Berry's transitionless quantum driving (2009) solves the 'reverse engineering' direction exactly: given a reference Hamiltonian H0(t), an explicit modified Hamiltonian H(t) = H0(t) + H1(t) drives the instantaneous eigenstates of H0(t) as exact evolving states, with zero transitions at any driving speed. The inverse question is left open: given a physically specified Hamiltonian H(t), determine the base Hamiltonian H0(t) whose instantaneous eigenstates are the exact evolving states of H, i.e. invert the relation H = H0 + H1[H0]. Berry notes that the natural adiabatic iteration for this inversion generates a divergent asymptotic series, and solves the inversion only for the two-state (spin-1/2) Landau-Majorana-Zener case, where it is multiply redundant. The subsequent literature has developed the forward construction (H0 to H) extensively, including recently convergent expansions; has substantially advanced the invariant and integrability connections and the practical-implementability questions raised in the same source passage; and, within the periodic (Floquet, fixed-period driving) class, realizes the inversion by a parallel-transport-gauge decomposition (a basis choice that removes the accumulated adiabatic phase from the Floquet eigenstates). No systematic inversion theory - existence and multiplicity of solutions of the self-consistent equation H = H0 + H1[H0], and a convergent or controlled-error approximate reconstruction scheme for general non-periodic Hamiltonians - has been found in the literature. The problem remains open, with the divergent-iteration obstacle and the two-state and Floquet special cases as its precise boundary.

Background

A quantum system driven by a time-dependent Hamiltonian is conventionally analyzed in the adiabatic basis: the instantaneous eigenstates ∣n(t)⟩|n(t)\rangle and energies En(t)E_n(t) defined by H^0(t)∣n(t)⟩=En(t)∣n(t)⟩\hat{H}_0(t)|n(t)\rangle=E_n(t)|n(t)\rangle. The adiabatic theorem states that if H^0(t)\hat{H}_0(t) changes slowly compared with the inverse spectral gaps and its eigenstates remain non-degenerate, a state prepared in ∣n(0)⟩|n(0)\rangle follows ∣n(t)⟩|n(t)\rangle up to nonadiabatic transitions that are exponentially small in the slowness parameter; the paradigmatic example is the Landau-Majorana-Zener (LMZ) two-level model of an avoided crossing, whose residual transition probability is finite for any finite sweep rate.

Transitionless (or counterdiabatic) driving eliminates these transitions exactly rather than asymptotically. Berry (2009) showed that for any H^0(t)\hat{H}_0(t) there is a nearby Hamiltonian for which the adiabatic-form states, carrying both dynamical and geometric phase, are exact solutions of the Schrodinger equation: H^(t)=H^0(t)+H^1(t)\hat{H}(t)=\hat{H}_0(t)+\hat{H}_1(t) with the counterdiabatic term

H^1(t)=iℏ∑m≠n∣m(t)⟩⟨m(t)∣ ∂tH^0(t) ∣n(t)⟩⟨n(t)∣En(t)−Em(t),\hat{H}_1(t)=\mathrm{i}\hbar\sum_{m\neq n}\frac{|m(t)\rangle\langle m(t)|\,\partial_t\hat{H}_0(t)\,|n(t)\rangle\langle n(t)|}{E_n(t)-E_m(t)},

equivalently H^=∑n∣n⟩En⟨n∣+iℏ∑n(∣∂tn⟩⟨n∣−⟨n∣∂tn⟩∣n⟩⟨n∣)\hat{H}=\sum_n|n\rangle E_n\langle n|+\mathrm{i}\hbar\sum_n(|\partial_t n\rangle\langle n|-\langle n|\partial_t n\rangle|n\rangle\langle n|), a form previously used in proofs of the adiabatic theorem and in an equivalent 'counterdiabatic field' construction in the chemical-physics literature (Demirplak and Rice, 2003). In the modern geometric formulation H^1\hat{H}_1 is the adiabatic gauge potential, the generator of parallel transport of the instantaneous basis. Berry illustrated the theory with spins in changing magnetic fields, where the modified field B(t)=B0(t)+γ−1b0(t)×∂tb0(t)\mathbf{B}(t)=\mathbf{B}_0(t)+\gamma^{-1}\mathbf{b}_0(t)\times\partial_t\mathbf{b}_0(t) (with b0\mathbf{b}_0 the unit field direction and γ\gamma the gyromagnetic ratio) prevents classical spin precession exactly.

