Ramanujan Complexes and Golden Gates in PU(3)

Abstract

The Clifford+T gate set is a topological generating set for P U (2), which has been wellstudied from the perspective of quantum computation on a single qubit. The discovery that it generates a full S-arithmetic subgroup of P U (2) has led to a fruitful interaction between quantum computation and number theory, leading in particular to a proof that words in these gates cover P U (2) in an almost-optimal manner.In this paper we study an analogue gate set for P U (3) called Clifford+D. We show that this set generates a full S-arithmetic subgroup of P U (3), and satisfies a slightly weaker almost-optimal covering property. Our proofs are different from those for P U (2): while both gate sets act naturally on a (Bruhat-Tits) tree, in P U (2) the generated group acts transitively on the vertices of the tree, and this is a main ingredient in proving both arithmeticity and efficiency. In the P U (3) (Clifford+D) case the action on the tree is far from being transitive. This makes the proof of arithmeticity considerably harder, and the study of covering rate by automorphic representation theory becomes more involved and results in a slower covering rate.

Cutoff on Ramanujan complexes and classical groups

“…Nevertheless, by considering the building of PGL d .F q ..t///, it was shown by Lubotzky, Samuels and Vishne ( [19], see also [25]) that the groups PSL d .F q / have explicit generators, for which the resulting Cayley graph is precisely the 1-skeleton of a Ramanujan complex of type z A d . For d D 3, such generators can also be given using the building of PGL 3 .Q p / [2,7]. We thus achieve:…”

“…Proof. Given r 1, the r-sphere S. ; r/ is shown in [7] to be of size jS. ; r/j D .r C 1/q 2r C 2rq 2r 1 C 2rq 2r 2 C .r 1/q 2r 3 :…”

On Sarnak’s Density Conjecture and Its Applications

“…This approach has obvious limitations, and in particular, it seems that one must assume that G is p -adic for it to succeed. We refer to [21] for some recent work on this subject and to an ongoing and yet unpublished work of Shai Evra.…”

A Hermitian TQFT from a non-semisimple category of quantum $${\mathfrak {sl}(2)}$$-modules

“…There are also notions of golden and super golden gate sets due to Sarnak [54] and developed in [19,50]. By definition, these gates cover the Lie group in an optimal way and there are efficient algorithms to write an arbitrary element in the Lie group in terms of the generating gate set.…”

“…By definition, these gates cover the Lie group in an optimal way and there are efficient algorithms to write an arbitrary element in the Lie group in terms of the generating gate set. For a precise definition see [19,Definition 2.8]. Proving that a gate set is golden or super golden uses some deep theorems in number theory.…”