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REVIEW 3 major objections 4 minor 13 references

Hamiltonian spectra in quantum computers through the generalized eigenvalue method

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper shows that applying the generalized eigenvalue method to real-time correlators yields eigenvalues $e^{-iE_k t}$ plus small corrections, so several low-lying Hamiltonian levels can be read off from short time evolution.

desk verdict A useful practical GEVP method for real-time quantum-computer correlators, with an honest hardware demo, but the error claim in Eq. (5) is not proven for approximate ground states. read the letter →

arxiv 2608.08181 v1 pith:I5UNTSGV submitted 2026-08-08 quant-ph hep-latnucl-th

Valery Simonyan , Greg Ridgway , Paulo Bedaque This is my paper · ORCID
classification quant-phhep-latnucl-th
keywords generalizedeigenvalueproblemreal-timecorrelatorsHamiltonianspectrumquantumhardwarefuzzysigmamodelFouriertransformTrotterizationspectroscopy
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

Real-time correlators produced by quantum computers do not decay, so the usual Euclidean trick of reading energies from late-time exponentials is not available. This paper argues that the generalized eigenvalue problem (GEVP), long used in lattice field theory, works just as well for these oscillating correlators: the eigenvalues of $C^{-1}(0)C(t)$ are approximately $e^{-iE_k t}$, and the energies $E_k$ can be extracted from $(i/t)\log\lambda_k(t)$ at relatively short times. The claim matters because Fourier analysis of the same data requires time extents roughly twenty times longer, which current noisy hardware cannot sustain. The paper supports the claim with exact simulations, noisy simulations, and runs on a trapped-ion quantum computer, all on the fuzzy $\sigma$-model, resolving several low-lying levels to sub-percent or few-percent accuracy.

What carries the argument

The load-bearing object is the split of the correlator matrix into $Q\,U_{\rm in}(t)\,Q^\dagger + P\,U_{\rm out}(t)\,P^\dagger$, where $Q$ is the $M\times M$ overlap of the imperfect operator basis with the first $M$ energy eigenstates and $P$ collects the overlaps with the remaining states. The assumption that $P$ is small relative to $Q$ makes $C^{-1}(0)C(t)$ approximately similar to the diagonal matrix $U_{\rm in}(t)$ of phase factors $e^{-iE_k t}$; diagonalizing it automatically forms the linear combinations of operators that isolate individual eigenstates. On the measurement side, the correlator elements are obtained with one ancilla qubit by entangling the system, evolving, and reading $\langle X\otimes P_a\rangle$ and $\langle Y\otimes P_a\rangle$.

What would settle it

For the $L=8$, $g=1.2$ fuzzy $\sigma$-model with the strong-coupling singlet product as $|\Omega\rangle$, compute the exact correlator matrix $C(t)$ by full diagonalization without truncating the intermediate-state sum, and compare $(i/t)\log\lambda_k(t)$ with the true $E_k$ over $t\in[1.6,5.6]$. If the deviations are not controlled by the size of $Q^{-1}P$ and do not shrink with time, the central approximation is false. A more targeted check is to evaluate the overlap matrix $\langle\Omega|m\rangle\langle n|\Omega\rangle$ and verify that its off-diagonal positive-frequency terms are numerically negligible.

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Extended reading notes

Core claim

The central discovery is a real-time analogue of the Euclidean GEVP. For a correlator matrix $C_{ij}(t)=\langle\Omega|O_i(t)O_j^\dagger(0)|\Omega\rangle$ built from $M$ imperfect operators, the paper shows that $C^{-1}(0)C(t)$ equals, up to corrections of order $(Q^{-1}P)^2$, a matrix with eigenvalues $e^{-iE_k t}$. Consequently $E_k$ can be recovered from $(i/t)\log\lambda_k(t)$, with the error decaying like $1/t$ or faster. The paper demonstrates this on the fuzzy $\sigma$-model at $L=8$ with exact time evolution, recovering 8, 36, and 276 levels with the three operator sets, and on $L=4$ hardware, where the four levels accessible to the single-Pauli basis come within a few percent of exact diagonalization.

