Decodable quantum LDPC codes beyond the $\sqrt{n}$ distance barrier using high dimensional expanders

Evra, Shai; Kaufman, Tali; Zémor, Gilles

doi:10.48550/arxiv.2004.07935

Cited by 11 publications

(26 citation statements)

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“…In this section, we introduce the locally minimal distance [24] [25] [12], and show that locally minimal distance is a lower bound of locally testable distance. This means that if we can show locally minimal distance is linear then locally testable distance is linear which implies constant soundness.…”

Section: B Locally Minimalmentioning

confidence: 99%

$c^3$-Locally Testable Codes from Lossless Expanders

Lin¹,

Hsieh²

2022

Preprint

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A locally testable code (LTC) is an error correcting code with a property tester. The tester tests if a word is codeword by reading constant random bits and rejects the word with probability proportional to the distance from the word to the closest codeword. An important open question until recently is whether there exist c 3 -LTCs which are LTCs with constant rate, constant relative distance and constant locality. In this work, we construct a new LTC family using 1-sided lossless expanders and balanced products.

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Section: B Locally Minimalmentioning

confidence: 99%

$c^3$-Locally Testable Codes from Lossless Expanders

Lin¹,

Hsieh²

2022

Preprint

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“…In fact, beating this bound turned out to be very challenging, even for general LDPC codes that need not be embeddable on a finitedimensional lattice. More precisely, an old construction due to Freedman, Meyer and Luo gives a minimum distance Θ(N 1/2 log 1/4 N) [FML02], and only very recently new constructions based on high-dimensional expanders have brought polylogarithmic improvements [EKZ20,KT20]. These were holding the record until the recent breakthrough of Ref.…”

Section: Quantum Ldpc Codesmentioning

confidence: 99%

Quantum XYZ Product Codes

Leverrier,

Apers,

Vuillot

2020

Preprint

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We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes might display a minimum distance as large as Θ(N 2/3 ) which would beat the current record for the minimum distance of quantum LDPC codes held by fiber bundle codes. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.

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“…We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a [ [4,2,2]] code, which we call a augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance.…”

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confidence: 99%

“…One application of this product has been to construct quantum codes with linear distance and rate and with stabilizers whose weight scales only as the square-root of the number of qubits [2]. Other applications include weight balancing [3,4] and the construction of some novel code families [5]. A special case of the homological product is the hypergraph product [6], which has been applied to construct quantum LDPC codes of linear rate and square-root distance.…”

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confidence: 99%

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Union-Find Decoders For Homological Product Codes

Delfosse,

Hastings

2020

Preprint

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Decodable quantum LDPC codes beyond the $\sqrt{n}$ distance barrier using high dimensional expanders

Cited by 11 publications

References 0 publications

$c^3$-Locally Testable Codes from Lossless Expanders

$c^3$-Locally Testable Codes from Lossless Expanders

Quantum XYZ Product Codes

Union-Find Decoders For Homological Product Codes

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