{"id":"2cfe1a71-aa9f-4ad0-b8e5-71b587f4f945","arxiv_id":"2412.16472","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Higher-order generalized spread complexities show a more pronounced pre-equilibration peak than standard spread complexity in random matrix quenches, quantifying chaos more sharply up to third order.","lead":"This paper studies higher-order spread complexity measures that track how a quantum state spreads across a Krylov subspace during time evolution, after a sudden quench in random matrix models. It finds that the second and third order moments show a sharper pre-equilibration peak than the standard complexity, making them potentially sharper diagnostics for quantum chaos.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The monotonic increase of P_m is contradicted by Table 4: for the GOE ground-state quench P_4=0.11 < P_3=0.14, so the unqualified abstract claim overstates the result.","rationale":"I read the paper in good faith as establishing a state-dependent, mostly monotonic trend for the peak parameter rather than a universal law. The single most load-bearing issue is internal: the paper's own Table 4 violates the unqualified monotonicity claim (P_4<P_3 for the GOE ground-state quench). This is not a disagreement with external consensus; it is a mismatch between the abstract/summary and the tabulated numerics. The continuum-limit calculation provides independent analytic support for the GUE TFD case, and the N-scaling in Fig. 5 is useful, but it does not cover the pre-quench ground-state GOE case where the trend fails. The reader's verdict of CONDITIONAL is appropriate: the central claim should be qualified by state and by m<=3. I agree with the reader's overall verdict, but my weakest-assumption focus differs from the reader's continuum-limit concern, hence 'partial' agreement. The proposed check—more realizations with error bars on P_3 and P_4—would settle whether the Table 4 violation is a finite-statistics fluctuation or a genuine counterexample; either way the abstract needs qualification.","tokens_in":27090,"tokens_out":3133,"duration_ms":27471,"concrete_test":"Recompute Tables 4 and 7 for the GOE and GUE quenches from the pre-quench ground state with N=1000 and at least 100 independent realizations (or N=2000), extracting P_3 and P_4 with bootstrap standard errors. If P_4-P_3 remains negative by more than the combined uncertainty for the GOE case, the monotonic-in-m claim must be explicitly restricted (e.g., to m<=3 and/or to the |TFD> and |TFD0> initial states) in the abstract and Section 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that P_m increases with m—is not supported by the paper's own tables. For the GOE quench from the pre-quench ground state, Table 4 gives P_3=0.14 and P_4=0.11, so P_4<P_3; the ratio column P_m/P_{m-1}=0.79 for m=4. This directly contradicts the abstract's unqualified assertion that higher-order generalized spread complexities show increased sensitivity to the peak, and it contradicts the Section 4 statement 'the peak parameter increases with increasing m' unless one adds the qualifications that appear only in the prose/footnotes ('at least up to m=3', 'typically'). Moreover, the analytic prediction in Table 1 (0.33, 0.47, 0.55, 0.60) is derived for a GUE Hamiltonian with the post-quench TFD state in the continuum limit, not for the quench initial states that are the paper's new arena; it therefore cannot by itself establish the robustness of the monotonic trend for the quench protocols. The numerical support for a universal monotonic trend thus rests on 10 realizations and is contradicted by one of the four tabulated cases.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies generalized spread complexities C_m(t) = sum_n n^m |<Ψ(t)|K_n>|^2 after sudden quenches between random matrix Hamiltonians drawn from GOE/GUE ensembles. For a one-parameter family H_r(h), the pre-quench Hamiltonian has h=-1 and the post-quench Hamiltonian has h=+1. The authors examine three initial states: the post-quench TFD state, the pre-quench TFD state, and the pre-quench ground state. They introduce the peak parameter P_m = (C_m(t_peak) - \\bar{C}_m)/C_m(t_peak), which measures the height of the chaotic peak relative to the saturation value, and claim that higher-order generalized spread complexities are more sensitive to the peak (P_m increases with m). The paper derives continuum-limit values for P_m (Table 1) for the GUE TFD state and presents numerical results for finite N=1000 matrices with 10 