A 2025 preprint by Beringer, Steinhuber, Richter, and Tomsovic argued that quantum chaos provides the resources for full quantum state controllability: in a chaotic system whose perturbed dynamics statistically mimics Wigner-Dyson random matrix ensembles, arbitrarily weak, finely tuned perturbations can steer any initial state to any target state (within the same symmetry sector) on times beyond a characteristic 'logtime', with the required perturbation strength vanishing as ℏ→0\hbar\to 0. The mechanism was demonstrated only for the single-degree-of-freedom quantum kicked rotor. The open question is whether this weak-perturbation controllability mechanism survives in many-body systems, where perturbations are few-body (body-rank restricted) operators and the relevant spectral statistics are those of embedded random-matrix ensembles rather than Wigner-Dyson ensembles, so that the ensemble isotropy underlying the argument is imperfect for an additional reason. Resolving it requires establishing the achievable isotropy and controllability under body-rank restrictions in many-body systems such as the Bose-Hubbard model or spin chains, where the quantities at issue are the achievable fidelities, the scaling of the required perturbation strength, and the control (scrambling) times.

Background

In classical mechanics, the extreme sensitivity to initial conditions of chaotic systems is not an obstacle to control but a resource: in the technique of targeting (Shinbrot, Ott, Grebogi, and Yorke, 1990), very weak perturbations direct a trajectory to a desired target state, with the required perturbation strength shrinking as the dynamics becomes more unstable. Quantum unitary evolution has no trajectories, no exponential divergence of neighboring states, and no ergodicity in the classical sense, so the classical argument does not transfer directly. Beringer, Steinhuber, Richter, and Tomsovic (2025) identified two quantum-chaotic properties that can take over these roles. The first is dynamical sensitivity to weak perturbations, as measured by Peres' fidelity decay: the overlap between the unperturbed and perturbed evolutions of a state decays at a rate set by the perturbation strength and the Lyapunov exponents, which fixes both the scale of the control parameters and the swiftness of the control time. The second is the statistical description of quantum chaotic spectra and eigenbases by random matrix theory: within the relevant Hilbert space, chaotic dynamics mimics the Wigner-Dyson ensembles, whose eigenbases are effectively uniformly (isotropically) distributed and uncorrelated with the spectrum. Isotropy here is meant operationally, in the sense of the source's characterisation of the Wigner-Dyson limit: within the relevant subspace all eigenbases are effectively equally probable (a Haar-like distribution of eigenvectors), the perturbed eigenbases are negligibly correlated with the spectrum, and every admissible nondegenerate spectrum carries nonzero weight; deviations from this ideal are what a body-rank restriction is expected to worsen. Together these properties imply that, beyond a characteristic time that grows only logarithmically with Hilbert-space dimension (the 'logtime'), finely tuned weak perturbations can steer any initial state to any target state lying in the same Hilbert space and respecting the same conserved symmetries, with the required perturbation strength ϵ∗\epsilon^{*} vanishing roughly linearly as ℏ→0\hbar\to 0 (here and below the effective Planck constant ℏ\hbar is the one fixed by the effective Hilbert-space dimension NN, as in the kicked-rotor demonstration where ℏ=1/(2πN)\hbar = 1/(2\pi N)). The authors demonstrated the mechanism on the quantum kicked rotor (a standard paradigm of quantum chaos: a rotor kicked periodically by a potential, whose classical limit is chaotic for strong kicking), producing revivals, cat-like states, and random-to-random state transfers with fidelities above 0.999 using perturbations below one percent of the kicking strength. Separately, the same group had earlier developed 'coherent quantum targeting', in which semiclassical control Hamiltonians guide wave packets along unstable classical trajectories, and extended it to Bose-Hubbard systems; that line controls dynamics along specially structured mean-field trajectories and is distinct from the arbitrary-state-to-arbitrary-state mechanism at issue here. Full controllability of many-body systems is also known by other means, namely Lie-algebraic conditions on local control Hamiltonians and optimal-control methods (Lloyd and Montangero, 2014), but those results say nothing about the chaos-based weak-perturbation mechanism or its resource scalings. The gap identified by the source paper is the following. In many-body systems, physically available perturbations are few-body operators, so the perturbed dynamics does not mimic a Wigner-Dyson ensemble but rather an embedded random-matrix ensemble: an ensemble of random few-body (body-rank-kk) interactions embedded in the many-particle Hilbert space, which has far fewer independent matrix elements and strong correlations between them, and which is known to describe the statistics of chaotic many-body Hamiltonians such as the Bose-Hubbard model (bosons hopping on a lattice with on-site repulsion) in its chaotic regime (Pausch et al., 2021, who also found departures from both the embedded ensemble and the Wigner-Dyson Gaussian orthogonal ensemble as Hilbert space grows). The perfect isotropy of Wigner-Dyson ensembles, which underlies the controllability argument and ties the control time to the Mandelstam-Tamm quantum speed limit (the lower bound on the time required to evolve between two states, set by the energy uncertainty of the evolving state), therefore fails in many-body systems for a second reason beyond dynamical continuity. Whether the weak-perturbation controllability mechanism, its logtime control scale, and the vanishing of ϵ∗\epsilon^{*} survive this loss of isotropy is unresolved.

Problem Statement

Determine whether the chaos-assisted weak-perturbation controllability mechanism established for the chaotic quantum kicked rotor — full state-to-state controllability by finely tuned, arbitrarily weak perturbations beyond a logarithmically growing control time, with required perturbation strength ϵ∗\epsilon^{*} vanishing as ℏ→0\hbar\to 0, resting on the near-perfect isotropy of Wigner-Dyson random-matrix statistics — extends to many-body quantum systems in which the physically available perturbations are few-body (body-rank-restricted) operators and the dynamical ensemble therefore has embedded random-matrix statistics with correlated matrix elements and imperfect isotropy. Concretely, for dynamical ensembles generated by weak body-rank-restricted perturbations of chaotic many-body Hamiltonians, the primary deliverable is the determination of whether full state controllability (within symmetry sectors imposed by conserved quantities) by arbitrarily weak perturbations, its logarithmically growing control or scrambling time, and the vanishing of the minimum perturbation strength ϵ∗\epsilon^{*} survive the loss of Wigner-Dyson isotropy; the Bose-Hubbard model and spin chains are illustrative examples of this many-body setting, not an exhaustive scope. The associated sub-questions — how the achievable isotropy, the control or scrambling time, and ϵ∗\epsilon^{*} scale with body rank, number of degrees of freedom, and effective ℏ\hbar, and whether the achievable fidelities and required perturbation strengths exhibit further subtleties, including deviations from the Wigner-Dyson-ensemble predictions of the single-degree-of-freedom theory — are the quantitative characterisation of that same determination rather than separate mandatory tasks: resolving any one of them settles that sub-question, while the primary deliverable is decided by the controllability determination itself. The objective is to establish whether quantum chaos remains an essential control resource once many-body body-rank restrictions break the random-matrix isotropy on which the mechanism rests.

References

  1. L. Beringer, M. Steinhuber, K. Richter, S. Tomsovic, "Quantum Chaos as an Essential Resource for Full Quantum State Controllability", arXiv:2512.13384 (2025), DOI 10.48550/arXiv.2512.13384, https://arxiv.org/abs/2512.13384 (source of the problem)
  2. L. Beringer, M. Steinhuber, J. D. Urbina, K. Richter, S. Tomsovic, "Controlling Many-Body Quantum Chaos: Bose-Hubbard systems", arXiv:2401.17744 (2024), DOI 10.48550/arXiv.2401.17744, https://arxiv.org/abs/2401.17744
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