Quantum expanders and the quantum entropy difference problem
Avraham Ben-Aroya, Amnon Ta-Shma
TL;DR
The paper develops a quantum-expander framework by translating classical expander/construction ideas to the quantum realm, proving constant-degree quantum expanders can arise from non-Abelian Cayley graphs such as PGL(2,q) via a careful basis change. It connects these expanders to quantum extractors, deriving bounds on spectral gaps and establishing a quantum entropy-Estimation toolkit. A central result is that the Quantum Entropy Difference (QED) problem is QS ZK-complete, obtained through reductions to Quantum State Distinguishability (QSD) and balanced quantum extractors built from expanders, linking graph-theoretic constructs to quantum complexity. The work thus bridges representation theory, group-based expanders, and quantum information to illuminate the complexity of estimating quantum entropies and to provide new quantum-conductor tools for entropy manipulation.
Abstract
We define quantum expanders in a natural way. We show that under certain conditions classical expander constructions generalize to the quantum setting, and in particular so does the Lubotzky, Philips and Sarnak construction of Ramanujan expanders from Cayley graphs of the group PGL. We show that this definition is exactly what is needed for characterizing the complexity of estimating quantum entropies.