realizations (Tables 2-7, Figs. 4-6), together with an N-dependence study for the TFD state (Fig. 5) and several appendices, including an exactly solvable harmonic-oscillator quench.","tokens_in":27305,"tokens_out":8302,"duration_ms":64617,"significance":"If the claimed monotonic increase of P_m with m were robust, generalized spread complexities C_2 and C_3 would offer sharper diagnostics of chaotic behavior in quench dynamics than the standard C_1, which would be a useful addition to the Krylov-complexity toolbox. The paper contains several valuable elements: a concrete quench protocol with validated IPR analysis (Fig. 2), an analytic continuum-limit calculation for the TFD state, an N-scaling study for the TFD state, and an exactly solvable oscillator example (Appendix A). The machine-checkable numerical data and the careful documentation of the Lanczos-coefficient distributions (Appendix B) are strengths. However, the central claim as stated in the abstract and Section 4 is stronger than the paper's own tables support, and the analytic derivation in Section 2.1 applies to a different initial state than the quench states that are the paper's new focus.","major_comments":[{"comment":"The claim that the peak parameter increases with m is stated without qualification in the abstract ('higher-order complexities show increased sensitivity to the peak') and in Section 4 ('the peak parameter increases with increasing m'). This is directly contradicted by Table 4 for the GOE quench from the pre-quench ground state |00>: P_3 = 0.14 and P_4 = 0.11, so P_4 < P_3. The manuscript only contains the needed qualification in Section 3.1 ('at least up to m = 3') and in footnote 14, which is not reflected in the abstract or in the unqualified Section 4 sentence. The authors should restrict the claim to m ≤ 3, or state explicitly which initial states and ensembles violate the trend, and adjust the abstract accordingly.","section":"Abstract; Section 4"},{"comment":"The analytic values P_m = 0.33, 0.47, 0.55, 0.60 in Table 1 are derived in the continuum limit for the infinite-temperature TFD state of the post-quench Hamiltonian, using the Lanczos-coefficient profile in eq. (2.9) and the GUE sine kernel. The pre-quench ground state and the pre-quench TFD state used in the quench protocols (Section 3) do not satisfy this setup, and the paper does not provide a derivation of the monotonic trend for these states. Presenting Table 1 as general corroboration of the quench numerics therefore goes beyond what the calculation supports. The text should explicitly state that the continuum-limit result applies to the TFD state and that its extension to the quench initial states is an assumption, not a derivation.","section":"Section 2.1 and Section 4"},{"comment":"The numerical support for the trend in P_m rests on averages over 10 realizations (stated in Section 3.1) reported to two significant digits in Tables 2-7, with no standard errors. The footnote accompanying Table 3 in Section 3.1 says the standard deviation of P_m was estimated and is small enough that P_m is incompatible with zero, but the values are not reported. In Table 4 the reversal P_4 < P_3 is a difference between 0.11 and 0.14, and without error bars or a larger number of realizations the reader cannot judge whether this difference (or the monotonic increases in the other tables) is statistically significant. The authors should report standard errors (or error bars) for P_m in the tables, or increase the number of realizations, so that the claim can be assessed quantitatively.","section":"Tables 2-7; Section 3.1"}],"minor_comments":[{"comment":"The text in Section 3.1 says the N-dependence study is 'averaged over 10 realizations' for GOE, while the caption of Fig. 5 says 'Each data point is the average of 20 realizations of GOE/GUE.' This inconsistency should be corrected.","section":"Section 3.1 and Fig. 5"},{"comment":"The definition of the peak parameter in Section 4 contains a typo: 'Pm =  Cm(tpeak)− ¯Cm)/Cm(tpeak)' is missing the opening parenthesis and should read 'P_m = (C_m(t_peak) − \\bar{C}_m)/C_m(t_peak).'","section":"Section 4 (summary)"},{"comment":"The symbol P is used both for the probability distribution P(j,t) in eq. (2.6) and for the peak parameter P_m defined in eq. (2.8); the notation should be disambiguated (for example, using lowercase p for the distribution) to avoid confusion.","section":"Section 2, eqs. (2.6)-(2.8)"},{"comment":"The statement 'as m increases, the peak occurs progressively