Optimal expansion in unitary groups I: Golden adelic groups and cutoff

Abstract

In this paper and its companion [DEP2] we construct and study special generating sets for $p$-arithmetic unitary Lie groups, extending the seminal works of Lubotzky, Phillips and Sarnak to higher dimensions. The real embeddings of our sets form optimal topological generators for compact unitary Lie groups, and their $p$-adic embeddings give rise to Cayley graphs and complexes over finite unitary groups, which exhibit the cutoff phenomenon. In the current paper we prove two main results: First, we construct $p$-arithmetic lattices in $P{U}_4$ and $P{U}_8$ which act transitively on their corresponding Bruhat-Tits buildings. These groups are used to construct Golden Gate sets on two and three qubits, which have optimal circuit length and efficient compiling algorithms. Second, we study the cutoff phenomenon for quotients of the Bruhat-Tits building of a general semisimple $p$-adic group. We show that cutoff occurs for quotients which satisfy either the Ramanujan property or the weaker Sarnak-Xue Density Hypothesis (SXDH). For this purpose we bound the operator norms of a large class of Hecke operators in terms of the rate of decay of unitary representations. In the companion paper [DEP2], we prove such a SXDH result for unitary groups.

arXiv commentThe previous version has been split to two papers, this is the first part