  Quantum algorithms have shown successful results in solving noisy linear
problems with quantum samples in which cryptographic hard problems are
relevant. In this paper the previous results are investigated in detail,
leading to new quantum and classical algorithms under the same assumptions as
in the earlier works. To be specific, we present a polynomial-time quantum
algorithm for solving the ring learning with errors problem with quantum
samples which was deemed to be infeasible in [12], as well as polynomial-time
classical algorithms that are more efficient than the corresponding quantum
algorithms in solving the short integer solution problem with quantum samples
and the learning with errors problem with size-reduced quantum samples.
