Quantum Expander Mixing Lemma and its Structural Converse
Ning Ning
TL;DR
The paper develops a quantum analogue of the expander mixing lemma (EML) and its structural converse for quantum expanders, framing expansion via the reduced spectral radius of a quantum channel $T_n(\rho)=\frac{1}{d}\sum_j u_j^{(n)}\rho u_j^{(n)*}$ with $T_n(I)=I$. It proves a quantum EML: for any orthogonal projections $P_1,P_2$, the HS inner product $\langle P_1,T_nP_2\rangle_{HS}$ concentrates around $\frac{1}{N_n}{\rm tr}(P_1){\rm tr}(P_2)$ with error controlled by $1-\tilde{\epsilon}$; and it establishes a quantum inverse EML: large $\rho(n)$ implies the existence of noncommutative witnesses (projections) with sizable correlations, mirroring classical inverse results. The work introduces intermediate constructs such as $T_n(U_1,U_2,V_1,V_2)$ and Schatten-height to translate classical inverse phenomena into the noncommutative setting, providing a rigidity statement for quantum expanders and a diagnostic tool for nonuniform behavior. These results deepen our understanding of noncommutative pseudorandomness and have implications for the design and analysis of quantum expander constructions and approximate unitary designs. Overall, the paper extends inverse principles from graph theory to quantum channels, highlighting that spectral non-expansion manifests as concrete noncommutative structure.
Abstract
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.