Good QLDPC code 

Description

Also called asymptotically good QLDPC codes. A family of QLDPC codes \([[n_i,k_i,d_i]]\) whose asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive.

The first good QLDPC codes are families constructed by applying the CSS construction to classical Tanner codes on expander graphs [1]. The three constructions are closely related, assigning qubits and check operators to vertices, edges, and faces of a particular graph called the left-right Cayley complex.

Code

vertices

edges

faces

expander lifted-product

qubits

\(X,Z\) checks

qubits

quantum Tanner

\(X,Z\) checks

qubits

Dinur-Hsieh-Lin-Vidick

\(X\) checks

qubits

\(Z\) checks

Table I: Assignment of qubits and checks for three asymptotically good QLDPC codes.

See [2; Fig. 12] for more relationships between the constructions.

Parent

Cousins

  • Lattice stabilizer code — Chain complexes describing some good QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [3,4]. Applying this procedure to good QLDPC codes yiels \([[n,n^{1-2/D},n^{1-1/D}]]\) lattice stabilizer codes in \(D\) spatial dimensions that saturate the BPT bound, up to corrections poly-logarithmic in \(n\) [5].
  • Layer code — Layer code parameters, \([[n,\Theta(n^{1/3}),\Theta(n^{1/3})]]\), achieve the BPT bound in 3D when asymptotically good QLDPC codes are used in the construction.
  • Dinur-Hsieh-Lin-Vidick (DHLV) code — DHLV code construction yields asymptotically good QLDPC codes.
  • Lossless expander balanced-product code — Taking a balanced product of two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [6].
  • Quantum Tanner code — Quantum Tanner code construction yields asymptotically good QLDPC codes.
  • Expander LP code — Lifted products of certain classical Tanner codes are the first asymptotically good QLDPC codes.

References

[1]
S. Hoory, N. Linial, and A. Wigderson, “Expander graphs and their applications”, Bulletin of the American Mathematical Society 43, 439 (2006) DOI
[2]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
[3]
M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
[4]
T.-C. Lin, A. Wills, and M.-H. Hsieh, “Geometrically Local Quantum and Classical Codes from Subdivision”, (2023) arXiv:2309.16104
[5]
E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
[6]
T.-C. Lin and M.-H. Hsieh, “Good quantum LDPC codes with linear time decoder from lossless expanders”, (2022) arXiv:2203.03581
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Zoo Code ID: good_qldpc

Cite as:
“Good QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/good_qldpc
BibTeX:
@incollection{eczoo_good_qldpc, title={Good QLDPC code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/good_qldpc} }
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“Good QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/good_qldpc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/stabilizer/qldpc/good_qldpc.yml.