{"id":"f75f32a0-4f7b-4a22-bad7-b15e1484e5a1","arxiv_id":"2607.10454","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Krylov spread complexity for time-dependent Hamiltonians can be computed through Floquet–Magnus approximations, with a piecewise Magnus scheme for non-periodic drives.","lead":"This paper gives a Magnus/Floquet-based recipe for computing Krylov spread complexity of states under time-dependent Hamiltonians, including explicit formulas for driven two-level systems and a piecewise Magnus algorithm for time-dependent oscillators. It matters because time-dependent drives are common in quantum engineering, and it offers a practical route to complexity growth when no static effective Hamiltonian exists in closed form.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central 'reliable method' claim for piecewise Magnus rests on an unproven linear error-accumulation assumption and is only validated in sudden/adiabatic limits; no intermediate or convergence test appears.","rationale":"I read the paper as making a practical-methods claim rather than a new analytic theorem: it proposes Floquet–Magnus and piecewise Magnus approximations as reliable tools for computing Krylov complexity in time-dependent systems. The reader's weakest assumption—that local errors in the piecewise product accumulate at most linearly—is exactly the load-bearing point of §5.3 and the abstract's reliability claim. I agree with the reader's assessment: the analytic parts (two-level Floquet results, squeezed-vacuum formula) are coherent, but the piecewise method is validated only in the sudden and adiabatic limits, which are precisely the regimes where the method is least needed. The intermediate regime, where the global Magnus expansion is claimed to lose accuracy and the piecewise method is supposed to help, is not benchmarked against an exact solution, and no Δ-convergence test is shown. I considered whether a deeper definitional issue—whether the even-Fock chain is truly the Krylov basis for a time-dependent quadratic Hamiltonian—could be more severe. However, the even-Fock basis is the natural su(1,1) Krylov basis for this class of systems, and the paper explicitly adopts it; this is a defensible choice and not the primary unresolved risk. Therefore, the central unresolved risk is the piecewise error accumulation and its numerical verification. Since the manuscript already receives CONDITIONAL, my read does not change the verdict: the claim remains plausible but insufficiently validated. The proposed intermediate benchmark with Δ-scaling would settle whether the concern lands.","tokens_in":14928,"tokens_out":18553,"duration_ms":182572,"concrete_test":"For the soft quench with ω0=1, ω1=2, τ=10, compute the exact Krylov complexity C_K(t) by high-precision numerical integration of the time-dependent symplectic flow (e.g., adaptive RK4/5 for M(t), or solving the Ermakov–Pinney equation) up to t=20 to machine precision. Then evaluate the piecewise-Magnus second-order method for Δ=0.1, 0.05, 0.025 and plot max_t |C_K^piecewise − C_K^exact|. If the error does not decrease as Δ² (or is not ≤1% of max C_K), the O(tΔ^m) accumulation assumption and the 'reliable' claim in the intermediate regime fail. Repeat for τ=1 to probe the non-adiabatic, non-sudden regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that a piecewise Magnus expansion 'provides a reliable method' when the global Magnus expansion loses convergence or accuracy (Abstract; §5.3; §6)—is not supported in the regime where it is most needed. The total-error estimate in §5.3, O(NΔ^{m+1}) = O(tΔ^m), relies explicitly on 'assuming that local errors accumulate at most linearly, up to stability constants.' This consistency-plus-stability argument is standard for one-step integrators, but it is neither proved nor verified by any convergence test. The numerical validation (§5.4) checks only two asymptotic limits: τ=0.1 against the sudden-quench formula and τ=100 against adiabatic intuition. The intermediate case τ=10 (Fig. 2) is shown with no comparison to an exact or high-accuracy solution, and no curve for computational error as a function of Δ is provided. Since the abstract claims reliability exactly when a global Magnus expansion fails, the absence of an intermediate benchmark or a Δ-scaling check leaves the central claim unsupported; code and data are not supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies Krylov (spread) complexity for states evolving under explicitly time-dependent Hamiltonians. For periodically driven two-level systems, the authors use Floquet theory and the Magnus/Floquet–Magnus expansion to obtain approximate stroboscopic complexity, including closed-form expressions in terms of Bessel and Struve functions. For non-periodic driving, they propose a piecewise Magnus expansion: the interval is split into small segments, each segment is approximated by a local effective Hamiltonian, and the evolution is composed from the local exponentials. For a harmonic oscillator with time-dependent frequency, the Krylov complexity is related to the Bogoliubov coefficient via C_K = (1/2) |ν(t)|^2, and the piecewise method is tested numerically on smooth frequency quenches. The central claim