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. 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.

Parent

Cousins

  • Topological code — Chain complexes describing some good QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [1]. Applying this procedure to good QLDPC codes yiels geometrically local \([[n,n^{1-2/D},n^{1-1/D}]]\) codes in \(D\) spatial dimensions, up to corrections poly-logarithmic in \(n\) [2].
  • Dinur-Hsieh-Lin-Vidick (DHLV) code — DHLV code construction yields asymptotically good QLDPC codes.
  • Quantum Tanner code — Quantum Tanner code construction yields asymptotically good QLDPC codes.
  • Expander lifted-product code — Lifted products of certain classical Tanner codes are the first asymptotically good QLDPC codes.

References

[1]
M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
[2]
E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
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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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Cite as:

“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/tree/main/codes/quantum/properties/block/qldpc/good_qldpc.yml.