Load-bearing premise

The entire derivation rests on the approximate ground state $|\Omega\rangle$ being close enough to the true ground state that the positive-frequency cross terms $\langle\Omega|m\rangle\langle n|\Omega\rangle$ can be dropped from the correlator matrix; if those overlaps are not small, the eigenvalues of $C^{-1}(0)C(t)$ are no longer $e^{-iE_k t}$ plus a small correction and the extracted energies are biased.

Editorial extensions

If this is right

  • Energy levels are extracted from time extents around $T=5.6$ lattice units, whereas the Fourier transform of the same correlators needs $T\approx100$ to resolve the same peaks.
  • The degeneracy pattern of the O(3) symmetric model emerges automatically from the diagonalization, so the method doubles as a symmetry diagnostic.
  • Trotterization errors enter at second order in the step size and can be removed by a quadratic extrapolation to $\Delta t\to0$.
  • On a current trapped-ion device, four low-lying levels are recovered within a few percent with bootstrap statistical errors; residual offsets are attributed to noise and Trotter effects.

Reading between the lines

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

  • Beyond the paper's tests, the same GEVP extraction should work with any approximate ground state that has good overlap with the true one, including weak-coupling or variational states; the gate cost would rise, but the spectral extraction itself is state-agnostic.
  • A direct numerical check of the paper's central approximation would be to compute the overlap matrix $\langle\Omega|m\rangle\langle n|\Omega\rangle$ for the strong-coupling singlet state at $g=1.2$; if these entries are not small, the claimed $O((Q^{-1}P)^2)$ bound is violated and extracted energies are biased.
  • The method's short-time efficiency makes it a natural candidate for excited-state spectroscopy in gauge theories at finite density, where Euclidean Monte Carlo fails because of the sign problem but real-time correlators remain measurable in principle.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript adapts the generalized eigenvalue problem (GEVP), standard for Euclidean lattice correlators, to real-time correlators produced on quantum computers. The central formal claim is that for a set of imperfect Pauli operators O_i and an approximate ground state |Ω⟩, the eigenvalues of C^{-1}(0)C(t), with C_ij(t)=⟨Ω|O_i(t)O_j†(0)|Ω⟩, are approximately e^{-iE_k t}+O((Q^{-1}P)^2), so energies can be read off as E_eff(t)=(i/t)log λ(t) at short time extents. An ancilla-qubit protocol is presented for measuring the off-diagonal correlators. The method is tested on the fuzzy σ-model via exact diagonalization (L=8), noiseless Trotterized simulation (L=4), and a run on IonQ Forte-Enterprise (L=4), and compared against Fourier-spectral extraction, which is reported to need much longer evolution times.

Significance. The proposed method is potentially valuable: real-time correlators are natural on quantum hardware, and a GEVP-based extraction could avoid the long time extents required by Fourier methods, where decoherence and Trotter error accumulate. The paper includes a clear exposition, a concrete measurement protocol with an ancilla, a nontrivial field-theory test with exact-diagonalization checks, and a hardware demonstration; the operator-set construction exploiting O(3) symmetry is a nice feature. However, the formal derivation contains a load-bearing approximation (the treatment of the approximate ground state as the exact vacuum) and the hardware evidence is partially weakened by one 12.6σ discrepancy with only statistical errors quoted. With these points addressed, the method would be a useful addition; as it stands, the central claim is not fully established.

