In this talk, I will describe a surprising connection between the Langlands program and designing efficient architectures for quantum computing. Constructions of quantum computers require finding finite sets of 2^n-by-2^n unitary matrices that efficiently and computably approximate arbitrary unitary matrices through short products. Extending ideas first used in Lubotzky-Phillips-Sarnak's construction of expander graphs, such an "optimal covering" property can be translated into a bound in the theory of automorphic representations. I will explain this translation and then broadly sketch how recent progress in automorphic theory and the Langlands program can be applied to prove the resulting bound in the cases most relevant to quantum computing.