  This paper investigates the problem of exact community recovery in the
symmetric $d$-uniform $(d \geq 2)$ hypergraph stochastic block model
($d$-HSBM). In this model, a $d$-uniform hypergraph with $n$ nodes is generated
by first partitioning the $n$ nodes into $K\geq 2$ equal-sized disjoint
communities and then generating hyperedges with a probability that depends on
the community memberships of $d$ nodes. Despite the non-convex and discrete
nature of the maximum likelihood estimation problem, we develop a simple yet
efficient iterative method, called the \emph{projected tensor power method}, to
tackle it. As long as the initialization satisfies a partial recovery condition
in the logarithmic degree regime of the problem, we show that our proposed
method can exactly recover the hidden community structure down to the
information-theoretic limit with high probability. Moreover, our proposed
method exhibits a competitive time complexity of $\mathcal{O}(n\log^2n/\log\log
n)$ when the aforementioned initialization condition is met. We also conduct
numerical experiments to validate our theoretical findings.
