{"id":"a7c357c3-e3ac-42d6-a27b-b0fbeb94d2fe","arxiv_id":"2505.16163","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Applying CRAB schedule optimization to adiabatic factorization Hamiltonians raises final-state fidelity for integers 21 to 2479, with a performance threshold near the quantum speed limit, and the improvement survives dephasing noise.","lead":"The authors show that a known quantum control trick, chopped random-basis optimization, can improve the accuracy of adiabatic quantum factorization for small numbers, especially when the run time is above a quantum speed limit. The result is a practical benchmark for making adiabatic quantum algorithms more robust on noisy near-term devices.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Local CD baseline in Fig. 4 is under-specified; without a faithful reimplementation the claim that CRAB outperforms local CD is not demonstrated.","rationale":"The reader correctly identifies the CD baseline as the weakest assumption. I agree: the paper gives the CD Hamiltonian but not enough detail to reproduce the blue curves. This is a concrete, testable gap. The QSL claim is secondary and qualitative, and the dephasing demonstration is for two instances only. The CD comparison is the key competitive claim of the paper, so it is the most load-bearing. If the baseline is faithful, the paper's main claims stand; if not, the comparative advantage is not demonstrated.","tokens_in":16835,"tokens_out":14670,"duration_ms":86249,"concrete_test":"Independently reimplement the local CD protocol from Ref. [45] for ω=21 and 91 with a stated digitization (e.g., first-order Trotter with time step dt=0.001 and linear schedule s(t)=t/T, using the α_i formula from Sec. III.D). Compute the infidelity-vs-T curves and overlay them on the blue points in Fig. 4. Also repeat with a second Trotter order (e.g., second-order Trotter with dt=0.01). If either reimplementation deviates from the published blue curves by more than the marker size, the baseline is not faithful and the comparison must be redone.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Figure 4 compares CRAB optimization against the approximate local CD method of Ref. [45]. The authors state the CD Hamiltonian (Sec. III.D) but omit the digitization step, Trotter order, time step, and the exact schedule s(t) used to evolve the CD Hamiltonian. The blue curves are central to the paper's claim that CRAB outperforms local CD methods (Conclusions). If the CD implementation is not faithful to Ref. [45] - for example, if a coarse Trotter step or a different digitization is used - the comparison would be unfair and the claim would not be established. This is a load-bearing gap because the paper's comparative advantage over existing CD-based factorization rests entirely on these curves.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Using exact Schrödinger and Lindblad evolution, the authors test whether chopped random-basis (CRAB) optimization of the time-dependent scheduling function s(t) improves the fidelity with which an adiabatic quantum factorization Hamiltonian reaches its problem ground state. The problem Hamiltonians are built by direct cost-function encoding for 21 and by binary-multiplication-table preprocessing with coefficient reweighting for 77, 91, 187, 703, and 2479. The manuscript reports that, for total evolution times beyond a problem-specific threshold, CRAB reduces final infidelity relative to the linear schedule and relative to the digitized local counter-diabatic method of Ref. [45], and that this improvement survives pure dephasing with rate γ=0.04. It also proposes that the empirical threshold time T_c tracks the two-level quantum speed limit T_QSL=π/Δ_min.","tokens_in":16953,"tokens_out":8750,"duration_ms":78709,"significance":"Within the small system sizes treated here (up to four qubits), the core observation that a finite-basis CRAB ansatz can significantly lower the final energy and infidelity is internally consistent and supported by exact simulation; the choice of cost function is explicit, and the open-data statement points to a reproducible resource. If the comparison to local CD were fully specified, the result would be a useful practical benchmark for small adiabatic factorization runs and for the claim that waveform optimization can beat simple CD schedules under dephasing. At present, however, the comparative and threshold claims are stronger than the provided evidence, so the significance is conditional on the requested clarifications.","major_comments":[{"comment":"The blue “local CD” data are the sole basis for the claim that CRAB outperforms counter-diabatic driving, yet the manuscript does not report the digitization step, Trotter order, number of time slices, or the scheduling function s(t) used to evolve the CD Hamiltonian; the text only gives the continuous H_CD formula and cites Ref. [45]. Since the infidelity of a digitized CD protocol depends strongly on these choices, the comparison is not reproducible and the “outperforms local CD” claim is not demonstrated. Please provide the full implementation details, including whether the same s(t) and time discretization were used for both methods, or identify