Cubic theory code[1]
Alternative names: Magic stabilizer code.
Description
A geometrically local commuting-projector code family defined on triangulations in arbitrary spatial dimensions. Its Hamiltonian contains Pauli-\(Z\) flux terms and non-Pauli Gauss-law terms built from products of Pauli-\(X\) operators and \(CZ\) gates. These commuting non-Pauli stabilizers realize higher-form \(\mathbb{Z}_2^3\) gauge theories with Abelian electric excitations and non-Abelian magnetic excitations.
For \(l=m=n=2\) in \(D=6\) spacetime dimensions, the code is a candidate non-Abelian self-correcting quantum memory with Abelian loop excitations and non-Abelian membrane excitations [1].
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.
Protection
On suitable triangulations of the Wu 5-manifold, a family of five-dimensional cubic theory codes has parameters \([[O(n),3,O(n^{2/5})]]\) [1].Gates
In five spatial dimensions, a constant-depth circuit built from physical \(CCZ\) and SWAP gates implements the logical gate \(\overline{\mathrm{SWAP}}_{1,2}\overline{\mathrm{CCZ}}_{1,2,3}\) on the three logical qubits supported by the Wu-manifold family [1].Decoding
Probabilistic local cellular-automaton decoder [1].Fault Tolerance
The Wu-manifold family combines the logical non-Clifford gate with code parameters \([[O(n),3,O(n^{2/5})]]\), giving \(O(d^{5/2})\) space-time overhead for this topological scheme [1].Cousins
- Self-correcting quantum code— A family of five-dimensional cubic theory codes with Abelian loop excitations and non-Abelian membrane excitations is argued to be self-correcting below a critical temperature via a Peierls argument [1].
- Color code— The cubic theory in \(D\) spacetime dimensions can be obtained by twisted compactification of a generalized color code in \(D+1\) spacetime dimensions; in particular, the five-dimensional cubic theory arises from a twisted compactification of the 6D color code [1].
- Dihedral \(G=D_m\) quantum-double code— For \(D=3\) with \(l=m=n=1\), the cubic theory is equivalent to the \(G=D_4\) quantum double, i.e. the non-Abelian Type-III \(\mathbb{Z}_2^3\) twisted quantum double [1].
- XP stabilizer code— The cubic theory code can be embedded into a larger codespace such that all diagonal logical operators are represented by XP operators [14; Sec. 4.3].
Primary Hierarchy
Parents
Cubic theory codes are joint eigenspaces of commuting non-Pauli stabilizers built from Pauli \(X\), Pauli \(Z\), and \(CZ\) operators, placing them at the second level of the Clifford hierarchy.
Cubic theory codes realize higher-form \(\mathbb{Z}_2^3\) gauge theories with non-Abelian excitations in arbitrary dimensions.
Cubic theory code
Children
References
- [1]
- P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the n \({}^{\text{1/3}}\) Distance Barrier”, PRX Quantum 6, (2025) arXiv:2405.11719 DOI
- [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
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- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
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- 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
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- D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
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- 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
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- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
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- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
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- 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
- [15]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 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