Small-distance qubit stabilizer code

Small-distance qubit stabilizer code

Description

A qubit stabilizer code that either detects or corrects errors on at most two subsystems, i.e., have distance \(\leq 5\).

All single error-correcting qubit stabilizer codes have been classified [1]. Qubit stabilizer codes for \(n < 10\) have been classified [2], all of which have small distance. Qubit stabilizer codes have been partially enumerated up to twelve qubits [3].

Cousins

Neural network quantum code— 13 inequivalent \([[9,3,3]]\) codes, along with others, have been found via reinforcement learning [4].
\([[23, 1, 7]]\) Quantum Golay code— The quantum Golay code can be punctured twice to obtain a \([[21,3,5]]\) code.
Hyperbolic color code— Many hyperbolic color codes have distance \(\leq 5\).

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Qubit stabilizer code

All single error-correcting qubit stabilizer codes have been classified [1]. Qubit stabilizer codes for \(n < 10\) have been classified [2], all of which have small distance. Qubit stabilizer codes have been partially enumerated up to twelve qubits [3].

Small-distance block quantum codeQECC Quantum

Small-distance qubit stabilizer code

Children

\([[6,1,3]]_{f}\) Vijay-Fu Majorana code

\([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code

Majorana box qubit

\([[6r,2r,2]]\) Ganti-Onunkwo-Young code

Quantum multi-dimensional parity-check (QMDPC) code

\([[n,n-2k,4]]\) Quantum cap code

Quantum cap codes tend to have a high rate and include codes with parameters \([[12,4,4]]\) [5], \([[12,2,4]]\), \([[20,10,4]]\), or \([[29,19,4]]\) [6].

\([[2^r, 2^r-r-2, 3]]\) Gottesman code

\([[54,6,5]]\) five-covered icosahedral code

Quantum repetition code

\([[2^r+r, 2^r-r-2, 3]]\) Ring CPC code

\([[10,1,2]]\) CSS code

\([[12,2,4]]\) carbon code

\([[12,2,2]]\) CSS code

\([[5,1,2]]\) rotated surface code

\([[5,1,3]]\) Five-qubit perfect code

\([[6,1,3]]\) Six-qubit stabilizer code

\([[6,4,2]]\) error-detecting code

\([[7,1,3]]\) bare code

\([[7,1,3]]\) twist-defect surface code

\([[8,2,2]]\) hyperbolic color code

\([[9,1,3]]\) Shor code

\([[9,3,3]]\) Quadric code

\([[16,4,3]]\) dodecahedral code

\([[11,1,5]]\) quantum dodecacode

\([[13,1,5]]\) cyclic code

Transverse-field Ising model (TFIM) code

\([[6k+2,3k,2]]\) Campbell-Howard code

\([[k+4,k,2]]\) H code

\([[3k + 8, k, 2]]\) triorthogonal code

\([[49,1,5]]\) triorthogonal code

\([[2^r-1,1,3]]\) simplex code

\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code

Ball color code

\([[16,6,4]]\) Tesseract color code

\([[17,1,5]]\) 4.8.8 color code

\([[30,8,3]]\) Bring code

\([[14,3,3]]\) Rhombic dodecahedron surface code

\([[9,1,3]]\) Surface-17 code

References

[1]: S. Yu, J. Bierbrauer, Y. Dong, Q. Chen, and C. H. Oh, “All the stabilizer codes of distance 3”, (2011) arXiv:0901.1968
[2]: A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
[3]: A. Dymarsky and A. Shapere, “Quantum stabilizer codes, lattices, and CFTs”, Journal of High Energy Physics 2021, (2021) arXiv:2009.01244 DOI
[4]: J. Olle, R. Zen, M. Puviani, and F. Marquardt, “Simultaneous Discovery of Quantum Error Correction Codes and Encoders with a Noise-Aware Reinforcement Learning Agent”, (2024) arXiv:2311.04750
[5]: J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini, and F. Pambianco, “The geometry of quantum codes”, Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial 6, 53 (2008) DOI
[6]: D. Bartoli, S. Marcugini, and F. Pambianco, “New quantum caps in PG(4,4)”, (2010) arXiv:0905.1059

Page edit log

Victor V. Albert (2025-01-09) — most recent

Cite as:

“Small-distance qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/small_distance_qubit_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small_distance_qubit_stabilizer.yml.

Small-distance qubit stabilizer code

Description

Cousins

Member of code lists

Primary Hierarchy

References

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