the exact code and version used for the blue curves.","section":"Sec. III.D and Fig. 4"},{"comment":"The claim that the empirical time-to-solution T_c “correlates well” with T_QSL=π/Δ_min is not quantitatively supported. The text describes T_c as extracted from convergence curves, but gives no extraction algorithm, no numerical values for most instances, and no uncertainty; the two-level QSL formula is applied to Hilbert spaces of dimension 8 or 16 without testing whether multi-level corrections matter. Please provide a table of T_c and T_QSL for all six integers, a definition of T_c with a reproducible extraction rule, and a test of the two-level approximation, for example by computing the exact minimal time to reach a target fidelity or an exact QSL bound. As written, the abstract’s “when the evolution time exceeds the quantum speed limit” is an interpretation, not an established result.","section":"Sec. III.D and Figs. 4–5"},{"comment":"The paper does not state whether the CRAB coefficients used in the noisy simulations were optimized on the closed-system cost function and then evaluated under dephasing, or optimized with the Lindblad dynamics included. The claim of robustness to dephasing is meaningful in either case, but its scope is different, and the on-device feedback motivation in Sec. I requires the latter or at least a clearly defined noise-aware optimization. Please state the optimization protocol for the noisy runs and report the dephasing rate relative to the Hamiltonian energy scale.","section":"Sec. III.D, Eq. (11), and Fig. 4(b,d)"},{"comment":"The comparison with Ref. [45] for 2479 is not controlled because the authors replace H_p by the reweighted Hamiltonian H'_p before running CRAB, while the cited 0.4-fidelity saturation of Ref. [45] refers to the published, unweighted construction. If this is the case, the reported advantage could be partly due to the enlarged gap from the reweighting rather than to CRAB. Please either re-run the local CD method on H'_p for 2479 (and, where applicable, for 77, 187, and 703) or explicitly state that the comparison is between different problem Hamiltonians and discuss the implications.","section":"Sec. III.C, Eq. (10), and Conclusions"}],"minor_comments":[{"comment":"“CRAB optimization becomes effective once T ≤ T_c” should read T ≥ T_c; this is opposite to the behavior described elsewhere and to Fig. 5.","section":"Sec. III.D, 2479 paragraph"},{"comment":"The two rows labeled r_k are ambiguous; since ω_k is defined with a single random number per k, clarify whether the second row corresponds to the same frequency or to an additional randomization for the B_k coefficients.","section":"Table II"},{"comment":"The bibliography entry [9] appears garbled (“Regev, o2306288”) and duplicates text from [10]; reference [73] repeats reference [61]. Please correct the citation database.","section":"References"},{"comment":"The Zenodo statement would be more useful if it specified whether the deposit contains raw data, scripts, or the full set of optimized coefficients for all instances.","section":"Data Availability"}],"recommendation":"major_revision","confidential_remarks":"Given that the main numerical claim is plausible and the deficiencies are about reproducibility and interpretation rather than a detected internal contradiction, I do not recommend rejection. The revision should be evaluated on whether the baseline and QSL issues are resolved. There is also a scope question: the paper is closer to a numerical case study than to a methodological advance, but for a quantum information journal the application-oriented contribution is acceptable if the comparison is made fully transparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a careful numerical demonstration that CRAB-optimized schedules improve ground-state fidelity for adiabatic factorization of small integers (21–2479, 2–4 qubits). The QSL-based threshold is suggestive, not proven. The one load-bearing gap is the local CD baseline: the blue curves in Fig. 4 are central to the 'outperforms CD' claim, but the implementation (Trotter step, digitization, schedule) is not specified. A referee must ask for it.\n\nWhat's new: applying CRAB to Hamiltonian-based factorization is new, and the specific numbers — optimized continuous schedules beating linear and local-CD driving on these few-qubit instances — are concrete results not in the cited literature. The paper is honest about its heuristics: the gap-enhancement weights in Eq. (10) are ad hoc, and T_c is extracted by eye from convergence curves. The Hamiltonians and CRAB ansatz are spelled out in the appendices, data is on Zenodo, and exact small-Hilbert-space simulation is the right tool for this size.\n\nSoft spots, in order: (1) The CD comparison. H_CD is written down, but not how it's digitized, integrated, or which schedule s(t) is used. If they reimplemented Ref. [45] verbatim, their claim may well hold; as written, no independent reader can verify the blue curves. (2) The abstract says 'when the evolution time exceeds the quantum speed limit' as if T_QSL were a demonstrated boundary. In the text they're more careful: T_QSL is a two-level approximation and T_c is a post-hoc observation. The abstract overstates. (3) Dephasing robustness is shown for 21 and 91 at one rate, γ = 0.04; that's a thin basis for 'resilience' in general. (4) Minor typo: in Sec. III.D for 2479, 'becomes effective once T ≤ Tc' contradicts the observed threshold; should be T ≥ Tc.