\([[16,6,4]]\) copy-cup code

\([[16,6,4]]\) copy-cup code[1]

Description

An even pure \([[16,6,4]]\) CSS code that admits a constant-depth logical \(CZ\) gate between two copies and is constructed as a balanced product of two weight-four cyclic group-algebra codes over \(\mathbb{F}_2[\mathbb{Z}_8]\). It is one of two inequivalent \([[16,6,4]]\) codes whose complete logical Clifford group is realizable by depth-one two-local circuits, the other being the tesseract color code.

A stabilizer tableau for the code, given by cyclic shifts of the generalized bicycle rows \(H_X=(A|B)\) and \(H_Z=(B^T|A^T)\), is [1] \begin{align} \begin{smallmatrix} X & X & X & X & I & I & I & I & X & X & I & X & I & I & X & I \\ I & X & X & X & X & I & I & I & I & X & X & I & X & I & I & X \\ I & I & X & X & X & X & I & I & X & I & X & X & I & X & I & I \\ I & I & I & X & X & X & X & I & I & X & I & X & X & I & X & I \\ I & I & I & I & X & X & X & X & I & I & X & I & X & X & I & X \\ Z & I & Z & I & I & Z & I & Z & Z & I & I & I & I & Z & Z & Z \\ Z & Z & I & Z & I & I & Z & I & Z & Z & I & I & I & I & Z & Z \\ I & Z & Z & I & Z & I & I & Z & Z & Z & Z & I & I & I & I & Z \\ Z & I & Z & Z & I & Z & I & I & Z & Z & Z & Z & I & I & I & I \\ I & Z & I & Z & Z & I & Z & I & I & Z & Z & Z & Z & I & I & I \end{smallmatrix}~, \tag*{(1)}\end{align} where \(A=a(P)\) and \(B=b(P)\) are the \(8\times 8\) circulants of \(a(x)=1+x+x^2+x^3\) and \(b(x)=1+x+x^3+x^6\), and \(P\) is the length-eight cyclic shift.

Transversal and Permutation-Based Gates

All logical Clifford gates can be realized as two-fold transversal gates, i.e., by depth-one circuits of two-local code-preserving physical gates [2].

Gates

Admits a constant-depth logical \(CZ\) gate between two copies of the code (an inter-code gate), the 2-copy-cup gate, built from a cup product on the balanced-product cochain complex subject to a pre-orientation condition on the two classical codes [1,3].

Cousin

\([[16,6,4]]\) Tesseract color code— The \([[16,6,4]]\) copy-cup code and the \([[16,6,4]]\) tesseract color code are the two inequivalent \([[16,6,4]]\) codes whose complete logical Clifford group is realizable by depth-one two-local circuits [2]; both are pure CSS codes whose stabilizer generators all have weight eight. The two are inequivalent in that they have different weight enumerators, and one is even while the other is doubly-even self-dual.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Small-distance qubit stabilizer codeSmall-distance block quantum QECC Quantum

Parents

Small-distance qubit stabilizer code

Generalized bicycle (GB) codeCSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

The \([[16,6,4]]\) copy-cup code is the generalized bicycle code \(\text{GB}(a,b)\) over \(\mathbb{Z}_8\) with \(a(x)=1+x+x^2+x^3\) and \(b(x)=1+x+x^3+x^6\), equivalently the balanced product of two weight-four cyclic group-algebra codes [1].

\([[16,6,4]]\) copy-cup code

References

[1]: R. Tiew and N. P. Breuckmann, “Copy-cup Gates in Tensor Products of Group Algebra Codes”, (2026) arXiv:2602.23307
[2]: V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
[3]: N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology Invariants and Logical Quantum Operations”, Communications in Mathematical Physics 407, (2026) arXiv:2410.16250 DOI

Page edit log

Victor V. Albert (2026-08-24) — most recent
Victor V. Albert (2026-08-19)

Cite as:

“\([[16,6,4]]\) copy-cup code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/copycup_16_6_4, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/16/copycup_16_6_4.yml.