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On Minimizing Krylov Complexity Using Higher-Order Generators

Saud Čindrak, Kathy Lüdge

Abstract

Krylov complexity provides a powerful framework for characterizing the dynamical evolution of quantum systems through the spreading of states in Krylov space. The motivation for this is rooted in the optimality of the Krylov basis for the analyzed cost function. In this work, we reinterpret the motivation for the Krylov basis from a dynamical perspective and show that it corresponds to a first-order approximation of the time-evolution operator. We extend this framework to higher-order generators and analytically disprove the optimality assumption by showing that an infinite-order generator can be constructed to exhibit smaller spread for arbitrary times. We propose a natural time scale for the construction of these higher-order generators and discuss results for matrices sampled from Gaussian Unitary Ensembles, demonstrating smaller Krylov complexity at all higher orders. These results extend the framework of Krylov complexity beyond the conventional Krylov basis by disproving the widely held assumption of optimality, extending the construction to higher-order generators, and introducing a physically motivated method for their construction. Our findings therefore suggest that previous statements and results concerning Krylov complexity may need to be reconsidered.

On Minimizing Krylov Complexity Using Higher-Order Generators

Abstract

Krylov complexity provides a powerful framework for characterizing the dynamical evolution of quantum systems through the spreading of states in Krylov space. The motivation for this is rooted in the optimality of the Krylov basis for the analyzed cost function. In this work, we reinterpret the motivation for the Krylov basis from a dynamical perspective and show that it corresponds to a first-order approximation of the time-evolution operator. We extend this framework to higher-order generators and analytically disprove the optimality assumption by showing that an infinite-order generator can be constructed to exhibit smaller spread for arbitrary times. We propose a natural time scale for the construction of these higher-order generators and discuss results for matrices sampled from Gaussian Unitary Ensembles, demonstrating smaller Krylov complexity at all higher orders. These results extend the framework of Krylov complexity beyond the conventional Krylov basis by disproving the widely held assumption of optimality, extending the construction to higher-order generators, and introducing a physically motivated method for their construction. Our findings therefore suggest that previous statements and results concerning Krylov complexity may need to be reconsidered.
Paper Structure (1 theorem, 23 equations, 2 figures)

This paper contains 1 theorem, 23 equations, 2 figures.

Key Result

Theorem 1

Let $H \in \mathbb{C}^{N\times N}$ be a Hamiltonian with Krylov grade $m=3$. Consider the orthonormal basis $\mathcal{B}_K = \{\ket{k_0}, \ket{k_1}, \ket{k_2}\}$ obtained by orthonormalizing the vectors $\{\ket{\psi_0},\, H\ket{\psi_0},\, H^2\ket{\psi_0}\}$, and fix a time $\tau$. The infinite-order

Figures (2)

  • Figure 1: Spread complexity for a $3\times 3$ GUE Hamiltonian using higher-order Krylov generators. a)$\mathcal{C}^{(p, \Delta t)}(t)$ for orders $p=1,2,3,\infty$. b) Differences $\Delta\mathcal{C}^{(p, \Delta t)}(t)$ with a zoomed inset. Krylov amplitudes $|\kappa^{(p)}_{1}(t)|^{2}$ and $|\kappa^{(p)}_{2}(t)|^{2}$ in c) and d).
  • Figure 2: Spread complexity $\mathcal{C}^{(p)}(t)$ (first column), its deviation from first order $\Delta \mathcal{C}^{(p)}(t)$ (second column), and the Hessian (third column) for generators of order $p=1$ (blue), $p=2$ (dashed light blue), $p=3$ (dashed orange), and $p=\infty$ (red) for a $50\times50$ GUE Hamiltonian. Rows correspond to $\Delta t = 0.2\cdot\Delta t_{\mathrm{scr}}$ (a–c), $\Delta t = \Delta t_{\mathrm{scr}}$ (d–f), and $\Delta t = 1.5\cdot\Delta t_{\mathrm{scr}}$ (g–i). The grey region marks $t>\tau_H$, where $\mathcal{C}^{(p)}(t)$ approximately saturates.

Theorems & Definitions (2)

  • Theorem 1
  • proof