major comments (3)
  1. [§II, Eq. (5)] The derivation of Eq. (5) replaces ⟨Ω|Ō_m(t)Ō_n†(0)|Ω⟩ with e^{-iE_n t}δ_{mn}. This is exact only when |Ω⟩ is the true ground state |0⟩. For a general state |Ω⟩=c_0|0⟩+Σ_{n≥1}c_n|n⟩, the exact matrix element for m,n≥1 is |c_0|^2 e^{-iE_n t}δ_{mn}+c_m^* c_n e^{iE_m t}; the second, positive-frequency term is not small unless the overlaps c_n are bounded. The paper uses the strong-coupling singlet product as |Ω⟩ (Section IV) and never reports its overlap with the exact ground state, nor any bound that would justify neglecting these terms. Consequently the central claim that eigenvalues of C^{-1}(0)C(t) equal e^{-iE_k t}+O((Q^{-1}P)^2) is not established for the tested states; the agreement in Figs. 2 and 5 may hold because the overlaps happen to be small, but that is an unverified assumption. This is the key load-bearing point and needs to be addressed, either by quantifying the overlaps for the models studied or by proving a bound under stated assumptions.
  2. [§V.B, Table II] The hardware demonstration is presented as successful agreement with exact diagonalization, but Table II shows the k=0 singlet extracted at 3.203(14) versus E_exact=3.380, a 12.6σ deviation. Only bootstrap (statistical) uncertainties are quoted; no systematic error budget from gate noise, readout, or Trotterization is provided. The text attributes the discrepancy to noise/decoherence based on the noiseless Trotter extrapolation, but that does not quantify the hardware systematic error. As written, the claim that all four levels are resolved 'within a few percent' overstates the evidence: one level is not resolved at the quoted precision. The authors should either report a systematic error estimate or qualify the hardware result as partially validated.
  3. [§V.A and Fig. 4] The relative-efficiency claim rests on comparing GEVP with a constant fit over t∈[1.6,5.6] against a Fourier transform with η=0.05 chosen to 'just barely smooth out' ringing artifacts. No sensitivity analysis is shown for η or the fit window, and the Fourier peaks in Fig. 4 are still not fully resolved at T=100. While the qualitative conclusion is plausible, the quantitative claim of being 'substantially more efficient' should be supported by a controlled comparison, for example by defining a resolution criterion and scanning over η and the fit-window endpoints.
minor comments (4)
  1. [§II, Eqs. (6)-(10)] The notation O((Q^{-1}P)^2) is used for matrix products without specifying a matrix norm; the scalar expression for λ_k^{(1)}(t) in Eq. (9) should be accompanied by an explicit norm and a statement of whether the bound is uniform in t.
  2. [§V.A, Fig. 3 caption] The caption states 'The bands show a quadratic fit and the bands indicate a one σ statistical (not systematic) error,' which is ambiguous: a quadratic fit line and an error band cannot both be represented by the same 'bands.' Please clarify what is plotted.
  3. [§V.A] The sentence 'The errors shown at bottom right of Fig. 2 arise entirely from the finite time extent used in the fits' is too strong; the residual pattern (underestimation of low energies, overestimation of high energies) is a systematic finite-window bias, but no error model is given for it.
  4. [§V.A, Fig. 4] The choice η=0.05 is described as tuned to 'just barely smooth out' ringing at T=100; a brief scan over η would show how the Fourier results depend on this subjective choice and strengthen the comparison with GEVP.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: GEVP energies are extracted from correlators and benchmarked against exact diagonalization; self-citations are contextual.

