Toward the Goldilocks blind compression of quantum states

TL;DR

This paper introduces a regime for quantum autoencoders that balances universality and efficiency, achieving optimal fidelity with minimal circuit width for compressing quantum states.

Contribution

It identifies the 'Goldilocks' regime where quantum autoencoders are both universal and resource-efficient, providing theoretical bounds and practical insights.

Findings

Existence of a quantum autoencoder with exactly k encoder ancillas achieving optimal fidelity.

Construction of source families requiring at least k encoder ancillas for optimal schemes.

Numerical evidence suggests negligible performance gap despite theoretical limitations.

Abstract

Quantum autoencoders (QAEs) are learning architectures that compress quantum data into a low-dimensional latent state while preserving the information needed for reconstruction. We study blind single-copy compression of quantum states through a $k$-qubit bottleneck and investigate the minimal circuit width required to attain the information-theoretic optimum under average infidelity. Between the conventional architecture, which is narrow but nonuniversal, and fully general \emph{completely positive and trace preserving} (CPTP) realizations, which are universal but overparameterized, we identify a \emph{Goldilocks} regime. We prove that for every distribution of pure $n$-qubit states, there exists a QAE with exactly $k$ encoder ancillas and $n$ decoder ancillas that achieves the optimal fidelity over all CPTP encoder--decoder pairs. The encoder-side statement is sharp in that we construct source families for which every optimal scheme necessarily uses at least $k$ encoder ancillas, thereby determining the universal encoder threshold exactly. On the decoder side, we show that isometric decoders are exactly optimal for several analytically tractable source families, but we also exhibit an explicit counterexample demonstrating that decoder isometry is not universally sufficient. Nevertheless, numerical experiments indicate that the performance gap is practically negligible.