An improved quantum algorithm for the quantum learning with errors problem

LWE (learning with errors)-based cryptography, whose security relies on the hardness of the underlying LWE problem, is one of the most promising candidates for post-quantum cryptography that is designed to be secure even when the adversaries can access the quantum computers. The quantum LWE problem is a quantum version of the LWE problem, where the solving algorithm can query the LWE oracle in a quantum way. For this quantum LWE problem, Grilo, Kerenidis and Zijlstra (GKZ) recently showed an efficient quantum solving algorithm, together with a test candidate algorithm which judges whether the solution returned by solving algorithm is correct. In this paper, we first present an improved version of the GKZ solving algorithm, which can support a higher error rate, and meanwhile achieve a higher success probability. In particular, for the high error rate case, our algorithm can almost increase the success probability by a factor q than the original GKZ solving algorithm, where q is the module of the LWE problem. Second, for the GKZ test algorithm, we show a quadratic speed-up in running time by introducing amplitude amplification.