Hyperbolic quantum color codes

Abstract

Current work presents a new approach to quantum color codes on compact surfaces with genus g\geq2 using the identification of these surfaces with hyperbolic polygons and hyperbolic tessellations. We show that this method may give rise to color codes with a very good parameters and we present tables with several examples of these codes whose parameters had not been shown before. We also present a family of codes with minimum distance d=4 and the encoding rate asymptotically going to 1 while n\rightarrow\infty.

Construction of New Quantum Color Codes

“…More information on hyperbolic geometry and shrunk lattices may be found in Refs. [7], [12,13,18,19]. Definition 2.1.…”

“…Given a regular polygon of {p, q}, the diameter of its circumscribed circle and an upper bound for an edge of the shrunk lattice are written, respectively as, [13], D(p, q) = 2arccosh cos(π/p)cos(π/q) sin(π/p)sin(π/q) , (2.14) and L(p, q) = l(p, q) + D(p, q).…”

“…(2.15) Also, a lower bound for the number of the reduced network edges in a non-trivial homology cycle belonging to a shrunk lattice is given as follows, [13],…”

“…Since the initial discovery of quantum code by Shor [2], much progress has been made in the development of quantum codes in Engineering [3] and Physics [4]. Over the last two decades, different types of quantum codes have been constructed from classical error-correcting code [5], which can be referred to Kitaev's toric code [6], color codes [7], surface codes [8], toric quantum codes [9] and others [10][11][12][13]. Also in the last two years, some binary quantum codes have been constructed in several ways (for instance [14][15][16][17]).…”

“…For these quantum codes, the encoding rate is such that k n tends 1 as n goes to infinity. Moreover, A table of quantum codes which are different parameters in relation to the classes previously presented in [12,13,17], among others is presented.…”