Operator complexity: a journey to the edge of Krylov space

TL;DR

The paper tackles how operators grow under Heisenberg evolution in finite-entropy quantum systems by analyzing Krylov space and Krylov complexity (K-complexity). It derives rigorous bounds on the Krylov dimension $K$ and the Lanczos sequence, and validates them with large-scale numerical simulations of chaotic SYK$_4$ and integrable SYK$_2$ models using refined Lanczos-based methods. A key finding is the Descent—a nonperturbative, slow decay of the Lanczos sequence—that accompanies chaotic dynamics, with chaotic SYK$_4$ saturating the $K$-space bound $K\le D^2-D+1$ while integrable SYK$_2$ remains exponentially below it. The results support K-complexity and K-entropy as robust, time-resolved diagnostics of chaos versus integrability and hint at connections to holographic phenomena and late-time universal features of quantum chaos.

Abstract

Heisenberg time evolution under a chaotic many-body Hamiltonian $H$ transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or `K-complexity', quantifies this growth with respect to a special basis, generated by $H$ by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time $t_s>\log (S)$. We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK$_4$ model, which is maximally chaotic, and compare the results with the SYK$_2$ model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.

Figures (8)

• Figure 1: Lanczos-sequence for $L=10$. Left: "The Descent" depicted with linear scale along the horizontal axis. After initial growth up to $n\sim S$, a slow decrease to zero with roughly constant negative non-perturbative slope of order $\sim -\frac{1}{K} \sim - e^{-2S}$. Fitted slope (in red) of the decaying part is $-2.21\cdot 10^{-5}\approx-1.58\cdot10^{-5}=-K^{-1}$. On a 1:1 scale, the horizontal axis should be at least $843$ meters long. Right: Logarithmic scale along the horizontal axis makes visible the initial ascent.
• Figure 2: K-complexity averaged over 5 random realizations with $L=10$ sites at half filling. Left: Full time range computed. Saturation occurs at time scales of order $t_K\sim K \sim e^{2S}$ with value near $K/2=31626.5$ (this is the time scale at which the wave-packet $\varphi_n(t)$, which propagates at roughly constant -- but slowly decreasing -- velocity, reaches the edge of the Krylov chain). Right: Zoom in at early times. Note the initial non-linear growth transforming into linear growth, as verified by the linear fit (in red). Relevant time scales are indicated in the plots, although their exact location may depend on a dimensionful prefactor (the Lyapunov exponent, as discussed in Barbon:2019wsy).
• Figure 3: K-entropy averaged over 5 random realizations with $L=10$ sites at half filling. Left: Full time range computed, with logarithmic scale along horizontal axis. A linear fit in the post-scrambling regime, where K-entropy is expected to grow logarithmically, is depicted in red. K-entropy grows linearly up to scrambling time, and then transitions to a logarithmic growth phase that continues until saturation around $S_K\sim L\sim S$ at exponentially late times (this is the time scale at which the wave-packet $\varphi_n(t)$ becomes fully dispersed). Right: K-entropy at early times with linear scale along the horizontal axis. The exact location of the time scales may depend on a dimensionful prefactor (see caption of Figure \ref{['KC-L10-main']}).
• Figure 4: Lanczos sequences for $L=8,\, N=4$ and $L=9,\, N=5$ and comparison of results for $L=8,9,10$. Top row: Results for $L=8$ in linear (left panel) and logarithmic (right panel) scale along the horizontal axis. The plots depict both the sequence of a single random realization and the average over $311$ realizations. A linear fit is included in the decaying tail, whose slope approaches numerically the naïve estimate $\sim - \frac{1}{K}\approx - 0.000206$. Middle row: Results for $L=9$ in linear (left panel) and logarithmic (right panel) scale along the horizontal axis. The plots depict both the sequence of a single random realization and the average over $50$ realizations. A linear fit is included in the decaying tail, whose slope is of the order of the naive estimate $\sim-\frac{1}{K}\approx -6.3\cdot10^{-5}$. Bottom row: Comparison of the Lanczos sequences for $L=8,9,10$ in linear (left panel) and logarithmic (right panel) scale along the horizontal axis.
• Figure 5: Results for K-complexity for $L=8,\, N=4$ and $L=9,\, N=5$ and comparison of results for $L=8,9,10$. Top row: Results for $L=8$; for exponentially long times (left panel) and for early times (right panel). Note the change in behaviour from very early times (inset) and later times. The value at saturation is near $\sim\frac{K}{2}=2415.5$. Middle row: Results for $L=9$; for exponentially long times (left panel) and for early times (right panel). The value at saturation is near $\sim\frac{K}{2}=7875.5$. Bottom row: Comparison of results for $L=8,9,10$ for exponentially long times (left panel) and for early times (right panel).