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REVIEW 2 major objections 4 minor 72 references

On a frustrated Ising ring with an exponentially small gap, DC-QAOA — QAOA with a variational counter-diabatic layer — beats plain QAOA and schedule-optimized annealing at equal circuit depth.

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On a frustrated Ising ring, CRAB-optimized DC-QAOA with variational counter-diabatic terms gives lower residual energy than analytical CD, optimized schedules, and plain QAOA.

T0 review reviewed 2026-08-01 challenge →

load-bearing objection Solid, well-documented numerics with an overstated layer-depth claim; the DC-QAOA advantage is real but not yet isolated from the extra variational parameter. the 2 major comments →

arxiv 2607.23074 v1 pith:DM2GMZ2S submitted 2026-07-25 quant-ph cond-mat.stat-mech

Digital techniques for the frustrated Ising ring: the role of counter-diabatic terms

Kiran Thengil , Vincenzo Roberto Arezzo , Giuseppe E. Santoro This is my paper
classification quant-ph cond-mat.stat-mech
keywords frustrated Ising ringcounter-diabatic drivingadiabatic gauge potentialQAOADC-QAOACRAB optimizationJordan-Wigner transformationquantum annealing
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Quantum annealing on a frustrated Ising ring stalls because the energy gap between ground and excited states closes exponentially with system size. The paper asks whether adding counter-diabatic (CD) corrections to a digitized version of annealing — where the evolution is a product of Trotterized unitaries — can overcome this bottleneck. Comparing six protocols, it finds that CD terms only help when their strength per layer is treated as a free variational parameter and optimized with the Fourier-based CRAB technique; this 'DC-QAOA' outperforms plain QAOA, fixed-schedule annealing, and schedule-optimized annealing on an equal-layer-depth basis, with the gap in performance growing for larger rings. The underlying mechanism is counterintuitive: the optimized CD unitaries drive the system into excited states early in the circuit and then guide it back to the ground state, acting as a shortcut to adiabaticity rather than a suppressor of excitations.

Core claim

The central discovery is that the lowest-order counter-diabatic operator H_xz = i[H_x, H_z] — a nearest-neighbor σ^x σ^y + σ^y σ^x term — is not useful when its coefficient is fixed by the standard analytic optimisation (which minimises an action), but becomes the key to the best performance when its coefficient is allowed to vary freely per layer. With CRAB-optimised parameters, this DC-QAOA variant yields lower residual energy than (i) fixed schedules with or without analytic CD, (ii) CRAB-optimised schedules with or without analytic CD, and (iii) plain QAOA without CD, when compared at the same circuit depth; the advantage is clear for N=35 rings and persists across frustration strengths.

What carries the argument

The central object is the first-order nested-commutator operator H_xz = i[H_x, H_z], added as an extra unitary exp(-i θ_xz H_xz) in each Trotter layer of the QAOA ansatz. The paper's crucial choice is to decouple θ_xz from the analytic expression θ_xz = s-dot α_opt (which minimises the action for the adiabatic gauge potential) and instead let a numerical optimiser (CRAB with two super-iterations) set θ_xz freely. This turns the counter-diabatic term from a fixed shortcut into a variational direction that the circuit can exploit. The other load-bearing machinery is the Jordan-Wigner mapping to free fermions, which lets the authors simulate the dynamics exactly and count the dimension of the s

Load-bearing premise

The paper equates 'counter-diabatic corrections' with the single first-order operator H_xz = i[H_x,H_z]; if higher-order or non-local AGP terms were included, the analytic CD constructions might become useful again and the DC-QAOA advantage could shrink or disappear.

