XP stabilizer code

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 [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

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

Qubit code

Children

XS stabilizer code — The XP stabilizer formalism reduces to the XS formalism at \(N=4\).
\([[2^D,D,2]]\) hypercube quantum code — The \(D\)th hypercube quantum code can be viewed as an XP stabilizer code with precision \(N = 2^D\) [1; Exam. 6.10].
Qubit stabilizer code — XP stabilizer codes reduce to qubit stabilizer codes for \(N=2\).
\([[2^r-1,1,3]]\) 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.
Tensor-network code — XP stabilizer codes can be understood through the Quantum Lego formalism [9].
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 [10,11].

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, L. Hartmann, M. Hein, M. Lewenstein, and H.-J. Briegel, “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, W. Dür, J. Eisert, R. Raussendorf, M. V. den Nest, and H.-J. Briegel, “Entanglement in Graph States and its Applications”, (2006) arXiv:quant-ph/0602096
[5]: S. Anders, M. B. Plenio, W. Dür, F. Verstraete, and H.-J. Briegel, “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, M. Huber, D. Bruß, and C. Macchiavello, “Quantum hypergraph states”, New Journal of Physics 15, 113022 (2013) arXiv:1211.5554 DOI
[7]: O. Gühne, M. Cuquet, F. E. S. Steinhoff, T. Moroder, M. Rossi, D. Bruß, B. Kraus, and C. Macchiavello, “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, D. J. Upchurch, S. N. Walck, and C. D. Yetter, “Local unitary symmetries of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 48, 095301 (2015) arXiv:1410.3904 DOI
[9]: R. Shen, Y. Wang, and C. Cao, “Quantum Lego and XP Stabilizer Codes”, (2023) arXiv:2310.19538
[10]: 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
[11]: 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

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