Effective Hamiltonians for Predictive Quantum Control

Hugo Ribeiro; Razvan Stanescu

REVIEW 3 major objections 8 minor 1 cited by

Effective Hamiltonians for Predictive Quantum Control

T0 review · 3 major / 8 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Correction pulses that work inside the Duffing model can fail when the drive is the real dressed charge operator of a transmon.

desk verdict Clean hybrid-transfer result: Duffing corrections that look great in-model can fail on a diag-calibrated baseline, with the main caveat that H_diag is still an idealized proxy. read the letter →

arxiv 2607.27111 v1 pith:M43U4EU2 submitted 2026-07-29 quant-ph

Razvan Stanescu , Hugo Ribeiro This is my paper · ORCID

classification quant-ph

keywords transmonquantumcontrolDuffingapproximationeffectiveHamiltonianleakageACStarkshiftMagnusexpansionsingle-qubitgates

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

High-fidelity open-loop gates need a model that predicts the errors that actually accumulate under the drive, not only the undriven spectrum. This paper compares the standard Duffing (weakly anharmonic oscillator) Hamiltonian with a more faithful low-energy description obtained by diagonalizing the static transmon and expressing the physical charge drive in that eigenbasis. The same Magnus-based correction construction is applied to both. Corrections that sharply cut error inside the Duffing model transfer poorly onto an independently calibrated diagonalized-transmon baseline; at short gate times they can even raise error above the uncorrected baseline. Small, individually justified differences in level spacings and drive matrix elements compound into different dynamical error generators, with AC Stark phase mismatch as one clear diagnostic. The model also dictates which control framework is adequate: a three-level truncation hides channels that a four-level model exposes, motivating a nonlinear extension that still uses only the same laboratory controls.

What carries the argument

Hybrid transfer test: Duffing-derived correction envelopes and detuning are combined with a baseline pulse calibrated only in the diagonalized-transmon model (via its own n0,1), then evaluated under the diagonalized-transmon Hamiltonian at fixed physical gate time. The test isolates multilevel model mismatch from trivial Rabi-rate mismatch.

What would settle it

On a device with known EJ/EC, apply the hybrid protocol (Duffing-derived corrections on a device-calibrated baseline) at short dimensionless gate times and compare measured average gate error to both the uncorrected baseline and a self-consistent correction designed in the diagonalized model; if the hybrid systematically improves as much as the self-consistent pulse, the claimed dynamical mismatch is not operative.

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Extended reading notes

Core claim

Using identical Magnus-based correction construction, Duffing-derived correction quadratures and constant detuning substantially reduce average gate error inside the Duffing model, but when added to an independently calibrated diagonalized-transmon baseline they remain far less effective and, in the fast-gate regime, can fail to improve—or can increase—error relative to the uncorrected diagonalized-transmon baseline.

Load-bearing premise

That a four-level, rotating-wave, distortion-free diagonalized-transmon Hamiltonian is faithful enough to the real driven device that hybrid failure under this model means Duffing-based predictive design would fail in practice.

Editorial extensions

If this is right

Pulse libraries designed only in the Duffing approximation cannot be assumed to transfer to hardware without re-deriving multilevel corrections in a dressed-charge model.
Model selection for control must be judged by driven error generators (e.g., AC Stark phase), not only static spectra or matrix elements.
Hilbert-space truncation is part of control design: too few levels can make a linear correction strategy look sufficient when a fuller model shows it is not.
The same physical controls (two quadratures plus constant detuning) can reach a distinct, better solution once higher-order channels are kept and a nonlinear Magnus strategy is used.
As EJ/EC decreases, the cost of using the Duffing model for fast-gate design rises.

