[Jump to code hierarchy]

Knill code[1]

Alternative names: 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 [1–3] 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]. Certain nice error bases have been classified and are related to the braid group [4].

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 [5]. A table of non-stabilizer Knill codes is available in Ref. [6]. An infinite family is constructed in Ref. [7].

Cousin

Member of code lists

Primary Hierarchy

Quantum Domain
Quantum code
Operator-algebra QECC (OAQECC)
Quantum error-correcting code (QECC)
Group-representation code
Parents
Group-representation code
Knill codes project onto a single irrep of the normalizer of a normal subgroup of the group formed by a nice error basis [1; Lemma 3.1].
Knill code
Children
Stabilizer codeBosonic stabilizer Analog stabilizer Quantum lattice CSS QLDPC Generalized homological-product Lattice stabilizer Fracton stabilizer BB Homological Fermion-into-qubit Color Twist-defect color CDSC Twist-defect surface Kitaev surface Quantum QR code Quantum Reed-Muller
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.

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]
Y. Zhang, “Abstract Error Groups Via Jones Unitary Braid Group Representations at q=i”, (2009) arXiv:0902.0383
[5]
A. Klappenecker and M. Roetteler, “Beyond Stabilizer Codes II: Clifford Codes”, (2001) arXiv:quant-ph/0010076
[6]
Klappenecker, Andreas, and Martin Rötteler. "On the structure of nonstabilizer Clifford codes." Quantum Information & Computation 4.2 (2004): 152-160.
[7]
H. Manabu and H. Imai, “Non stabilizer Clifford codes with qupits”, (2004) arXiv:quant-ph/0402060
[8]
A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

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} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/knill

Cite as:

“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.