Modular-qudit cluster-state code

Modular-qudit cluster-state code [1]

Alternative Names: Modular-qudit graph-state code.

Description

A code based on a modular-qudit cluster state.

Modular-qudit cluster states are modular-qudit stabilizer states defined on a graph. There is one modular-qudit stabilizer generator $S_v$ per graph vertex $v$ of the form [1; Eq. (1)] \begin{align} S_v = X^{\dagger}_{v} \prod_{w\in N(v)} Z_w~,\tag*{$(1)$} \end{align}where the neighborhood $N(v)$ is the set of vertices which share an edge with $v$.

Encoding

Operators forming the information group can be used to track how logical information is encoded [2].

Gates

1D modular-qudit cluster states [1,3] are resources for universal MBQC.

Realizations

Quantum computation with cluster states has been realized using photons in the time and frequency domains [4].

Cousins

Symmetry-protected topological (SPT) code— Qudit cluster states defined on 1D lattices are representatives of various SPT phases [5].
1D lattice stabilizer code— Qudit cluster states defined on 1D lattices are representatives of various 1D SPT phases [5].
Perfect-tensor code— MDS codes can be used to obtain cluster states that are AME with minimal support [6–11].
$[[11,1,5]]_3$ qutrit Golay code— The qutrit Golay code can be realized as a modular-qudit cluster-state code [12; Fig. 2].

Member of code lists

Primary Hierarchy

Quantum Domain

Modular-qudit Kingdom

Modular-qudit codeQECC Quantum

Modular-qudit USt code

Modular-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

Parents

Modular-qudit stabilizer code

Modular-qudit cluster-state codes are particular modular-qudit stabilizer codes. Any modular-qudit stabilizer code is equivalent to a graph quantum code for $G=\mathbb{Z}_q$ via a single-modular-qudit Clifford circuit [13] (see also [14]).

Modular-qudit CWS codeQECC Quantum

A type of modular-qudit cluster-state code can be built from a modular-qudit cluster state by applying the modular-qudit CWS construction using a linear $q$-ary code, in which codewords are obtained by applying modular-qudit $Z$-type operators defined by the code to the modular-qudit cluster state; see, e.g., Ref. [15].

Graph quantum codeStabilizer Hamiltonian-based QECC Quantum

Graph quantum codes for $G=\mathbb{Z}_q$ reduce to modular-qudit cluster-state codes.

Hopf-algebra cluster-state codeQECC Quantum

Hopf-algebra cluster-state codes reduce to modular-qudit cluster-state codes when the Hopf algebra reduces to the group $\mathbb{Z}_q$.

Modular-qudit cluster-state code

Children

Cluster-state code

Modular-qudit cluster-state codes reduce to cluster-state codes for $q=2$.

References

[1]
D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, “Quantum computation based ond-level cluster state”, Physical Review A 68, (2003) arXiv:quant-ph/0304054 DOI
[2]
V. Gheorghiu, S. Y. Looi, and R. B. Griffiths, “Location of quantum information in additive graph codes”, Physical Review A 81, (2010) arXiv:0912.2017 DOI
[3]
S. Clark, “Valence bond solid formalism ford-level one-way quantum computation”, Journal of Physics A: Mathematical and General 39, 2701 (2006) arXiv:quant-ph/0512155 DOI
[4]
C. Reimer et al., “High-dimensional one-way quantum processing implemented on d-level cluster states”, Nature Physics 15, 148 (2018) DOI
[5]
L. Tsui, H.-C. Jiang, Y.-M. Lu, and D.-H. Lee, “Quantum phase transitions between a class of symmetry protected topological states”, Nuclear Physics B 896, 330 (2015) arXiv:1503.06794 DOI
[6]
A. V. Thapliyal, Multipartite maximally entangled states, minimal entanglement generating states and entropic inequalities unpublished presentation (2003)
[7]
W. Helwig and W. Cui, “Absolutely Maximally Entangled States: Existence and Applications”, (2013) arXiv:1306.2536
[8]
W. Helwig, “Absolutely Maximally Entangled Qudit Graph States”, (2013) arXiv:1306.2879
[9]
D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski, “Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices”, Physical Review A 92, (2015) arXiv:1506.08857 DOI
[10]
Z. Raissi, C. Gogolin, A. Riera, and A. Acín, “Optimal quantum error correcting codes from absolutely maximally entangled states”, Journal of Physics A: Mathematical and Theoretical 51, 075301 (2018) arXiv:1701.03359 DOI
[11]
D. Alsina, “PhD thesis: Multipartite entanglement and quantum algorithms”, (2017) arXiv:1706.08318
[12]
S. Prakash, “Magic state distillation with the ternary Golay code”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, (2020) arXiv:2003.02717 DOI
[13]
D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
[14]
M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, 45 arXiv:quant-ph/0703112 DOI
[15]
Z. Wu, S. Cheng, and B. Zeng, “A ZX-Calculus Approach for the Construction of Graph Codes”, (2024) arXiv:2304.08363

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2024-06-13)

Cite as:

“Modular-qudit cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qudit_cluster_state, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/qudit_cluster_state.yml.