XP stabilizer code[1] 

Also known as Weighed hypergraph code.

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 [35] and hypergraph states [68]. 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

Initialization of all qubits in the \(|+\rangle\) state and action of generalied controlled \(Z\) gates on multi-edges of the underlying hypergraph [1,2].

Parent

Children

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 [35] and hypergraph states [68]. 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
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Zoo Code ID: xp_stabilizer

Cite as:
“XP stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xp_stabilizer
BibTeX:
@incollection{eczoo_xp_stabilizer, title={XP stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/xp_stabilizer} }
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“XP stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xp_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/nonstabilizer/xp_stabilizer/xp_stabilizer.yml.