Krylov Dynamics and Operator Growth in Time-Dependent Systems via Lie Algebras

• URL: http://arxiv.org/abs/2605.05290v1
• Date: Wed, 06 May 2026 18:00:00 GMT
• Title: Krylov Dynamics and Operator Growth in Time-Dependent Systems via Lie Algebras
• Abstract summary: We study quantum dynamics generated by time-dependent Hamiltonians in Krylov space, the minimal subspace in which the evolution takes place.<n>We develop a general framework in which the dynamics in the time-dependent Krylov subspace is generated by ladder operators of the associated Lie algebra.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study quantum dynamics generated by time-dependent Hamiltonians in Krylov space, the minimal subspace in which the evolution takes place. We establish a direct link between dynamics in the time-dependent Krylov subspace and the underlying Lie-algebraic structure of the Hamiltonian. We develop a general framework in which the dynamics in the time-dependent Krylov subspace is generated by ladder operators of the associated Lie algebra. In particular, we identify the minimal conditions under which the exact time-dependent Krylov dynamics is naturally determined by the interaction-picture Hamiltonian and governed by an embedded $\mathfrak{sl}(2,\mathbb{C})$ subalgebra. We further show that an exact single-exponential representation of the time-evolution operator gives rise to a distinct time-independent Krylov dynamics in a unitarily related basis, from which the exact time-dependent Krylov dynamics can nevertheless be recovered. We also extend the framework to the oscillator algebra as the simplest extension of the nilpotent Heisenberg--Weyl algebra, and provide further examples, including the translated and dilated harmonic oscillator, systems governed by closed Virasoro subalgebras, a spin in a rotating magnetic field, and higher-dimensional generalizations for multi-level systems. In addition, we introduce a new quantum speed limit to the complexity growth rate generated by a time-dependent generator and show that, for evolutions governed by a Lie algebra, it retains the same functional form as in the time-independent case. Remarkably, saturation of this bound is strongly affected by temporal driving and persists only when the Hamiltonian commutes with itself at different times. These results establish a unified framework for characterizing operator growth and Krylov complexity in time-dependent quantum systems with underlying Lie-algebraic structures.

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