Cubic theory code

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

Two-gauge theory code— Both families realize higher gauge theories. Cubic theory codes allow gauge fields of arbitrary degree and are not restricted to finite two-group gauge theory [1].
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 Hilbert space such that all diagonal Clifford-hierarchy logical operators are represented by XP operators [14; Sec. 4.3].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Clifford-hierarchy stabilizer code

Parents

Clifford-hierarchy stabilizer code

Cubic theory codes are joint eigenspaces of undressed Gauss-law and flux operators built from Pauli \(X\), Pauli \(Z\), and \(CZ\) operators, placing them at the second level of the Clifford hierarchy. The Gauss-law operators commute only within the zero-flux sector [1].

Topological codeHamiltonian-based QECC Quantum

Cubic theory codes realize higher-form \(\mathbb{Z}_2^3\) gauge theories with non-Abelian excitations in arbitrary dimensions [1].

Commuting-projector Hamiltonian codeHamiltonian-based QECC Quantum

Projecting the Gauss-law terms to zero flux gives a local commuting-projector Hamiltonian with the same codespace [1].

Monomial stabilizer codeHamiltonian-based QECC Quantum

Cubic theory code Hamiltonians contain Gauss-law terms dressed by zero-flux projectors, and the dressed terms are non-unitary and hence non-monomial [1]. However, the codespace coincides with the joint \(+1\) eigenspace of the diagonal Pauli-\(Z\) flux operators together with the undressed Gauss-law operators. The latter are products of Pauli-\(X\) strings and diagonal gates built from Pauli-\(Z\) and \(CZ\) operators, so the codespace is a joint \(+1\) eigenspace of a finite monomial group [15].

Cubic theory code

Children

Hexagonal \(CZ\) code

The \(2+1\)D cubic theory with \(l=m=n=1\) realizes the same topological order as the Type-III \(\mathbb{Z}_2^3\) twisted quantum double / \(G=D_4\) quantum double, and the hexagonal \(CZ\) code is a hexagonal-lattice realization of this phase [1,16].

Square-lattice TQD code

The square-lattice TQD code is the \(D=3\), \(l=m=n=1\) hypercubic specialization of the cubic theory code [1].

References

[1]
P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the n \({}^{1/3}\) 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
[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”, Nature Physics 22, 598 (2026) arXiv:2410.02213 DOI
[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]
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
[16]
B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI

Page edit log

Victor V. Albert (2026-08-26) — most recent
Victor V. Albert (2026-06-08)
Guanyu Zhu (2024-05-23)
Victor V. Albert (2024-05-23)

Cite as:

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

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