Learning to erase quantum states: thermodynamic implications of quantum learning theory

The energy cost of erasing quantum states depends on our knowledge of the states. We show that learning algorithms can acquire such knowledge to erase many copies of an unknown state at the optimal energy cost. This is proved by showing that learning can be made fully reversible and has no fundamental energy cost itself. With simple counting arguments, we relate the energy cost of erasing quantum states to their complexity, entanglement, and magic. We further show that the constructed erasure protocol is computationally efficient when learning is efficient. Conversely, under standard cryptographic assumptions, we prove that the optimal energy cost cannot be achieved efficiently in general. These results also enable efficient work extraction based on learning. Together, our results establish a concrete connection between quantum learning theory and thermodynamics, highlighting the physical significance of learning processes and enabling provably-efficient learning-based protocols for thermodynamic tasks.