Tradeoffs for reliable quantum information storage in surface codes and color codes

Nicolas Delfosse (1, 2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 GRACE - Geometry, arithmetic, algorithms, codes and encryption

The family of hyperbolic surface codes is one of the rare families of quantum LDPC codes with non-zero rate and unbounded minimum distance. First, we introduce a family of hyperbolic color codes. This produces a new family of quantum LDPC codes with non- zero rate and with minimum distance logarithmic in the blocklength. Second, we show that the parameters [[n, k, d]] of surface codes and color codes satisfy kd^2 ≤ C(log k)^2n, where C is a constant that depends only on the row weight of the parity-check matrix. Our results prove that the best asymptotic minimum distance of LDPC surface codes and color codes with non-zero rate is logarithmic in the length.

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]; Physique Quantique [quant-ph]

Nicolas Delfosse. Tradeoffs for reliable quantum information storage in surface codes and color codes. IEEE International Symposium on Information Theory, ISIT 2013, Jul 2013, Istanbul, Turkey. ⟨10.1109/ISIT.2013.6620360⟩. ⟨hal-00798030⟩