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
The codes'' rate and distance are both separated from zero as block length goes to infinity. Rains shadow enumerators can be used to show that the distance of an asymptotically good QLDPC code should be bounded as \(d\leq n/3\) [3]; see Ref. [4]. AEL distance amplification [5,6] can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound [7; Corr. 5.3].Cousins
- Lattice stabilizer code— Chain complexes describing some QLDPC codes [8,9], and, more generally, CSS codes [10] can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality. 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 [9,11,12].
- Lattice subsystem code— An \([[n,k,d]]\) qubit stabilizer code can be converted into an order \([[O(\ell \delta n),k,\Omega(d/w)]]\) subsystem qubit stabilizer code with weight-three gauge operators via the wire-code mapping [13], which uses weight reduction. Here, \(w\) and \(\delta\) are the weight and degree of the input code's Tanner graph, while \(\ell\) is the length of the longest edge of a particular embedding of that graph. Applying this procedure to good QLDPC codes and using an embedding into \(D\)-dimensional Euclidean space yields \([[n^{1-1/D},\Theta(n),\Theta(n)]]\) lattice subsystem codes that saturate the subsystem BT bound [13].
- 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, of order \((10,40,4)\) , 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 [14].
- 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.
Primary Hierarchy
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]
- V. Guemard, “Lifts of quantum CSS codes”, (2024) arXiv:2404.16736
- [11]
- E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
- [12]
- X. Li, T.-C. Lin, and M.-H. Hsieh, “Transform Arbitrary Good Quantum LDPC Codes into Good Geometrically Local Codes in Any Dimension”, (2024) arXiv:2408.01769
- [13]
- N. Baspin and D. Williamson, “Wire Codes”, (2024) arXiv:2410.10194
- [14]
- 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