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arXiv:2607.28383

Counterdiabatic Driving under Variational Frame Dressing

Boxi Li, Franco Nori, Felix Motzoi

quant-phSubmitted Jul 30, 2026• Announced Jul 31, 2026• 0 citations
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Abstract

Counterdiabatic (CD) driving accelerates adiabatic protocols by prescribing auxiliary control fields, but often fails to map them to physically available operations. We derive a general formalism for such a mapping. We formulate CD driving in a variational dressed frame, where an unconstrained auxiliary generator reshapes the effective adiabatic problem while simultaneously forcing the applied correction to stay restricted to the native laboratory controls. This yields a laboratory-frame commutator equation that can be solved without constructing instantaneous eigenstates or the adiabatic and dressed-frame unitaries. The additional dressed-frame freedom reveals solutions that are inaccessible in the conventional adiabatic frame. We illustrate this mechanism in three settings: suppression of spectator errors in a multi-qubit chain driven by a single quadrature, an analytical acceleration of adiabatic Bell-state preparation with a fixed entangling interaction, and implementation of a fast adiabatic holonomic gate in a degenerate tripod manifold with correction pulses confined to the native couplings. Our results provide a systematic nonperturbative framework for constructing implementable counterdiabatic protocols beyond the conventional adiabatic frame.

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Counterdiabatic (CD) driving accelerates adiabatic protocols by prescribing auxiliary control fields, but often fails to map them to physically available operations. We derive a general formalism for such a mapping. We formulate CD driving in a variational dressed frame, where an unconstrained auxiliary generator reshapes the effective adiabatic problem while simultaneously forcing the applied correction to stay restricted to the native laboratory controls. This yields a laboratory-frame commutator equation that can be solved without constructing instantaneous eigenstates or the adiabatic and dressed-frame unitaries. The additional dressed-frame freedom reveals solutions that are inaccessible in the conventional adiabatic frame. We illustrate this mechanism in three settings: suppression of spectator errors in a multi-qubit chain driven by a single quadrature, an analytical acceleration of adiabatic Bell-state preparation with a fixed entangling interaction, and implementation of a fast adiabatic holonomic gate in a degenerate tripod manifold with correction pulses confined to the native couplings. Our results provide a systematic nonperturbative framework for constructing implementable counterdiabatic protocols beyond the conventional adiabatic frame.
Counterdiabatic (CD) driving presents a way of generating adiabatic dynamics at an arbitrary pace, where excitations due to nonadiabaticity are exactly compensated by adding an auxiliary driving term to the Hamiltonian. While this CD term is theoretically known and given by the adiabatic gauge potential, obtaining and implementing this potential in many-body systems is a formidable task, requiring knowledge of the spectral properties of the instantaneous Hamiltonians and control of highly nonlocal multibody interactions. We show how an approximate gauge potential can be systematically built up as a series of nested commutators, remaining well defined in the thermodynamic limit. Furthermore, the resulting CD driving protocols can be realized up to arbitrary order without leaving the available control space using tools from periodically driven (Floquet) systems. This is illustrated on few- and many-body quantum systems, where the resulting Floquet protocols significantly suppress dissipation and provide a drastic increase in fidelity.
We introduce an inertial counterdiabatic (CD) protocol that reformulates shortcut-to-adiabaticity design as a moving-frame diagonalization problem. The method employs the Magnus expansion to construct a variational propagator generated by an accessible CD ansatz. In the new rotating frame, the controls are optimized to suppress the off-diagonal Hamiltonian terms, and the CD terms are defined to counteract the corresponding inertial contribution. The propagator generated with the Magnus expansion naturally exploits additional control directions generated through its nested commutator, while the sequential design of the propagator introduces temporal nonlocality into the optimization. We benchmark the method on a two-qubit system and on the one-dimensional quantum Ising chain, where it consistently outperforms variational CD protocols based on action minimization. The largest improvements are observed near gap-closing regions, demonstrating the advantage of combining temporal and algebraic nonlocality for suppressing diabatic excitations.
Counterdiabatic (CD) protocols enable fast driving of quantum states by invoking an auxiliary adiabatic gauge potential (AGP) that suppresses transitions to excited states throughout the driving process. Usually, the full spectrum of the original unassisted Hamiltonian is a prerequisite for constructing the exact AGP, which implies that CD protocols are extremely difficult for many-body systems. Here, we apply a variational CD protocol recently proposed by Claeys et al. [Phys. Rev. Lett. 123, 090602 (2019)] to a two-component fermionic Hubbard model in one spatial dimension. This protocol engages an approximated AGP expressed as a series of nested commutators. We show that the optimal variational parameters in the approximated AGP satisfy a set of linear equations whose coefficients are given by the squared Frobenius norms of these commutators. We devise an exact algorithm that escapes the formidable iterative matrix-vector multiplications and evaluates the nested commutators and the CD Hamiltonian in analytic representations. We then examine the CD driving of the one-dimensional Hubbard model up to L=14L=14L=14 sites with driving order $l\ensuremath{\leqslant}3$. Our results demonstrate the usefulness of the variational CD protocol to the Hubbard model and permit a possible route towards fast ground-state preparation for many-body systems.
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.
Variational counterdiabatic (CD) driving is a disciplined and widely used method to robustly control quantum many-body systems by mimicking adiabatic processes with high fidelity and reduced duration. Central to this technique is a universal structure of the adiabatic gauge potential (AGP) over a parameterized Hamiltonian. Here, we reveal that introducing a new degree of freedom into the theory of the AGP can significantly improve variational CD driving. Specifically, we find that the algebraic characterization of the AGP is not unique, and we exploit this nonuniqueness to develop the for deriving a refined driving protocol. This approach extends the conventional method in two aspects: it assigns customized weights to matrix elements relevant to specific problems, and it effectively incorporates nonlocal information into local driving coefficients. We also develop an efficient numerical algorithm to compute the refined driving protocol using computer algebra. Our framework is broadly applicable and, in principle, it can replace any previous use of variational CD driving. We demonstrate its practicality by applying it to adiabatic evolution along the ground state of a parameterized Hamiltonian. This proposal outperforms the conventional method in terms of fidelity, as confirmed by extensive numerical simulations on quantum Ising models.
Approximate counterdiabatic (CD) protocols are a powerful tool to enhance quantum adiabatic processes that allow to reliably manipulate quantum systems on short time scales. However, implementing CD protocols entails the introduction of additional control fields in the Hamiltonian, often associated with highly non-local multi-body interactions. Here, we introduce a novel variational rotated ansatz (RA) to systematically generate experimentally accessible approximate CD protocols. We numerically benchmark our approach on state preparation and adiabatic quantum computing algorithms, and find that using RA protocols significantly enhances their performances.