Small-distance block quantum code

Small-distance block quantum code

Description

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

See Refs. [1–3] for small-distance codes.

Cousins

Small-distance block code— Small-distance block quantum codes are quantum analogues of small-distance block codes.
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 \(((27,2,5))\).
Quantum plane-curve code— The quantum plane-curve code for the Hermitian curve \(y^3 + y = x^4\) is a \([[27,13,4]]_3\) qutrit code.

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Operator-algebra QECC (OAQECC)

Quantum error-correcting code (QECC)

Block quantum code

Parents

Block quantum code

Small-distance block quantum code

Children

Kitaev current-mirror qubit code

\([[3,1,2]]_{\mathbb{Z}}\) Three-rotor code

Zero-pi qubit code

\([[5,1,3]]_{\mathbb{Z}}\) 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.

Self-complementary qubit code

Self-complementary quantum codes are at most distance-two since the minimal number of computational basis states in a logical state is two [5; Thm. 4, contrapositive].

\(((2m+1,3 \times 2^{2m-3},2))\) Rains code

\(((9,2,3))\) Ruskai code

\(((n,2,2))\) Bravyi-Lee-Li-Yoshida PI code

\(((10,24,3))\) qubit code

The \(((10,24,3))\) qubit code can be combined to form an infinite family of distance-three qubit codes whose logical dimension is \(50\%\) larger than that of the optimal stabilizer code [6].

Numerically optimized four-qubit AD code

\(((6,2,3))\) transversal-\(\mathbb{Z}_{10}\) code

\(((8,2,3))\) Plenio-Vedral-Knight CE code

\(((9,12,3))\) qubit code

The \(((9,12,3))\) qubit code can be combined to form an infinite family of distance-three qubit codes whose logical dimension is \(50\%\) larger than that of the optimal stabilizer code [6].

Small-distance qubit stabilizer code

Subset-Sum-Linear-Programming (SS-LP) 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

\([[3,1,2]]_3\) Three-qutrit code

\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code

\([[9,1,5]]_3\) quantum Glynn code

\(((3,2,2))_3\) Three-qutrit single-deletion 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 \(d \geq 2\).

References

[1]: E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[2]: D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C. H. Oh, “Graphical nonbinary quantum error-correcting codes”, Physical Review A 78, (2008) arXiv:0801.0831 DOI
[3]: W.-T. Yen and L.-Y. Hsu, “Optimal Nonadditive Quantum Error-Detecting Code”, (2009) arXiv:0901.1353
[4]: E. Kubischta and I. Teixeira, “Quantum Codes and Irreducible Products of Characters”, (2024) arXiv:2403.08999
[5]: S. Bravyi, D. Lee, Z. Li, and B. Yoshida, “How Much Entanglement Is Needed for Quantum Error Correction?”, Physical Review Letters 134, (2025) arXiv:2405.01332 DOI
[6]: S. Yu, Q. Chen, and C. H. Oh, “Two infinite families of nonadditive quantum error-correcting codes”, (2009) arXiv:0901.1935

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/small_distance_quantum.yml.

Small-distance block quantum code

Description

Cousins

Member of code lists

Primary Hierarchy

References

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