Berry's construction runs in the 'reverse engineering' direction: the reference Hamiltonian H^0(t)\hat{H}_0(t) is given, and the transitionless H^(t)\hat{H}(t) is built from it. Section 6 of the same paper poses the opposite, 'direct' direction: given a physically specified H^(t)\hat{H}(t), one usually wants the evolution it drives, which is equivalent to finding the H^0(t)\hat{H}_0(t) whose instantaneous eigenstates are the exact evolving states of H^\hat{H}. Determining H^0\hat{H}_0 amounts to inverting the relation above, and one natural approach is the iteration of the self-consistent equation H^0(t)=H^(t)−iℏ∑m≠n∣m⟩⟨m∣ ∂tH^0 ∣n⟩⟨n∣/(En−Em)\hat{H}_0(t)=\hat{H}(t)-\mathrm{i}\hbar\sum_{m\neq n}|m\rangle\langle m|\,\partial_t\hat{H}_0\,|n\rangle\langle n|/(E_n-E_m), which generates a version of the familiar adiabatic (superadiabatic) series, each step introducing higher time derivatives of H^\hat{H}. This is the same iteration studied in the superadiabatic-tracking literature (iterating the adiabatic basis so that residual transitions are suppressed order by order; Lim and Berry, 1991), and it diverges: the coupling terms decrease only up to an optimal order and then grow, as is standard for asymptotic series. Berry demonstrates the inversion exactly only in the two-state LMZ case, by solving the Schrodinger equation in parabolic cylinder functions (the standard special functions solving the two-level linear-sweep Schrodinger equation); the construction is highly non-unique (a two-parameter family of solutions), and he points out, without pursuing, connections between the two-state reflectionless-like structure and Ermakov-Lewis invariants (dynamical invariants of time-dependent Hamiltonians, in the sense of Lewis and Riesenfeld).

The subsequent literature on shortcuts to adiabaticity is large and must be read carefully, because its central terminology points in the opposite direction. 'Inverse engineering' in the sense of the Muga-Chen school means designing a Hamiltonian (or a Lewis-Riesenfeld invariant) that meets prescribed boundary conditions, not inverting a given one. The forward direction (H^0\hat{H}_0 to H^\hat{H}, or approximations to it) has been developed through superadiabatic iterative interaction pictures (Ibáñez, Chen and Muga, 2013), variational and local counterdiabatic driving (Sels and Polkovnikov, 2017; Kolodrubetz et al., 2017; Čepaitė et al., 2023), and, recently, non-variational nested-commutator expansions that converge with guarantees in the expansion order (van Vreumingen, 2024; Finžgar et al., 2025). Chen, Torrontegui and Muga (2011) showed that Berry's transitionless tracking and invariant-based inverse engineering are strongly related and potentially equivalent, and Okuyama and Takahashi (2016) obtained counterdiabatic terms exactly from the Lax pairs of classical integrable systems (pairs of matrices whose time-independent spectral data encode the conserved quantities of an integrable evolution), partially realizing the integrability connection Berry left open. For periodically driven systems, Schindler and Bukov (2025) showed that the lab-frame periodic Hamiltonian itself generates transitionless counterdiabatic driving for its Floquet eigenstates (the eigenstates of the one-period evolution operator that organize the dynamics of periodically driven systems), with the parallel-transport gauge (the basis choice in which the accumulated adiabatic phase is removed from the states) fixing the corresponding base Hamiltonian. The authoritative field review (Guéry-Odelin et al., 2019) documents the divergence of the superadiabatic iterations and the whole forward toolbox, but contains no treatment of the inversion of H^\hat{H} into H^0\hat{H}_0 for general, non-periodic Hamiltonians; the exact inversion in general remains, as in Berry's paper, equivalent to solving the time-dependent Schrodinger equation, which is precisely what makes a systematic approximate reconstruction theory the open scientific content.

