  Recently, Krylov complexity was proposed as a measure of complexity and
chaoticity of quantum systems. We consider the stadium billiard as a typical
example of the quantum mechanical system obtained by quantizing a classically
chaotic system, and numerically evaluate Krylov complexity for operators and
states. Despite no exponential growth of the Krylov complexity, we find a clear
correlation between variances of Lanczos coefficients and classical Lyapunov
exponents, and also a correlation with the statistical distribution of adjacent
spacings of the quantum energy levels. This shows that the variances of Lanczos
coefficients can be a measure of quantum chaos. The universality of the result
is supported by our similar analysis of Sinai billiards. Our work provides a
firm bridge between Krylov complexity and classical/quantum chaos.
