  We consider the community detection problem in a sparse $q$-uniform
hypergraph $G$, assuming that $G$ is generated according to the Hypergraph
Stochastic Block Model (HSBM). We prove that a spectral method based on the
non-backtracking operator for hypergraphs works with high probability down to
the generalized Kesten-Stigum detection threshold conjectured by Angelini et
al. (2015). We characterize the spectrum of the non-backtracking operator for
the sparse HSBM and provide an efficient dimension reduction procedure using
the Ihara-Bass formula for hypergraphs. As a result, community detection for
the sparse HSBM on $n$ vertices can be reduced to an eigenvector problem of a
$2n\times 2n$ non-normal matrix constructed from the adjacency matrix and the
degree matrix of the hypergraph. To the best of our knowledge, this is the
first provable and efficient spectral algorithm that achieves the conjectured
threshold for HSBMs with $r$ blocks generated according to a general symmetric
probability tensor.