What would settle it

Run the same CRAB-optimised comparison on the N=35 frustrated ring with a DC-QAOA ansatz that includes a second-order variational CD term built from double nested commutators. If the added term yields no residual-energy improvement over the first-order DC-QAOA, or if an analytically optimised second-order CD protocol matches or beats DC-QAOA, then the paper's claim that the advantage comes from flexible variational control of the local first-order CD operator would be falsified.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • On an equal-layer-depth basis, DC-QAOA with variational CD terms is the best of the six digitized strategies tested on the frustrated Ising ring, and its advantage over plain QAOA increases with system size.
  • For this model, adding an analytically optimised CD term to a fixed or CRAB-optimised schedule does not improve — and often worsens — performance, so analytic CD constructions are not a reliable shortcut in the digital setting.
  • The circuit becomes fully controllable with about N^2/2 parameters, independent of the exponential gap, implying that the spin-glass bottleneck of continuous-time quantum annealing can be bypassed by digital variational optimisation.
  • The optimal DC-QAOA circuits work by deliberately exciting the system early and returning it to the ground state late, a shortcut mechanism that differs fundamentally from continuous-time counter-diabatic driving.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • All conclusions rely on the first-order CD operator H_xz only; if second-order or more non-local AGP terms were included, analytical CD constructions might become competitive again, and the ranking could change. The paper does not test this.
  • The mechanism of 'shortcuts through excited states' is likely to generalise beyond the free-fermion ring to other hard optimisation problems, but that is an extrapolation: the paper's numerics are limited to the frustrated Ising ring.
  • Because the paper excludes longitudinal bias fields due to the Jordan-Wigner mapping, the method's performance on problems where bias fields are needed (e.g., MaxCut or spin-glass ground states) remains open; extending DC-QAOA to those settings is a testable next step.
  • The quadratic controllability threshold is shown for the first-order CD term; one may conjecture that higher-order terms lower the threshold further, but the paper does not address this.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies digital quantum optimization protocols for a frustrated Ising ring with an exponentially small spectral gap. Using a Jordan-Wigner and reflection-symmetry reduction, the authors simulate exact free-fermion dynamics for six protocols: fixed schedules, fixed schedules with first-order analytical counter-diabatic (CD) terms, CRAB-optimized schedules with and without analytical CD terms, plain QAOA, and DC-QAOA (QAOA with variational CD terms). Performance is quantified by residual energy as a function of total parameter count Nθ, and the paper reports three main findings: analytical lowest-order CD terms do not improve dQA schedules; CRAB-optimized DC-QAOA outperforms all other tested strategies, especially at larger sizes; and the DC-QAOA dynamics reach the target through intermediate excited states rather than by preserving instantaneous ground-state population. The paper also numerically verifies the quadratic-in-N controllability threshold for the DC-QAOA ansatz.

Significance. If the conclusions hold, the paper provides a useful controlled comparison of CD-based and QAOA-based digital methods on a hard gap-bottleneck model, and offers a specific mechanistic picture of how variational CD unitaries can act as shortcut dynamics in a digitized setting. The authors' exact free-fermion simulations, detailed CRAB/dCRAB description, and archived code/data are notable strengths: the simulations are technically sound and the method is reproducible. The main caveat is that the headline performance ranking and the 'analytical CD terms are not useful' conclusion rest on comparisons at fixed Nθ rather than fixed layer count or fixed total evolution time, which introduces confounds that must be resolved before the conclusions can be accepted.

major comments (2)
  1. [Fig. 5; §IV; Eqs. (13), (21)] The x-axis of Fig. 5 is Nθ, not P. For QAOA, Nθ=2P; for DC-QAOA, Nθ=3P. Therefore at every fixed Nθ in the figure, DC-QAOA runs with fewer layers than QAOA (P=Nθ/3 vs P=Nθ/2). The concluding claim in §IV that DC-QAOA is best performing 'on an equal layer-depth basis' is not supported by the presented data. Moreover, because the structural difference in a DC-QAOA layer is the extra θ_xz H_xz unitary, the gain at fixed Nθ could be due either to the CD operator or simply to the larger number of variational parameters per layer (3P vs 2P). Please add comparisons at fixed P (e.g., ε_res vs P with matched layer counts), and/or a control ansatz with an additional non-CD variational unitary per layer, before making the layer-depth claim.
  2. [Fig. 5 caption; §III.C; methods 2)–3)] For methods 1)–3) the caption states Δt=1. Hence method 2) (no CD) has P=Nθ/2 and total evolution time τ=Nθ/2, while method 3) (analytical CD) has P=Nθ/3 and τ=Nθ/3. At the same plotted Nθ, method 3 has fewer Trotter steps and a shorter total time than method 2. The statement that 'the addition of analytical CD terms appears to be counterproductive' is therefore confounded: the residual-energy increase could be an artifact of the shorter evolution time rather than a property of the CD term. The same confound affects comparisons involving method 1). Please compare methods 1)–3) at matched P or matched total time (e.g., by adjusting Δt for the CD methods) before concluding that analytical CD terms are not useful.
minor comments (4)
  1. [Fig. 5 caption; Eqs. (28)–(31)] The text calls Nθ the 'circuit depth', but Nθ is defined as the total number of angles in the Ansatz, not the number of layers P and not the number of CRAB-optimized coefficients. For methods 4) and 5) with Nc=P, the numbers of optimized Fourier coefficients are 2Nc+2 and 3Nc+3, respectively, not 2P and 3P. Please define Nθ precisely and avoid conflating 'depth' with parameter count.
  2. [Fig. 4; §III.B] The histogram shows that some optimization runs reach the 10^-12 threshold at P=48, but the text does not state the number or percentage of runs below threshold. Please report the success count for each P and Jf so that the controllability claim is quantitatively verifiable.
  3. [Abstract and §IV] The conclusions are mostly careful to say 'lowest-order CD terms', but the abstract's general phrase 'local counter-diabatic (CD) terms' and some discussion of 'CD terms' may overgeneralize. Please add an explicit caveat that only first-order nested-commutator AGP terms (ℓ=1 in Eq. (19)) were tested, and that higher-order AGP terms could alter the ranking.
  4. [§IV] The discussion correctly acknowledges that local longitudinal bias fields are not included. Since the central comparison is restricted to the no-bias-field family, this limitation should be stated in the abstract or conclusions to avoid overgeneralization to general DCQO/DC-QAOA settings.

