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 [4–13], 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
- Guanyu Zhu (2024-05-23) — most recent
- Victor V. Albert (2024-05-23)
Cite as:
“Cubic theory code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cubic_theory