Expansion of Higher-Dimensional Cubical Complexes With Application To Good ...
summary
Constructs quantum locally testable codes achieving near-optimal distance (up to polylog factors) and constant weight LDPC checks after amplification, by building a 4D cubical geometry and placing four local tensor/Tanner codes on its cells. Core ingredients are spectral expansion of abelian Cayley complexes, a sheaf-style chain complex, and a local robustness condition that reduces global soundness to local tests; proofs adapt DHLV techniques and a transpose reduction for the X-checks. Surprising point: the distance ≈ n / polylog(n) guarantee and the reliance on an 4D cubical (abelian Cayley) complex + 4 local tensor codes.
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chapters (10)
- 0:00Quantum locally testable codes
- 5:22New code parameters
- 6:52Why LTCs are interesting
- 9:00History of LTCs
- 12:36Check relations in codes
- 18:27Tanner codes construction
- 23:234D cubical complex overview
- 30:49Sheaf and chain complexes
- 33:25Proof techniques for distance
- 35:57Results and future work
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