Quantum Expander Mixing Lemma and its Structural Converse

Expander graphs are fundamental in both computer science and mathematics, with a wide array of applications. With quantum technology reshaping our world, quantum expanders have emerged, finding numerous uses in quantum information theory, quantum complexity, and noncommutative pseudorandomness. The classical expander mixing lemma plays a central role in graph theory, offering essential insights into edge distribution within graphs and aiding in the analysis of diverse network properties and algorithms. This paper establishes the quantum analogue of the classical expander mixing lemma and its structural converse for quantum expanders.