  In this paper, we present a quantum singular value decomposition algorithm
for third-order tensors inspired by the classical algorithm of tensor singular
value decomposition (t-svd) and then extend it to order-$p$ tensors. It can be
proved that the quantum version of the t-svd for a third-order tensor
$\mathcal{A} \in \mathbb{R}^{N\times N \times N}$ achieves the complexity of
$\mathcal{O}(N{\rm polylog}(N))$, an exponential speedup compared with its
classical counterpart. As an application, we propose a quantum algorithm for
recommendation systems which incorporates the contextual situation of users to
the personalized recommendation. We provide recommendations varying with
contexts by measuring the output quantum state corresponding to an
approximation of this user's preferences. This algorithm runs in expected time
$\mathcal{O}(N{\rm polylog}(N){\rm poly}(k)),$ if every frontal slice of the
preference tensor has a good rank-$k$ approximation. At last, we provide a
quantum algorithm for tensor completion based on a different truncation method
which is tested to have a good performance in dynamic video completion.
