  Quantum auto-encoder (QAE) is a powerful tool to relieve the curse of
dimensionality encountered in quantum physics, celebrated by the ability to
extract low-dimensional patterns from quantum states living in the
high-dimensional space. Despite its attractive properties, little is known
about the practical applications of QAE with provable advantages. To address
these issues, here we prove that QAE can be used to efficiently calculate the
eigenvalues and prepare the corresponding eigenvectors of a high-dimensional
quantum state with the low-rank property. With this regard, we devise three
effective QAE-based learning protocols to solve the low-rank state fidelity
estimation, the quantum Gibbs state preparation, and the quantum metrology
tasks, respectively. Notably, all of these protocols are scalable and can be
readily executed on near-term quantum machines. Moreover, we prove that the
error bounds of the proposed QAE-based methods outperform those in previous
literature. Numerical simulations collaborate with our theoretical analysis.
Our work opens a new avenue of utilizing QAE to tackle various quantum physics
and quantum information processing problems in a scalable way.
