Groups and dynamics seminar

Time: Thursday, 30/1, 10AM
Place: Ross 70

Speaker: Shai Evra (HUJI)



 Title: Arithmeticity, thinness and efficiency of gate sets in PU(3)


Abstract:

Let Σ be a finite subset of a compact unitary Lie group, whose elements have coefficients in the ring of S -integers of a number field. It is natural to ask: Does Σ generates the full S -arithmetic subgroup (or a finite index subgroup of it)? 
This question arises in both arithmetic group theory, as well as in the theory of quantum computations (under the term synthesis). An special case of interest, is the Clifford+T (or C+T for short) gate set in PU(2) , whose elements have coefficients in Z[ζ8,12] , where ζn=e2πin , for which Kliuchnikov, Maslov, and Mosca gave a positive answer to the above question. Recently, two analogues of the C+T gate set for PU(3) were raised by Kalra, Valluri, and Mosca, which we call C+T and C+D (the latter contains the former), both have elements with coefficients in Z[ζ9,13] . 
In this talk we will describe a recent joint work which gives a positive answer to the above question for the C+D gates and a negative answer for the C+T gates. The proof involves a detailed study of the action of the gates and the 3 -arithmetic group on a 4 -regular Bruhat-Tits tree. If time permits, we will discuss the follow up question of how fast the words of Σ cover the group PU(3) .
This is based on a recent joint work with Ori Parzanchevski.