  Quantum algorithms have demonstrated promising speed-ups over classical
algorithms in the context of computational learning theory - despite the
presence of noise. In this work, we give an overview of recent quantum
speed-ups, revisit the Bernstein-Vazirani algorithm in a new learning problem
extension over an arbitrary cyclic group and discuss applications in
cryptography, such as the Learning with Errors problem.
  We turn to post-quantum cryptography and investigate attacks in which an
adversary is given quantum access to a classical encryption scheme. In
particular, we consider new notions of security under non-adaptive quantum
chosen-ciphertext attacks and propose symmetric-key encryption schemes based on
quantum-secure pseudorandom functions that fulfil our definitions. In order to
prove security, we introduce novel relabeling techniques and show that, in an
oracle model with an arbitrary advice state, no quantum algorithm making
superposition queries can reliably distinguish between the class of functions
that are randomly relabeled at a small subset of the domain.
  Finally, we discuss current progress in quantum computing technology,
particularly with a focus on implementations of quantum algorithms on the
ion-trap architecture, and shed light on the relevance and effectiveness of
common noise models adopted in computational learning theory.
