{"id":"a1370d5e-e0d7-4183-a665-a78b77c6ceba","arxiv_id":"2501.11846","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Using the size of an approximate counterdiabatic term as a geometric metric yields quantum annealing schedules that beat linear ramps in numerical tests on Ising and ANNNI models, with a known failure regime.","lead":"Quantum annealing schedules that avoid errors are usually hard to design because they require knowing the system's energy levels. This paper builds a schedule from an approximate correction term that can be computed without that information, and shows it helps in several test models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The protocol is well-defined and spectrum-free, but its central premise—that the truncated Krylov AGP amplitude g* is a valid nonadiabaticity metric—is asserted, not derived, and the paper's own antiferromagnetic ANNNI result (Fig. 6) shows it can fail for T>3.5.","rationale":"The paper is honest: it reports the antiferromagnetic ANNNI failure and discusses it, and the numerical improvements in the transverse Ising chain and ferromagnetic ANNNI are genuine evidence for a useful heuristic. I do not see internal inconsistency in the Lanczos and variational construction; Eq. (5) follows from minimizing ∥G∥^2, and Eq. (7) defines a well-defined schedule. The central claim, however, is conditional on the unproven correspondence between the truncated AGP amplitude and the nonadiabaticity that actually controls the final relative error. The reader's weakest_assumption points to exactly this, and the paper's own Figs. 4 and 6 reinforce the concern: performance is non-monotonic in dA, and the full-basis result is worse than linear for T>3.5 in the antiferromagnetic phase. Therefore no change to the CONDITIONAL verdict is warranted, but the condition should explicitly require either a derivation or bound linking g* to transition probability, or a benchmark documenting in which regimes the method beats linear.","tokens_in":11313,"tokens_out":9531,"duration_ms":109248,"concrete_test":"For the L=6 ANNNI model at k=0.7, solve Eq. (1) with the exact ground-state quantum geometric tensor g^(0)(λ) from Eq. (D1) instead of g*(λ), and compare the resulting final relative error against the linear schedule and the paper's g* schedules for T = 1.75, 3.5, 5.91, and 10. If the g^(0)-schedule beats linear for T>3.5 while the g*-schedule does not, the failure is specifically in the total-spectral truncated metric choice; if no metric-based schedule beats linear, then geodesic metrics of this form do not capture the relevant dynamics and the scope claim must be narrowed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the identification of g*(λ)=Σ_k α_k(λ)^2 (Eq. 6), the squared Hilbert-Schmidt amplitude of the truncated Krylov AGP, with the metric in the brachistochrone action. No theorem or bound in the paper connects g* to the probability of leaving the instantaneous ground state; Eq. (6) is introduced as an 'expectation' in Sec. II A. This matters because the exact counterdiabatic norm (D1) contains all eigenstates, and the paper itself argues that including all spectral information can be harmful (Sec. IV). The counterexample in Fig. 6 is not a minor blemish: for the antiferromagnetic ANNNI phase, every g*-based schedule, including the full-basis dA=88 curve, has higher relative error than linear for T≳3.5. The explanation offered (fidelity recovery through nonadiabatic transitions) shows that the metric does not measure the quantity relevant to the final ground-state energy. Additionally, the single-parameter action (A1) is reparametrization invariant, so Eq. (7), λdot=C/√g*, is an affine-parameter choice rather than a consequence of minimizing a well-defined transition error. Thus the central claim that the method 'improves performance' is currently supported only for particular models and regimes, not by a general principle.