Problem Statement

Berry's transitionless quantum driving gives an exact solution of the reverse-engineering problem: for a reference Hamiltonian H^0(t)\hat{H}_0(t) with instantaneous eigenstates ∣n(t)⟩|n(t)\rangle and energies En(t)E_n(t), the Hamiltonian

H^(t)=H^0(t)+iℏ∑m≠n∣m(t)⟩⟨m(t)∣ ∂tH^0(t) ∣n(t)⟩⟨n(t)∣En(t)−Em(t)\hat{H}(t)=\hat{H}_0(t)+\mathrm{i}\hbar\sum_{m\neq n}\frac{|m(t)\rangle\langle m(t)|\,\partial_t\hat{H}_0(t)\,|n(t)\rangle\langle n(t)|}{E_n(t)-E_m(t)}

drives the instantaneous eigenstates of H^0(t)\hat{H}_0(t) as exact evolving states, with no transitions between them, for any driving speed. The open problem is the inversion of this relation.

Given a physically specified time-dependent Hamiltonian H^(t)\hat{H}(t) - with no restriction to two-state, integrable, or periodically driven systems beyond a finite-dimensional Hilbert space with a discrete, non-degenerate instantaneous spectrum - determine the base Hamiltonian H^0(t)\hat{H}_0(t) whose instantaneous eigenstates are the exact evolving states of H^(t)\hat{H}(t); that is, characterize and solve the self-consistent (fixed-point) equation

H^(t)=H^0(t)+iℏ∑m≠n∣m(t)⟩⟨m(t)∣ ∂tH^0(t) ∣n(t)⟩⟨n(t)∣En(t)−Em(t)\hat{H}(t)=\hat{H}_0(t)+\mathrm{i}\hbar\sum_{m\neq n}\frac{|m(t)\rangle\langle m(t)|\,\partial_t\hat{H}_0(t)\,|n(t)\rangle\langle n(t)|}{E_n(t)-E_m(t)}

where ∣n(t)⟩|n(t)\rangle and En(t)E_n(t) are now the instantaneous eigenstates and energies of the unknown H^0(t)\hat{H}_0(t). The scientific targets are: (i) the existence and multiplicity structure of the solution set, including which gauge freedoms generate the non-uniqueness (Berry's two-state construction already exhibits infinitely many H^0\hat{H}_0 for one H^\hat{H}) and which solutions preserve physically relevant structure such as locality or the operator class of H^\hat{H}; (ii) a systematic constructive or approximate inversion scheme that is convergent, or optimally truncated with controlled error, replacing the adiabatic iteration of the equation H^0(t)=H^(t)−iℏ∑m≠n∣m⟩⟨m∣∂tH^0∣n⟩⟨n∣/(En−Em)\hat{H}_0(t)=\hat{H}(t)-\mathrm{i}\hbar\sum_{m\neq n}|m\rangle\langle m|\partial_t\hat{H}_0|n\rangle\langle n|/(E_n-E_m), whose generated adiabatic series is known to diverge, together with an analysis of what this divergence implies for practical reconstruction of H^0\hat{H}_0 from H^\hat{H}; and (iii) explicit demonstration of the inversion for nontrivial classes beyond the two-state Landau-Majorana-Zener case solved by Berry and beyond the periodically driven (Floquet) class - Hamiltonians repeating with a fixed period, whose evolving states are organized by the eigenstates of the one-period evolution operator - for which a gauge-fixed decomposition is already available. A general scheme satisfying (ii) also establishes the existence part of (i) and enables (iii). The companion questions raised in the same source passage - the connection of two-state transitionlessness to Ermakov-Lewis invariants and integrability, and the physical implementability, robustness, and energetic cost of transitionless control fields - have been substantially developed in the later literature, and are not part of the open inversion problem posed here.

