Computational complexity in quantum learning tasks

Abstract

Quantum learning theory is a burgeoning field at the intersection of quantum information and machine learning theory. However, current research for learning quantum objects typically focuses on efficiency with respect to sample complexity, rather than computational complexity. In this thesis, we study classes of quantum processes and classical functions for which computationally efficient quantum learning is achievable. On the other hand, it is known that there exist classes of quantum states that are provably hard to learn in polynomial time, under standard cryptographic assumptions. We consider the converse: can we design useful cryptography from a given class of quantum states that is hard to learn efficiently? We answer this question affirmatively by constructing fundamental cryptographic primitives from the computational hardness of learning. This reflects the deep connection between learning and cryptography in the classical world, which has been widely unexplored in the quantum setting. First, as an example of a computationally efficient quantum learning algorithm, we prove a rigorous quantum advantage against classical gradient methods for learning periodic neurons. Next, we develop a more general formalism, called agnostic process tomography, for approximating an unknown quantum channel by a simpler one in a given class. In this setting, we prove that the correct measure of efficiency is computational complexity and design efficient quantum algorithms for learning a wide variety of processes. Finally, having understood when quantum algorithms can learn efficiently, we study the hardness of learning classes of quantum states and its applications to cryptography