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 (F₂(m×m))ⱽ −→^(δ0) (F₂ᵐ)ᴱ −→^(δ¹) F₂^F 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 C_A,C_B : F₂ᵐ → F₂^Δ where Δ is 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.