Abstract: Any construction of a quantum computer would require finding good sets of quantum logic gates: finite sets of 2^n-by-2^n unitary matrices that efficiently and computably approximate arbitrary unitary matrices through short products. We explain a connection between constructing these gate sets and automorphic representations (extending ideas from the Lubotzky-Phillips-Sarnak construction of expander graphs). Using this, we explain how to input analytic bounds proven using the endoscopic classification to produce the first provable constructions of optimal "golden" gate sets for more than one qubit.