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arxiv: 2605.18237 · v1 · pith:TMISOLAQnew · submitted 2026-05-18 · 🪐 quant-ph

Regularized Counterdiabatic Driving for the Quantum Rabi Model

Juli\'an Ferreiro-V\'elez , Pablo Garc\'ia-Azor\'in , F. A. C\'ardenas-L\'opez , Xi Chen This is my paper

Pith reviewed 2026-05-20 10:29 UTC · model grok-4.3

classification 🪐 quant-ph
keywords counterdiabatic drivingquantum Rabi modelvariational optimizationregularizationlight-matter interactionFloquet engineeringquantum optimal controlunbounded Hilbert space
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The pith

A variational framework with subspace regularization enables counterdiabatic driving in the quantum Rabi model.

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

The paper develops a variational optimization method for counterdiabatic driving in systems with unbounded Hilbert spaces, such as the quantum Rabi model, where standard trace-based approaches lead to divergences. By applying renormalization schemes that restrict the metric to displaced and low-energy subspaces, the authors obtain finite control terms that suppress diabatic excitations during finite-time evolution. These terms take the form of couplings between the atomic degree of freedom and the position and momentum quadratures of the field, and they remain effective from strong to deep-strong light-matter interaction strengths. The work also presents a fidelity-based optimal-control alternative and demonstrates implementation through Floquet engineering via parametric modulation of the Hamiltonian.

Core claim

The authors introduce a variational optimization framework equipped with physically motivated renormalization schemes that regularize the trace-based metric by restricting it to relevant displaced and low-energy subspaces. Applied to the quantum Rabi model beyond the dispersive approximation, this yields two distinct counterdiabatic contributions that couple the atomic degree of freedom to the position and momentum quadratures of the field and suppress diabatic excitations across coupling regimes ranging from strong to deep-strong light-matter interaction.

What carries the argument

The renormalized trace-based variational metric restricted to displaced and low-energy subspaces, which produces approximate counterdiabatic Hamiltonians for the Rabi model.

If this is right

  • Counterdiabatic driving extends consistently to continuous-variable systems that possess unbounded Hilbert spaces.
  • Two specific counterdiabatic contributions can be derived for the Rabi model that remain effective beyond the dispersive approximation.
  • A fidelity-based quantum optimal-control strategy provides a practical bypass when trace-based variational methods encounter limitations.
  • The resulting control terms can be realized experimentally through Floquet engineering by parametric modulation of the native Hamiltonian.

Where Pith is reading between the lines

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

  • The regularization approach may extend to other bosonic models encountered in quantum optics or superconducting circuits.
  • Experimental verification in circuit-QED platforms could test whether the derived terms maintain high fidelity during fast state preparation.
  • The subspace-restriction technique offers a route toward scalable control protocols for preparing states in regimes of very strong light-matter coupling.

Load-bearing premise

Restricting the trace-based functional to displaced and low-energy subspaces eliminates divergences while still capturing the essential diabatic suppression dynamics across the full range of coupling strengths.

What would settle it

Numerical simulations of the Rabi model in which the regularized counterdiabatic protocol produces significantly lower state-preparation fidelity than exact methods in the deep-strong coupling limit would show that the subspace restriction fails to capture the required dynamics.

Figures

Figures reproduced from arXiv: 2605.18237 by F. A. C\'ardenas-L\'opez, Juli\'an Ferreiro-V\'elez, Pablo Garc\'ia-Azor\'in, Xi Chen.

Figure 1
Figure 1. Figure 1: FIG. 1. Regime map of the QRM across the normalized parameter space. The diagram divides the landscape into representative regions for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration of the CD corrections in the QRM. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic representation of the effective low-energy mani [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Final ground-state fidelity versus normalized coupling [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Simulated CD dynamics across the Rabi parameter space a dimensionless evolution time [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Infidelity of the analytically derived CD protocol (dashed [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison between exact CD evolution (dashed red) and [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Action landscape and corresponding gradients for different variational CD cost functionals, evaluated as functions of the AGP coeffi [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison between the accumulated normalized action and the final ground-state fidelity for different regularized metrics. The panels [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

Counter-diabatic (CD) driving provides a powerful route to fast and robust state preparation by suppressing diabatic excitations during finite-time evolution. Yet, deriving analytical CD protocols for complex systems remains challenging, motivating the development of variational approaches. These methods typically rely on minimizing trace-based functionals to construct approximate control Hamiltonians. However, in unbounded systems, such functionals can become ill-defined because of the unbounded bosonic Hilbert space, leading to divergent cost functions and unphysical variational coefficients. Here, we introduce a variational optimization framework equipped with physically motivated renormalization schemes that regularize the trace-based metric by restricting it to relevant displaced and low-energy subspaces. As a paradigmatic example, we apply our method to the quantum Rabi model beyond the dispersive approximation and identify two distinct CD contributions that couple the atomic degree of freedom to the position and momentum quadratures of the field. These terms suppress diabatic excitations across coupling regimes ranging from strong to deep-strong light--matter interaction. We further formulate a fidelity-based quantum optimal-control strategy that bypasses the limitations of trace-based variational methods. Finally, we show that the resulting CD terms can be implemented via Floquet engineering through parametric modulation of the native Hamiltonian. Our results demonstrate that CD driving can be consistently extended to continuous-variable systems with unbounded Hilbert spaces, providing a controlled and scalable framework for quantum control in strongly interacting light-matter platforms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a variational optimization framework for counterdiabatic (CD) driving in the quantum Rabi model. It regularizes the trace-based metric by restricting it to relevant displaced and low-energy subspaces to handle divergences in the unbounded bosonic Hilbert space. The approach identifies two distinct CD terms that couple the atomic degree of freedom to the position and momentum quadratures of the field, which are claimed to suppress diabatic excitations from strong to deep-strong coupling regimes. The work also formulates a fidelity-based quantum optimal-control strategy and demonstrates implementation of the CD terms via Floquet engineering through parametric modulation of the native Hamiltonian.

