[Jump to code hierarchy]

Cubic theory code[1]

Alternative names: Magic stabilizer code.

Description

A geometrically local commuting-projector code defined on triangulations of lattices in arbitrary spatial dimensions. Its code Hamiltonian terms include Pauli-\(Z\) operators and products of Pauli-\(X\) operators and \(CZ\) gates. The Hamiltonian realizes higher-form \(\mathbb{Z}_2^3\) gauge theories whose excitations obey non-Abelian Ising-like fusion rules.

The construction is based on first constructing a model for an SPT phase [2,3], gauging its symmetries [413], and making all terms commute outside of the ground-state subspace by projecting them to zero flux.

Decoding

Probabilistic local cellular-automaton decoder [1].

Cousins

  • Self-correcting quantum code— A family of five-dimensional cubic theory codes with non-Abelian excitations is argued to be self-correcting below a critical temperature via a Peierls argument [1].
  • XP stabilizer code— The cubic theory code can be embedded into a larger codespace such that all \(CZ\) operators are represented by XP operators [14].

Primary Hierarchy

Parents
Cubic theory codes realize higher-form \(\mathbb{Z}_2^3\) gauge theories with non-Abelian excitations in arbitrary dimensions.
Cubic theory code
Children
The cubic theory code defined on a hexagonal lattice reduces to the hexagonal \(CZ\) code [1].

References

[1]
P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
[2]
X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group”, Physical Review B 87, (2013) arXiv:1106.4772 DOI
[3]
L. Tsui and X.-G. Wen, “Lattice models that realize Zn -1 symmetry-protected topological states for even n”, Physical Review B 101, (2020) arXiv:1908.02613 DOI
[4]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[5]
J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
[6]
S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[7]
D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[8]
L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
[9]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[10]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[11]
K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[12]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[13]
D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[14]
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
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: cubic_theory

Cite as:
“Cubic theory code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cubic_theory
BibTeX:
@incollection{eczoo_cubic_theory, title={Cubic theory code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/cubic_theory} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/cubic_theory

Cite as:

“Cubic theory code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cubic_theory

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