is that the piecewise Magnus expansion provides a reliable method when a global Magnus expansion loses convergence or accuracy.","tokens_in":15245,"tokens_out":12424,"duration_ms":136055,"significance":"If fully established, the paper offers a practical tool for computing Krylov complexity in time-dependent systems without the need for an exact Floquet Hamiltonian. The analytical results for the two-level systems—especially the dressed-frame Floquet–Magnus derivation and the Bessel/Struve expressions—are useful and internally consistent. The exact relation C_K = (1/2)|ν(t)|^2 for the squeezed-vacuum Krylov chain is derived in Appendix C without free parameters fitted to the predicted quantity; h_x and h_z are computed from the Magnus integrals, and the complexity is read off afterward. The abrupt-quench limit and the adiabatic limit serve as honest consistency checks. However, the paper's main new contribution—the claim that the piecewise Magnus expansion is reliable in regimes where the global expansion fails—is not yet supported by the numerical evidence or by a rigorous error-accumulation argument.","major_comments":[{"comment":"The central claim that the piecewise Magnus expansion 'provides a reliable method' rests on the total-error estimate O(N Δ^{m+1}) = O(t Δ^m) stated after Eq. (101). This estimate follows from the sentence 'Assuming that these local errors accumulate at most linearly, up to stability constants.' For an ordered product of local exponentials S_{N-1} ⋯ S_0, the linear-accumulation premise is not a trivial consequence of the local error bound; it requires a stability argument for the error propagation through the composition map. No such proof is given, and no numerical test of the Δ-dependence of the global error is provided. Because the reliability claim is the main new contribution, this is a load-bearing gap. Please either supply a stability/convergence proof for the composition of local Magnus exponentials, or provide a numerical convergence study that measures the global error as a func","section":"§5.3, Eq. (101)"},{"comment":"The numerical validation covers only two asymptotic regimes: τ = 0.1 against the sudden-quench formula Eq. (89), and τ = 100 against the adiabatic expectation. The intermediate regime τ = 10, which is precisely the regime where a global Magnus expansion may lose accuracy and where the piecewise method is claimed to be useful, is shown in Fig. 2 with no comparison to an exact or high-accuracy solution. No error-versus-Δ curve is presented, and the statements 'agrees well' and 'approaches a nearly constant final value' are not quantified. The abstract and §6 claim reliability when the global expansion 'loses convergence or accuracy'; the current evidence does not support that claim in the intermediate regime. Please add an intermediate-time benchmark (e.g., a direct high-accuracy integration of the time-dependent Schrödinger equation) and a Δ-scaling error plot.","section":"§5.4, Figs. 2–4"}],"minor_comments":[{"comment":"The second term is written as 'Ω(t)' but should be 'Ω_2(t)'.","section":"Eq. (16)"},{"comment":"The vertical-axis label appears garbled: 'ln( (t) (N)(t) )' should be ln ||Ω(t)−Ω^{(N)}(t)|| or similar.","section":"Fig. 1"},{"comment":"The caption 'Sudden Approximation' is vague; please state the parameters and specify what the plotted curve is compared against.","section":"Fig. 3"},{"comment":"The phrase 'agrees well with the exact analytical expression' is not quantified. Please state the maximum deviation or show an error plot.","section":"§5.4"},{"comment":"The manuscript contains numerical results but no code/data availability statement. For reproducibility, please provide the code or a sufficiently detailed description of the numerical implementation and partition choices.","section":"General"},{"comment":"Reference [31] is listed as 'Unpublished'. If it is not publicly available, consider removing it or citing a preprint/arXiv version.","section":"Ref. [31]"},{"comment":"Please clarify explicitly how the even-Fock basis |K_n⟩=|2n⟩ arises as the Lanczos basis for the piecewise product of local effective Hamiltonians, connecting it to the general construction in Appendix A. The basis is natural for each local effective Hamiltonian, but the product of different local generators deserves a sentence of justification.","section":"§5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a mix of review and new results. The Floquet–Magnus analytical part is solid and publishable in principle. The main concern is the gap between the advertised 'reliable method' claim and the evidence supplied: the linear-accumulation assumption is unproven, and the numerical validation misses the intermediate regime and lacks convergence tests. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection. The authors should either prove a global error bound for the piecewise composition or provide a convincing numerical convergence study with an intermediate-time benchmark and a Δ-scaling plot."