Optimized Quantum Autoencoder

Quantum autoencoder (QAE) compresses a bipartite quantum state into its subsystem using a self-checking mechanism. How to characterize and minimize the lost information in this process is essential for understanding the compression mechanism of QAE. Here, we investigate how to minimize the lost information in QAE for any input mixed state. We theoretically show that the lost information is the quantum mutual information between the remaining subsystem and the discarded one; the encoding unitary transformation is designed to minimize this mutual information. Furthermore, we show that the optimized unitary transformation can be decomposed as the product of a permutation unitary transformation and a disentanglement unitary transformation, and the permutation unitary transformation can be searched by a regular Young tableau algorithm. When the search can be made exhaustive in lower-dimensional systems, the lost information is minimized numerically, which is shown theoretically to be a global minimum. When the dimension of the system becomes larger such that an exhaustive search is impossible, we adopt an approximate search algorithm to numerically identify that our compression scheme gives lower lost information than that from the quantum variational circuit based QAE.