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

TL;DR

This paper introduces new hyperbolic color codes with non-zero rate and logarithmic minimum distance, and analyzes fundamental tradeoffs between code parameters in surface and color codes, establishing upper bounds on their minimum distance.

Contribution

It presents a new family of hyperbolic color codes and derives bounds on the tradeoff between code length, rate, and distance, highlighting limitations of LDPC surface and color codes.

Findings

Hyperbolic color codes with logarithmic minimum distance.

Upper bound on the product of distance and number of encoded qubits.

Logarithmic asymptotic minimum distance for LDPC surface and color codes.

Abstract

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 study the tradeoff between the length n, the number of encoded qubits k and the distance d of surface codes and color codes. We prove that kd^2 is upper bounded by 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.