XS stabilizer code[1]
Description
A type of stabilizer code where stabilizer generators are elements of the group \( \{\alpha I, X, \sqrt{Z}]\}^{\otimes n} \), with \( \sqrt{Z} = \text{diag} (1, i)\). The codespace is a joint \(+1\) eigenspace of a set of stabilizer generators, which need not commute to define a valid codespace.
The phases in the qubit-basis expansion of an XS stabilizer state are polynomials of degree three or below [1].
Parent
- XP stabilizer code — The XP stabilizer formalism reduces to the XS formalism at \(N=4\).
Children
- \([[8,3,2]]\) CSS code — The \([[8,3,2]]\) code can be viewed as an XS stabilizer code [2; Exam. 6.10].
- \([[15,1,3]]\) quantum Reed-Muller code — The \([[15,1,3]]\) code can be viewed as an XS stabilizer code [2; Exam. 6.4].
Cousins
- Abelian TQD stabilizer code — TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [1].
- 3D color code — The 3D color code on a particular lattice admits XS stabilizers; see talk by M. Kesselring at the 2020 FTQC conference.
References
- [1]
- X. Ni, O. Buerschaper, and M. Van den Nest, “A non-commuting stabilizer formalism”, Journal of Mathematical Physics 56, (2015) arXiv:1404.5327 DOI
- [2]
- M. A. Webster, B. J. Brown, and S. D. Bartlett, “The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases”, Quantum 6, 815 (2022) arXiv:2203.00103 DOI
Page edit log
- Victor V. Albert (2022-04-19) — most recent
Cite as:
“XS stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xs_stabilizer