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REVIEW 4 major objections 4 minor 92 references

Improving adiabatic quantum factorization via chopped random-basis optimization

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Pith's one-line read 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…

desk verdict Solid numerical demo that CRAB improves adiabatic factorization for small integers, but the comparative claim against local CD rests on an under-specified baseline. read the letter →

arxiv 2505.16163 v1 pith:4YWGQ5ED submitted 2025-05-22 quant-ph

Tianlai Yang , Mo Xiong , Ming Xue , Xinwei Li , Jinbin Li This is my paper · ORCID
classification quant-ph
keywords adiabaticquantumcomputingintegerfactorizationchoppedrandom-basisoptimizationoptimalcontrolspeedlimitdephasingnoisecounter-diabaticdrivingground-statepreparation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $T > T_{\mathrm{QSL}}$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

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.

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 (4)
  1. [Sec. III.D and Fig. 4] 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.
  2. [Sec. III.D and Figs. 4–5] 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.
  3. [Sec. III.D, Eq. (11), and Fig. 4(b,d)] 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.
  4. [Sec. III.C, Eq. (10), and Conclusions] 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.
minor comments (4)
  1. [Sec. III.D, 2479 paragraph] “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.
  2. [Table II] 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.
  3. [References] The bibliography entry [9] appears garbled (“Regev, o2306288”) and duplicates text from [10]; reference [73] repeats reference [61]. Please correct the citation database.
  4. [Data Availability] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: CRAB coefficients are optimized against an explicit cost function, and the QSL comparison is a post-hoc spectral estimate rather than a fitted prediction.

full rationale

The paper's central derivation chain is self-contained and does not reduce to its own inputs. The CRAB schedule is defined as s_CRAB(t) = s0(t) f(t), and the coefficients {A_k, B_k} are optimized by minimizing the explicit cost function E({A_k},{B_k}) = <ψ(T)|H_p|ψ(T)> (Sec. III.B, Eq. 8). The reported infidelity I = 1 - |<ψ_p|ψ(T)>|^2 is then computed from the evolved state under the optimized schedule, so the improvement over the linear guess is a genuine simulation result rather than a quantity forced by the optimization objective. The quantum speed limit estimate T_QSL = π/Δ_min is obtained from the energy spectrum of the Hamiltonian, independently of the CRAB optimization, and the empirical threshold T_c is extracted from convergence behavior of the same runs. The observed agreement between T_c and T_QSL is a post-hoc correlation, not a fitted prediction; no parameter is adjusted to make the two timescales coincide. This slightly weakens the predictive claim but does not constitute circularity. The local CD comparison is taken from Ref. [45]; while the implementation details (Trotter order, digitization, schedule) are under-specified, under-specification is a reproducibility and correctness concern, not circularity. The modified Hamiltonian H'_p in Eq. (10) is a designed Hamiltonian whose gap is enlarged, and the QSL computed from it is again spectral, not derived from CRAB performance. Finally, there is no load-bearing self-citation: the method citations Refs. [45], [58], [59], [62], [63] are external works, and the only self-citation is the Zenodo data repository [93], which is not used as evidence for any claim. Accordingly, no step in the derivation chain is equivalent to its input by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central results depend on hand-chosen Hamiltonian weights, a single dephasing rate, a fixed CRAB truncation, and a two-level approximation for the quantum speed limit. These choices are not benchmarked parametrically, so the quantitative claims inherit their uncertainty.

free parameters (4)
  • Gap-enhancement coefficients (10, 5) = 10 and 5 for 2479 (Eq. 10); similar per-instance weights in Appendices B-D
    Hand-chosen multipliers on squared constraint terms to enlarge the minimum gap; affect the QSL estimate and achievable fidelity.
  • Initial transverse field strength g = 10 (stated for 21, likely used throughout)
    Sets H0 = -g sum sigma_x; influences gap and dynamics; value taken from prior work or chosen by hand.
  • CRAB component count Nc = 4
    Number of frequency groups in the CRAB ansatz; a larger Nc could change results.
  • Dephasing rate gamma = 0.04
    Single noise strength used in Lindblad simulations; robustness at other rates is not shown.
assumptions (5)
  • standard math Adiabatic theorem and Schrodinger evolution
    Underpins the mapping between ground state of Hp and factorization solution.
  • domain assumption Lindblad master equation for pure dephasing
    Noise model used in Eq. (11); assumes Markovian dephasing only.
  • domain assumption Two-level approximation for quantum speed limit
    T_QSL = pi / Delta_min is derived from a two-level reduction of the full spectrum; no justification that dynamics are dominated by the ground and first excited states.
  • domain assumption Binary multiplication table preprocessing equations are correct and complete
    The reduced constraint sets in Appendices A-D must be logically equivalent to factorization; errors here would change the ground state encoding.
  • ad hoc to paper The CRAB ansatz with Nc=4 can represent schedules close to optimal
    No convergence guarantee or comparison with larger Nc; success is measured on the same objective being optimized.

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Pith. "Pith review of Improving adiabatic quantum factorization via chopped random-basis optimization." pith.science (2026). https://pith.science/paper/4YWGQ5ED

@misc{pith2026250516163,
  author       = {Pith},
  title        = {Pith review of: Improving adiabatic quantum factorization via chopped random-basis optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4YWGQ5ED}},
  note         = {Machine review of arXiv:2505.16163}
}
read the original abstract

Integer factorization remains a significant challenge for classical computers and is fundamental to the security of RSA encryption. Adiabatic quantum algorithms present a promising solution, yet their practical implementation is limited by the short coherence times of current NISQ devices and quantum simulators. In this work, we apply the chopped random-basis (CRAB) optimization technique to enhance adiabatic quantum factorization algorithms. We demonstrate the effectiveness of CRAB by applying it to factor the integers ranging from 21 to 2479, achieving significantly improved fidelity of the target state when the evolution time exceeds the quantum speed limit. Notably, this performance improvement shows resilience in the presence of dephasing noise, highlighting CRAB's practical utility in noisy quantum systems. Our findings suggest that CRAB optimization can serve as a powerful tool for advancing adiabatic quantum algorithms, with broader implications for quantum information processing tasks.

Figures

Figures reproduced from arXiv: 2505.16163 by the authors.

Figure 1
Figure 1. FIG. 1. The factorization of 21 before and after CRAB optimization: [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A comparison of the energy spectrum of the total Hamilto [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Fidelity of obtaining the ground state under different total [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Infidelity [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Infidelity [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The energy spectrum of Hamiltonian for factoring 77, 187 and 703 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

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