later' is made for general chaotic systems, but the quench tables (Tables 2-7) do not report peak times, so this part of the claim is not numerically documented for the quench protocols; the authors should either add the peak times or phrase the statement more cautiously.","section":"Section 2, after eq. (2.8)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of JHEP and contains useful numerical results and a clean analytic setup for the TFD state. The main problem is that the abstract and Section 4 overstate the central claim: the monotonic increase of P_m is violated by the authors' own Table 4 for the GOE ground-state quench, and the continuum-limit derivation in Table 1 applies to a different initial state. These issues are fixable by careful revision, but they are load-bearing and require more than cosmetic changes. If revised to qualify the claim properly, report error bars, and clarify the scope of the analytic result, the paper could be acceptable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: the paper's main new tool, the peak parameter P_m, is a sensible quantifier, and the numerical trend P_1 < P_2 < P_3 holds across all their initial states for both GOE and GUE quenches. The fourth order breaks it in one case (Table 4, GOE ground state: P_4=0.11 < P_3=0.14), so the abstract's unqualified claim that higher-order complexities are more sensitive to the peak is too strong. The authors themselves qualify this in the text ('at least up to m=3', 'typically'), so this is an abstract/table mismatch, not a load-bearing flaw.\n\nWhat is actually new: P_m as a quantifier, the systematic extension of the quench protocol from GOE to GUE and to pre-quench ground/TFD initial states, and the analytic distribution of ratios of successive Lanczos coefficients for the Gaussian beta-ensemble (Appendix B), which is a clean, independent result. The continuum-limit P_m values (Table 1) come from the same group's prior work, but they are honestly used as a benchmark and compared against numerics; that's not circular.\n\nSoft spots, in order of severity: the abstract overstates the result, and the qualifier 'up to m=3' or 'for the first few orders' needs to be in the abstract. Table 4's P_4 < P_3 is a direct counterexample to the unqualified claim; it is in the paper's own data. There are no error bars on P_m in the tables. The text says the standard deviation is small enough that P_m is not compatible with zero, which is relevant, but it would be better to report it. The analytic Table 1 is derived for the post-quench TFD state, not for the quench initial states that are the paper's new arena; the numerics carry the argument there, and 10 realizations at N=1000 is modest but adequate given the N-dependence check.\n\nThe citation pattern is fine; the heavy reliance on [44] is legitimate because that is where the framework was set up.\n\nThis paper deserves a serious referee. I would encourage acceptance after a modest revision: qualify the abstract, either fix or caveat Table 4, and add error bars.","headline":"A useful refinement of spread-complexity diagnostics, with a real but fixable overstatement in the abstract about monotonic P_m.","tokens_in":27864,"tokens_out":1994,"would_cite":true,"duration_ms":16255,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.45.Mt"],"model":"deepseek-v4-flash","headline":"The peak parameter $P_m$ increases with moment order $m$, making $C_2$ and $C_3$ sharper probes of level repulsion than standard spread complexity.","keywords":["Krylov complexity","spread complexity","generalized spread complexity","quantum quench","random matrix theory","thermofield double state","level repulsion","Lanczos coefficients"],"falsifier":"Perform an exact finite-$N$ GUE quench with $N \\geq 1000$ and many more than ten realizations, and compute $P_m$ for $m = 1$ through $6$: if $P_m$ stops increasing at any order, or $P_m/P_{m-1} \\leq 1$, the central claim is falsified. Alternatively, measure $C_1$ through $C_4$ in a tunable chaotic quantum simulator such as a kicked top or trapped-ion quench; if higher-order peak parameters do not exceed $P_1$ for states with large inverse participation ratio, the hierarchy claimed here fails.","tokens_in":26876,"feed_emoji":"🎲","tokens_out":12510,"duration_ms":96982,"temperature":0.7,"pith_summary":"The paper studies how quickly a time-evolved state spreads through its Krylov subspace—the part of Hilbert space it actually visits—by tracking not only the mean position $C_1$ on the Krylov chain but the higher