# Knill code[1]

## 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].

## Notes

## Parent

- 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].

## 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

- Subsystem QECC — Subsystem Knill codes can be formulated [7].

## 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

## Page edit log

- Victor V. Albert (2024-04-04) — most recent

## Cite as:

“Knill code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/knill