  We study a sparse version of the Sachdev-Ye-Kitaev (SYK) model defined on
random hypergraphs constructed either by a random pruning procedure or by
randomly sampling regular hypergraphs. The resulting model has a new parameter,
$k$, defined as the ratio of the number of terms in the Hamiltonian to the
number of degrees of freedom, with the sparse limit corresponding to the
thermodynamic limit at fixed $k$. We argue that this sparse SYK model recovers
the interesting global physics of ordinary SYK even when $k$ is of order unity.
In particular, at low temperature the model exhibits a gravitational sector
which is maximally chaotic. Our argument proceeds by constructing a path
integral for the sparse model which reproduces the conventional SYK path
integral plus gapped fluctuations. The sparsity of the model permits larger
scale numerical calculations than previously possible, the results of which are
consistent with the path integral analysis. Additionally, we show that the
sparsity of the model considerably reduces the cost of quantum simulation
algorithms. This makes the sparse SYK model the most efficient currently known
route to simulate a holographic model of quantum gravity. We also define and
study a sparse supersymmetric SYK model, with similar conclusions to the
non-supersymmetric case. Looking forward, we argue that the class of models
considered here constitute an interesting and relatively unexplored sparse
frontier in quantum many-body physics.
