Coherent Quantum Inference and Quantum Purity Amplification

We present a general theory of coherent quantum learning, a framework for extracting structured quantum information directly into usable quantum states without relying on measurement or tomography. We formalize the coherent analogue of classical learning theory and show that coherent learning can reduce sample complexity in certain tasks. As a central example, we study Quantum Purity Amplification (QPA), which transforms multiple noisy copies of a quantum state into high-quality outputs using only coherent operations. We introduce a family of optimal QPA channels that generalize the original one-output setting to multiple outputs and arbitrary eigenstates. Their asymptotic optimality under general noise is established using representation theoretic tools. We derive fidelity bounds that hold uniformly across system dimensions, with scaling that is either polynomial or exponential depending on the regime. We also construct efficient quantum circuits based on generalized quantum phase estimation and Clebsch Gordan initialization, which implement these channels with only polynomial overhead in the number of inputs and the system size. In addition, we study the connection between coherent and traditional quantum learning, and show that the two approaches become equivalent in the entanglement breaking limit, where no quantum coherence is preserved. These results position QPA as a foundational example of coherent quantum learning, with broad applications in quantum state preparation, noise mitigation, and tomography.