  The growth of simple operators is essential for the emergence of chaotic
dynamics and quantum thermalization. Recent studies have proposed different
measures, including the out-of-time-order correlator and Krylov complexity. It
is established that the out-of-time-order correlator serves as the signature of
quantum many-body chaos, while the Krylov complexity provides its upper bound.
However, there exist non-chaotic systems in which Krylov complexity grows
exponentially, indicating that the Krylov complexity itself is not a witness of
many-body chaos. In this letter, we introduce the missing ingredient, named as
the Krylov metric $K_{mn}$, which probes the size of the Krylov basis. We
propose that the universal criteria for fast scramblers include (i) the
exponential growth of Krylov complexity, (ii) the diagonal elements $K_{nn}\sim
n^h$ with $h\in(0,1]$, and (iii) the negligibility of off-diagonal elements
$K_{mn}$ with $m\neq n$. We further show that $h=\varkappa / 2\alpha$ is a
ratio between the quantum Lyapunov exponent $\varkappa$ and the Krylov exponent
$\alpha$. This proposal is supported by both generic arguments and explicit
examples, including solvable SYK models, Luttinger Liquids, and many-body
localized systems. Our results provide a refined understanding of how chaotic
dynamics emerge from the Krylov space perspective.
