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.

The first example of an error basis based on a non-Abelian error group is due to S. Egner and consists of products of \(S\), Pauli, and Hadamard gates [1].

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.