Circularity Check

0 steps flagged

No significant circularity; residual-energy rankings are direct numerical results. Self-citation of the controllability criterion is validated, and the equal-layer-depth caveat is a comparison-fairness issue, not a circular derivation.

full rationale

The central claims—fixed-schedule dQA is poor, CRAB-optimized schedules improve over fixed schedules, analytical CD terms are ineffective, and DC-QAOA gives the lowest residual energies—are obtained by explicit numerical simulation of the Trotterized unitary circuits and the residual energy ϵ_res^P, not by assuming the conclusion or by fitting the target answer into the variational ansatz. The analytical CD coefficient α_1^opt is taken from an external action-minimization principle (Refs. [17,20], App. A1), not from the frustrated-ring data. The controllability criterion Eq. (23) is imported from the same group's Ref. [28], but it is not used as an input to generate the performance rankings; it is independently verified by random-initialization histograms in Fig. 4, so this self-citation is not load-bearing. The omission of higher-order AGP terms is a scope limitation, not a circular step. The main caveat is a comparison-fairness issue rather than circularity: Fig. 5 plots residual energy versus total parameter count Nθ, where QAOA has Nθ=2P and DC-QAOA has Nθ=3P, so a fixed Nθ compares different layer counts; and at equal layer depth P, the DC-QAOA family contains QAOA by setting θ_xz=0 in Eq. (21), so a non-strict advantage at equal P is guaranteed by construction. This does not invalidate the numerical evidence at fixed Nθ, but it means the 'equal layer-depth basis' wording in Section IV overstates what the plotted comparison shows. These are confounds/claims-precision issues, not circular derivations.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The central performance claim rests on standard imported results (Jordan-Wigner mapping, Gaussian-state dimension, Sels-Polkovnikov AGP action) and on numerical hyperparameters (CRAB frequency sampling, Δt, threshold). The main ad-hoc-to-paper assumption is that the first-order H_xz CD operator is a sufficient representative of CD physics. No new physical entities are introduced; the AGP/H_xz operator is standard, and the 'shortcut via excited states' is a numerical observation, not a new entity.

