Geometric picture behind quantum expanders
[This answer was copied from my answer on the now-defunct theoreticalphysics stackexchange site.]
For classical expanders, the spectral definition can be expressed in terms of the second-smallest eigenvalue of the graph Laplacian, which can be thought of as the minimum of a quadratic form over all unit vectors orthogonal to the all-ones vector.  If we restrict this minimization to vectors of the form (a,a,...,a, b,b, ..b), then this yields the edge expansion of the graph.  here is a discussion.  The rough equivalence of these two definitions is known as Cheeger's inequality.

This suggests that for the quantum case we should consider the action of the channel (formed by applying a random unitary from the expander) on projectors.  A result analogous to Cheeger's inequality is derived in Appendix A of arXiv:0706.0556.

On the other hand, while this is mathematically analogous, we do still know of many fewer applications of quantum expanders than are known for classical expanders.