  The saturation of a recently proposed universal bound on the Lyapunov
exponent has been conjectured to signal the existence of a gravity dual. This
saturation occurs in the low temperature limit of the dense Sachdev-Ye-Kitaev
(SYK) model, $N$ Majorana fermions with $q$-body ($q>2$) infinite-range
interactions. We calculate certain Out of Time Order Correlators (OTOC) for
$N\le 64$ fermions for a highly sparse SYK model and find no significant
dependence of the Lyapunov exponent on sparsity up to near the percolation
limit where the Hamiltonian breaks up into blocks. This suggests that in the
sparse case, the Lyapunov exponent also saturates the low-temperature bound. A
key ingredient to reaching $N = 64$ is the development of a novel quantum spin
model simulation library that implements highly-optimized matrix-free Krylov
subspace methods on Graphical Processing Units (GPUs). This leads to a
significantly lower simulation time as well as vastly reduced memory usage over
previous approaches, while using modest computational resources. Strong
sparsity-driven statistical fluctuations require both the use of a vastly
larger number of disorder realizations with respect to the dense limit and a
careful finite size scaling analysis. Our results potentially broadens the
landscape of theories that may have a gravity analogue.