free parameters (5)
  • dCRAB frequency-distribution parameters (Gamma shape/scale) = α=3/2, β=4 (1st iteration), β=20 (2nd iteration)
    Hand-chosen sampling distribution for random Fourier frequencies; all residual-energy curves depend on this randomness, and only 10 samples are used per experiment.
  • dCRAB protocol settings = 2 super-iterations; N_c=P; 10 random-frequency trials; best selected
    These choices define the reported residual energy and its error bars; no sensitivity or convergence study is reported.
  • Trotter time step Δt = 0.1 (Fig. 3); 1 (methods 1–3)
    Digitization step; authors state that optimizing Δt gives values close to 1 but do not show supporting data.
  • controllability residual threshold = 10^-12
    Runs with residual below this threshold are counted as exact ground-state solutions in the histograms of Fig. 4.
  • model couplings J_w and J_f = J_w=0.5, J_f∈[0.2501,0.45]
    Problem parameters inherited from Refs. [25–29]; chosen to satisfy J J_f > J_w^2, but not fitted to the target results.
axioms (5)
  • standard math Jordan-Wigner mapping converts H_x and H_z to quadratic fermionic Hamiltonians in a single parity sector; evolution stays in Gaussian states (Thouless parametrization).
    Used in Sec. II to justify exact simulation and the controllability dimension. Standard for 1D transverse-field Ising models.
  • domain assumption The reflection-symmetric Gaussian-state subspace has dimension (N^2−1)/2 for odd N, and the accessible algebra for the QAOA/CD unitaries is the corresponding Lie algebra.
    Imported from same-group Ref. [28] via Eqs. (16)–(17); the numerical histograms in Fig. 4 give indirect support, but the paper provides no proof.
  • ad hoc to paper The first-order nested-commutator AGP H_xz=i[H_x,H_z] is the only counter-diabatic operator considered for all CD methods.
    Eq. (20); the variational CD parameter in DC-QAOA also multiplies H_xz. This truncation is the weakest premise for generalizing the 'role of CD terms' conclusions.
  • domain assumption The analytical CD coefficient α_1^opt is obtained by minimizing the Sels-Polkovnikov action and is inserted via θ_xz=ṡ α_1^opt Δt.
    Eqs. (25)–(26) and App. A1; assumes the approximate AGP action captures the relevant non-adiabatic transitions in the digitized protocol.
  • domain assumption Closed-system noiseless unitary dynamics; no decoherence or measurement feedback.
    All evolution operators in Eqs. (12) and (21) are unitary and all simulations are noiseless; hardware implications are not tested.

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Cite this review

Pith. "Pith review of Digital techniques for the frustrated Ising ring: the role of counter-diabatic terms." pith.science (2026). https://pith.science/paper/DM2GMZ2S

@misc{pith2026260723074,
  author       = {Pith},
  title        = {Pith review of: Digital techniques for the frustrated Ising ring: the role of counter-diabatic terms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DM2GMZ2S}},
  note         = {Machine review of arXiv:2607.23074}
}
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read the original abstract

We investigate the role of local counter-diabatic (CD) terms in enhancing the performance of discrete-time digital protocols for a frustrated Ising ring, a system with an exponentially small spectral gap that acts as a bottleneck for conventional quantum annealing. The techniques investigated range from a digitised version of a fixed-schedule protocol, including lowest-order analytical CD terms, all the way to a full-fledged Quantum Approximate Optimisation Algorithm (QAOA), including variational CD terms (DC-QAOA). By analyzing the resulting residual energy, we show that DC-QAOA combined with Chopped RAndom Basis (CRAB) quantum control techniques for optimizing the circuit parameters, can outperform all the other strategies. By monitoring the ground-state population during the dynamics we learn that DC-QAOA finds effective shortcuts towards the target state through intermediate excited states, a process that is unexpectedly enhanced by the inclusion of local CD unitaries. Our results highlight the importance of flexible variational control of CD dynamics and demonstrate that digital optimization can explore operator dynamics that remain inaccessible to standard analytical CD constructions.

Figures

Figures reproduced from arXiv: 2607.23074 by Giuseppe E. Santoro, Kiran Thengil, Vincenzo Roberto Arezzo.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic diagram of the frustrated Ising ring model [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The energy gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Residual energy calculated following nearly [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The histogram of the residual energy for a fixed [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The residual energies obtained from different opti [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The schedule parameter [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The counterdiabatic parameter CD [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The optimal “time-steps” ∆ [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The log-gain of residual energies, defined in Eq. (36), [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The instantaneous ground state fidelity of different [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Residual energy obtained for a frustrated Ising ring of size (a) N = 17 and (b) N = 35 using 4) QAOA and 5) [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

72 extracted references · 3 canonical work pages

  1. [1]

    [17], which lead to α opt 1 (sp) = ℏTr([ ˆH(s p), ∂s ˆH]) 2 Tr([ ˆH(s p),[ ˆH(s p), ∂s ˆH]]) 2 .(25) This leads setting in Eqs

    Fixed schedule + analytical CD:A second pos- sibility is to use a fixed schedule with a fixed small ∆t, for instances cub(t), and include the CD term by analytically optimizing the coefficientα 1 via the techniques explained in Ref. [17], which lead to α opt 1 (sp) = ℏTr([ ˆH(s p), ∂s ˆH]) 2 Tr([ ˆH(s p),[ ˆH(s p), ∂s ˆH]]) 2 .(25) This leads setting in E...

