A Family of Asymptotically Good Quantum Codes Based on Code Concatenation

Abstract

We explicitly construct an infinite family of asymptotically good concatenated quantum stabilizer codes where the outer code uses CSS-type quantum Reed-Solomon code and the inner code uses a set of special quantum codes. In the field of quantum error-correcting codes, this is the first time that a family of asymptotically good quantum codes is derived from bad codes. Its occurrence supplies a gap in quantum coding theory. As in classical coding theory, we want to construct quantum codes with large minimum distance. More generally, we want to construct asymptotically good 1

Sparse Quantum Codes From Quantum Circuits

“…Work on codes with large k initially focused on random codes, where it was shown that random stabilizers have k, d ∝ n [23][24][25], and more recently that short random circuits generate good codes (meaning that k and d are both proportional to n) [26]. There are also known constructive examples of good stabilizer codes such as those constructed by Ashikhmin, Litsyn, and Tsfasman [27] and others [28][29][30][31]. All of these codes have stabilizer generators with weight ∝ n, however.…”

“…= spackle(K) ∩ G 4 (32b) (30) = spackle(K ∩ ∆) ∪ (spackle(K) ∩ G 2 ) (32c) (31) = spackle(K ∩ ∆) (32d)…”

Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes

“…The quantum codes obtained in this way are called stabilizer codes. After the establishment of the above connection between quantum codes and classical codes, the construction of stabilizer codes can be converted to that of classical codes with symplectic, Euclidean or Hermitian selforthogonal property (see [1], [7], [15], [16], [21], [25], etc).…”

Sparse Quantum Codes from Quantum Circuits

“…Work on codes with large k initially focused on random codes, where it was shown that random stabilizers have k, d ∝ n [16,15,3], and more recently that short random circuits generate good codes [14]. There are also known constructive examples of good stabilizer codes such as those constructed by Ashikhmin, Litsyn, and Tsfasman [2] and others [17,18,34,31]. All of these codes have stabilizer generators with weight ∝ n, however.…”