Quantum Tanner code[1]

Description

Stub.

Protection

Minimum distance bound obtained using robustness of dual tensor-product codes [2].

Rate

Asymptotically good QLDPC codes.

Decoding

Linear-time decoder similar to the small-set-flip decoder for quantum expander codes [3].Linear-time decoder [4].Logarithmic-time decoder [2].

Realizations

Used to obtain explicit lower bounds in the sum-of-squares game [5].States that, on average, achieve small violations of check operators for quantum Tanner codes require a circuit of non-constant depth to make. They are used in the proof [6] of the No low-energy trivial states (NLTS) conjecture [7].

Notes

For details, see talk by A. Leverrier.

Parent

Child

  • Rotated surface code — Specializing the quantum Tanner construction to the surface code yields the rotated surface code [8][9].

Cousins

  • Good QLDPC code — Quantum Tanner code construction yields asymptotically good QLDPC codes.
  • Tanner code — Tanner codes are used in constructing quantum Tanner codes.
  • Tensor-product code — Tensor codes are used in constructing quantum Tanner codes.
  • Expander lifted-product code — Quantum Tanner codes are an attempt to construct asymptotically good QLDPC codes that are similar to but simpler than expander lifted-product codes; see Ref. [4] for connection between the codes.
  • CSS classical product code — A \([[512,174,8]]\) classical-product code performed well [10] against erasure and depolarizing noise when compared to a member of an asymptotically good quantum Tanner code family.
  • Left-right Cayley complex code — Applying the CSS construction to two left-right Cayley complex codes yields quantum Tanner codes, and one can simultaneously prove a linear distance for the quantum code and local testability for one of its constituent classical codes [1].

References

[1]
Anthony Leverrier and Gilles Zémor, “Quantum Tanner codes”. 2202.13641
[2]
Anthony Leverrier and Gilles Zémor, “Decoding quantum Tanner codes”. 2208.05537
[3]
Shouzhen Gu, Christopher A. Pattison, and Eugene Tang, “An efficient decoder for a linear distance quantum LDPC code”. 2206.06557
[4]
Anthony Leverrier and Gilles Zémor, “Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes”. 2206.07571
[5]
Max Hopkins and Ting-Chun Lin, “Explicit Lower Bounds Against $Ω(n)$-Rounds of Sum-of-Squares”. 2204.11469
[6]
Anurag Anshu, Nikolas P. Breuckmann, and Chinmay Nirkhe, “NLTS Hamiltonians from good quantum codes”. 2206.13228
[7]
M. H. Freedman and M. B. Hastings, “Quantum Systems on Non-$k$-Hyperfinite Complexes: A Generalization of Classical Statistical Mechanics on Expander Graphs”. 1301.1363
[8]
Nikolas P. Breuckmann, private communication, 2022
[9]
Anthony Leverrier, Mapping the toric code to the rotated toric code, 2022.
[10]
Dimiter Ostrev et al., “Classical product code constructions for quantum Calderbank-Shor-Steane codes”. 2209.13474
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Zoo code information

Internal code ID: quantum_tanner

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Zoo Code ID: quantum_tanner

Cite as:
“Quantum Tanner code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_tanner
BibTeX:
@incollection{eczoo_quantum_tanner, title={Quantum Tanner code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_tanner} }
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Permanent link:
https://errorcorrectionzoo.org/c/quantum_tanner

Cite as:

“Quantum Tanner code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_tanner

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/qldpc/quantum_tanner.yml.