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. In a 2015 letter, Sarnak developed a notion of "golden" gate sets as the most efficient approximators that could reasonably be hoped for.
We present the first proven construction of golden gates sets for n > 1 qubits, discussing the connection to the theory of automorphic representations and some key input bounds derived from the endoscopic classification. This is joint work with Shai Evra and Ori Parzanchevski.