full rationale

The paper's central claim—that eigenvalues of C^{-1}(0)C(t) approximate e^{-iE_k t}—is a mathematical derivation from the definition of the correlator matrix, not an input fitted to the target energies. The extracted energies are read off from the time dependence of the measured/full-evolution correlators and compared with exact diagonalization as an external benchmark; no energy level is used as an input to the GEVP. The strong-coupling singlet |Omega> and Pauli operator sets define the trial subspace, but the time evolution is generated by the full Hamiltonian, so the resulting phases are not put in by hand. The main caveat is that Eq. (5) replaces <Omega|Obar_m(t)Obar_n^dag|Omega> by e^{-iE_n t}delta_{mn}, which is exact only for the true ground state; for the approximate Omega used in the numerics this is an uncontrolled approximation (a correctness/validity concern), but it is not circular because the energies are not assumed in the construction. Self-citations to Refs. [4]–[7] identify the fuzzy sigma model and prior hardware work, but Eq. (17) defines the Hamiltonian in the paper itself, and the standard GEVP references [1]–[3] are external. Minor tuning parameters (eta for the Fourier comparison, the constant-fit window) affect the comparison but do not enter the GEVP derivation. Accordingly no circular step is present; score reflects only non-load-bearing self-citation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on three main assumptions: the operator basis is complete enough that out-of-basis overlaps P are small, the approximate ground state behaves like the exact vacuum in the correlator formula, and the Pauli operators chosen from strong coupling span the low-lying sectors. The first two are not quantified. The free parameters are tuning choices for the comparison and the extrapolation, not for the GEVP derivation itself. No new particles, forces, or entities are introduced.

free parameters (3)
  • Fourier damping eta = 0.05
    Chosen by hand in Sec. V.A to 'just barely smooth out' ringing artifacts in the T=100 Fourier spectrum. Affects the comparison but not the GEVP extraction.
  • GEVP constant-fit window = t in [1.6, 5.6]
    Selected for the plateau fits in Fig. 2. The extracted energies depend on this choice, and the residual plot shows a systematic bias.
  • Trotter extrapolation coefficients = c0, c1, c2 (not quoted numerically)
    A quadratic fit over dt = 0.05, 0.1, 0.2, 0.4 is used to extrapolate to dt -> 0 in Fig. 3. This is a fitting step, not a prediction.
assumptions (4)
  • domain assumption The imperfect operator basis spans the low-lying states of interest, and states outside the basis have a small overlap matrix P relative to Q.
    Used in Eqs. (6) through (9) to justify dropping P U_out P^dagger terms. Not derived from the Hamiltonian; it is assumed to hold for the chosen operator sets.
  • domain assumption The approximate ground state |Omega> has negligible overlap with low-lying excited states, so <Omega|Obar_m(t)Obar_n^dagger(0)|Omega> is approximately e^{-iE_n t} delta_{mn} in Eq. (5).
    Required to make the GEVP derivation exact. The paper does not quantify the overlap or bound the resulting positive-frequency contamination.
  • domain assumption The Pauli operators in Table I generate states that approximately connect |Omega> to the energy eigenstates in each symmetry sector.
    Based on strong-coupling intuition for the fuzzy sigma model. It is assumed, not proven, that these operators are sufficient for the low-lying spectrum.
  • domain assumption Trotter error is small and quadratic in the step size dt.
    Used for the dt -> 0 extrapolation in Fig. 3. Tested at four step sizes only, so the quadratic form is an assumption.

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Cite this review

Pith. "Pith review of Hamiltonian spectra in quantum computers through the generalized eigenvalue method." pith.science (2026). https://pith.science/paper/I5UNTSGV

@misc{pith2026260808181,
  author       = {Pith},
  title        = {Pith review of: Hamiltonian spectra in quantum computers through the generalized eigenvalue method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5UNTSGV}},
  note         = {Machine review of arXiv:2608.08181}
}
read the original abstract

Quantum computers can generate real-time correlators of field theories. By adapting the generalized eigenvalue problem to these correlators, energy eigenvalues can be extracted directly. The method is tested using both classical simulations and quantum hardware, successfully resolving several low-lying energy levels in agreement with exact diagonalization. Comparison with an alternative spectrum determination based on the Fourier transform of correlators shows that the proposed approach is substantially more efficient.

Figures

Figures reproduced from arXiv: 2608.08181 by the authors.

Figure 1
Figure 1. FIG. 1: Visualization of the Hamiltonian in Eq. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Relative difference between extracted low-lying [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The convergence of the discrete Fourier transform method from Eq. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Result of run on IonQ’s Forte Enterprise quantum computer using the smallest (set 1) of operators, [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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Reference graph

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