XP stabilizer code[1]
Description
The XP Stabilizer formalism is a generalization of the XS and Pauli stabilizer formalisms, with stabilizer generators taken from the group \( \mathsf{BD}_{2N}^{\otimes n} = \{\omega I, X, P\}^{\otimes n} \), which is the tensor product of the binary dihedral group of order \(8N\). Here, \(N\) is called the precision, \( \omega \) is a \( 2N \)th root of unity, and \( P = \text{diag} ( 1, \omega^2) \). The codespace is a \(+1\) eigenspace of a set of XP stabilizer generators, which need not commute to define a valid codespace.
XP stabilizer codes are classified into XP-regular and XP-non-regular, where the former can be mapped to a CSS code with similar logical operator structure.
Parent
Children
- XS stabilizer code — The XP stabilizer formalism reduces to the XS formalism at \(N=4\).
- \([[2^D,D,2]]\) hypercube code — The \(D\)th hypercube code can be viewed as an XP stabilizer code with precision \(N = 2^D\) [1; Exam. 6.10].
- Qubit stabilizer code — The XP stabilizer formalism reduces to the Pauli formalism at \(N=2\).
- \([[2^r-1, 1, 3]]\) simplex code — Each \([[2^r-1, 1, 3]]\) simplex code can be viewed as an XP stabilizer code with precision \(N = 2^{r-2}\) [1; Exam. 6.4].
Cousins
- Qubit CSS code — Each XP-regular code can be mapped to a CSS code with a similar logical operator structure [1].
- Codeword stabilized (CWS) code — The orbit representatives of XP codes play a similar role to the word operators of CWS codes.
- Binary dihedral permutation-invariant code — Binary dihedral permutation invariant codewords form error spaces of XP stabilizer codes.
References
- [1]
- 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
- Muhammad Junaid Aftab (2022-04-15)
Cite as:
“XP stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xp_stabilizer