# Computer Science/Discrete Mathematics Seminar II

## Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification

Expander graphs are fundamental objects in theoretical computer science and mathematics. They have numerous applications in diverse fields such as algorithm design, complexity theory, coding theory, pseudorandomness, group theory, etc.

In this talk, we will describe an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, $d \leq 1/\lambda^{2+o(1)}$)

trade-off between (any desired) spectral expansion $\lambda$ and degree $d$. The optimal quadratic trade-off is known as the Ramanujan bound, so our construction gives almost Ramanujan expanders from arbitrary expanders.

This transformation preserves structural properties of the original graph, and thus has many consequences. Applied to Cayley graphs, our transformation shows that any expanding finite group has almost Ramanujan expanding generators. Similarly, one can obtain almost optimal explicit constructions of quantum expanders, dimension expanders, monotone expanders, etc., from existing (suboptimal) constructions of such objects.

Our results generalize Ta-Shma's technique in his breakthrough paper [STOC 2017] used to obtain explicit almost optimal binary codes. Specifically, our spectral amplification extends Ta-Shma's analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way.

Joint work with: Tushant Mittal, Sourya Roy and Avi Wigderson