  2. [2]

    0th-order

    Fixed schedule:Definitely the simplest “0th-order” approach consists in using a fixed schedules(t), for instances lin(t) as in standard QA approaches, or improved versions with vanishing derivatives att= 0 andt=τ, likes cub(t) ors sin(t), to deduces p = s(tp) at the discrete timest p = (p− 1 2 )∆t with a fixed small ∆ t, and set in Eqs. (21)-(22):  ...

  3. [3]

    (28) and ˙s p calculated using the same CRAB schedules CRAB(t)

    Optimized schedule + analytical CD:As a fur- ther possible improvement, one might consider 6 adding an analytical CD term, thereby writing    sp =s CRAB(t)|t=tp ˙sp = ˙sCRAB(t)|t=tp θz p =s p∆t θx p = (1−s p)∆t θxz p = ˙spα opt 1 (sp)∆t ,(29) withs p given by Eq. (28) and ˙s p calculated using the same CRAB schedules CRAB(t)

  4. [4]

    Optimized schedule without CD:Alternatively, one might consider a CRAB-based variational optimization of the schedules p without any CD term, leading to:    sp =s CRAB(t)|t=tp θz p =s p∆t θx p = (1−s p)∆t θxz p = 0 .(27) Here the schedules CRAB(t) is obtained by adding a finite number N c of Fourier modes to the linear ramps (lin)(t) =t/τ: ...

  5. [5]

    dec- orate

    QAOA with variationally optimized CD terms (a.k.a. DC-QAOA[21]). All these techniques are consistently in- terpreted within a single framework of repeatedly applied Trotterized unitary operations. For the numerical op- timization of variational parameters in methods 2)–5), we use a Fourier-based quantum control technique, the Chopped RAndom Basis (CRAB) [...

  6. [6]

    QAOA without CD terms:A further alternative approach without CD is a full QAOA with an- gles smoothly optimized with CRAB [27]. Here we write:   θz p = tp τ Cz 0 + NcX n=1 Cz n sin(ωn(tp −τ)) θx p = 1− tp τ Cx 0 + NcX n=1 Cx n sin(ωn(tp −τ)) θxz p = 0 , (30) with possible further dressed-CRAB re- finements [31] detailed in App. A. T...

  7. [7]

    (31) ThisAnsatzinvolves 3N c + 3 free parametersC= (Cx 0 , ...,Cx Nc ,C z 0, ...,Cz Nc ,C xz 0 , ...,Cxz Nc )

    QAOA with CD terms:Finally, we can further add a variational CD term without relying on the analytic expressionθ xz p = ˙s pα opt 1 (sp)∆t with a choice    θz p = tp τ Cz 0 + NcX n=1 Cz n sin(ωn(tp −τ)) θx p = 1− tp τ Cx 0 + NcX n=1 Cx n sin(ωn(tp −τ)) θxz p = C xz 0 + NcX n=1 Cxz n sin(ωn(tp −τ)) . (31) ThisAnsatzinvolves 3N ...

  8. [8]

    DC-QAOA 50 100 150 Nθ 10−13 10−10 10−7 10−4 10−1 ϵres P N = 35 Jf = 0.45

  9. [9]

    DC-QAOA FIG. 5. The residual energies obtained from different opti- mization techniques for N = 17 (top) and N = 35 (bottom) with circuits of different depthN θ. For the methods involv- ing dCRAB, we fix the value of N c = P. Furthermore, for each iteration of dCRAB, the optimization of the parameters Cis performed 10 times using randomly generated freque...

  10. [10]

    DC-QAOA 0.0 0.2 0.4 0.6 0.8 1.0 p/P 0.0 0.5 1.0 sp N = 35 Jf = 0.45

  11. [11]

    DC-QAOA FIG. 6. The schedule parameters p =θ z p/(θx p +θ z p) of the various methods, for N = 17 (top) and N = 35 (bottom), with a fixed total number of parametersN θ = 162. Figures 6, 7 and 8 give some insight on the variational parametersθbehind the various methods, showing the effective schedules p and time-step ∆ p sp = θz p ∆p ,∆ p =θ x p +θ z p ,(3...

