Efficient Decoding of Random Errors for Quantum Expander Codes

We show that quantum expander codes, a constant-rate family of quantum low-density parity check (LDPC) codes, with the quasi linear time decoding algorithm of Leverrier, Tillich and Zemor 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 alpha-percolation: for a random subset E of vertices of a given graph, we consider the size of the largest connected alpha-subset of E, where Xis an alpha-subset of E if vertical bar X boolean AND E vertical bar >= alpha vertical bar X vertical bar.