References

  1. M. V. Berry, 'Transitionless quantum driving', J. Phys. A: Math. Theor. 42, 365303 (2009). DOI 10.1088/1751-8113/42/36/365303, https://doi.org/10.1088/1751-8113/42/36/365303
  2. M. Demirplak, S. A. Rice, 'Adiabatic Population Transfer with Control Fields', J. Phys. Chem. A 107, 9937 (2003). DOI 10.1021/jp030708a, https://doi.org/10.1021/jp030708a
  3. D. A. Garanin, R. Schilling, 'Inverse problem for the Landau-Zener effect', Europhys. Lett. 59, 7 (2002). DOI 10.1209/epl/i2002-00152-9, https://doi.org/10.1209/epl/i2002-00152-9
  4. R. Lim, M. V. Berry, 'Superadiabatic tracking of quantum evolution', J. Phys. A: Math. Gen. 24, 3255 (1991). DOI 10.1088/0305-4470/24/14/014, https://doi.org/10.1088/0305-4470/24/14/014
  5. Xi Chen, E. Torrontegui, J. G. Muga, 'Lewis-Riesenfeld invariants and transitionless quantum driving', Phys. Rev. A 83, 062116 (2011). DOI 10.1103/PhysRevA.83.062116, https://doi.org/10.1103/PhysRevA.83.062116
  6. Kazutaka Takahashi, 'Transitionless quantum driving for spin systems', Phys. Rev. E 87, 062117 (2013). DOI 10.1103/PhysRevE.87.062117, https://doi.org/10.1103/PhysRevE.87.062117 (arXiv:1209.3153)
  7. S. Ibáñez, Xi Chen, J. G. Muga, 'Improving shortcuts to adiabaticity by iterative interaction pictures', Phys. Rev. A 87, 043402 (2013). DOI 10.1103/PhysRevA.87.043402, https://doi.org/10.1103/PhysRevA.87.043402
  8. Manaka Okuyama, Kazutaka Takahashi, 'From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity', Phys. Rev. Lett. 117, 070401 (2016). DOI 10.1103/PhysRevLett.117.070401, https://doi.org/10.1103/PhysRevLett.117.070401 (arXiv:1603.01053)
  9. Dries Sels, Anatoli Polkovnikov, 'Minimizing irreversible losses in quantum systems by local counterdiabatic driving', Proc. Natl. Acad. Sci. USA 114 (20) (2017). DOI 10.1073/pnas.1619826114, https://doi.org/10.1073/pnas.1619826114
  10. Michael Kolodrubetz, Dries Sels, Pankaj Mehta, Anatoli Polkovnikov, 'Geometry and non-adiabatic response in quantum and classical systems', Phys. Rep. 697, 1 (2017). DOI 10.1016/j.physrep.2017.07.001, https://doi.org/10.1016/j.physrep.2017.07.001
  11. D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, J. G. Muga, 'Shortcuts to adiabaticity: concepts, methods, and applications', Rev. Mod. Phys. 91, 045001 (2019). DOI 10.1103/RevModPhys.91.045001, https://doi.org/10.1103/RevModPhys.91.045001 (arXiv:1904.08448)
  12. Alan C. Santos, 'Adiabatic Dynamics and Shortcuts to Adiabaticity: Fundamentals and Applications' (2021). arXiv:2107.11798, https://arxiv.org/abs/2107.11798
  13. I. Čepaitė, A. Polkovnikov, A. J. Daley, C. W. Duncan, 'Counterdiabatic Optimized Local Driving', PRX Quantum 4, 010312 (2023). DOI 10.1103/PRXQuantum.4.010312, https://doi.org/10.1103/PRXQuantum.4.010312
  14. Dyon van Vreumingen, 'Gate-based counterdiabatic driving with complexity guarantees', Phys. Rev. A 110, 052419 (2024). DOI 10.1103/PhysRevA.110.052419, https://doi.org/10.1103/PhysRevA.110.052419
  15. Jernej Rudi Finžgar, Simone Notarnicola, Madelyn Cain, Mikhail D. Lukin, Dries Sels, 'Counterdiabatic Driving with Performance Guarantees', Phys. Rev. Lett. 135, 180602 (2025). DOI 10.1103/pqhl-nbtk, https://doi.org/10.1103/pqhl-nbtk (arXiv:2503.01958, https://arxiv.org/abs/2503.01958)
  16. Paul M. Schindler, Marin Bukov, 'Geometric Floquet Theory', Phys. Rev. X 15, 031037 (2025). DOI 10.1103/7l91-gw77, https://doi.org/10.1103/7l91-gw77 (arXiv:2410.07029, https://arxiv.org/abs/2410.07029)