Multi-Qubit Golden Gates

Our goal in this paper is to construct optimal topological generators for compact unitary Lie groups, extending the work of a letter of Sarnak and arXiv:1704.02106 on golden and super-golden gates to higher dimensions. To do so we consider a variant of the Sarnak--Xue Density Hypotheses in the weight aspect for definite projective unitary groups and prove it using the endoscopic classification of automorphic representations. Our main motivation is to construct efficient multi-qubit universal gate sets for quantum computers. For example, we find a set of universal gates that, for a given accuracy, can heuristically approximate arbitrary unitary operations on 2 qubits with $\approx$10 times fewer ``expensive'' $T$-type gates than the standard Clifford+$T$ set. Our framework also covers the 2-qubit Clifford+CS gate set, well-known for being particularly friendly to fault-tolerant implementation. We thereby prove tight upper bounds on the required CS count for approximations (specifically, $4.8$x fewer non-Clifford gates than Clifford+$T$).