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arXiv:2411.18529 · Journal of Physics A Mathematical and Theoretical

Robustness of quantum symmetries against perturbations

Paolo Facchi A, Marilena Ligabò, Vito Viesti

Published Mar 12, 2025• 2 citations
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Abstract

Abstract We investigate quantum symmetries in terms of their large-time stability with respect to perturbations of the Hamiltonian. We find a complete algebraic characterization of the set of symmetries robust against a single perturbation and we use such result to characterize their stability with respect to arbitrary sets of perturbations.

DOI
10.1088/1751-8121/adbfe5

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Abstract We investigate quantum symmetries in terms of their large-time stability with respect to perturbations of the Hamiltonian. We find a complete algebraic characterization of the set of symmetries robust against a single perturbation and we use such result to characterize their stability with respect to arbitrary sets of perturbations.
We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics. To prove this result, we introduce a resummation of a perturbation series, which generalizes the Hamiltonian of the quantum Zeno dynamics.
This paper considers the problem of robust stability for a class of uncertain quantum systems subject to unknown perturbations in the system Hamiltonian. Some general stability results are given for different classes of perturbations to the system Hamiltonian. Then, the special case of a nominal linear quantum system is considered with either quadratic or non-quadratic perturbations to the system Hamiltonian. In this case, robust stability conditions are given in terms of strict bounded real conditions.
Abstract Symmetries play important roles in physical systems. We study the symmetries of a Hamiltonian system by investigating the asymmetry of the Hamiltonian with respect to certain algebras. We define the asymmetry of an operator with respect to an algebraic basis in terms of their commutators. Detailed analysis is given to the Lie algebra and its q -deformation. The asymmetry of the q -deformed integrable spin chain models is calculated. The corresponding geometrical pictures with respect to such asymmetry is presented.
We develop a mathematical framework for quantum time transfer based on commuting families of Hamiltonians and synchronization observables. The synchronization subspace is defined as the kernel of a difference operator between local clocks, and we show that this subspace is preserved exactly by a commutative ∗*∗-subalgebra of Hamiltonians compatible with the clocks. Our first main result establishes \emph{perturbative stability}: for ϵ\epsilonϵ-compatible dynamics, where the commutator with the synchronization operator is bounded in norm by ϵ\epsilonϵ, we prove quantitative drift bounds showing that timing correlations degrade at most linearly in time with slope proportional to ϵ\epsilonϵ. Our second main result provides a \emph{representation-theoretic classification}: in the presence of a finite group symmetry, the synchronization subspace coincides with the diagonal isotypic component in the tensor product decomposition, and synchronization preservation is characterized by the commutant algebra of the group action. These results identify synchronization as a structural invariant of operator algebras, connecting approximate commutation, kernel-preserving dynamics, and symmetry protection. Beyond quantum time transfer, the framework suggests categorical and resource-theoretic generalizations and contributes to the broader study of operator-algebraic invariants in multipartite quantum dynamics.
For given control channels, quantum control theory determines whether quantum gates can be realized, but not their resilience to errors. We address this problem by establishing existence criteria for robust gate implementation in ddd-level systems. Our criteria reveal that implementability of all gates is not enough: a universal family of robust gates exists if and only if the driven Hamiltonians share no common symmetry. When these criteria hold, we can make gates resilient to any combination of static perturbations, uniform Hamiltonian rescaling, perturbations at a finite set of selected frequencies, and perturbations with arbitrary selected waveforms. We can impose robustness conditions broadly with limited information about errors, e.g., against any static perturbations; we can also tailor these conditions to the platform, e.g., against environment-induced noise with dominant spectral peaks, requiring no spectral details beyond the peak frequencies. We numerically construct explicit protocols for robust gates, reducing average gate infidelity by approximately two to seven orders of magnitude under Hamiltonian perturbations and environment-induced noise. Together, our existence criteria and numerical constructions provide practical guidance for evaluating the potential of quantum platforms and designing robust quantum gates.