  Building upon recent research in spin systems with non-local interactions,
this study investigates operator growth using the Krylov complexity in
different non-local versions of the Ising model. We find that the non-locality
results in a faster scrambling of the operator to all sites. While the
saturation value of Krylov complexity of local integrable and local chaotic
theories differ by a significant margin, this difference is much suppressed
when non-local terms are introduced in both regimes. This results from the
faster scrambling of information in the presence of non-locality. In addition,
we investigate the behavior of level statistics and spectral form factor as
probes of quantum chaos to study the integrability breaking due to non-local
interactions. Our numerics indicate that in the non-local case, late time
saturation of Krylov complexity distinguishes between different underlying
theories, while the early time complexity growth distinguishes different
degrees of non-locality.
