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

Abstract

International audienceThe 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