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a spectrum-free scheduling protocol for quantum annealing. Starting from the Krylov/Lanczos variational construction of an approximate adiabatic gauge potential, the authors define g*(lambda) = sum_k alpha_k(lambda)^2, the squared Hilbert-Schmidt amplitude of the truncated counterdiabatic Hamiltonian, and use it as the metric in the single-parameter quantum adiabatic brachistochrone equation. Solving lambda-dot = C / sqrt(g*(lambda)) yields a schedule that spends more time near parameter values where the approximate counterdiabatic amplitude is large. The method is benchmarked numerically on the transverse-field Ising chain and on the axial next-nearest-neighbor Ising (ANNNI) model. For the Ising chain and the ferromagnetic ANNNI phase, the resulting schedules give lower relative ground-state-energy errors than the linear schedule. For the antiferromagnetic-like ANNNI phase, the method improves on linear scheduling only for short annealing times and is worse for T larger than about 3.5; this failure is presented and discussed honestly. The authors conclude that their protocol removes the need for energy-spectrum information in adiabatic scheduling.","tokens_in":11650,"tokens_out":5794,"duration_ms":69229,"significance":"The proposal is concrete, reproducible in structure, and addresses a practical bottleneck: quantum adiabatic brachistochrone schedules normally require spectral data, which is unavailable or costly in quantum-annealing-scale problems. The numerical evidence that a truncated Krylov adiabatic gauge potential can produce useful schedules without gap information is interesting, and the authors deserve credit for explicitly reporting the antiferromagnetic ANNNI counterexample instead of selecting only favorable models. The paper also makes a clear algorithmic statement: coefficients are obtained by solving a tridiagonal linear system, and the schedule follows from a simple ODE. However, the central premise—that sqrt(g*(lambda)) is a valid nonadiabaticity metric—is asserted rather than derived, and the manuscript's own antiferromagnetic results show that this premise can fail in a way that is directly relevant to the claimed figure of merit. The current evidence base is therefore suggestive but not yet a general principle, and the variational/brachistochrone framing needs careful qualification before the paper can support its broader claims.","major_comments":[{"comment":"The central premise that the Hilbert-Schmidt amplitude sqrt(g*(lambda)) of the truncated counterdiabatic Hamiltonian is an appropriate measure of nonadiabaticity is introduced with the phrase \"we expect\" and is never derived or bounded. The exact counterdiabatic norm in Eq. (D1) is a sum over all energy eigenstates, while g* is the squared norm of a truncated Krylov approximation; no result connects g* to the probability of leaving the instantaneous ground state or to the final ground-state fidelity. Because Eq. (7) and all schedules in the paper follow from this premise, this is a load-bearing assumption rather than a harmless technical choice.","section":"Section II A, Eq. (6)"},{"comment":"For a single parameter with fixed endpoints, the action in Eq. (A1) reduces to epsilon = integral sqrt(2 g(lambda)) |lambda-dot| dt = integral sqrt(2 g(lambda)) d-lambda, which is independent of the time dependence lambda(t) for any monotone schedule. Therefore the variational problem does not select a schedule, and Eq. (7), lambda-dot = C / sqrt(g(lambda)), is an affine-parameter choice rather than a consequence of minimizing a well-defined transition error. The paper should either derive Eq. (7) from a genuine minimization over schedules, for example using a cost functional tied to transition probability, or present the rule explicitly as a heuristic for dwelling where g* is large.","section":"Appendix A and Section II B, Eq. (7)"},{"comment":"The antiferromagnetic-like ANNNI result is not a peripheral blemish. For T larger than about 3.5, every g*-based schedule, including the full-basis dA = 88 curve, has larger relative error than the linear schedule. The authors' own explanation—that fidelity recovery through nonadiabatic transitions improves the final ground-state energy—shows that g* does not measure the quantity relevant to the final figure of merit in this regime. The claim that the method \"improves performance\" should therefore be restricted to the regimes where the metric is aligned with the ground-state-energy objective, and the paper should state this limitation in the abstract and conclusion.","section":"Section IV, Figs. 6-8"},{"comment":"The scalability claim that the method works for large quantum-annealing systems is asserted rather than demonstrated. For disordered spin-glass problems the Lanczos basis count generically grows rapidly, and the symbolic computation in Step 1 is not analyzed for its cost or termination behavior. All numerical benchmarks are translation-invariant chains or the L = 6 ANNNI model; the statement in Section IV that \"our method works even for large systems\" is a conjecture unsupported by complexity analysis or random-instance tests. If the quantum-annealing motivation is to be retained, the authors should either provide such an analysis or substantially soften this claim.","section":"Section II B, Step 1, and Section IV"}],"minor_comments":[{"comment":"The phrase \"The horizontal line indicates the normalized time\" should read \"The horizontal axis indicates the normalized time.