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 \( \langle\omega I, X, P\rangle^{\otimes n} \). Here, \(N\) is called the precision, \( \omega \) is a \( 2N \)th root of unity, and \( P = diag ( 1, \omega^2) \). 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 admit weighted-hypergraph representations on their affine computational-basis supports [1,2]. Conversely, weighted hypergraph states can be represented as XP stabilizer states, generally after embedding into a larger Hilbert space [1]. Weighted hypergraph states 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 admit logical dimension \(K=2^k\) (for some integer \(k\)) and can be mapped to a CSS code with the same diagonal logical operators and similar non-diagonal logical operators, while the latter can have arbitrary dimension, are non-additive, and resemble CWS codes [1]. Measurement of diagonal Pauli operators can be classically simulated efficiently on any XP code, but estimating outcome probabilities for general precision-\(4\) XP operators is NP-complete and measurement can take one outside the XP formalism [1].

Encoding

XP stabilizer states can be prepared by applying generalized controlled phase gates to a uniform superposition over their affine computational-basis support [1,2].

Cousins

Clifford-hierarchy stabilizer code— XP stabilizer codes at precision \(N=2^t\) are Clifford-hierarchy stabilizer codes, but the XP formalism also allows non-dyadic precision [1].
Qubit CSS code— Each XP-regular code can be mapped to a CSS code with the same diagonal logical operators and similar non-diagonal logical operators [1].
Codeword stabilized (CWS) code— The orbit representatives of XP codes play a similar role to the word operators of CWS codes, and non-XP-regular codes have a similar structure [1].
Tensor-network code— XP stabilizer codes can be understood through the Quantum Lego formalism [9].
Cubic theory code— The cubic theory code can be embedded into a larger Hilbert space such that all diagonal Clifford-hierarchy logical operators are represented by XP operators [10; Sec. 4.3].
Chen-Hsin invertible-order code— The Chen-Hsin invertible-order code can be embedded into a larger Hilbert space such that all diagonal Clifford-hierarchy logical operators are represented by XP operators [10; Sec. 4.3].
Binary dihedral PI code— Binary dihedral permutation invariant codewords form error spaces of XP stabilizer codes.
\([[2^D,D,2]]\) 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].
\([[8,1,2]]\) Shen-Wang-Cao code— The \([[8,1,2]]\) code is an XP-regular code that can be obtained via the XP stabilizer formalism applied to the \([[15,1,3]]\) Reed-Muller code [9].
\([[8,3,2]]\) Smallest interesting color code— As the \(D=3\) member of the hypercube-code family, the \([[8,3,2]]\) code can be viewed as an XP stabilizer code with precision \(N=8\) [1; Exam. 6.10].
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 [11,12].
\([[2^r-1,1,3]]\) 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].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Parents

Qubit code

Monomial stabilizer codeHamiltonian-based QECC Quantum

XP stabilizer generators are products of \(X\)-type Pauli strings and diagonal phase operators, so they are monomial matrices that generate finite groups [1,13]. Orbit representatives of XP codes and the coset structure of XP codewords realize the monomial stabilizer orbit basis [1].

XP stabilizer code

Children

XS stabilizer code

The XP stabilizer formalism reduces to the XS formalism at \(N=4\).

Qubit stabilizer codeFermion-into-qubit Quantum RM code QLDPC BB Homological Color Twist-defect color CDSC Twist-defect surface Kitaev surface

XP stabilizer codes reduce to qubit stabilizer codes for \(N=2\).

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]
M. Webster, “The XP Stabilizer Formalism”, PhD thesis, 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”, Quantum 9, 1876 (2025) arXiv:2310.19538 DOI
[10]
M. A. Webster, A. O. Quintavalle, and S. D. Bartlett, “Transversal diagonal logical operators for stabiliser codes”, New Journal of Physics 25, 103018 (2023) arXiv:2303.15615 DOI
[11]
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
[12]
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
[13]
M. V. den Nest, “A monomial matrix formalism to describe quantum many-body states”, New Journal of Physics 13, 123004 (2011) arXiv:1108.0531 DOI

Page edit log

Victor V. Albert (2026-08-26) — most recent
Victor V. Albert (2026-06-08)
Victor V. Albert (2022-04-19)
Muhammad Junaid Aftab (2022-04-15)

Cite as:

“XP stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xp_stabilizer, arXiv:2606.11484

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