  12. [12]

    DC-QAOA 0.0 0.2 0.4 0.6 0.8 1.0 p/P 0.0 0.5 1.0 1.5 2.0 ∆ p N = 35 Jf = 0.45

  13. [13]

    time-steps

    DC-QAOA FIG. 7. The optimal “time-steps” ∆ p =θ x p +θ z p of QAOA and DC-QAOA, for N = 17 (top) and N = 35 (bottom). Here Nθ = 162. eters related toθ xz p : CDan 1 (sp) = θxz p ∆t = ˙spα opt 1 (sp), CDvar 1 = θxz p ∆p .(35) Figure 6 shows thats p is strongly modified from the fixed cubic schedule used in method1). A finer comparison of the two QAOA-metho...

  14. [14]

    DC-QAOA 0.0 0.2 0.4 0.6 0.8 1.0 p/P 0.0 0.5 1.0 CDan/var 1 N = 35 Jf = 0.45

  15. [15]

    DC-QAOA FIG. 8. The counterdiabatic parameter CD an 1 or CDvar 1 of the various methods, see Eq. (35), for N = 17 (top) and N = 35 (bottom), with a fixed total number of parametersN θ = 162. 17 23 29 35 41 47 53 59 N 0 1 2 3 4Log-gainDC-QAOA Jf = 0.45 Nθ = 30 Nθ = 90 Nθ = 150 FIG. 9. The log-gain of residual energies, defined in Eq. (36), which compares t...

  16. [16]

    DC-QAOA 0.0 0.2 0.4 0.6 0.8 1.0 p/P 0.0 0.2 0.4 0.6 0.8 1.0 P0(p) N = 35 Jf = 0.45

  17. [17]

    DC-QAOA FIG. 10. The instantaneous ground state fidelity of different strategies with smooth schedules for systems of size N = 17 (top) and N = 35 (bottom). The methods illustrated and the corresponding specifications of the numerical experiments are same as that of Fig. 5 for a fixed total number of parameters Nθ = 162. The ground-state fidelity shown is...

  18. [18]

    (20) by minimizing the action [20, 50] S(A (1) s ) = Tr G2 s ,(A1) whereG s =∂ s ˆH(s) + i ℏ h A(1) s , ˆH(s) i and ˆH(s) is given by Eq

    Analytical CD terms We analytically optimize the parameterα 1 of Eq. (20) by minimizing the action [20, 50] S(A (1) s ) = Tr G2 s ,(A1) whereG s =∂ s ˆH(s) + i ℏ h A(1) s , ˆH(s) i and ˆH(s) is given by Eq. (6). The minimization ofSresults in an optimal value ofα 1: α opt 1 (sp) = ℏTr([ ˆH(s p), ∂s ˆH]) 2 Tr([ ˆH(s p),[ ˆH(s p), ∂s ˆH]]) 2 =− P j J 2 j 16...

  19. [19]

    (28), we adopt the dressed-CRAB (dCRAB) [31] technique with two super-iterations

    The dressed-CRAB schedule For the optimization of the schedule in Eq. (28), we adopt the dressed-CRAB (dCRAB) [31] technique with two super-iterations. In the first dCRAB iteration, the schedule is initialized with Fourier frequenciesω (1) n : s(1) p = tp τ + tp τ NcX n=1 Cn sin(ω(1) n (tp −τ)).(A3) We optimize the parametersC= (C 1, ....,CNc ) by minimiz...

  20. [20]

    QAOA with dressed CRAB We use dCRAB to optimizeθgiven by Eq. (30), where in the first iteration, they are projected onto Fourier modes of frequenciesω (1) n : θz,1 p = Cz 0 tp τ + tp τ NcX n=1 Cz n sin(ω(1) n (tp −τ)), θx,1 p = Cx 0 1− tp τ + 1− tp τ NcX n=1 Cx n sin(ω(1) n (tp −τ)), (A7) and optimizeC= (C x 0 , ...,Cx Nc ,C z 0, ...,Cz Nc ) through minim...

  21. [21]

    DC-QAOA (a) 50 100 150 Nθ 10−15 10−12 10−9 10−6 10−3 100 ϵres P N = 35

  22. [22]

    DC-QAOA (b) FIG. 11. Residual energy obtained for a frustrated Ising ring of size (a) N = 17 and (b) N = 35 using 4) QAOA and 5) DC-QAOA, where the frustration is induced by randomly fixing 30 different values ofJ f from the range [0.2501,0.45]. The dCRAB optimization routine and parameters are same as those of Fig. 5, except that only 1 trial of the nume...

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This paper was first reviewed by deepseek-v4-flash on August 1, 2026.