\"","section":"Figure 3 caption"},{"comment":"The full-basis ANNNI calculation is identified by dA = 88 because b_177 converges numerically to 0; the convergence tolerance or criterion used to declare this termination should be stated so that the \"full basis\" characterization is reproducible.","section":"Section III B"},{"comment":"The normalization conventions connecting the multi-parameter action in Eq. (A1), the factor of 2, and the single-parameter connection Gamma(lambda) = (1/2g) partial_lambda g should be spelled out, since g* is defined without the factor of 2 and the geodesic equation is sensitive to this normalization.","section":"Equation (A1) and Eq. (1)"},{"comment":"The relative-error curves appear to be single deterministic runs; stating whether any averaging over initial states or disorder realizations was performed, or noting that the results are deterministic, would improve interpretability.","section":"Figures 2, 4, and 6"}],"recommendation":"major_revision","confidential_remarks":"The paper's advertised contribution—a spectrum-free adiabatic brachistochrone—rests on an unproven heuristic for the nonadiabaticity metric, and the single-parameter action does not actually select the schedule through minimization. That said, the numerical results are interesting and honestly reported, and the method could be reframed as a heuristic scheduling rule supported by more extensive benchmarking. I do not think rejection is warranted, because the load-bearing assumption is fixable in principle: the authors can either supply a derivation or bound linking g* to transition probability, or explicitly downgrade the claim to a numerical heuristic and test it on generic disordered instances. For the current version, the gap between the variational language and what is actually demonstrated is too large for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Shingu and Hatomura's paper is a solid, honest heuristics paper. The new thing is the specific move: take the Krylov-space approximate adiabatic gauge potential, use its Hilbert-Schmidt amplitude as the metric in the quantum adiabatic brachistochrone, and solve the geodesic equation to get a schedule that does not require energy spectra. That combination is new as far as the cited literature goes, and the method is well-defined and cheap to compute classically.\n\nWhat it does well: the numerical evidence is consistent in two models—transverse-field Ising chain and ferromagnetic ANNNI—and the improvement over linear schedules is real in those regimes. The authors also do something rare: they report a clear failure. In the antiferromagnetic ANNNI phase, their schedules are worse than linear for T ≳ 3.5, and they dig into why (fidelity recovery through nonadiabatic transitions). That transparency earns trust.\n\nSoft spots, in proportion. The load-bearing assumption is that g*(λ) is an appropriate nonadiabaticity metric. The paper introduces this as 'we expect,' and there is no derivation or bound connecting g* to transition probability. That is a genuine gap, not a manufactured one. The antiferromagnetic counterexample matters: it shows the metric is not universal. The paper's own framing in the abstract and conclusion is a bit more optimistic than the data warrant. Also, the systems are small (6 qubits for ANNNI), and no code or data are given, so independent reproduction is not immediate. The reparametrization point from the stress test is less concerning: Eq. (7) is indeed an affine-parameter choice, but the schedule shape still comes from the metric; the real question is whether that shape minimizes transitions.\n\nWho gets value: anyone working on quantum annealing schedules, counterdiabatic driving, or shortcuts to adiabaticity. It's a useful data point and a plausible new heuristic.