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Annual Symposium on Foundations of Computer Science

Decodable quantum LDPC codes beyond the square root distance barrier using high dimensional expanders

Shai Evra, Tali Kaufman, Gilles Zémor

Published Nov 1, 2020• 34 citations
Publisher

Abstract

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 X-distance but a much smaller Z-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 n1/2logn, 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 n1/2log1/2n. 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.

DOI
10.1109/focs46700.2020.00029

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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 X-distance but a much smaller Z-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 n1/2logn, 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 n1/2log1/2n. 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.
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.
Decodable Quantum LDPC Codes beyond the n\sqrt{n}n​ Distance Barrier Using High-Dimensional Expanders
Constructing quantum low-density parity-check (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 generalizes a construction of Hastings and also the Tillich–Zémor 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 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. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance n1/2log⁡nn^{1/2}\log nn1/2logn, which sets a new record for quantum LDPC codes.
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.
Quantum low-density parity-check (qLDPC) codes are an important component in the quest for quantum fault tolerance. Dramatic recent progress on qLDPC codes has led to constructions which are asymptotically good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. These constructions, while theoretically explicit, rely on inner codes with strong properties only shown to exist by probabilistic arguments, resulting in lengths that are too large to be practically relevant. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice. A previous construction of qLDPC codes based on the lifted product of an expander-based classical LDPC code with a repetition code (Panteleev and Kalachev, 2020) achieved a near-linear distance, and avoids the need for such intractable inner codes. Our main result is an efficient decoding algorithm for these codes that corrects a near-linear number of adversarial errors. En route, we give a similar algorithm for the hypergraph product version these codes, which are simpler but have distance growing only as the square root of the block length. Our decoding algorithms leverage the fact that the codes we consider are quasi-cyclic, meaning that they respect a cyclic group symmetry. Since the repetition code is not based on expanders, previous approaches to decoding expander-based qLDPC codes, which typically worked by greedily flipping code bits to reduce some potential function, do not apply in our setting. Instead, we reduce our decoding problem (in a black-box manner) to that of decoding classical expander-based LDPC codes under noisy parity-check syndromes. For completeness, we also include a treatment of such classical noisy-syndrome decoding that is sufficient for our application to the quantum setting.
We present an efficient decoding algorithm for constant rate quantum hyper graph-product LDPC codes which provably corrects adversarial errors of weight proportional to the code minimum distance, or equivalently to the square-root of the block length. The algorithm runs in time linear in the number of qubits, which makes its performance the strongest to date for linear-time decoding of quantum codes. The algorithm relies on expanding properties, not of the quantum code's factor graph directly, but of the factor graph of the original classical code it is constructed from.