{"id":"9b2720f9-0609-453f-8a67-4fcd0361de61","arxiv_id":"2608.08181","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Energy eigenvalues of a Hamiltonian can be extracted from real-time correlator matrices via a generalized eigenvalue problem, and the method outperforms Fourier analysis on a quantum computer.","lead":"This paper adapts the generalized eigenvalue problem, a standard lattice field theory tool, to the real-time correlators produced by quantum computers, and uses it to extract several low-lying energy levels of the fuzzy sigma model. Classical simulations and an IonQ ion-trap run show that the method needs much shorter time evolution than Fourier analysis of the same data, which matters for near-term devices with limited coherence.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The derivation of Eq. (5) requires |Ω> to be the true ground state; the strong-coupling singlet used in tests is not, so the claimed O((Q^{-1}P)^2) error bound lacks support without quantifying overlaps.","rationale":"The reader identifies the same weakest assumption: Eq. (5) requires the true ground state, and no bound is given for the approximate singleton product. This is indeed the load-bearing step; if it fails, the GEVP eigenvalues can be biased and the extracted energies unreliable. No other concern (e.g., Fourier comparison, Trotter extrapolation, branch cuts) is as fundamental, because the central novelty is the eigenvalue formula itself. The paper's numerical results show the method works in practice, but they do not quantify the overlap error, and the hardware run uses an even smaller basis and larger Trotter error. Thus the verdict should remain CONDITIONAL: the method is promising and supported by experiments, but the theoretical justification is incomplete without an overlap bound or a numerical demonstration that the neglected terms are small. The proposed test would settle this directly.","tokens_in":74,"tokens_out":7746,"duration_ms":78684,"concrete_test":"Using exact diagonalization of the L=4 (or L=8) fuzzy σ-model at g=1.2, compute the overlaps c_n=<Ω|n> for the strong-coupling singlet product |Ω> and the low-lying eigenstates. Then compute the matrix M_mn(t)=<Ω|Obar_m(t)Obar_n†|Ω> - δ_mn e^{-iE_n t} for the operator set 1. If ||M|| / ||diag(e^{-iE_n t})|| is not small (e.g., >10%), Eq. (5) is violated. Also compute the exact eigenvalues λ_k(t) of C^{-1}(0)C(t) and compare to e^{-iE_k t}; check whether the maximum deviation scales as ||Q^{-1}P||^2, where P and Q are constructed from the overlaps. This directly tests whether the central claim holds for the prepared state.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central result—that eigenvalues of C^{-1}(0)C(t) are e^{-iE_k t}+O((Q^{-1}P)^2)—rests on Eq. (5), where <Ω|Obar_m(t)Obar_n†|Ω> is replaced by δ_mn e^{-iE_n t}. This identity holds only for |Ω>=|0>, the exact ground state. The numerical and hardware tests use |Ω> as the strong-coupling singlet product (Section IV), which is not the ground state at g=1.2. For a general |Ω>=Σ_n c_n|n>, the exact matrix element includes positive-frequency terms such as e^{iE_m t}<Ω|m><n|Ω> and a ground-state contribution e^{-iE_m t}|<0|Ω>|^2 δ_mn; these are not captured by the derivation. The paper neither estimates c_n nor bounds the resulting error, so the stated O((Q^{-1}P)^2) correction is not established for the tested state. The empirical agreement in Figs. 2 and 5 may hold because the overlaps are small, but that is an unverified assumption, and the claim of a general method is therefore conditional.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8466,"tokens_out":13857,"duration_ms":134363,"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":[{"comment":"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.","section":"§II, Eq. (5)"},{"comment":"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.","section":"§V.B, Table II"},{"comment":"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.","section":"§V.A and Fig. 4"}],"minor_comments":[{"comment":"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.","section":"§II, Eqs. (6)-(10)"},{"comment":"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.","section":"§V.A, Fig. 3 caption"},{"comment":"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.","section":"§V.A"},{"comment":"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.","section":"§V.A, Fig. 4"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the paper is in scope for quant-ph and builds naturally on the authors' prior work [5]. The main technical gap is the treatment of the approximate ground state in Eq. (5); this is fixable with overlap estimates or a restricted theorem. The hardware discrepancy at k=0 is concerning but also fixable with an honest systematic error budget. I would not reject at this stage; the method is promising and the numerical demonstrations are valuable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, honest methods paper, but the central derivation has a gap that matters. The authors adapt the standard Euclidean GEVP