Sathyawageeswar Subramanian

Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum mutual information quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.

Shallow-depth quantum circuits with gates of bounded fan-in have been demonstrated to achieve computational advantages over shallow-depth classical circuits, even allowing for unbounded fan-in (AC0). Despite their versatility, these computational models are known to be less powerful than Polynomial Threshold Function (PTF) circuits, which serve as a natural model for neural networks and exhibit enhanced expressivity and computational capabilities. We prove that PTF circuits with a constant number of layers, when biased (having the activation region of their gates limited), fail to solve certain computational (relational) problems that quantum circuits of constant depth can solve. Furthermore, we prove such a separation for a family of problems, one for each prime qudit dimension. We prove all of these separations via correlation bounds for average-case hardness. We also establish a tight lower bound on the size of biased PTF circuits that can solve a specific relational problem *exactly*. This allows us to significantly reduce the estimated resource requirements for potential demonstrations of quantum advantage. The main challenges in this area of research arise in establishing the classical lower bounds, and in designing non-local games with quantum-classical gaps in the winning strategy in order to go beyond qubits to higher dimensions. To address the former, we have formulated novel switching lemmas specifically designed for multi-output biased PTF circuits, and have developed a way to assess the difficulty of deriving exact solutions. Our contribution towards the latter is grounded in a novel assortment of non-local games, characterized by an exponential difference between their classical and quantum success probabilities. Finally, our technical developments could be of more general and independent interest.