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].

Cousins

Neural network quantum code— 13 inequivalent \([[9,3,3]]\) codes, along with others, have been found via reinforcement learning [2].
\([[23, 1, 7]]\) 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

Small-distance block quantum codeQECC Quantum

Small-distance qubit stabilizer code

Children

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

Majorana box qubit

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

Quantum multi-dimensional parity-check (QMDPC) code

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

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

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

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

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

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

Ball color code

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

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

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

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

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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