  The Sachdev-Ye-Kitaev (SYK) model is a system of $N$ Majorana fermions with
random interactions and strongly chaotic dynamics, which at low energy admits a
holographically dual description as two-dimensional Jackiw-Teitelboim gravity.
Hence the SYK model provides a toy model of quantum gravity that might be
feasible to simulate with near-term quantum hardware. Motivated by the goal of
reducing the resources needed for such a simulation, we study a sparsified
version of the SYK model, in which interaction terms are deleted with
probability $1{-p}$. Specifically, we compute numerically the spectral form
factor (SFF, the Fourier transform of the Hamiltonian's eigenvalue pair
correlation function) and the nearest-neighbor eigenvalue gap ratio $r$
(characterizing the distribution of gaps between consecutive eigenvalues). We
find that when $p$ is greater than a transition value $p_1$, which scales as
$1/N^3$, both the SFF and $r$ match the values attained by the full
unsparsified model and with expectations from random matrix theory (RMT). But
for $p<p_1$, deviations from unsparsified SYK and RMT occur, indicating a
breakdown of holography in the highly sparsified regime. Below an even smaller
value $p_2$, which also scales as $1/N^3$, even the spacing of consecutive
eigenvalues differs from RMT values, signaling a complete breakdown of spectral
rigidity. Our results cast doubt on the holographic interpretation of very
highly sparsified SYK models obtained via machine learning using teleportation
infidelity as a loss function.
