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

TL;DR

This paper introduces a high-dimensional cubical complex for designing quantum locally testable codes, generalizing previous square complex constructions, and establishes bounds on their expansion properties to create new quantum LTCs.

Contribution

It generalizes existing square complex constructions to higher dimensions and applies them to develop quantum LTCs with provable expansion and testing properties.

Findings

Constructed a high-dimensional cubical complex for quantum LTCs

Proved lower bounds on cycle and co-cycle expansion

Developed a new family of quantum LTCs with constant rate and inverse-polylogarithmic distance

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.