Quantum tensor singular value decomposition with applications to recommendation systems

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 extend it to order-$p$ tensors. It can be proved that the quantum version of t-svd for the tensor $\mathcal{A} \in \mathbb{C}^{N\times N \times N}$ achieves the complexity of $\mathcal{O}({\rm polylog}(N))$, an exponential speedup compared with its classical counterpart. As an application, we propose a quantum algorithm for context-aware recommendation systems which incorporates the contextual situation of users to the personalized recommendation. Since a user's preference in a certain context still influences his recommendation in other contexts, our quantum algorithm first uses quantum Fourier transform to merge the preference of a user in different contexts together, then project the quantum state with the preference information of a user into the space spanned by the singular vectors corresponding to the singular values greater than the given thresholds. We provides recommendations varying with contexts by measuring the output quantum state corresponding to the approximation of this user's preference. This quantum recommendation system algorithm runs in expected time $\mathcal{O}({\rm polylog}(N){\rm poly}(k),$ which is exponentially faster than previous classical algorithms. At last, we provide another quantum algorithm for third-order tensor compression based on a different truncate method which is tested to have better performance in dynamic video completion.