\n\nThe internals are consistent: cost functions are sums of squared equations, spectra are computed, no step reduces to itself. I believe the numerical claims as far as they go.\n\nWho this is for: people in quantum optimal control and adiabatic optimization, especially those building small NISQ testbeds. It deserves a serious referee — the CD baseline question is exactly what a referee should chase. With that resolved and the abstract hedged, this is publishable.\n\nRecommendation: send it to peer review.","headline":"Solid numerical demo that CRAB improves adiabatic factorization for small integers, but the comparative claim against local CD rests on an under-specified baseline.","tokens_in":17490,"tokens_out":3965,"would_cite":false,"duration_ms":33406,"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":"CRAB-optimized scheduling significantly improves the fidelity of adiabatic quantum factorization for integers 21 through 2479, especially once the evolution time exceeds the quantum speed limit, and the improvement persists under…","keywords":["adiabatic quantum computing","integer factorization","chopped random-basis optimization","quantum optimal control","quantum speed limit","dephasing noise","counter-diabatic driving","ground-state preparation"],"falsifier":"Faithfully re-implement the digitized local CD protocol of Ref. [45] for factoring 21 and compare its infidelity curve with Fig. 4(a); the central performance claim fails if the blue CD curve falls below the red CRAB curve for times $T > T_{\\mathrm{QSL}}$.","tokens_in":16619,"feed_emoji":"⚛️","tokens_out":9961,"duration_ms":74452,"temperature":0.7,"pith_summary":"The paper claims that CRAB—chopped random-basis optimization—can substantially improve the fidelity of adiabatic quantum factorization for composite integers from 21 up to 2479. By optimizing a few Fourier coefficients in the time-dependent schedule, the final ground-state infidelity drops from about 0.3 to about $10^{-3}$ for factoring 21 in a short evolution time, and similar gains appear for larger numbers once the evolution time exceeds a threshold that tracks the quantum speed limit. The improvement persists under pure dephasing noise, and for the largest instance (2479) the achieved fidelity exceeds what the authors report for higher-order local counter-diabatic methods. If true, this gives a practical control lever that does not require knowing the Hamiltonian spectrum for improving ground-state preparation on near-term devices.","feed_headline":"CRAB control lifts quantum factoring fidelity to 0.999","feed_subtitle":"A few optimized schedule coefficients beat counter-diabatic methods past the quantum speed limit, even with noise.","key_machinery":"The central object is the CRAB-optimized scheduling function $s_{\\mathrm{CRAB}}(t) = s_0(t) f(t)$, where $f(t) = 1 + \\sum_{k=1}^{N_c} [A_k \\sin(\\omega_k t)+B_k \\cos(\\omega_k t)]/\\lambda(t)$, with random frequencies $\\omega_k = 2\\pi k(1+r_k)/T$ and boundary-enforcing weight $\\lambda(t)=1/\\sin(\\pi t/T)$. This converts the infinite-dimensional problem of choosing the best drive into a few-parameter minimization of the final energy $\\langle \\psi(T)|\\hat{H}_p|\\psi(T)\\rangle$, and the paper demonstrates via instantaneous-eigenstate populations that the optimized schedule suppresses nonadiabatic transitions. The companion machinery is the problem Hamiltonian encoding factorization as a ground state, built either by direct cost-function translation or by binary-multiplication-table preprocessing that reduces the qubit count to 2--4 for the studied numbers; for 2479, an additional coefficient reweighting enlarges the minimal gap and further helps the control.","core_discovery":"On the paper's own terms, the central discovery is that a low-dimensional CRAB ansatz for the adiabatic schedule $s(t) = s_0(t)[1 + \\sum_{k=1}^{N_c} (A_k \\sin(\\omega_k t)+B_k \\cos(\\omega_k t))/\\lambda(t)]$ turns a poorly performing adiabatic factorization into a high-fidelity one at fixed total time $T$. The optimized schedule suppresses excitations out of the instantaneous ground state, so the final state's overlap with the target factor-encoding ground state rises from $\\sim 0.7$ to $>0.99$ for $T=0.5$ in the 21 case, and the threshold time for this efficient regime matches the two-level quantum speed limit $T_{\\mathrm{QSL}} = \\pi/\\Delta_{\\min}$ (e.g., $\\sim 0.176$ for 21). Using the binary-multiplication-table preprocessing, the same control strategy works for 2479 with only four qubits, and the fidelity there is reported to exceed that of higher-order CD methods. The robustness under Lindblad dephasing (rate $\\gamma=0.04$) in the 21 and 91 cases supports the claim's practical relevance.","pith_inferences":["An untested but plausible corollary is that the correlation between the threshold time and $T_{\\mathrm{QSL}} = \\pi/\\Delta_{\\min}$ holds for any gap-structured adiabatic problem, making