\n\nRecommendation: This deserves a serious referee. It is not a fundamental advance, but it is a clear, honest, and testable proposal with a new combination and numerical support. I would send it to peer review and ask for a derivation (or at least a stronger empirical argument) for the metric, a fuller benchmark set, and code. Conditional accept, not reject.","headline":"A spectrum-free scheduling heuristic that works in some models and honestly fails in another; the metric is a guess, but the paper is worth refereeing.","tokens_in":12109,"tokens_out":2831,"would_cite":true,"duration_ms":29577,"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":"A spectrum-free geometric metric can set the pace of quantum annealing schedules.","keywords":["quantum adiabatic brachistochrone","quantum annealing","adiabatic gauge potential","Krylov subspace","counterdiabatic driving","parameter scheduling","transverse-field Ising chain","ANNNI model"],"falsifier":"In a system small enough for exact diagonalization, compute the schedule that solves the geodesic equation for $g^*$, then run the Schr\\\"odinger equation and compare final ground-state fidelity with a second schedule whose integrated $g^*$ is larger; if the $g^*$-optimal schedule is not at least as good, the metric is not a faithful nonadiabaticity gauge. The paper's own antiferromagnetic ANNNI data at $T \\simeq 5.9$ already approach this test, since fidelity recovers only through nonadiabatic returns.","tokens_in":11131,"feed_emoji":"⚛️","tokens_out":10006,"duration_ms":94125,"temperature":0.7,"pith_summary":"This paper proposes a way to schedule adiabatic quantum control without ever diagonalizing the Hamiltonian. The idea is to replace the usual metric that requires energy spectra with the squared amplitude of an approximate counterdiabatic term, $g^*(\\lambda)=\\sum_k \\alpha_k(\\lambda)^2$, and then solve the geodesic equation of the quantum adiabatic brachistochrone using that metric. In numerical tests on a transverse-field Ising chain and on the ANNNI model in its ferromagnetic phase, the resulting schedules give lower relative ground-state energy errors than the commonly used linear schedule, even when the counterdiabatic term is truncated to a few basis operators. The paper also reports an antiferromagnetic ANNNI case in which the linear schedule wins at long annealing times, because nonadiabatic transitions can return the system to the ground manifold. If correct, the method provides a classically precomputable, spectrum-free path to improved annealing schedules.","feed_headline":"A metric made without spectra sets the annealing pace","feed_subtitle":"The squared size of a truncated counterdiabatic term, used as a metric, beats the linear ramp in Ising benchmarks.","key_machinery":"The central object is the squared norm of an approximate adiabatic gauge potential computed in a truncated Krylov basis. The paper generates odd basis operators $\\hat{O}_{2k-1}$ by the Lanczos recurrence, fixes their coefficients $\\alpha_k$ through the variational equation, and defines $g^*=\\sum_k \\alpha_k^2$. This positive, real, symmetric quantity plays the role that the quantum geometric tensor plays in the standard brachistochrone, but it is computable from the operator structure of $H(\\lambda)$ alone. Solving the geodesic equation with this metric yields the schedule $\\lambda(t)$.","core_discovery":"The central claim is that the brachistochrone metric can be built from the truncated variational adiabatic gauge potential instead of from the quantum geometric tensor. The coefficients $\\alpha_k$ come from expanding the potential in odd operators generated by the Lanczos recurrence, and the paper uses $g^*(\\lambda)=\\sum_k \\alpha_k(\\lambda)^2$ as the nonadiabaticity measure in the geodesic equation $\\ddot{\\lambda} + (1/2g^*)(\\partial_\\lambda g^*)\\dot{\\lambda}^2=0$, equivalently $\\dot{\\lambda}=C/\\sqrt{g^*}$. Because the Lanczos recurrence and the variational equations use only commutators and Hilbert\\textendash Schmidt inner products, no energy eigenvalues or eigenstates enter the construction. The benchmarks show lower relative errors than the linear schedule in the transverse-field Ising chain and the ferromagnetic ANNNI model, with truncated schedules often close to the full-basis result; in the antiferromagnetic ANNNI phase the advantage holds only at short annealing times.","pith_inferences":["A natural next step, not pursued in the paper, is to use $g^*$ as an instantaneous monitor during a run and slow the schedule only where it spikes, instead of solving the global geodesic problem.","Because $g^*$ is classical to compute, its time integral could rank problem instances by expected annealing difficulty before hardware time is spent; this is an editorial inference.","The antiferromagnetic failure suggests a hybrid schedule: follow the geodesic where direct transitions dominate and switch toward linear where returns dominate; this is not analyzed in the paper."],"forward_implications":["Schedules can be computed classically before any quantum run, because the construction needs only the operator form of the annealing Hamiltonian.","Truncation