Reading between the lines

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

Closed-loop or data-driven recalibration may partly mask the hybrid failure, so the practical risk is largest for purely open-loop or first-shot pulse deployment.
The same compounding of small spectrum and matrix-element errors should appear in other weakly anharmonic platforms whenever the control operator is replaced by a harmonic proxy.
Including measured transfer functions or decoherence would likely widen, not close, the gap between Duffing-designed and dressed-charge-designed corrections.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The useful result here is the hybrid protocol. Same Magnus correction machinery on both sides, baseline calibrated separately via each model’s n0,1, and only the Duffing gx, gy, and constant Δ transferred onto the diagonalized-transmon baseline. Inside Duffing the corrections crush the error; on H_diag they are much weaker and at short gates can sit at or above the uncorrected baseline (Fig. 4). That isolates multilevel error-generator mismatch rather than a trivial Rabi-rate offset. The AC Stark phase diagnostic (Fig. 5) and the three- vs four-level linear/nonlinear story (Figs. 7–8) are solid supporting pieces.

What is actually new is not “models matter for open-loop control”—that is already in the citations—but the controlled transfer test, the quantified dynamical diagnostic, and the explicit link from truncation to which correction ansatz is even viable. Numerics are internally consistent: truncation checks for uncorrected and corrected dynamics, sequential-transition estimate backed by numerics in the appendix, and visibly different nonlinear pulse shapes. Math and citation pattern look fine; prior Magnus papers are tools, not the claim.

Soft spot, in proportion: H_diag is labeled “more faithful” but still RWA, adjacent-only drive, four-level, ideal V(t), coherent only. Figs. 2 and 6 only vary level count inside that same rotating-frame object, so they cannot catch common-mode omissions. Corrected floors go to ~10^-5–10^-7; counter-rotating or nonadjacent pieces that are small vs uncorrected leakage could still reorder hybrid vs baseline. The authors mostly own this scope, so it conditions the hardware warning rather than killing the paper. No experiment, no code—moderate confidence is right.

For people who design open-loop pulses or pick effective models for transmons, this is worth reading. I would send it to referees. Engage if you care about model choice vs control ansatz; skip if you only want device-level validation.

Referee Report

3 major / 8 minor

Summary. The manuscript studies how the choice of effective Hamiltonian affects open-loop, model-based pulse design for single-qubit gates in a capacitively driven transmon. Two four-level models are compared: a Hamiltonian obtained by diagonalizing the static transmon and expressing the charge drive in the dressed eigenbasis, and the standard Duffing (Kerr-oscillator) approximation. The same Magnus-based constrained-control construction (from the authors' prior work) is applied to both models, with the baseline pulse calibrated model-specifically through n0,1 to remove the trivial Rabi-rate mismatch. The central result (Fig. 4) is a hybrid test: Duffing-derived correction quadratures and constant detuning, added to an independently calibrated diagonalized-transmon baseline and evaluated at fixed physical gate time, underperform the self-consistent correction and at short gate times can increase the error over the uncorrected baseline. A relative mismatch in the accumulated AC Stark phase (Fig. 5), growing as EJ/EC decreases and tf shortens, is presented as an illustrative diagnostic. A sequential-transition estimate (Eq. 21, App. A, validated in Fig. 9) plus four- vs ten-level checks (Figs. 2, 6) support the truncation. Finally, three- vs four-level truncations change which correction strategy appears sufficient (Fig. 7), and a nonlinear construction yields a distinct, improved control solution (Fig. 8).

Significance. If the results hold, the paper makes a useful, quantitatively documented point for the superconducting-circuit control community: accuracy of an effective Hamiltonian for static low-energy quantities does not certify its predictions for driven error generators, and the discrepancy becomes acute precisely in the fast-gate regime the field is pursuing. The comparison design is careful and worth explicit credit: identical correction machinery in both models, model-specific n0,1 calibration that removes a trivial confound, and hybrid evaluation at fixed physical tf rather than fixed |α2|tf. The analytic truncation criterion (Eq. 21) is validated against direct propagation (Fig. 9), and the EJ/EC and gate-time trends in Figs. 4–5 constitute falsifiable predictions. The Sec. V–VI demonstration that Hilbert-space truncation determines which correction framework appears sufficient — and that a nonlinear construction finds a genuinely distinct pulse with the same controls — is a constructive methodological contribution beyond the headline negative result.

major comments (3)

