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arXiv:2111.03654

Asymptotically Good Quantum and Locally Testable Classical LDPC Codes

Pavel Panteleev, Gleb Vyacheslavovich Kalachev

Published Nov 5, 2021• 9 citations
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Abstract

We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups. We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. Moreover, we show that the produced classical LDPC codes are also asymptotically good and locally testable with constant query and soundness parameters, which proves a well-known conjecture in the field of locally testable codes.

DOI
10.48550/arxiv.2111.03654

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We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups. We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. Moreover, we show that the produced classical LDPC codes are also asymptotically good and locally testable with constant query and soundness parameters, which proves a well-known conjecture in the field of locally testable codes.
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In this paper we study classical and quantum LDPC codes with constant rate obtained using tensor product operation over noncommutative group algebra for chain complexes. It is shown that these families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. The existence of asymptotically good families of locally testable codes with constant parameters of locality and soundness is also shown, which gives the solution of the known conjecture in the field of classical locally testable codes.
We construct quantum locally testable codes (LTCs) with constant rate, distance, soundness and locality under a variant of a product expansion conjecture of Bafna and Vyas about Reed-Solomon codes. In particular, we use the high-dimensional expansion framework of Dinur, Lin and Vidick for constructing quantum LTCs, instantiated with the non-Abelian cubical complexes of Rungtanapirom, Stix and Vdovina. Our code is obtained by equipping the complex with carefully chosen Reed-Solomon local codes whose symmetries are compatible with those of the complex.
We give a construction of quantum LDPC codes of dimensionΘ(log⁡N)\Theta (\log N)Θ(logN)and distanceΘ(N/log⁡N)\Theta (N/\log N)Θ(N/logN)as the code length$N\to \infty .UsingaproductofchaincomplexesthisconstructionalsoprovidesafamilyofquantumLDPCcodesofdistance. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance.UsingaproductofchaincomplexesthisconstructionalsoprovidesafamilyofquantumLDPCcodesofdistance\Omega (N^{1-\alpha /2}/\log N)anddimensionand dimensionanddimension\Omega (N^\alpha \log N)$, where0≤α<10 \le \alpha < 10≤α<1. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixedR<1R < 1R<1there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at leastRRRwith, in some sense, optimal circulant sizeΩ(N/log⁡N)\Omega (N/\log N)Ω(N/logN)as the code length$N\to \infty $.