Algebraic and Geometric Methods for Construction of Topological Quantum Codes from Lattices

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices Lambda '/Lambda of index m combined with geometric technique of the projections of vector basis Lambda ' over vector basis Lambda, we reproduce surface codes found in the literature with parameter [[2m,2,|a|+|b|]] for the case Lambda=Z2 and m=a2+b2, where a and b are integers that are not null, simultaneously. We also obtain a new class of surface code with parameters [[2m,2,|a|+|b|]] from the Lambda=A2-lattice when m can be expressed as m=a2+ab+b2, where a and b are integer values. Finally, we will show how this technique can be extended to the construction of color codes with parameters [[18m,4,6(|a|+|b|)]] by considering honeycomb lattices partition A2/Lambda ' of index m=9(a2+ab+b2) where a and b are not null integers.