Algebraic and Geometric Methods for Construction of Topological Quantum Codes from Lattices - Carvalho, Edson Donizete de [UNESP]; Soares Jr, Waldir Silva; Copatti, Douglas Fernando; Martins, Carlos Alexandre Ribeiro; Silva, Eduardo Brandani da (Mdpi, 2024)
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.
Keywords: surface codes; color codes; flat torus; lattice