Good Quantum LDPC Codes with Linear Time Decoders

Abstract

We construct a new explicit family of good quantum low-density parity-check codes which additionally have linear time decoders. Our codes are based on a three-term chain (2m× m)V →-0 (2m)E →-1 2F where V (X-checks) are the vertices, E (qubits) are the edges, and F (Z-checks) are the squares of a left-right Cayley complex, and where the maps are defined based on a pair of constant-size random codes CA,CB:2m→2-"where Δis the regularity of the underlying Cayley graphs. One of the main ingredients in the analysis is a proof of an essentially-optimal robustness property for the tensor product of two random codes.