Quantum Tanner codes

• URL: http://arxiv.org/abs/2202.13641v3
• Date: Fri, 16 Sep 2022 10:36:37 GMT
• Title: Quantum Tanner codes
• Authors: Anthony Leverrier, Gilles Z\'emor
• Abstract summary: We prove a theorem that simultaneously gives a growing minimum distance for the quantum code and recovers the local testability of the Dinur et al. code.
• Score: 0.38073142980732994
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Decoding Quasi-Cyclic Quantum LDPC Codes [23.22566380210149]
  Quantum low-density parity-check (qLDPC) codes are an important component in the quest for fault tolerance. Recent progress on qLDPC codes has led to constructions which are quantumally good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice.
  arXiv  Detail & Related papers  (2024-11-07T06:25:27Z)
• List Decodable Quantum LDPC Codes [49.2205789216734]
  We give a construction of Quantum Low-Density Parity Check (QLDPC) codes with near-optimal rate-distance tradeoff. We get efficiently list decodable QLDPC codes with unique decoders.
  arXiv  Detail & Related papers  (2024-11-06T23:08:55Z)
• SSIP: automated surgery with quantum LDPC codes [55.2480439325792]
  We present Safe Surgery by Identifying Pushouts (SSIP), an open-source lightweight Python package for automating surgery between qubit CSS codes. Under the hood, it performs linear algebra over $mathbbF$ governed by universal constructions in the category of chain complexes. We show that various logical measurements can be performed cheaply by surgery without sacrificing the high code distance.
  arXiv  Detail & Related papers  (2024-07-12T16:50:01Z)
• Small Quantum Codes from Algebraic Extensions of Generalized Bicycle Codes [4.299840769087443]
  Quantum LDPC codes range from the surface code, which has a vanishing encoding rate, to very promising codes with constant encoding rate and linear distance. We devise small quantum codes that are inspired by a subset of quantum LDPC codes, known as generalized bicycle (GB) codes.
  arXiv  Detail & Related papers  (2024-01-15T10:38:13Z)
• Improved rate-distance trade-offs for quantum codes with restricted connectivity [34.95121779484252]
  We study how the connectivity graph associated with a quantum code constrains the code parameters. We establish a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph.
  arXiv  Detail & Related papers  (2023-07-06T20:38:34Z)
• Spatially-Coupled QDLPC Codes [3.6622737533847936]
  We describe toric codes as quantum counterparts of classical spatially-coupled (2D-SC) codes. We introduce spatially-coupled quantum LDPC (SC-QLDPC) codes as a class of convolutional LDPC codes. This paper focuses on QLDPC codes with rate less than 1/10, but we construct 2D-SC HGP codes with small memories, higher rates (about 1/3), and superior thresholds.
  arXiv  Detail & Related papers  (2023-04-29T00:57:57Z)
• Homological Quantum Rotor Codes: Logical Qubits from Torsion [51.9157257936691]
  homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code. We show that the $0$-$pi$-qubit as well as Kitaev's current-mirror qubit are indeed small examples of such codes.
  arXiv  Detail & Related papers  (2023-03-24T00:29:15Z)
• Neural Belief Propagation Decoding of Quantum LDPC Codes Using Overcomplete Check Matrices [60.02503434201552]
  We propose to decode QLDPC codes based on a check matrix with redundant rows, generated from linear combinations of the rows in the original check matrix. This approach yields a significant improvement in decoding performance with the additional advantage of very low decoding latency.
  arXiv  Detail & Related papers  (2022-12-20T13:41:27Z)
• Decoding quantum Tanner codes [0.38073142980732994]
  We introduce sequential and parallel decoders for quantum Tanner codes. Our decoders provably correct arbitrary errors of weight linear in the code length. The same decoders are easily adapted to the expander lifted product codes of Panteleev and Kalachev.
  arXiv  Detail & Related papers  (2022-08-10T19:50:18Z)
• Morphing quantum codes [77.34726150561087]
  We morph the 15-qubit Reed-Muller code to obtain the smallest known stabilizer code with a fault-tolerant logical $T$ gate. We construct a family of hybrid color-toric codes by morphing the color code.
  arXiv  Detail & Related papers  (2021-12-02T17:43:00Z)
• KO codes: Inventing Nonlinear Encoding and Decoding for Reliable Wireless Communication via Deep-learning [76.5589486928387]
  Landmark codes underpin reliable physical layer communication, e.g., Reed-Muller, BCH, Convolution, Turbo, LDPC and Polar codes. In this paper, we construct KO codes, a computationaly efficient family of deep-learning driven (encoder, decoder) pairs. KO codes beat state-of-the-art Reed-Muller and Polar codes, under the low-complexity successive cancellation decoding.
  arXiv  Detail & Related papers  (2021-08-29T21:08:30Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.