Fault tolerance using constant space overhead against long-range correlated errors is an important practical question. In the pioneering works of Terhal and Burkhard [Phys. Rev. A 71, 012336 (2005)], Aliferis, Gottesman, and Preskill [Quant. Inf. Comput. 6, 97 (2006)], and Aharonov, Kitaev, and Preskill [Phys. Rev. Lett. 96, 050504 (2006)], fault tolerance using polylogarithmic overhead against long-range correlation modeled by a pairwise joint Hamiltonian was proven when the total correlation of an error at a qubit location with errors at other locations was $O\left(1\right)$, 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 $\frac{1}{{\mathrm{dist}}^{\dim }}$. However, the pairwise Hamiltonian model remained intractable for constant overhead quantum low density parity check (qLDPC) codes. Recently, Bagewadi and Chatterjee [Phys. Rev. A 112, 022422 (2025)] introduced and analyzed the generalized hidden Markov random field model, which provably captures all stationary distributions, including long-range correlations [Kunsch, Geman, and Kehagias, Ann. Appl. Prob. 5, 577 (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 Zémor, in IEEE 63rd Annual Symposium on the Foundations of Computer Science (FOCS) (IEEE, 2022)] for superpolynomial time. In this paper, we prove a similar result for square-root distance qLDPC codes, and we provide an explicit expression for the noise threshold in terms of the code rate, for up to $o\left(\sqrt{\#\mathrm{qubits}}\right)$ scaling of the total correlation of error at a location with errors at other locations.

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• Quantum error correction