Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes

• URL: http://arxiv.org/abs/2402.07476v2
• Date: Thu, 11 Apr 2024 13:55:30 GMT
• Title: Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes
• Authors: Irit Dinur, Ting-Chun Lin, Thomas Vidick,
• Abstract summary: We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to quantum locally testable codes. 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.
• Score: 5.224344210588583
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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