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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<sup>2</sup> &#x2264; C(log k)<sup>2</sup>n, 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.

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