Knill code[1] 

Also known as Clifford code.

Description

A group representation code whose projection is onto an irrep of a normal subgroup of the group formed by a nice error basis. Knill codes yield stabilizer-like codes based on error bases that are non-Pauli but that nevertheless maintain many of the useful features of Pauli-type bases.

Nice error basis: A nice error basis [13] for an \(q\)-dimensional vector space is a set \(\{E_g~,~g\in G\}\) of unitary operators, where \(G\) is a (not necessarily Abelian) group of order \(q^2\), and where \begin{align} \text{tr}(E_{g})&=q\delta^{G}_{g,1}\tag*{(1)}\\ E_{g}E_{h}&=\omega_{g,h}E_{gh} \tag*{(2)}\end{align} for all group elements \(g,h\). Above, \(\delta^{G}_{g,1}\) is the group Kronecker-delta function. A basis is called very nice if \(\omega\) is a root of unity. This definition can naturally be extended to continuous groups.

Notes

Catalogue of nice error bases, managed by A. Klappenecker and M. Rotteler, is available on this website.Many Knill codes are qubit stabilizer codes [4]. A table of non-stabilizer Knill codes is available in Ref. [5]. An infinite family is constructed in Ref. [6].

Parent

Child

  • Stabilizer code — Stabilizer codes are Knill codes whose nice error basis is either the Pauli strings, modular-qudit Pauli strings, Galois-qudit Pauli strings, oscillator displacement operators, or rotor generalized Pauli strings.

Cousin

References

[1]
E. Knill, “Group Representations, Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608049
[2]
E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608048
[3]
A. Klappenecker and M. Roetteler, “Beyond Stabilizer Codes I: Nice Error Bases”, (2001) arXiv:quant-ph/0010082
[4]
A. Klappenecker and M. Roetteler, “Beyond Stabilizer Codes II: Clifford Codes”, (2001) arXiv:quant-ph/0010076
[5]
Klappenecker, Andreas, and Martin Rötteler. "On the structure of nonstabilizer Clifford codes." Quantum Information & Computation 4.2 (2004): 152-160.
[6]
H. Manabu and H. Imai, “Non stabilizer Clifford codes with qupits”, (2004) arXiv:quant-ph/0402060
[7]
A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
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Zoo Code ID: knill

Cite as:
“Knill code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/knill
BibTeX:
@incollection{eczoo_knill, title={Knill code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/knill} }
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“Knill code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/knill

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/group_rep/knill.yml.