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

Evra, Shai; Kaufman, Tali; Zémor, Gilles

doi:10.1109/focs46700.2020.00029

Cited by 34 publications

(37 citation statements)

References 27 publications

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“…2 which is slightly worse than Equation (6). It is obtained by dropping the second term in Equation ( 7) which implicitly assumes that Equation (8) holds.…”

Section: Properties Of Expander Codes 1) Parametersmentioning

confidence: 96%

“…they are roots of normed polynomials with integer coefficients. 6 In that case we can adjoin the missing roots to Z giving a ring A and obtain G + r,s ≤ PSL(2, A). Reducing A modulo a prime p via π p maps the matrix coefficients into a finite field F p m .…”

Section: ) Lps-expandermentioning

confidence: 99%

“…The goal of this section is to prove the following theorem. The distance balancing of [16] and [6] respects the splitting of the homology group described in Section IV-G. Applying the distance balancing procedure of [6] using classical [N 0018-9448 (c) 2021 IEEE.…”

Section: Explicit Examplesmentioning

confidence: 99%

“…The distance balancing of [16] and [6] respects the splitting of the homology group described in Section IV-G. Applying the distance balancing procedure of [6] using classical [N 0018-9448 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.…”

Section: Explicit Examplesmentioning

confidence: 99%

“…The fibers are hyperbolic surfaces twisted along geodesics, giving rise to a quantum code family with distance 4 √ log N √ N . Evra-Kaufman-Zémor [6] and Kaufman-Tessler [7] further improved this to √ log N √ N and log k (N ) √ N , respectively. Recently, the apparent polylog(N ) √ N distance barrier was broken.…”

Section: Introductionmentioning

confidence: 99%

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Balanced Product Quantum Codes

Breuckmann

Eberhardt²

2021

IEEE Trans. Inform. Theory

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This work provides the first explicit and nonrandom family of [[N, K, D]] LDPC quantum codes which encode K ∈ Θ(N 4 5 ) logical qubits with distance D ∈ Ω(N 3 5 ). The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called balanced product.Recently, Hastings-Haah-O'Donnell and Panteleev-Kalachev were the first to show that there exist families of LDPC quantum codes which break the polylog(N ) √ N distance barrier. However, their constructions are based on probabilistic arguments which only guarantee the code parameters with high probability whereas our bounds hold unconditionally.Further, balanced products allow for non-abelian twisting of the check matrices, leading to a construction of LDPC quantum codes that can be shown to have K ∈ Θ(N ) and that we conjecture to have linear distance D ∈ Θ(N ).

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“…2 which is slightly worse than Equation (6). It is obtained by dropping the second term in Equation ( 7) which implicitly assumes that Equation (8) holds.…”

Section: Properties Of Expander Codes 1) Parametersmentioning

confidence: 96%

Section: ) Lps-expandermentioning

confidence: 99%

Section: Explicit Examplesmentioning

confidence: 99%

Section: Explicit Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Balanced Product Quantum Codes

Breuckmann

Eberhardt²

2021

IEEE Trans. Inform. Theory

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Explicit Lower Bounds Against Ω(n)-Rounds of Sum-of-Squares

Hopkins

Lin

2022

2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)

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Quantum Tanner codes

Leverrier

Zémor

2022

2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)

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Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes.In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a c 3 locally testable code by Dinur et al. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et al. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et al. code.

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

Cited by 34 publications

References 27 publications

Balanced Product Quantum Codes

Balanced Product Quantum Codes

Explicit Lower Bounds Against Ω(n)-Rounds of Sum-of-Squares

Quantum Tanner codes

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