[§II.A.1 (RWA paragraph after Eq. (10)) and Fig. 4(b)] The RWA justification compares counter-rotating amplitudes (~1/(ωd tf)) to the *uncorrected* leakage error (~1/(|α2| tf)). But the paper's central quantity in Fig. 4(b) is the corrected error floor, ~10^-5–10^-7, and both models share the RWA and the adjacent-only charge-matrix-element truncation, so the convergence tests of Figs. 2 and 6 are blind to these common-mode omissions by construction. A contribution negligible against uncorrected leakage need not be negligible against the corrected floor, and it could in principle reorder baseline/hybrid/self-consistent curves at short tf. Concrete test: at the shortest |α2| tf and EJ/EC=30, propagate Eq. (6) in the lab frame (or a rotating frame retaining counter-rotating and nonadjacent terms, N=8–10) for the baseline, self-consistent, and hybrid protocols, and report whether the ordering and floors survive. Alternatively, provide perturbati
[§III.A, Eq. (36) and surrounding text] The hybrid construction recalibrates n0,1 on the evaluation model but transfers Δ_Duff 'without further adjustment.' Calibrating the drive frequency against the (dressed) qubit transition is as routine experimentally as the amplitude calibration the paper does allow, so the asymmetry needs justification. This matters quantitatively: Fig. 5 identifies an accumulated-phase mismatch as a contributor, and a single-scalar Δ recalibration could absorb part of it, potentially changing the fast-gate regime where the hybrid 'can fail to improve... and can even increase' error. Please either (a) show the hybrid curve with Δ recalibrated by one scalar parameter on H_diag, demonstrating the fast-gate failure persists, or (b) argue explicitly why frequency recalibration falls outside the paper's definition of predictive design while amplitude calibration does not.
[§V–VI, Figs. 7(b) and 8(a)] The nonlinear framework is motivated by the short-gate-time saturation of the linear strategy (Fig. 7(b): 'especially at short gate times'), yet Fig. 8(a) shows the fourth-order nonlinear advantage (~1 order of magnitude) only for |α2| tf ≳ 7, with linear and nonlinear errors 'comparable' at shorter times. The claimed 'systematic route' to the shortest-time regime via higher-order nonlinear corrections is an expectation, not a demonstrated result. Please reconcile: either include a higher-order (e.g., sixth-order) nonlinear data point at short tf, or revise the §V framing so the demonstrated regime and the anticipated regime are clearly distinguished. This is load-bearing for the paper's second result, not the first.

minor comments (8)

[Figs. 4, 7, 8 captions] Neither caption states EJ/EC for the plotted traces (Fig. 5 varies it; these figures presumably fix one value). Also give the physical tf range in ns for context.
[§II and Fig. 4] Report absolute device parameters (ωT/2π, EC/h, and the drive frequency). The RWA hierarchy |α2|/ωd ≪ 1 invoked in §II.A.1 cannot be checked from the dimensionless quantities alone, and experimental relevance (which tf in ns corresponds to |α2| tf = 5.74) requires them.
[§III, after Eq. (35)] State the Fourier basis size and number of free coefficients used for gx, gy, and comment on conditioning/solution selection for the algebraic system from Eq. (35) (uniqueness, minimal-norm choice if underdetermined). A code/data availability statement would substantially strengthen reproducibility of the 10^-7-scale error floors.
[Eq. (37)] The second matrix element is written ⟨0|Σ Ω(tf)|0⟩ with the Magnus-order subscript l missing on Ω. Also state which order m is used for the Fig. 5 curves ('up to fourth order' — is m=4 for all traces?).
[Eq. (15)] Check the parenthesization of the n1,2 sector: as typeset, the grouping of the cos[θ/2](...) + sin[θ/2](...) terms is ambiguous about which σx/σy factors belong to which trigonometric prefactor.
[Fig. 8(d)] §III restricts the detuning to be constant, and §VI states the nonlinear scheme uses 'the same constant detuning control,' yet panel (d) plots Δ/ωC against time. If each trace is a constant, say so in the caption; if the nonlinear construction relaxes the constant-detuning constraint, reconcile with §III.
[§I or §III.A] The hybrid failure is closely related to the known model sensitivity of DRAG coefficients and optimal detunings in transmons (e.g., higher-derivative and FAST DRAG calibrations in Ref. [6]). A short paragraph connecting the present mechanism to that literature would help experimental readers place the result.
[Eq. (23), 'complex Hadamard gate'] Consider noting explicitly that this is the √X-type gate used as the benchmark in Refs. [21, 25], to orient readers before Eq. (22)–(23).

