Expander qLDPC Codes against Long-range Correlated Errors in Memory
Yash Deepak Kashtikar, Pranay Mathur, Sudharsan Senthil, Avhishek Chatterjee
TL;DR
This paper proves fault-tolerance using constant space-overhead against long-range correlated errors for square-root distance qLDPC codes and provides an explicit expression for the noise threshold in terms of the code rate, for up to $o(\sqrt{\text{\#qubits}})$ scaling of the total correlation of error at a location with errors at other locations.
Abstract
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 $\frac{1}{\text{dist}^{\text{dim}}}$. 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 $o(\sqrt{\text{\#qubits}})$ scaling of the total correlation of error at a location with errors at other locations.