Expander qLDPC Codes against Long-range Correlated Errors in Memory

Fault-tolerance using constant space-overhead against long-range correlated errors is an important practical question. In the pioneering works [Terhal and Burkard, PRA 2005], [Aliferis et al, PRA 2005], [Aharonov et al, PRL 2006], fault-tolerance using poly-logarithmic overhead against long-range correlation modeled by pairwise joint Hamiltonian was proven when the total correlation of an error at a qubit location with errors at other locations was $O(1)$, i.e., the total correlation at a location did not scale with the number of qubits. This condition, under spatial symmetry, can simply be stated as the correlation between locations decaying faster than 1distᵈⁱᵐ. However, the pairwise Hamiltonian model remained intractable for constant overhead codes. Recently, [Bagewadi and Chatterjee, PRA 2025] introduced and analyzed the generalized hidden Markov random field (MRF) model, which provably captures all stationary distributions, including long-range correlations [Kunsch et al, Ann. App. Prob. 1995]. It resulted in a noise threshold in the case of long-range correlation, for memory corrected by the linear-distance Tanner codes [Leverrier and Zemor, FOCS 2022] for super-polynomial time. In this paper, we prove a similar result for square-root distance qLDPC codes and provide an explicit expression for the noise threshold in terms of the code rate, for up to o(√#qubits) scaling of the total correlation of error at a location with errors at other locations.