moments $C_m = \\langle n^m \\rangle$. It examines sudden quenches between random matrices drawn from the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), where the post-quench Hamiltonian is obtained by flipping the sign of the off-diagonal blocks, with initial states given by the post-quench thermofield double state or pre-quench ground and TFD states. The central claim is that the peak parameter $P_m = (C_m(t_{\\mathrm{peak}}) - \\bar{C}_m)/C_m(t_{\\mathrm{peak}})$ increases with $m$, so higher-order generalized spread complexities are more sensitive to the pre-equilibration peak that signals level repulsion. In the continuum limit the paper computes $P_1 = 0.33$, $P_2 = 0.47$, $P_3 = 0.55$, $P_4 = 0.60$ for GUE TFD states, and finite-size numerics for both ensembles show the same monotonic trend, with $C_2$ and $C_3$ the most efficient diagnostic orders.","feed_headline":"Higher moments of quantum spread sharpen the chaos peak","feed_subtitle":"C2 and C3 detect level repulsion more clearly than the standard spread complexity C1.","key_machinery":"The engine of the paper is the Krylov-chain representation of time evolution: starting from an initial state $|\\psi_0\\rangle$, the Hamiltonian generates an orthonormal Krylov basis $|K_n\\rangle$, and the time-evolved state has amplitudes $\\phi_n(t) = \\langle K_n | \\psi(t) \\rangle$. The generalized spread complexity $C_m(t) = \\sum_n n^m |\\phi_n(t)|^2$ is the $m$-th moment of the particle's position on the chain, with $C_1$ the ordinary spread complexity. The quantitative probe is the peak parameter $P_m = (C_m(t_{\\mathrm{peak}}) - \\bar{C}_m)/C_m(t_{\\mathrm{peak}})$, which measures the height of the pre-equilibration peak relative to the late-time average. The analytic calculation uses the continuum limit, in which the discrete chain index becomes a continuous coordinate and the ensemble-averaged Lanczos coefficients take the semicircle profile $\\langle a_n \\rangle = 0$, $\\langle b_n \\rangle = \\sqrt{1 - n/N}$, with GUE spectral correlations given by the sine kernel; these ingredients turn the Krylov dynamics into solvable first-order equations whose solutions fix the peak location and height.","core_discovery":"The central discovery is that the peak parameter $P_m = (C_m(t_{\\mathrm{peak}}) - \\bar{C}_m)/C_m(t_{\\mathrm{peak}})$ grows with the order $m$ for quenches between chaotic random-matrix Hamiltonians, so that higher-order generalized spread complexities probe the chaotic peak more sharply than the usual spread complexity. After a sudden quench that flips the sign of the off-diagonal blocks of a GOE or GUE matrix, all initial states studied—the post-quench TFD state, the pre-quench TFD state, and the pre-quench ground state—show the rise-peak-plateau pattern, but the peak is more prominent for GUE and for higher $m$. Analytically, in the continuum limit with the ensemble-averaged Lanczos profile $\\langle a_n \\rangle = 0$, $\\langle b_n \\rangle = \\sqrt{1 - n/N}$ and the sine-kernel two-point function, the paper obtains $P_1 = 0.33$, $P_2 = 0.47$, $P_3 = 0.55$, $P_4 = 0.60$ for the post-quench TFD state; the finite-$N$ numerics reproduce the monotonic increase of $P_m$ with $m$ for both ensembles, with the continuum values systematically overestimating the GUE numbers. The paper concludes that $C_2$ and $C_3$ are the most efficient orders for detecting the peak, since the ratio $P_m/P_{m-1}$ shrinks and the saturation values drop as $m$ grows.","pith_inferences":["One could test the same $P_m$ hierarchy in SYK or chaotic spin chains after a symmetry-breaking quench; if it persists, higher moments of the Krylov-position distribution become a generic order parameter for chaotic-integrable transitions.","The Appendix distribution of $r_n = b_{n+1}/b_n$ for Gaussian $\\beta$-ensembles suggests that tuning $\\beta$ (for instance through disordered spin chains) should interpolate the peak parameter between its GOE and GUE values, yielding a continuous chaos diagnostic.","The systematic gap between continuum and numerical values of $P_m$ points to a calculable $1/N$ correction; extracting it would let experiments use the peak height as a quantitative distance-from-random-matrix measure.","Because $C_m$ weights large chain positions increasingly heavily, higher moments should also sharpen other late-time chaos signatures such as the spectral form factor ramp, although the paper does not address this."],"forward_implications":["For GUE-based quenches the peak parameter grows from $P_1 \\approx 0.18$ to $P_4 \\approx 0.35$, so computing $C_2$ or $C_3$ instead of $C_1$ alone makes the chaotic peak substantially easier to detect.","Because $t_{\\mathrm{peak}} \\sim N$ and $C_m(t_{\\mathrm{peak}}) \\sim N^m$ for both ensembles, the normalized peak parameter $P_m$ is essentially independent of system size, so the diagnostic remains meaningful for large systems.","The continuum-limit values overestimate the finite-size GUE numerics for every $m$, so the analytic peak parameter acts as a systematic upper bound that finite-size effects shift downward.","The pre-quench ground state and pre-quench TFD state produce nearly identical complexity curves, indicating that the peak's presence depends mainly on the mutual randomness of the two energy bases—quantified by the inverse participation ratio—rather than on the detailed form of the initial state."],"supporting_citations":[{"why":"Defines the TFD-state spread complexity and the rise-peak-plateau behavior that the paper extends to higher moments.","marker":"[2]"},{"why":"Supplies the tridiagonal random-matrix construction and the average Lanczos coefficients used in the continuum limit.","marker":"[15]"},{"why":"Derives the continuum-limit expression for ensemble-averaged C1 from the semicircle Lanczos profile, the backbone of the analytic peak-parameter calculation.","marker":"[16]"},{"why":"Provides the characteristic-function formalism and the analytic C2 expression and saturation values from which the higher-order peak parameters are computed.","marker":"[44]"},{"why":"Introduces the block-sign-flip quench protocol and the inverse-participation-ratio scaling used to quantify randomness between pre- and post-quench bases.","marker":"[45]"}],"fun_headline_variants":["Higher-order spread complexity sharpens chaos peak","C2 and C3 reveal chaos peak more clearly than C1","Peak parameter grows with moment order in quenches","Higher moments of spread complexity probe chaos sharper"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that after the quench the hopping coefficients of the Krylov chain are well described by their ensemble-averaged semicircle profile $\\langle a_n \\rangle = 0$, $\\langle b_n \\rangle = \\sqrt{1 - n/N}$, so that the continuum-limit peak parameters $P_1 = 0.33$, $P_2 = 0.47$, $P_3 = 0.55$, $P_4 = 0.60$ apply to the initial states studied; if that profile is inaccurate for the pre-quench ground or TFD states, the claim rests on only ten numerical realizations.","fun_headline_variants_meta":{"raw":{"variants":["Higher-order spread complexity sharpens chaos peak","C2 and C3 reveal chaos peak more clearly than C1","Peak parameter grows with moment order in quenches","Higher moments of spread complexity probe chaos sharper"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1359,"prompt_tokens":1130,"completion_tokens":229,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":746,"completion_tokens_details":{"reasoning_tokens":166}},"tokens_in":746,"tokens_out":229,"duration_ms":2951,"temperature":1.0,"reasoning_tokens":166,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:32:39.959214+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform an exact finite-$N$ GUE quench with $N \\geq 1000$ and many more than ten realizations, and compute $P_m$ for $m = 1$ through $6$: if $P_m$ stops increasing at any order, or $P_m/P_{m-1} \\leq 1$, the central claim is falsified. Alternatively, measure $C_1$ through $C_4$ in a tunable chaotic quantum simulator such as a kicked top or trapped-ion quench; if higher-order peak parameters do not exceed $P_1$ for states with large inverse participation ratio, the hierarchy claimed here fails.","supporting_citations":[{"cited_title":"Balasubramanian, P","cited_arxiv_id":null,"evidence_quote":"Defines the TFD-state spread complexity and the rise-peak-plateau behavior that the paper extends to higher moments."},{"cited_title":"Balasubramanian, J","cited_arxiv_id":null,"evidence_quote":"Supplies the tridiagonal random-matrix construction and the average Lanczos coefficients used in the continuum limit."},{"cited_title":"Erdmenger, S.-K","cited_arxiv_id":null,"evidence_quote":"Derives the continuum-limit expression for ensemble-averaged C1 from the semicircle Lanczos profile, the backbone of the analytic peak-parameter calculation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the block-sign-flip quench protocol and the inverse-participation-ratio scaling used to quantify randomness between pre- and post-quench bases."}],"review_version":1}