Efficient decoding of random errors for quantum expander codes

We show that quantum expander codes, a constant-rate family of quantum LDPC codes, with the quasi-linear time decoding algorithm of Leverrier, Tillich and Zémor can correct a constant fraction of random errors with very high probability. This is the first construction of a constant-rate quantum LDPC code with an efficient decoding algorithm that can correct a linear number of random errors with a negligible failure probability. Finding codes with these properties is also motivated by Gottesman’s construction of fault tolerant schemes with constant space overhead. In order to obtain this result, we study a notion of α-percolation: for a random subset W of vertices of a given graph, we consider the size of the largest connected α-subset of W , whereX is an α-subset ofW if |X ∩W | ≥ α|X |.