Circularity Check

1 steps flagged · score 1.0 of 10

No load-bearing circularity: hybrid Duffing-vs-diagonalized failure is an independent numerical outcome, not forced by the Magnus framework or by baseline calibration.

self citation load bearing [Sec. III, opening; Refs. [21, 25]]
"Rather than developing a new control protocol, we use the Magnus-based constrained-control framework introduced in Refs. [21, 25] and apply it separately to the two effective Hamiltonians derived above. This allows us to isolate how the choice of effective Hamiltonian affects the pulse predicted by an otherwise fixed control strategy."

The correction construction is taken from prior papers with overlapping authorship (Ribeiro). This is ordinary methodological self-citation and is not load-bearing for the central scientific claim: the hybrid transfer failure and AC Stark mismatch are numerical comparisons between two Hamiltonians under that fixed recipe, not consequences forced by the cited framework’s definitions or uniqueness results. Flagged only as minor self-citation, not as a reduction of the result to its inputs.

full rationale

The paper’s central claim—that Duffing-derived correction fields can fail when transferred onto an independently calibrated diagonalized-transmon baseline—is established by direct numerical propagation of two different effective Hamiltonians under a fixed control construction (Fig. 4 hybrid protocol; AC Stark diagnostic in Fig. 5). Model-specific normalization of the baseline envelope by each model’s n0,1 is an intentional isolation step that removes only the trivial computational-subspace Rabi mismatch; residual hybrid error is then a free dynamical outcome, not an algebraic identity. Hilbert-space truncation is justified by a sequential-transition Magnus estimate and checked against multi-level Schrödinger evolution (Figs. 2, 6, 9), not assumed by definition. The only self-citation is adoption of the Magnus constrained-control method from the authors’ prior work [21, 25]; that method supplies the pulse-construction tool but does not force the model-dependence result, which would be meaningful under any shared correction recipe. No fitted parameter is renamed a prediction, no uniqueness theorem is imported to forbid alternatives, and no static approximation is smuggled in as a dynamical identity. Score 1 reflects only that minor methodological self-citation; the derivation chain is otherwise self-contained.

Assumptions & free parameters 5 free parameters · 8 assumptions · 2 invented entities

The paper is a coherent-model comparison under standard circuit-QED and Magnus-control assumptions. Load-bearing inputs are the ideal drive map, RWA and matrix-element truncations, sufficiency of four levels, chosen device ratios EJ/EC, the fixed cosine-like baseline envelope, and the authors’ prior constrained Magnus cancellation conditions. No new physical entity is postulated; free choices are numerical/model parameters, not fits to experimental gate data.

free parameters (5)

EJ/EC ratio set {30, 40, 50} = 30, 40, 50
Device working points chosen to scan distance from deep-transmon regime; not fitted to experiment but selected by hand and used to show growing model mismatch.
Dimensionless gate-time window |α2|tf (e.g. down to ~5.74) = approximately 5.7–20 (figures)
Defines the fast-gate regime where hybrid failure is claimed; range is a study choice, not derived.
Baseline envelope shape fx(t) ∝ 1−cos(2πt/tf) = Eq. (16) form
Fixed smooth pulse family with s=3 boundary cancellation; results on leakage hierarchy and corrections are conditional on this family.
Magnus cancellation order m and Fourier basis size for gx, gy = m=2 primary; higher orders in Secs. V–VI
Perturbative order (m=2 in main hybrid comparison; up to 4th/6th in later sections) and finite Fourier parametrization are implementation choices that set achievable error floors.
Constant-detuning restriction Δ = constant Δ during gate
Drive-frequency correction limited to a single constant rather than a chirp; shapes the reachable correction manifold.

