Z2-systolic freedom and quantum codes

Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength

“…Toric codes have minimum distances which grow like the square root of the blocklength and parity-check equations of weight 4 but unfortunately have fixed dimension which is 2, and hence zero rate asymptotically. It turns out that by taking appropriate surfaces of large genus, quantum LDPC codes of non vanishing rate can be constructed with minimum distance logarithmic in the blocklength, this has actually been achieved in [17,Th. 12.4], see also [40], [26].…”

A Construction of Quantum LDPC Codes From Cayley Graphs

“…, φ(w 2 n ) provides the same adjacency matrix. Another way to prove the assertion is to look at (11) and observe that it is true for n = 3. Then to prove the result by induction on n using Lemma 19.…”

“…[5,11,3,4,23]) and higher-dimensional objects. These constructions exhibit minimum distances that scale at most as a square root of the blocklength N (to be precise, N 1/2 log N is achieved in [11]) though this often comes at the cost of a very low dimension (recall that the dimension of the toric code is 2).…”

Framework for classifying logical operators in stabilizer codes

“…However, this is often difficult, both analytically or computationally. This is a challenge which underlies broad questions in quantum coding theory, including upper bounds on code distances of locally defined stabilizer codes [14,15] and the feasibility of self-correcting memory [16,17].…”