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 |
Parent
Cousins
- Topological code — Chain complexes describing some good QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [2]. 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 \(n\) [3].
- 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 [4].
- 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]
- M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
- [3]
- E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
- [4]
- 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