A new approximation for solving the time-dependent Schrödinger equation is proposed. It improves the time-dependent Hartree approximation by including time-dependent unitary operators acting on the Hartree product. This allows for the approximate description of the correlation between the various degrees of freedom. The mathematical structure of the new approach is analyzed and an alternative Lie algebraic derivation is presented. By adopting two different time-dependent variational principles two different sets of equations are obtained. Differences between the two resulting methods are discussed.A new approximation for solving the time-dependent Schrödinger equation is proposed. It improves the time-dependent Hartree approximation by including time-dependent unitary operators acting on the Hartree product. This allows for the approximate description of the correlation between the various degrees of freedom. The mathematical structure of the new approach is analyzed and an alternative Lie algebraic derivation is presented. By adopting two different time-dependent variational principles two different sets of equations are obtained. Differences between the two resulting methods are discussed.

Journal of Mathematical PhysicsJun 1, 198843 citations

Generalization in Data Re-uploading Quantum Neural Networks: A Dynamical Lie Algebraic and Harmonic Analysis Perspective

Ayad Alhusseiny

Commutation failure controls quantum generalization. When data encoding gates fail to commute with trainable parameter gates across consecutive re-uploading layers, the effective dynamical Lie algebra becomes data-dependent; this dependency invalidates the parameter-isolation assumption underpinning existing Lie-algebraic frameworks (Caro et al., 2022; Ohno, 2025). We resolve this structural gap by proving that the statistical generalization capacity of L-layer data re-uploading circuits is tightly bounded by the effective hypothesis dimension: d_eff(L) = min(L * p_layer, |Ω_L|) <= min(L * p_layer, (2L + 1)^d), where d is the input data dimension, p_layer the number of trainable parameters per layer, and |Ω_L| the cardinality of accessible Fourier frequencies generated through non-commuting Lie brackets in the joint dynamical Lie algebra g_joint = Lie_R{i H_j, i G_k}. Harmonic analysis combined with metric entropy chaining establishes matching upper and lower bounds; together, they settle the exact minimax estimation rate R*(F_L, M) = \tilde{\Theta}(\sqrt{d_eff(L)/M}). Two distinct orbit concepts govern learning. The point-wise adjoint orbit dimension dim(O_O) <= dim(g_joint) at fixed x quantifies single-point state variation, whereas the global function-space Fourier capacity |Ω_L| governs sample complexity across the full data domain; conflating these quantities yields vacuous bounds. An initial basis rotation unlocks harmonic modes up to k_max = L; without this rotation, eigenstate initialization constrains the cutoff to k_max = L - 1. Bandwidth dictates depth. For target functions with harmonic bandlimit k_target, the network underfits when L < k_target and achieves full expressivity convergence once L >= k_target. Permutation-equivariant data encoding compresses the joint DLA to an su(2) subalgebra of dimension 3. This algebraic confinement suppresses the 4^n - 1 dimensional explosion across arbitrary qubit counts and shields the circuit from the curse of dimensionality. Numerical benchmarks in PennyLane across L in {1, 2, 4, 8} and sample sizes M in [10, 200] corroborate this harmonic-algebraic scaling and its associated uniform convergence rates.Commutation failure controls quantum generalization. When data encoding gates fail to commute with trainable parameter gates across consecutive re-uploading layers, the effective dynamical Lie algebra becomes data-dependent; this dependency invalidates the parameter-isolation assumption underpinning existing Lie-algebraic frameworks (Caro et al., 2022; Ohno, 2025). We resolve this structural gap by proving that the statistical generalization capacity of L-layer data re-uploading circuits is tightly bounded by the effective hypothesis dimension: d_eff(L) = min(L * p_layer, |Ω_L|) <= min(L * p_layer, (2L + 1)^d), where d is the input data dimension, p_layer the number of trainable parameters per layer, and |Ω_L| the cardinality of accessible Fourier frequencies generated through non-commuting Lie brackets in the joint dynamical Lie algebra g_joint = Lie_R{i H_j, i G_k}. Harmonic analysis combined with metric entropy chaining establishes matching upper and lower bounds; together, they settle the exact minimax estimation rate R*(F_L, M) = \tilde{\Theta}(\sqrt{d_eff(L)/M}). Two distinct orbit concepts govern learning. The point-wise adjoint orbit dimension dim(O_O) <= dim(g_joint) at fixed x quantifies single-point state variation, whereas the global function-space Fourier capacity |Ω_L| governs sample complexity across the full data domain; conflating these quantities yields vacuous bounds. An initial basis rotation unlocks harmonic modes up to k_max = L; without this rotation, eigenstate initialization constrains the cutoff to k_max = L - 1. Bandwidth dictates depth. For target functions with harmonic bandlimit k_target, the network underfits when L < k_target and achieves full expressivity convergence once L >= k_target. Permutation-equivariant data encoding compresses the joint DLA to an su(2) subalgebra of dimension 3. This algebraic confinement suppresses the 4^n - 1 dimensional explosion across arbitrary qubit counts and shields the circuit from the curse of dimensionality. Numerical benchmarks in PennyLane across L in {1, 2, 4, 8} and sample sizes M in [10, 200] corroborate this harmonic-algebraic scaling and its associated uniform convergence rates.

Oct 21, 20260 citations

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

András Grabarits, E. Medina-Guerra, Adolfo del Campo

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.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.

quant-pharXiv:2605.05290May 8, 20260 citations

Time-dependent Hartree-Fock Theory in a two-dimensional rotating frame

Hans Feldmeier

Oct 1, 19770 citations

\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.