Sparsity dependence of Krylov state complexity in the SYK model

We study the Krylov state complexity of the Sachdev-Ye-Kitaev (SYK) model for $N \le 28$ Majorana fermions with $q$-body fermion interaction with $q=4,6,8$ for a range of sparse parameter $k$. Using the peak of the Krylov complexity as a probe, we find change in behavior as we vary $k$ for various $q$. We argue that this captures the change from holographic to non-holographic behavior in the sparse SYK-type models such that model is holographic for all $k \ge k_{\text{min}}$.