Significance. If the subspace regularization is shown to be robust, the results would provide a practical route to extend CD driving to continuous-variable systems with unbounded spectra, which is relevant for fast and robust state preparation in strongly coupled light-matter platforms. The explicit identification of quadrature couplings and the Floquet implementation offer concrete, potentially scalable protocols for quantum control experiments.

major comments (2)
  1. [Abstract and variational framework section] The central regularization procedure (abstract and the section introducing the variational framework): the claim that restricting the trace-based functional to displaced and low-energy subspaces eliminates divergences while still capturing the essential diabatic suppression dynamics is load-bearing for the main result. In the deep-strong coupling regime the ground-state displacement grows and higher Fock states mix strongly; an explicit convergence test with respect to subspace size or a direct comparison of the resulting variational coefficients against exact time-dependent numerics is required to confirm that the identified position- and momentum-quadrature CD terms do not leave residual excitations outside the chosen subspace.
  2. [Rabi model application section] Application to the Rabi model (section presenting the identified CD terms): the two quadrature couplings are stated to work across the full range of coupling strengths, yet no quantitative fidelity or excitation-suppression metrics are referenced for the deep-strong limit. Without such data it remains unclear whether the variational solution fully suppresses the relevant transitions or only a subset, which directly affects the claimed generality of the method.
minor comments (2)
  1. [Abstract] The abstract would benefit from a brief explicit statement of the functional form of the two identified CD terms (e.g., the operators multiplying the atomic Pauli matrices).
  2. [Method section] Notation for the displaced and low-energy subspaces should be introduced with a clear definition (e.g., a projector or cutoff parameter) at first use to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to incorporate additional validation and quantitative metrics as requested.

read point-by-point responses

  1. Referee: The central regularization procedure (abstract and the section introducing the variational framework): the claim that restricting the trace-based functional to displaced and low-energy subspaces eliminates divergences while still capturing the essential diabatic suppression dynamics is load-bearing for the main result. In the deep-strong coupling regime the ground-state displacement grows and higher Fock states mix strongly; an explicit convergence test with respect to subspace size or a direct comparison of the resulting variational coefficients against exact time-dependent numerics is required to confirm that the identified position- and momentum-quadrature CD terms do not leave residual excitations outside the chosen subspace.

    Authors: We agree that explicit convergence validation is essential to substantiate the regularization scheme, especially in the deep-strong regime. In the revised manuscript, we have added a new subsection and figure presenting the variational coefficients as a function of subspace size (both displacement cutoff and Fock-state truncation) for g/ω = 2. We also include a direct comparison of the approximate CD evolution against exact time-dependent numerics for accessible Hilbert-space dimensions, confirming that residual excitations outside the regularized subspace remain negligible and that the quadrature couplings suppress the dominant diabatic transitions. revision: yes

  2. Referee: Application to the Rabi model (section presenting the identified CD terms): the two quadrature couplings are stated to work across the full range of coupling strengths, yet no quantitative fidelity or excitation-suppression metrics are referenced for the deep-strong limit. Without such data it remains unclear whether the variational solution fully suppresses the relevant transitions or only a subset, which directly affects the claimed generality of the method.

    Authors: We thank the referee for highlighting this point. While the original manuscript emphasized qualitative behavior across regimes, the revised version now includes quantitative fidelity and residual-excitation data specifically for the deep-strong limit (g/ω > 1). These metrics, shown in an updated figure, demonstrate that the two quadrature CD terms achieve high fidelity and strong suppression of excitations relative to the undriven case, supporting the claimed generality. The new results are referenced in the Rabi-model application section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; regularization framework is independently motivated and applied.

full rationale

The paper proposes a variational framework that regularizes the trace-based metric via explicit restriction to displaced and low-energy subspaces, then applies it to derive two quadrature CD terms for the Rabi model. This restriction is introduced as a physically motivated choice to handle the unbounded bosonic space, not derived from or equivalent to the target CD coefficients by construction. No load-bearing self-citations, uniqueness theorems from prior author work, or fitted inputs renamed as predictions appear in the derivation chain. The fidelity-based optimal control is presented as a separate bypass, and Floquet implementation is a downstream application. The central claims remain independent of the inputs and are externally falsifiable via simulation or experiment.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the physical motivation for restricting the trace functional; no explicit free parameters or invented entities are stated in the abstract.

axioms (1)
  • domain assumption Restricting the trace-based metric to displaced and low-energy subspaces yields a well-defined and physically relevant cost function for unbounded bosonic systems.
    Invoked to handle divergences in the quantum Rabi model Hilbert space.

pith-pipeline@v0.9.0 · 5798 in / 1205 out tokens · 43661 ms · 2026-05-20T10:29:48.723050+00:00 · methodology

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