Percolation bounds for decoding thresholds with correlated erasures in quantum LDPC codes

Correlations between errors can dramatically affect decoding thresholds, in some cases eliminating the threshold altogether. We analyze the existence of a threshold for quantum low-density parity-check (LDPC) codes in the case of correlated erasures. When erasures are positively correlated, the corresponding multi-variate Bernoulli distribution can be modeled in terms of cluster errors, where qubits in clusters of various size can be marked all at once. In a code family with distance scaling as a power law of the code length, erasures can be always corrected below percolation on a qubit adjacency graph associated with the code. We bound this correlated percolation transition by weighted (uncorrelated) percolation on a specially constructed cluster connectivity graph, and apply our recent results [1] to construct several bounds for the latter.% \smallskip\\[0pt] [1] K. E. Hamilton and L. P. Pryadko, ``\emph{Algebraic bounds for weighted percolation on directed and undirected graphs},'' arXiv:1505.03963 (2015).