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know before reading: the genuinely new material is the dressed-frame second-order Floquet–Magnus term for the linearly driven two-level system and the clean derivation that Krylov complexity for a squeezed vacuum is (1/2)|ν|². The piecewise-Magnus 'reliable method' claim is softer than the abstract makes it sound. I would send it to a referee, but the referee should push hard on §5.3–5.4.\n\nThe strong parts are the concrete formulas. Equation (51) with Bessel and Struve functions is a real calculation that I have not seen in the cited literature, and the derivation in Appendix B checks out. The symplectic formalism and the C_K = (1/2)|ν|² relation are clean and useful; the proof in Appendix C is correct. The sudden-quench benchmark is exact and gives a nice closed form (89). No parameter is fitted to the predicted quantity: h_x and h_z come from Magnus integrals, and the Bogoliubov coefficient is then translated into complexity. That is a legitimate derivation.\n\nThe weak spot is exactly where the stress-test note lands. The claim in §5.3 that piecewise Magnus 'provides a reliable method' rests on an unproven assumption that local errors accumulate at most linearly. That is a standard assumption for one-step integrators and probably true here, but the paper neither proves it nor cites a standard error bound. The numerics in §5.4 check only the two asymptotic limits: τ=0.1 against the sudden formula and τ=100 against adiabatic intuition. The intermediate case τ=10 has no exact or high-accuracy comparison, and there is no plot of error versus step size Δ. So the abstract's claim about reliability when the global expansion fails is not supported in the regime it is most relevant. A referee should ask for either a convergence test or a softened wording.\n\nMinor issues: the comparison with the extended-Hilbert-space and diabatic-Magnus approaches is brief and does not benchmark them side by side. The 'Unpublished' reference [31] should be flagged or filled in. No code or data is supplied, which makes the numerical validation hard to verify independently.\n\nWho should read this: people working on Krylov complexity in driven or time-dependent quadratic systems will find the formulas useful. I would cite the (1/2)|ν|² relation and the dressed-frame Floquet–Magnus term, but not the piecewise reliability claim. It deserves a serious referee, with a request to focus on §5.3–5.4. With modest additional numerical work it could be a solid methods paper. Reading group? Maybe—it would prompt a good discussion about what counts as reliable in approximate time-evolution methods.","headline":"A solid methods paper with a genuinely new dressed-frame Floquet-Magnus term and a clean C_K = 1/2|ν|^2 relation; the piecewise-Magnus reliability claim is the weak spot and needs referee attention.","tokens_in":15693,"tokens_out":6231,"would_cite":true,"duration_ms":61493,"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":"This paper establishes that Krylov spread complexity for explicitly time-dependent Hamiltonians is computable through Floquet–Magnus approximations, and that a piecewise Magnus expansion remains reliable when the global expansion loses accu","keywords":["Krylov complexity","spread complexity","Floquet theory","Magnus expansion","time-dependent Hamiltonian","Bogoliubov coefficient","squeezed vacuum","harmonic oscillator"],"falsifier":"For the soft-quench oscillator with ω0=1, ω1=2 and an intermediate duration such as τ=10, compute the exact Krylov complexity by high-precision numerical integration of the oscillator's symplectic propagator, then compare with the piecewise Magnus result for successively smaller Δ. If the difference does not decrease as O(t Δ^m) and instead saturates or grows with the number of steps, the linear-error-accumulation assumption is false and the method is not reliable in the regime where the paper claims it.","tokens_in":14852,"feed_emoji":"⚛️","tokens_out":9625,"duration_ms":91203,"temperature":0.7,"pith_summary":"Krylov complexity measures how broadly a quantum state spreads across Hilbert space during evolution. The usual Krylov construction assumes a time-independent generator, and this paper's goal is to make the measure usable when the Hamiltonian depends explicitly on time. It shows that periodic drives can be handled by Floquet theory with the Magnus expansion supplying an effective static Floquet Hamiltonian, and that beyond periodicity a piecewise Magnus expansion—time sliced into small intervals, each exponentiated locally—remains reliable even when the global expansion loses accuracy. For a harmonic oscillator with time-dependent frequency, the paper derives the exact relation C_K(t)=1/2 |ν(t)|^2, so Krylov complexity is just half the squared Bogoliubov coefficient. A reader should care because time ordering was the main obstacle, and the paper reduces it to controlled local approximations.","feed_headline":"Magnus expansion makes Krylov complexity computable for driven systems","feed_subtitle":"For a driven oscillator, complexity equals half the squared Bogoliubov coefficient; the method works when the global Magnus expansion fails.","key_machinery":"The central object is the Magnus expansion, which represents the time-ordered evolution as U=exp(Ω1+Ω2+…), with Ωn built from nested commutators of H(t) at different times; this turns time ordering into a series of