assumptions (8)

domain assumption Rotating-wave approximation: counter-rotating terms at ~2ωd are negligible versus leakage ~1/(|α2|tf) for the parameters studied.
Invoked in Sec. II A with a parametric hierarchy |α2|/ωd ≪ 1; not re-derived from full lab-frame numerics for every corrected pulse.
domain assumption Only adjacent charge matrix elements need be retained; non-adjacent symmetry-allowed elements are negligible.
Stated after Eq. (8); authors say numerically verified but do not show the check.
domain assumption Ideal, distortion-free map from specified envelopes to port voltage V(t); no decoherence, drift, or transfer-function uncertainty.
Explicit isolation choice in Sec. II; central ‘predictive control’ claims are conditional on coherent Hamiltonian mismatch only.
domain assumption Four-level truncation captures relevant uncorrected and corrected dynamics over the studied gate times (sequential-transition η4/η3 ≪ 1 plus numerical checks).
Sec. II A–B, Fig. 2, Sec. IV Fig. 6; underpins all control comparisons.
domain assumption Diagonalized-transmon eigenbasis plus dressed n_k,l is the more faithful low-energy model of the cosine circuit Hamiltonian under the same RWA/truncation.
Framing throughout Introduction and Sec. II; hybrid evaluation treats H_diag as the reference truth.
domain assumption Magnus-based constrained cancellation conditions of Refs. [21,25] define a valid open-loop correction when residual Magnus terms through order m are set to zero.
Sec. III; framework imported rather than re-proved; numerical propagation is used as operational validation.
standard math Standard Mathieu/charge-basis spectrum and circuit Hamiltonian of the Cooper-pair box / transmon (Koch et al.).
Eq. (1) and eigenbasis construction; textbook starting point.
standard math Average gate fidelity error formula of Pedersen et al. on the computational subspace projector.
Eq. (22); standard metric choice.

invented entities (2)

Hybrid correction protocol (Duffing g_x,g_y,Δ on diag-calibrated baseline)
purpose: Isolate transferability of multilevel corrections without confounding baseline Rabi calibration.
Methodological construction, not a physical object; defined in Sec. III and Fig. 1(b).
Sequential-transition leakage estimate η_k
purpose: Analytic ranking criterion for Hilbert-space cutoff from products of first-order Magnus adjacent probabilities.
Eq. (21) and Appendix A; heuristic upper-bound product, validated numerically against Schrödinger leakage hierarchy.

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Cite this review

Pith. "Pith review of Effective Hamiltonians for Predictive Quantum Control." pith.science (2026). https://pith.science/paper/M43U4EU2

@misc{pith2026260727111,
  author       = {Pith},
  title        = {Pith review of: Effective Hamiltonians for Predictive Quantum Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M43U4EU2}},
  note         = {Machine review of arXiv:2607.27111}
}

read the original abstract

High-fidelity quantum control relies on accurate models of driven dynamics. We examine this re- quirement for single-qubit gates in superconducting transmons by comparing control pulses derived from the standard Duffing approximation and from a Hamiltonian constructed by diagonalizing the transmon eigenbasis. Using the same correction-pulse construction for both models, we show that correction fields derived from the Duffing approximation can substantially reduce the gate error pre- dicted by that model while remaining less effective when combined with an independently calibrated baseline pulse in the diagonalized-transmon model. In the fast-gate regime, such transferred correc- tions can even fail to improve over the uncorrected diagonalized-transmon baseline. We show that small model-dependent differences in both the energy spectrum and the representation of the drive operator can compound during driven evolution, resulting in different predicted error generators and correction pulses. A mismatch in the accumulated AC Stark phase provides one illustrative di- agnostic of this dynamical model dependence. We further demonstrate that the model Hamiltonian informs the choice of control framework: Omitting relevant leakage pathways or higher-order error channels can lead to an overly restricted correction strategy. Including these channels motivates an extended correction framework that improves the gate performance using the same physical control resources.

