Effect of Correlated Errors on Quantum Memory

Recent improvements in LDPC code based fault-tolerance for memory against i.i.d. errors naturally lead to the question of fault-tolerance against errors with long-range correlations. We introduce a classical correlation model based on hidden random fields for modeling such errors. We show that this proposed model can capture certain correlation patterns not captured by the joint (system and bath) Hamiltonian model with pairwise terms. Towards that, we derive a converse result for retention time in the presence of an error distribution, which is from the proposed class and exhibits quadratically small correlations. On the other hand, we show that for a broad subclass of error distributions within the proposed model, Tanner codes can ensure exponential retention time when the error rate is sufficiently low. The proposed model is analytically tractable due to the existence of a rich probability literature and thus, can offer insights complementary to the joint Hamiltonian model with pairwise terms.