Super-Golden-Gates for PU(2)

Abstract

To each of the symmetry groups of the Platonic solids we adjoin a carefully designed involution yielding topological generators of PU (2) which have optimal covering properties as well as efficient navigation. These are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups. The generators give super efficient 1-qubit quantum gates and are natural building blocks for the design of universal quantum gates.

High Dimensional Expanders

“…On the other hand the study of the cut-off phenomenon of Ramanujan complexes in [LLP17] did use the full power of the Ramanujan property. The same can be said about the application of Ramanujan graphs and Ramanujan complexes to the study of "golden gates" for quantum computation (see [PS17] and [PS18]), where the Ramanujan bounds give a distribution of elements in SU(2) with "optimal entropy".…”

From Ramanujan graphs to Ramanujan complexes

“…). This means that any gate can be approximated with arbitrary precision as a product of elements of S. The notion of Golden Gates is a much stronger one, loosely requiring the following (see [4,33] for precise definitions):…”

“…This also solves the compiling problem: by writing any A ∈ S p in p-adic coordinates, one recovers its decomposition in S p by following the (unique) path leading from A to the root of the tree (cf. [33]).…”

Optimal topological generators of U(1)

“…* For a nonabelian consideration, see e.g. Parzanchevski-Sarnak [6]. † This is in contrast to [m], which denotes {1, .…”