A survey on the complexity of learning quantum states

Quantum learning theory is a new and very active area of research at the intersection of quantum computing and machine learning. Important breakthroughs in the past two years have rapidly solidified its foundations and led to a need for an encompassing survey that can be read by seasoned and early-career researchers in quantum computing. In this Perspective, we survey various results that rigorously study the complexity of learning quantum states. These include progress on quantum tomography, learning physical quantum states, alternative learning models to tomography, and learning classical functions encoded as quantum states. We highlight how these results are leading towards a successful theory with a range of exciting open questions, some of which we list throughout the text. Quantum learning theory is a new and very active area of research at the intersection of quantum computing and machine learning. This Perspective surveys the progress in this field, highlighting a number of exciting open questions.