XP stabilizer code

XP stabilizer code[1]

Alternative names: Weighted 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].

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].

Member of code lists

Quantum codes

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Parents

Qubit code

XP stabilizer 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 codeBB Homological Fermion-into-qubit Quantum Reed-Muller Color Twist-defect color CDSC Twist-defect surface Kitaev surface

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].

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.