Robust many-body localized and error-correcting phases of matter in low-density parity-check codes

Abstract:  Low-density parity-check (LDPC) codes play an important role in everyday electronic devices, and in quantum error correction where “good” quantum LDPC codes are able to achieve extremely robust quantum error correction with only a constant overhead. In this talk, I will show that from a physics point of view, many LDPC codes represent stable phases of matter living in infinite dimensions. First, certain classical LDPC codes, in the presence of local quantum perturbations, form the first provably robust example of many-body localization [1], where low-energy-density eigenstates cluster near one codeword and violate the eigenstate thermalization hypothesis. Similarly, the error-correcting ground states of quantum LDPC codes are stable against perturbations [2], which robustly violate the third law of thermodynamics. Our result also implies stability of the 2d toric code under spatially nonlocal perturbations.

[1] Chao Yin, Rahul Nandkishore, Andrew Lucas. PRL 133, 137101, Editor's Suggestion (2024)
[2] Chao Yin, Andrew Lucas. To appear