# 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 \( \mathsf{BD}_{2N}^{\otimes n} = \langle\omega I, X, P\rangle^{\otimes n} \), 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 states are in one-to-one correspondence with weighted hypergraph states [1,2], which generalize both weighted graph states [3–5] and hypergraph states [6–8]. XP stabilizer codes are classified into XP-regular and XP-non-regular, where the former admits logical dimension \(K=2^k\) (for some integer \(k\)) and can be mapped to a CSS code with similar logical operator structure.

## Encoding

## 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 PI code — Binary dihedral permutation invariant codewords form error spaces of XP stabilizer codes.
- Cluster-state code — XP stabilizer states are in one-to-one correspondence with weighted hypergraph states [1,2], which generalize both weighted graph states [3–5] and hypergraph states [6–8]. The latter can also be utilized in MBQC schemes [9,10].

## 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
- [2]
- Webster, Mark. The XP Stabilizer Formalism. Dissertation, University of Sydney, 2023.
- [3]
- W. Dür et al., “Entanglement in Spin Chains and Lattices with Long-Range Ising-Type Interactions”, Physical Review Letters 94, (2005) arXiv:quant-ph/0407075 DOI
- [4]
- M. Hein et al., “Entanglement in Graph States and its Applications”, (2006) arXiv:quant-ph/0602096
- [5]
- S. Anders et al., “Ground-State Approximation for Strongly Interacting Spin Systems in Arbitrary Spatial Dimension”, Physical Review Letters 97, (2006) arXiv:quant-ph/0602230 DOI
- [6]
- M. Rossi et al., “Quantum hypergraph states”, New Journal of Physics 15, 113022 (2013) arXiv:1211.5554 DOI
- [7]
- O. Gühne et al., “Entanglement and nonclassical properties of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 47, 335303 (2014) arXiv:1404.6492 DOI
- [8]
- D. W. Lyons et al., “Local unitary symmetries of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 48, 095301 (2015) arXiv:1410.3904 DOI
- [9]
- M. Gachechiladze, O. Gühne, and A. Miyake, “Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states”, Physical Review A 99, (2019) arXiv:1805.12093 DOI
- [10]
- Y. Takeuchi, T. Morimae, and M. Hayashi, “Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements”, Scientific Reports 9, (2019) 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