Quantum Dynamics in Krylov Space: Methods and Applications

TL;DR

This paper surveys Krylov subspace methods as a powerful framework for describing quantum dynamics, including operator growth, Krylov complexity, and their connections to chaos, scrambling, and quantum technologies. It develops two Lanczos formulations, analytic approaches (moment method and Toda chain) to compute Lanczos coefficients, and a universal operator growth hypothesis, with detailed studies in large-q SYK and related models. The work extends to finite-temperature, density matrices, and open quantum systems via Arnoldi and bi-Lanczos methods, and explores implications for quantum field theory, holography, integrability, and quantum control. Together, these developments provide a cohesive toolkit for diagnosing chaotic dynamics, bounding complexity growth, and informing quantum technologies and holographic interpretations. The framework offers both deep theoretical insights and practical computational strategies for nonequilibrium quantum many-body systems across closed and open settings.

Abstract

The dynamics of quantum systems unfolds within a subspace of the state space or operator space, known as the Krylov space. This review presents the use of Krylov subspace methods to provide an efficient description of quantum evolution and quantum chaos, with emphasis on nonequilibrium phenomena of many-body systems with a large Hilbert space. It provides a comprehensive update of recent developments, focused on the quantum evolution of operators in the Heisenberg picture as well as pure and mixed states. It further explores the notion of Krylov complexity and associated metrics as tools for quantifying operator growth, their bounds by generalized quantum speed limits, the universal operator growth hypothesis, and its relation to quantum chaos, scrambling, and generalized coherent states. A comparison of several generalizations of the Krylov construction for open quantum systems is presented. A closing discussion addresses the application of Krylov subspace methods in quantum field theory, holography, integrability, quantum control, and quantum computing, as well as current open problems.

Figures (27)

• Figure 1: Example of the dip-ramp-plateau structure of the SFF for characterizing quantum chaotic systems. The Hamiltonian average is taken over $1000$$\mathrm{GUE}$ matrices with $\sigma=1$, $d=128$, setting $\hbar=1$ and $\beta=0$.
• Figure 2: Infinite-temperature OTOC in the Lipkin-Meshkov-Glick (LMG) model with Hamiltonian $H = \hat{x} + 2 \hat{z}^2$, see Sec. \ref{['sec:linearGrowth_chaos?']} for further details. The quantity shows a period of exponential growth between the dissipation $t_d$ and Ehrenfest $t_E$ times. The system shows saddle-dominated scrambling with the quantum Lyapunov exponent given by $\lambda_{\rm OTOC}=\sqrt{3}$. The operators are chosen as $V(0)=W(0)=\hat{z}$, the spin is $s = 100$ and the effective Planck's constant is $\hbar_{\rm eff} = 1/S$. The propagator $U(t)$ is constructed from the rescaled Hamiltonian $H/\hbar_{\rm eff}$. The figure is adapted from Xu:2019lhc with different system parameters.
• Figure 3: The growth of an initial operator $\mathcal{O}_0 \equiv \mathcal{O}$ (left) is mapped to a single particle hopping problem in a one-dimensional Krylov chain (right). Here, $b_{n+1}$ and $b_n$ denote the hopping rates from the $n$-th site to the $(n+1)$-th and $(n-1)$-th sites, respectively. Adapted from parker2019.
• Figure 4: Schematic diagrams of the generic behavior of Lanczos coefficients (left), the associated Krylov complexity (middle), and Krylov entropy (right). The region of Lanczos ascent, Lanczos plateau, Lanczos descent, and the relevant timescales for Krylov complexity and Krylov entropy are shown. Here, $S \sim O(\log N)$ is the thermodynamic entropy of the system, directly proportional to the degrees of freedom $N$. The plots are schematic and do not have a proper numerical scale. For example, the entropy plots are zoomed in, hence the linear region seems longer compared to the Krylov complexity. Also, the difference between the plateau and the descent regime of the Lanczos coefficients is often subtle and may not be separated clearly. The generic structure of these figures is adapted from barbon2019Kar2022Bhattacharjee:2022vltRabinovici:2022beuRabinovici:2020ryf.
• Figure 5: (Left) Behavior of the Lanczos coefficients $\overline{b}_n = b_n s$ in the LMG model for the initial operator $\hat{z}$ for spins $s = 25$, $50$, and $75$ respectively. The black dashed line indicates the linear growth with coefficient $\alpha \simeq \sqrt{3}/2$. (Right) The entire Lanczos spectrum for $s = 25$. The Krylov dimension $D_K \sim 1250$ is much smaller than the maximum dimension $d^2 - d + 1 = 2551$, where the Hilbert space dimension is $d = 2s+1 = 51$. Adapted from Bhattacharjee:2022vlt.