Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates

Zeng, Weilei; Pryadko, Leonid P.

doi:10.1103/physrevlett.122.230501

Cited by 35 publications

(36 citation statements)

References 47 publications

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“…We can also use the qSCPC concatenation scheme to derive qLDPC codes with positive rates and minimum distance which is the square root of the block length. Compared with the HP qLDPC codes in [11,30], the minimum distance upper bound of our codes can beat that of the HP qLDPC codes. Further, we show that qSCPC codes can be decoded in linear time and correct a much larger fraction of adversarial errors than quantum expander codes in [23,24].…”

Section: Introductionmentioning

confidence: 92%

From Generalization of Bacon-Shor Codes to High Performance Quantum LDPC Codes

Fan¹,

Li²,

Wang³

et al. 2021

Preprint

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We utilize a concatenation scheme to construct new families of quantum error correction codes that include the Bacon-Shor codes. We show that our scheme can lead to asymptotically good quantum codes while Bacon-Shor codes cannot. Further, the concatenation scheme allows us to derive quantum LDPC codes of distance Ω(N2/3/loglogN) which can improve Hastings’s recent result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman- Zémor distance balancing construction, our concatenation scheme can yield quantum LDPC codes with non-vanishing code rates and better minimum distance upper bound than the hypergraph product quantum LDPC codes. Finally, we derive a family of fast encodable and decodable quan- tum concatenated codes with parameters Q = [[N,Ω(√N),Ω(√N)]] and they also belong to the Bacon-Shor codes. We show that Q can be encoded very efficiently by circuits of size O(N) and depth O(√N), and can correct any adversarial error of weight up to half the minimum distance bound in O(√N) time. To the best of our knowledge, they are the most powerful quantum codes for correcting so many adversarial errors in sublinear time by far.

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Section: Introductionmentioning

confidence: 92%

From Generalization of Bacon-Shor Codes to High Performance Quantum LDPC Codes

Fan¹,

Li²,

Wang³

et al. 2021

Preprint

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“…The special case where where C and D are 1-complexes are also known as hypergraph products which were introduced in [12]. This was generalized for C being a complex of arbitrary finite length in [23].…”

Section: Double Complexes and Total Complexesmentioning

confidence: 99%

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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“…1) Compared to surface codes and color codes of similar code lengths, QLDPC codes boast better code rates with faulttolerant thresholds supported by low-complexity iterative decoding algorithms [1], [3], [10], [40].…”

Section: Quantum Ldpc Codes and Syndrome-based Iterative Decodingmentioning

confidence: 99%

“…Q UANTUM low-density parity-check (QLDPC) codes form a promising family of quantum error-correcting codes (QECCs) since they involve stabilizer checks of bounded and low weight, have minimum distance scaling better than the square root of the code length, and can be decoded efficiently using iterative decoding methods [1]- [10]. A specific construction of QLDPC codes is the hypergraph product construction [11], which combines two arbitrary classical linear codes and produces a QECC.…”

Section: Introductionmentioning

confidence: 99%

Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound

Raveendran,

Rengaswamy,

Rozpędek

et al. 2021

Preprint

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Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. In particular, several researchers have considered the individual code qubits being encoded with the continuous variable Gottesman-Kitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder -a hardwarefriendly min-sum algorithm (MSA) -utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.N. Raveendran, N. Rengaswamy and B. Vasić are with the

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Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates

Cited by 35 publications

References 47 publications

From Generalization of Bacon-Shor Codes to High Performance Quantum LDPC Codes

From Generalization of Bacon-Shor Codes to High Performance Quantum LDPC Codes

Balanced Product Quantum Codes

Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound

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