Quantum Learning Algorithms and Post-Quantum Cryptography

Abstract: 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.