Minimal distances for certain quantum product codes and tensor products of chain complexes

Distance bounds for generalized bicycle codes

“…20, are particularly suited for constructing short codes, as a GB code can be constructed from a pair of linear cyclic codes which are only a factor of two shorter. Second, as we show in this work, a subset of codes from several well-studied families, most notably, quantum hypergraph-product (QHP) codes in two and higher dimensions [13,16,17], including the codes with finite asymptotic rates and power-law distance scaling, can be mapped to bicycle codes. At the same time, the distance bound d ≤ n 1/2 which limits the parameters of all QHP codes, does not apply to GB codes; we show in this work that this family includes codes with linear distances.…”

“…As a result, even though the GB code with a (x) = 1 + x + x 2 + x 4 and b (x) = 1 + x 14 has the same parameters [ [42,8,3]], this is coincidental. Indeed, replacing the polynomial h2(x) with h 2 (x) = 1 + x + x 2 gives the QHP code [[42, 16, 2]] and an equivalent code using the map (17), but the present map gives b (x) = h 2 (x 7 ) which is mutually prime with a(x), resulting in an empty GB code.…”

“…where I i are the identity matrices of size n i , i ∈ {1, 2}. Such a code has the parameters [13,17] […”

“…Unlike in the case of classical LDPC codes [9,10] where sparse random matrices can be used to define the code, due to a commutativity constraint, an algebraic ansatz is required in the case of quantum LDPC codes. For over a decade, no construction was known to give distances larger than a square root of the block size n, up to a polylogarithmic factor [1,6,[11][12][13][14][15][16][17][18]. The O √ n polylog n barrier was broken by Hastings, Haah, and O'Donnell [2] who demonstrated a code family with the distance O(n 3/5 / polylog n).…”

“…1. (Color online) (a)A map(17) of an n1 × n2 square lattice with periodic boundary conditions along the vectors L1 = (n1, 0) and L2 = (0, n2) to a chain of length = n1n2, with n1 = 7 and n2 = 3. Red digits below and to the left of the axes show the column i and row j indices; the index t is placed above and to the right of the corresponding vertex.…”