# From Ramanujan Graphs to Ramanujan Complexes

## TL;DR

This paper surveys the development of Ramanujan graphs and complexes, highlighting their spectral properties, applications in combinatorics, computer science, quantum computation, and their connection to the Ramanujan conjecture.

## Contribution

It provides a comprehensive overview of recent advances in high-dimensional Ramanujan objects and their applications, including new results on random walks and Euclidean spheres.

## Key findings

- Spectral bounds of Ramanujan complexes
- Applications in quantum computation via golden gates
- Connections to the Ramanujan conjecture

## Abstract

Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments are surveyed. After explaining their connection to the Ramanujan conjecture we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to "golden gates" which are of importance in quantum computation.