commutator corrections. Over one driving period it becomes the Floquet–Magnus expansion and produces an effective time-independent Floquet Hamiltonian. The paper's piecewise variant splits the time axis and applies the same local expansion on each segment, giving an ordered product of local exponentials. For the oscillator, each local exponential is a quadratic symplectic generator; the resulting Bogoliubov coefficient ν(t) then directly yields Krylov complexity through C_K=1/2|ν|","core_discovery":"On its own terms, the paper establishes that the time-ordering obstruction to Krylov complexity for time-dependent Hamiltonians can be bypassed with Magnus exponentials. In the periodic case, a Magnus expansion over one driving period yields an approximate Floquet Hamiltonian; for the linearly polarized two-level drive this produces a high-frequency stroboscopic complexity written in terms of Bessel and Struve functions. In the non-periodic case, the paper introduces the piecewise Magnus expansion: partition the interval, exponentiate a local effective Hamiltonian on each piece, and multiply the local propagators in time order. Applied to a time-dependent oscillator, the method gives C_K(t)=","pith_inferences":["Beyond the paper: the linear-error-accumulation assumption can be tested directly with an intermediate-duration quench (e.g., ω0=1, ω1=2, τ≈10), comparing piecewise Magnus against high-precision numerical integration; the promised O(t Δ^m) scaling is checkable.","Beyond the paper: because C_K equals half the squared Bogoliubov coefficient for Gaussian evolution, measuring an oscillator mode's squeezing in the lab would directly measure its Krylov complexity, connecting an abstract complexity measure to quadrature statistics.","Beyond the paper: the same piecewise strategy should extend to multi-mode quadratic systems and to operator complexity, where local Magnus generators and Bogoliubov transformations play the same role.","Beyond the paper: accuracy could be improved without changing the structure by including higher local Magnus orders or replacing each step with a higher-order symplectic integrator."],"forward_implications":["For periodically driven two-level systems, stroboscopic Krylov complexity can be written analytically: the circular drive gives Rabi-like sin² oscillations, and the high-frequency linear drive gives Bessel- and Struve-function expressions.","For a time-dependent oscillator, Krylov spread complexity is exactly the squeezed-vacuum measure C_K(t)=1/2 sinh² r(t)=1/2|ν(t)|², so complexity growth is nothing but squeezing.","The piecewise Magnus method only needs the local convergence condition δ h_max Δ<1 on each slice, so it applies when the global condition δ h_max t<1 fails.","The method reproduces the sudden-quench result and the adiabatic limit, and smooth finite-time quenches show bounded post-ramp oscillations.","The framework is not restricted to periodic drives; it offers a practical route to spread complexity for a broad class of time-dependent quantum systems."],"fun_headline_variants":["Piecewise Magnus expansion computes Krylov complexity for driven systems","Krylov complexity from Magnus exponentials, beyond periodic drives","Driven oscillator: complexity equals half squared Bogoliubov coefficient","Floquet Magnus method extends Krylov complexity to time-dependent Hamiltonians","Magnus expansion generalizes Krylov complexity to arbitrary time dependence"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise, stated as an assumption in Section 5.3, is that errors made on each small time slice accumulate at most linearly with the number of slices; the paper does not prove this for the ordered product of local exponentials, and if the errors compound faster, the O(t Δ^m) accuracy claim and the method's reliability conclusion collapse.","fun_headline_variants_meta":{"raw":{"variants":["Piecewise Magnus expansion computes Krylov complexity for driven systems","Krylov complexity from Magnus exponentials, beyond periodic drives","Driven oscillator: complexity equals half squared Bogoliubov coefficient","Floquet Magnus method extends Krylov complexity to time-dependent Hamiltonians","Magnus expansion generalizes Krylov complexity to arbitrary time dependence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1201,"prompt_tokens":602,"completion_tokens":599,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":346,"completion_tokens_details":{"reasoning_tokens":511}},"tokens_in":346,"tokens_out":599,"duration_ms":7251,"temperature":1.0,"reasoning_tokens":511,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:13:45.014219+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the soft-quench oscillator with ω0=1, ω1=2 and an intermediate duration such as τ=10, compute the exact Krylov complexity by high-precision numerical integration of the oscillator's symplectic propagator, then compare with the piecewise Magnus result for successively smaller Δ. If the difference does not decrease as O(t Δ^m) and instead saturates or grows with the number of steps, the linear-error-accumulation assumption is false and the method is not reliable in the regime where the paper claims it.","supporting_citations":[],"review_version":2}