Clifford-hierarchy stabilizer code[1]
Description
A qubit code whose codespace is a joint eigenspace of a subset of operators in the Clifford hierarchy. The stabilizing set, which need not be a group, contains Pauli strings and operators at any level \(m\) of the Clifford hierarchy, generalizing qubit stabilizer codes (\(m=1\)) and Clifford stabilizer codes (\(m=2\)).
Clifford-hierarchy codes in \(D\) spatial dimensions include \((D+1)\)-dimensional Dijkgraaf-Witten gauge theories with non-Abelian topological order [1]. A \(D\)-dimensional code can be constructed from a twisted \(\mathbb{Z}_2^{D+1}\) gauge theory with Dijkgraaf-Witten twist \((-1)^{\int a_1 \cup a_2 \cup \cdots \cup a_{D+1}}\), where the stabilizers include gates at the \(D\)th level of the Clifford hierarchy in addition to Pauli \(X\) operators.
Transversal and Permutation-Based Gates
A transversal logical \(\text{diag}(1, e^{i2\pi/2^D})\) gate at the \(D\)th level of the Clifford hierarchy in \((D-1)\) spatial dimensions [1].Cousins
- Clifford group— Clifford-hierarchy codes are joint eigenspaces of subsets of the Clifford hierarchy, whose second level is the Clifford group.
- Two-gauge theory code— Clifford-hierarchy codes in \(D\) spatial dimensions include \((D+1)\)-dimensional Dijkgraaf-Witten gauge theories with non-Abelian topological order [1]. A \(D\)-dimensional code can be constructed from a twisted \(\mathbb{Z}_2^{D+1}\) gauge theory with Dijkgraaf-Witten twist \((-1)^{\int a_1 \cup a_2 \cup \cdots \cup a_{D+1}}\), where the stabilizers include gates at the \(D\)th level of the Clifford hierarchy in addition to Pauli \(X\) operators.
Member of code lists
Primary Hierarchy
References
- [1]
- Anonymous, “Clifford hierarchy stabilizer codes: Transversal non-Clifford gates and magic”, Physical Review Letters (2026) arXiv:2511.02900 DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2026-03-25)
Cite as:
“Clifford-hierarchy stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/clifford_hierarchy, arXiv:2606.11484