Abstract: One of the frontiers in our understanding of quantum many-body physics is the study of geometrically non-local but sparsely interacting systems. For example, there are major open questions related to the existence of high performance quantum error correcting codes and the robustness of many-body entanglement to thermal noise. Another interesting question is whether the emergence of gravity in the sense of AdS/CFT requires all-to-all interactions. We use a probabilistic pruning procedure to define new sparsely interacting quantum Hamiltonians from all-to-all connected "mean-field-like" Hamiltonians. In the sparse limit, the number of terms in the Hamiltonian is proportional to the number of degrees of freedom with proportionality constant k. Applied to the Sachdev-Ye-Kitaev (SYK) model, we obtain a sparse SYK (s-SYK) model which in the large N limit admits a 1/k expansion with the fully connected solution as the leading approximation. Applying the procedure to a supersymmetric version of SYK yields a model in which the fully connected physics is recovered down to the lowest temperatures. Hence, we exhibit a sparse quantum Hamiltonian which has a sector dual to a supersymmetric gravity theory. The sparsity gives additional technical benefits: classical and quantum simulation of the model is easier. In the context of classical simulation, I will show numerical results for real time dynamics and thermodynamics for up to 52 majorana fermions. In the context of quantum simulation, the current best algorithm to simulate SYK has complexity cost per unit time proportional to N^7/2; the analogous cost in the sparse model is of order sqrt(k) N^2. Hence, if the goal is to perform the simplest possible simulation of a model with a partial gravity dual, the sparse model is preferable. I may also briefly comment on applications to error correcting codes, possible glassy physics, and sparse models without randomness.