Geometry of Krylov Complexity

TL;DR

The paper builds a geometric framework for operator growth and Krylov complexity by exploiting generalized coherent states and information geometry. It shows that Liouvillian dynamics, when organized by symmetry, corresponds to displacements on coherent-state manifolds, turning operator growth into geodesic motion with Krylov complexity proportional to a volume in the associated geometry. The authors provide explicit realizations for SL(2,R), SU(2), Heisenberg-Weyl, and 2d CFTs, deriving simple, interpretable Lanczos coefficients and revealing how complexity bounds emerge from algebra closure. They also connect this picture to quantum information tools and to geometric notions of circuit complexity, hinting at observable connections in quantum optics and holography. The framework offers a unified language for chaos, integrability, and holographic ideas, while outlining future directions in Virasoro extensions and OPE-based growth.

Abstract

We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues towards computation of Lanczos coefficients and sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy and capacity of entanglement.

Figures (4)

• Figure 1: Cartoon of the operator growth and Krylov complexity for SL(2,R). Coherent states allow us to map the operator evolution to a geodesic (in orange) on the hyperbolic disc. Volume (in yellow) of the region enclosed by the particle's position $\rho=2\alpha t$ at $\phi=\pi/2$ is proportional to the Krylov complexity.
• Figure 2: Distribution of the SU(2) Lanczos coefficients. Sample plot for $j=20$.
• Figure 3: All the 11 wavefunctions $\varphi_n(t)$ for spin $j=5$ plotted between $\alpha t\in(0,\pi/2)$. Different wavefunctions are peaked at later values of $\alpha t$ symmetrically, reflecting the symmetry of $b_n$'s.
• Figure 4: Wavefunctions from the Weyl-Heisenberg coherent states as functions of $n$ for $\alpha t=1,2,4,6,8,10$, from left to right.