy shift for H2 (unreported)
Required to prevent VQE from converging to the zero-eigenvalue ground state when the first excited-state energy is positive (large R); the paper does not state what shift, if any, was used for H2.
Identity coefficient lambdaI for LiH = range [-10,10], step 0.2, optimized to satisfy E0<=E1<=E2
Added to complete the Cz set; the value is swept and selected to enforce energy-ordering constraints, and then subtracted at the end (S4).

assumptions (5)

standard math The problem Hamiltonian can be written as a sum of Pauli string operators with real coefficients (Eq. 1).
Standard mapping of fermionic Hamiltonians via Jordan-Wigner or Bravyi-Kitaev.
domain assumption A variational algorithm (VQE) can approximate the ground state of the Hamiltonian at each iteration.
The method's inputs are the optimized parameters from VQE; the paper does not run VQE, it uses exact diagonalization.
ad hoc to paper The covariance assertion: only string operators already present in the Hamiltonian are retained from the projection operator and the rest are discarded (Eqs. 5-6).
This is the paper's central simplification; exact only for abelian group Hamiltonians with stabilizer ground states.
ad hoc to paper The projection operator can be represented as a projection onto the stabilizing subspace of a set of commuting operators (Eq. S4), without guaranteeing the projector matches the true ground-state projection.
State in S1: 'we simply assume that the projection operator can be equivalently replaced by the projection to the stabilizing subspace of a set of commuting independent operators without further assumptions.'
ad hoc to paper The perturbation terms in a perturbed-Cz Hamiltonian are small enough that the covariance assertion remains a good approximation.
Used to justify applying the method to H2 and LiH; no quantitative measure of 'small' is provided.

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Cite this review

Pith. "Pith review of A Method of Determining Excited-States for Quantum Computation." pith.science (2026). https://pith.science/paper/J3VU72NU

@misc{pith2026190805238,
  author       = {Pith},
  title        = {Pith review of: A Method of Determining Excited-States for Quantum Computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J3VU72NU}},
  note         = {Machine review of arXiv:1908.05238}
}

read the original abstract

A method is presented in which the ground-state subspace is projected out of a Hamiltonian representation. As a result of this projection, an effective Hamiltonian is constructed where its ground-state coincides with an excited-state of the original problem. Thus, low-lying excited-state energies can be calculated using existing hybrid-quantum classical techniques and variational algorithm(s) for determining ground-state. The method is shown to be fully valid for the H2 molecule. In addition, conditions for the method's success are discussed in terms of classes of Hamiltonians.

Figures

Figures reproduced from arXiv: 1908.05238 by the authors.

**Figure 1.** FIG. 1. (Color online) Molecular hydrogen (H [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. (Color online) The H [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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