A set of $d\times d$ unitary matrices $\{U_1,\cdots,U_n\}$ is called a quantum expander if the identity matrice is the only eigenvector of the linear map $\mathcal{M}(X)=\frac{1}{n} U_i X U_I^{+}$ corresponding to eigenvalue 1. For any $tr(X)=0$, $|\mathcal{M}(X)|\leq \lambda |X|$ where $|X|$ denoting the sum of the absolutionof the singular values.
Is there an explicit construction, matrix presentation, of a quantum expander with $n<< d$ and $\lambda<1/10$.
