## 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 |
---|---|---|---|

qubits | \(X,Z\) checks | qubits | |

\(X,Z\) checks | qubits | ||

\(X\) checks | qubits | \(Z\) checks |

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

## Rate

## 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 [8,9]. 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 \(n\) [10].
- Quantum maximum-distance-separable (MDS) code — AEL distance amplification [5,6] can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound [7; Corr. 5.3].
- 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 [11].
- 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]
- E. M. Rains, “Quantum shadow enumerators”, (1997) arXiv:quant-ph/9611001
- [4]
- D. Miller et al., “Experimental measurement and a physical interpretation of quantum shadow enumerators”, (2024) arXiv:2408.16914
- [5]
- N. Alon, J. Edmonds, and M. Luby, “Linear time erasure codes with nearly optimal recovery”, Proceedings of IEEE 36th Annual Foundations of Computer Science DOI
- [6]
- N. Alon and M. Luby, “A linear time erasure-resilient code with nearly optimal recovery”, IEEE Transactions on Information Theory 42, 1732 (1996) DOI
- [7]
- T. Bergamaschi, L. Golowich, and S. Gunn, “Approaching the Quantum Singleton Bound with Approximate Error Correction”, (2022) arXiv:2212.09935
- [8]
- M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
- [9]
- T.-C. Lin, A. Wills, and M.-H. Hsieh, “Geometrically Local Quantum and Classical Codes from Subdivision”, (2024) arXiv:2309.16104
- [10]
- E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
- [11]
- T.-C. Lin and M.-H. Hsieh, “Good quantum LDPC codes with linear time decoder from lossless expanders”, (2022) arXiv:2203.03581

## Page edit log

- Victor V. Albert (2022-06-24) — most recent

## Cite as:

“Good QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/good_qldpc