Figures

Figures reproduced from arXiv: 2607.27111 by the authors.

**Figure 1.** Model dependence of quantum-control predictions. (a) For a fixed target unitary Uˆtarget, the same control framework can predict different control protocols when applied to different effective Hamiltonians. Two descriptions, Hˆeff,1(t) and Hˆeff,2(t), may therefore yield different controls, f1(t) and f2(t), designed to implement the same target operation. (b) When evaluated using a more faithful description of the … view at source ↗

**Figure 2.** Hilbert-space convergence of the uncorrected single-qubit gate. Average fidelity error ϵ for the target complex-Hadamard gate as a function of the dimensionless gate time |α2|tf. (a) Results obtained from the diagonalizedtransmon Hamiltonian truncated to three, four, and eight energy levels. (b) Corresponding results for the Duffing Hamiltonian truncated to the same number of levels. In both models, the four- an… view at source ↗

**Figure 3.** Comparison of the two effective transmon models used for pulse design. The diagonalized-transmon model, Hˆdiag(t), retains the transition frequencies and dressed charge matrix elements nk,l obtained from the eigenbasis of the static transmon Hamiltonian. The Duffing model, HˆDuff(t), replaces the spectrum by that of a Kerr oscillator with constant adjacent-transition anharmonicity and represents the charge-drive … view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Model dependence of correction-pulse performance. Average fidelity error ϵ as a function of the dimensionless gate time |α2|tf. (a) Results calculated with the Duffing Hamiltonian. The gray trace shows the calibrated baseline protocol, while the blue trace shows the…

**Figure 5.** Figure 5: shows the relative difference up to fourth order in the Magnus expansion, in percent, between the time-averaged AC Stark shifts predicted by the diagonalized-transmon and Duffing Hamiltonians for EJ /EC = 50, 40, and 30. For all three values, the mismatch is largest i…

**Figure 6.** Figure 6: Validity of the four-level truncation for corrected dynamics. Average fidelity error for the diagonalizedtransmon Hamiltonian using four- and ten-level Hilbert-space truncations. The uncorrected four- and ten-level evolutions agree over the gate-time range shown, con…

**Figure 7.** Figure 7: demonstrates the consequence for linear corrections [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: compares fourth-order linear and nonlinear corrections in the four-level diagonalized-transmon model [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: Validation of the leakage hierarchy underlying Eq. (21). The plot compares normalized higher-level leakage quantities for pathways originating from |1⟩. The curves labeled “sequential” show the corresponding normalized sequential-transition estimates, whereas the cur…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Error-Generator-Level Compression for Fast Quantum Gates
quant-ph 2026-07 conditional novelty 6.0 of 10

A three-parameter Magnus-based pulse, chosen by minimizing a spectral penalty on the projected error generator, suppresses fast transmon-gate errors and beats grid-optimized leading-order DRAG in simulation.

Reference graph

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Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.

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( 6) to a frame rotating at the drive frequency

Rotating Frame and Interaction Picture We first transform Eq. ( 6) to a frame rotating at the drive frequency. The unitary change-of-frame operator is ˆSrot(t) = exp " −iωdt ∞X k=0 k|k⟩ ⟨k| # = ∞X k=0 e−ikωdt|k⟩ ⟨k|. (7) Using this transformation, the Hamiltonian in the rotat- ing frame is ˆHrot(t) = ˆS† rot(t) ˆHdiag(t) ˆSrot(t) −i ˆS† rot(t)∂t ˆSrot(t) ...

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sequential

Magnus Expansion and Transition Probabilities to Leakage States We now use Eq. ( 15) to estimate which leakage levels must be retained in the effective Hilbert space. The time- 6 evolution operator generated by ˆVI(t) can be written as ˆUI(t) = exp[ ˆΩ(t)], (17) where ˆΩ(t) = P∞ l=1 ˆΩl(t) is the Magnus expansion [ 22, 23]. The first two terms are ˆΩ1(t) ...

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2024