Good QLDPC code 


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.





expander lifted-product


\(X,Z\) checks


quantum Tanner

\(X,Z\) checks



\(X\) checks


\(Z\) checks

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

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



  • 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.


S. Hoory, N. Linial, and A. Wigderson, “Expander graphs and their applications”, Bulletin of the American Mathematical Society 43, 439 (2006) DOI
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
T.-C. Lin, A. Wills, and M.-H. Hsieh, “Geometrically Local Quantum and Classical Codes from Subdivision”, (2023) arXiv:2309.16104
E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
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.
@incollection{eczoo_good_qldpc, title={Good QLDPC code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Good QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.