Description
A block quantum code on \(n\) subsystems 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]. See Refs. [2,3] for other small-distance stabilizer codes.
Parent
Children
- Kitaev current-mirror qubit code
- Three-rotor code
- Zero-pi qubit code
- Five-rotor code
- \([[10,1,4]]_{G}\) tenfold code
- \([[4,2,2]]_{G}\) four group-qudit code
- \([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code
- \([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code
- Perfect quantum code — All non-trivial perfect codes have distance three.
- Majorana box qubit
- \(((10,24,3))\) qubit code
- \(((9,12,3))\) qubit code
- \(((5+2r,3\times 2^{2r+1},2))\) Rains code
- Smolin-Smith-Wehner (SSW) code
- \([[7, 1:1, 3]]\) hybrid stabilizer code
- \([[8, 2:1, 3]]\) hybrid stabilizer code
- \(((9,2,3))\) Ruskai code
- \(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code
- Quantum multi-dimensional parity-check (QMDPC) code
- \([[2^r, 2^r-r-2, 3]]\) Gottesman code
- \([[12,2,4]]\) carbon code
- \([[10,1,2]]\) CSS code
- \([[11,1,5]]\) quantum dodecacode
- \([[13,1,5]]\) cyclic code
- \([[5,1,2]]\) rotated surface code
- \([[6,1,3]]\) Six-qubit stabilizer code
- \([[6,2,2]]\) \(C_6\) code
- \([[6,4,2]]\) error-detecting code
- \([[8,2,2]]\) hyperbolic color code
- \([[7,1,3]]\) twist-defect surface 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
- \([[30,8,3]]\) Bring code
- Surface-17 code
- Rhombic dodecahedron surface code
- \(((3,6,2))_{\mathbb{Z}_6}\) Euler code
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
- \([[11,1,5]]_3\) qutrit Golay code
- Three-qutrit code
- \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
- \([[9,1,5]]_3\) quantum Glynn code
- \([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code
- \([[9m-k,k,2]]_3\) triorthogonal code
- \(((n,1+n(q-1),2))_q\) union stabilizer code
- \([[5,1,3]]_q\) Galois-qudit code
- \([[6,2,3]]_{q}\) code
- \([[7,3,3]]_{q}\) code
- Twisted \(1\)-group code — All twisted \(1\)-group codes have a distance \(d \geq 2\).
Cousins
- Small-distance block code
- Group-representation code — See Ref. [4] for tables of distance-two codes with various families of transversal gates.
- Binary dihedral PI code — The first and second families of binary dihedral PI codes have distance three, and the third family has the member \(((27,2,5))\).
- Hyperbolic color code — Many hyperbolic color codes have distance \(\leq 6\).
References
- [1]
- S. Yu et al., “All the stabilizer codes of distance 3”, (2011) arXiv:0901.1968
- [2]
- E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
- [3]
- D. Hu et al., “Graphical nonbinary quantum error-correcting codes”, Physical Review A 78, (2008) arXiv:0801.0831 DOI
- [4]
- E. Kubischta and I. Teixeira, “Quantum Codes and Irreducible Products of Characters”, (2024) arXiv:2403.08999
Page edit log
- Victor V. Albert (2023-02-01) — most recent
Cite as:
“Small-distance block quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/small_distance_quantum