Information-theoretic generalization bounds for learning from quantum
  data
Matthias Caro Tom Gur Cambyse Rouzé Daniel Stilck França 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 information-theoretic 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.