to real-time correlators, with an ancilla-based scheme for measuring off-diagonal Pauli correlators and a hardware demonstration on IonQ. The method works in their benchmark, and the comparison against Fourier is fair and compelling. But Eq. (5) treats the approximate ground state |Ω> as the exact vacuum. For a general |Ω> = c0|0> + Σ_{m≥1} c_m |m>, the matrix element <Ω|Obar_m(t)Obar_n†|Ω> is |c0|^2 e^{-iE_m t} δ_{mn} + c_m* c_n e^{iE_m t}, not just e^{-iE_m t} δ_{mn}. The second term is a positive-frequency contamination that is not O((Q^{-1}P)^2), and the paper neither bounds it nor estimates the overlaps c_m. In the tests |Ω> is the strong-coupling singlet product at g=1.2, which is not the true ground state. That the results come out right suggests the overlaps are small, but this is an unverified numerical coincidence, not a proven property.\n\nWhat's genuinely new: the ancilla protocol for the off-diagonal correlators, the O(3) symmetry-aware operator sets, and the real-hardware run. The hardware section is refreshingly candid: the k=0 level is 12.6σ off, and they attribute it to noise rather than hiding it. The Fourier comparison uses a hand-chosen damping η, but that's a minor tuning issue. Lack of code/data is a reproducibility soft spot.\n\nNet: the paper is worth a serious referee. It's a practical advance, not a deep theoretical one, and the GEVP relation itself is a textbook continuation. The referee should focus on Eq. (5) and demand either a bound on the c_m overlaps or a numerical estimate for the states used. If that gap is closed, it's a publishable methods paper for the quantum-simulation and lattice communities. I'd accept it for peer review as is.","headline":"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.","tokens_in":8968,"tokens_out":6069,"would_cite":true,"duration_ms":55391,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["generalized eigenvalue problem","real-time correlators","Hamiltonian spectrum","quantum hardware","fuzzy sigma model","Fourier transform","Trotterization","spectroscopy"],"falsifier":"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.","tokens_in":8019,"feed_emoji":"⚛️","tokens_out":9811,"duration_ms":84484,"temperature":0.7,"pith_summary":"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.","feed_headline":"Generalized eigenvalues resolve spectra from short real-time runs","feed_subtitle":"Pulls low-lying energies from time extents where Fourier analysis fails, on exact, noisy, and real hardware runs.","key_machinery":"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 $\\langle Y\\otimes P_a\\rangle$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Established the generalized eigenvalue method for extracting energies from correlator matrices, which this paper adapts to real time.","marker":"[1]"},{"why":"Provided an early application of GEVP-style extraction to lattice gauge theory, grounding the method.","marker":"[2]"},{"why":"Gave the systematic treatment of GEVP for energies and matrix elements in lattice field theory, including the error structure carried over here.","marker":"[3]"},{"why":"Defined the fuzzy sigma model used as the testbed and supplied the strong-coupling ground state and operator intuition.","marker":"[4]"},{"why":"Reported the prior attempt to extract the mass gap on quantum hardware that motivates this work.","marker":"[5]"},{"why":"Supplied the bootstrap procedure used to estimate statistical errors from finite shot counts on the hardware runs.","marker":"[13]"}],"fun_headline_variants":["GEVP extracts spectra from short correlator runs","Real-time GEVP beats Fourier for spectra","Generalized eigenvalues yield spectra from real-time data","Spectra from quantum hardware via GEVP"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["GEVP extracts spectra from short correlator runs","Real-time GEVP beats Fourier for spectra","Generalized eigenvalues yield spectra from real-time data","Spectra from quantum hardware via GEVP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000183,"raw_usage":{"total_tokens":1234,"prompt_tokens":782,"completion_tokens":452,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":398,"completion_tokens_details":{"reasoning_tokens":395}},"tokens_in":398,"tokens_out":452,"duration_ms":4973,"temperature":1.0,"reasoning_tokens":395,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:18:51.837794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Sigma models on quantum computers","cited_arxiv_id":"1903.06577","evidence_quote":"Defined the fuzzy sigma model used as the testbed and supplied the strong-coupling ground state and operator intuition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reported the prior attempt to extract the mass gap on quantum hardware that motivates this work."},{"cited_title":"Efron, Bootstrap methods: another look at the jackknife, The Annals of Statistics7, 1 (1979)","cited_arxiv_id":null,"evidence_quote":"Supplied the bootstrap procedure used to estimate statistical errors from finite shot counts on the hardware runs."}],"review_version":1}