to a few Krylov basis operators captures most of the full-basis benefit in the benchmarks, so the classical cost can be kept low.","The metric's positivity, reality, and symmetry mean the same construction extends, in principle, to multiple control parameters.","In regimes where nonadiabatic transitions actively help, the geodesic schedule can be worse than a linear ramp; the method should be aimed at suppressing direct transitions rather than at exploiting returns."],"supporting_citations":[{"why":"Introduces the quantum adiabatic brachistochrone and the action whose minimization yields the geodesic equation used here.","marker":"[10]"},{"why":"Supplies the metric-based geodesic equation and the constant-velocity rewriting $\\dot{\\lambda}=C/\\sqrt{g}$ that the paper solves.","marker":"[11]"},{"why":"Provides the variational principle for approximate counterdiabatic driving that fixes the coefficients $\\alpha_k$ without energy spectra.","marker":"[24]"},{"why":"Establishes that variational adiabatic gauge potentials carry spectral information, supporting the paper's use of their amplitude as a nonadiabaticity measure.","marker":"[31]"},{"why":"Introduces the Lanczos approach to the adiabatic gauge potential that generates the Krylov basis operators.","marker":"[32]"},{"why":"Provides the Krylov-space construction of shortcuts that the paper follows in truncating the Lanczos recurrence.","marker":"[33]"},{"why":"Defines the adiabatic gauge potential and its geometric role, the object whose truncated norm becomes the metric.","marker":"[36]"},{"why":"Describes the ANNNI model used as the second numerical benchmark.","marker":"[38]"}],"fun_headline_variants":["Brachistochrone without spectra: geometric control for annealing","Spectra-free metric speeds up adiabatic scheduling","Quantum annealing: a metric from counterdiabatic terms","No eigenvalues needed: geometric brachistochrone for Ising"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the squared size of the approximate counterdiabatic term, $g^*(\\lambda)$, faithfully measures how much nonadiabatic error a schedule will cause; the paper introduces this as an expectation rather than proving a bound, and the antiferromagnetic ANNNI case is a regime where it breaks down.","fun_headline_variants_meta":{"raw":{"variants":["Brachistochrone without spectra: geometric control for annealing","Spectra-free metric speeds up adiabatic scheduling","Quantum annealing: a metric from counterdiabatic terms","No eigenvalues needed: geometric brachistochrone for Ising"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001008,"raw_usage":{"total_tokens":4292,"prompt_tokens":1008,"completion_tokens":3284,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":624,"completion_tokens_details":{"reasoning_tokens":3225}},"tokens_in":624,"tokens_out":3284,"duration_ms":24904,"temperature":1.0,"reasoning_tokens":3225,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:47:30.429269+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In a system small enough for exact diagonalization, compute the schedule that solves the geodesic equation for $g^*$, then run the Schr\\\"odinger equation and compare final ground-state fidelity with a second schedule whose integrated $g^*$ is larger; if the $g^*$-optimal schedule is not at least as good, the metric is not a faithful nonadiabaticity gauge. The paper's own antiferromagnetic ANNNI data at $T \\simeq 5.9$ already approach this test, since fidelity recovers only through nonadiabatic returns.","supporting_citations":[{"cited_title":"del Campo, M","cited_arxiv_id":null,"evidence_quote":"Provides the variational principle for approximate counterdiabatic driving that fixes the coefficients $\\alpha_k$ without energy spectra."},{"cited_title":"Passarelli, V","cited_arxiv_id":null,"evidence_quote":"Establishes that variational adiabatic gauge potentials carry spectral information, supporting the paper's use of their amplitude as a nonadiabaticity measure."},{"cited_title":"A general method to construct mean field counter diabatic driving for a ground state search","cited_arxiv_id":"2305.08352","evidence_quote":"Provides the Krylov-space construction of shortcuts that the paper follows in truncating the Lanczos recurrence."},{"cited_title":"Takahashi and A","cited_arxiv_id":null,"evidence_quote":"Defines the adiabatic gauge potential and its geometric role, the object whose truncated norm becomes the metric."},{"cited_title":"Pandey, P","cited_arxiv_id":null,"evidence_quote":"Describes the ANNNI model used as the second numerical benchmark."}],"review_version":1}