the two-level estimate a simple design rule for choosing evolution times in other optimization Hamiltonians.","The paper leaves implicit that CRAB and local CD driving are complementary; combining them (e.g., along the lines of the cited COLD-CRAB and Floquet-CD work) could yield even shorter evolution times than either method alone.","A hardware demonstration with trapped-ion N-body interactions, as cited for the 2479 Hamiltonian, would test the dephasing-robustness claim beyond the Lindblad model used here, which assumes pure dephasing and no other error channels.","Extending the noise model beyond pure dephasing to include amplitude damping or measurement errors could change the relative ranking of CRAB versus CD, so a benchmarking study with a realistic error model would be informative."],"forward_implications":["Outside the context of factoring, the same CRAB schedule optimization should improve the final ground-state fidelity of other adiabatic ground-state preparation tasks at fixed run time, without requiring spectral knowledge.","The empirical time-to-solution $T_c$ extracted from convergence curves can serve as a practical benchmark for when control becomes effective, and it correlates with the QSL across the six instances studied.","Because the improvement persists under pure-dephasing noise at $\\gamma = 0.04$, the method can plausibly be applied directly on NISQ devices without explicit error mitigation for that noise channel.","The binary-multiplication-table preprocessing reduces the qubit count to $O(n \\log n)$ for $n$-bit numbers, so the CRAB approach may extend to larger composites with modest qubit overhead.","For the number 2479, the achieved fidelity surpasses the reported higher-order CD result, indicating that CRAB can be a better control strategy for small ground-state-search problems."],"supporting_citations":[{"why":"Supplies the adiabatic factorization Hamiltonian construction and the local counter-diabatic baseline that CRAB is compared against.","marker":"[45]"},{"why":"Introduces the optimal control technique for many-body quantum dynamics that forms the basis of CRAB.","marker":"[58]"},{"why":"Defines the chopped random-basis quantum optimization ansatz with random frequencies and boundary conditions used here.","marker":"[59]"},{"why":"Provides the two-level quantum speed limit estimate $T_{\\mathrm{QSL}} = \\pi/\\Delta_{\\min}$ used for the threshold analysis.","marker":"[62]"},{"why":"Gives the initial Hamiltonian choice ($g=10$) and the adiabatic factorization formulation used for the 21 case.","marker":"[63]"},{"why":"Establishes the $O(n \\log n)$ qubit scaling for factorization as optimization, used to argue the method's extensibility.","marker":"[79]"}],"fun_headline_variants":["CRAB optimizer boosts quantum factoring to 0.999 fidelity","CRAB control lifts factoring fidelity to 0.999, noise-resistant","Optimized CRAB schedules beat speed limit for factoring","Quantum factoring gets 0.999 fidelity via CRAB optimization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim that CRAB beats local CD methods rests on an unspecified implementation of the CD baseline—the paper does not state the digitization step, Trotter order, or coefficient optimization for the local CD Hamiltonian it attributes to Ref. [45]—and the claimed threshold time rests on a two-level approximation of the spectrum that is not independently validated.","fun_headline_variants_meta":{"raw":{"variants":["CRAB optimizer boosts quantum factoring to 0.999 fidelity","CRAB control lifts factoring fidelity to 0.999, noise-resistant","Optimized CRAB schedules beat speed limit for factoring","Quantum factoring gets 0.999 fidelity via CRAB optimization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1575,"prompt_tokens":945,"completion_tokens":630,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":559}},"tokens_in":561,"tokens_out":630,"duration_ms":5759,"temperature":1.0,"reasoning_tokens":559,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:05:38.784554+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Faithfully re-implement the digitized local CD protocol of Ref. [45] for factoring 21 and compare its infidelity curve with Fig. 4(a); the central performance claim fails if the blue CD curve falls below the red CRAB curve for times $T > T_{\\mathrm{QSL}}$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the adiabatic factorization Hamiltonian construction and the local counter-diabatic baseline that CRAB is compared against."},{"cited_title":"Doria, T","cited_arxiv_id":null,"evidence_quote":"Introduces the optimal control technique for many-body quantum dynamics that forms the basis of CRAB."},{"cited_title":"Caneva, T","cited_arxiv_id":null,"evidence_quote":"Provides the two-level quantum speed limit estimate $T_{\\mathrm{QSL}} = \\pi/\\Delta_{\\min}$ used for the threshold analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the initial Hamiltonian choice ($g=10$) and the adiabatic factorization formulation used for the 21 case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the $O(n \\log n)$ qubit scaling for factorization as optimization, used to argue the method's extensibility."}],"review_version":1}