Quantum Recommendation Systems.

A recommendation system uses the past purchases or ratings of $n$ products by a group of $m$ users, in order to provide personalised recommendations to individual users. The information is modelled as an $m \times n$ preference matrix which is assumed to have a good $k$-rank approximation, for a small constant $k$. In this work, we present a quantum algorithm for recommendation systems that has running time $O(\text{poly}(k)\text{polylog}(mn))$. All classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides a real world application of quantum algorithms in machine learning.