Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes
<|reference_start|>Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes: We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively.   We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes $h_1,\ldots,h_t$ as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex.   For t=4 our construction gives a new family of "almost-good" quantum LTCs -- with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.<|reference_end|>
@article{dinur2024expansion,
    title={Expansion of higher-dimensional cubical complexes with application to
  quantum locally testable codes},
    author={Irit Dinur, Ting-Chun Lin, Thomas Vidick},
    journal={arXiv preprint arXiv:2402.07476},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.07476},
    primaryClass={quant-ph cs.CC cs.IT math.IT}
}