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ACM Symposium on Theory of Computing (STOC)

New cosystolic expanders from tensors imply explicit Quantum LDPC codes with Ω(√ n log k n ) distance

Tali Kaufman, Ran J. Tessler

Published Jun 15, 2021• 16 citations
Publisher

Abstract

In this work we introduce a new notion of expansion in higher dimensions that is stronger than the well studied cosystolic expansion notion, and is termed Collective-cosystolic expansion.
DOI
10.1145/3406325.3451029

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New Cosystolic Expanders from Tensors Imply Explicit Quantum LDPC Codes with Ω(nlog⁡kn)Ω(\sqrt{n}\log^kn)Ω(n​logkn) Distance
In this work we introduce a new notion of expansion in higher dimensions that is stronger than the well studied cosystolic expansion notion, and is termed {\em Collective-cosystolic expansion}. We show that tensoring two cosystolic expanders yields a new cosystolic expander, assuming one of the complexes in the product, is not only cosystolic expander, but rather a collective cosystolic expander. We then show that the well known bounded degree cosystolic expanders, the Ramanujan complexes are, in fact, collective cosystolic expanders. This enables us to construct new bounded degree cosystolic expanders, by tensoring of Ramanujan complexes. Using our new constructed bounded degree cosystolic expanders we construct {\em explicit} quantum LDPC codes of distance nlog⁡kn\sqrt{n} \log^k nn​logkn for any kkk, improving a recent result of Evra et. al. \cite{EKZ}, and setting a new record for distance of explicit quantum LDPC codes. The work of \cite{EKZ} took advantage of the high dimensional expansion notion known as cosystolic expansion, that occurs in Ramanujan complexes. Our improvement is achieved by considering tensor product of Ramanujan complexes, and using their newly derived property, the collective cosystolic expansion.
Quantum LDPC codes with Ω(nlog⁡kn)Ω(\sqrt{n}\log^kn)Ω(n​logkn) distance, for any kkk
In this work we construct quantum LDPC codes of distance nlog⁡kn\sqrt{n} \log^k nn​logkn for any kkk, improving a recent result of Evra et. al. \cite{EKZ}. The work of \cite{EKZ} took advantage of the high dimensional expansion notion known as cosystolic expansion, that occurs in Ramanujan complexes. Our improvement is achieved by considering tensor product of Ramanujan complexes. The main conceptual contribution of our work is the following: a tensor product of a cosystolic expander with a complex with a linear cosystole has a linear cosystole.
In classical coding theory, the quest for explicit codes achieving list decoding capacity has been an important driving force. While random codes are easily shown to achieve capacity, an explicit construction was only discovered decades later in the seminal work [Guruswami and Rudra, STOC 2006]. In quantum coding theory, it is highly desirable that a code family be LDPC. While good quantum codes were known for decades, obtaining the additional LDPC property was elusive. In fact, only very recently that good quantum LDPC codes were discovered in a breakthrough work [Panteleev and Kalachev, STOC 2022]. In this context, a natural question is to ask for an explicit family of quantum codes achieving list decoding capacity while also possessing the important LDPC property. In this work, we provide (to the best of our knowledge) the first explicit constructions of quantum LDPC codes achieving list decoding capacity, namely, with a list decoding radius approaching the quantum Singleton bound with constant list sizes. Furthermore, we provide near-linear time (in the block-length) list decoding algorithms approaching capacity. Our explicit code construction are based on expander graphs via the quantum analogue of Alon-Edmonds-Luby (AEL) amplification [Bergamaschi, Golowich and Gunn, STOC 2024], and this enables the important LDPC property. Our efficient list decoding algorithms are obtained by generalizing the classical expander-based weak-regularity list decoding algorithms [Srivastava and Tulsiani, FOCS 2025] [Jeronimo and Singh, 2025] to suitable instantiations of quantum AEL.
Constructing quantum LDPC codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large XXX-distance but a much smaller ZZZ-distance. However, together with a classical expander LDPC code and a tensoring method that generalises a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance n1/2log⁡nn^{1/2}\log nn1/2logn, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance n1/2log⁡1/2nn^{1/2}\log^{1/2}nn1/2log1/2n. We then exploit the expansion properties of the complex to devise the first polynomial time algorithm that decodes above the square root barrier for quantum LDPC codes.