\([[6k+2,3k,2]]\) Campbell-Howard code[1]
Description
Family of \([[6k+2,3k,2]]\) qubit stabilizer codes with quasi-transversal \(CCZ^{\otimes k}\) gates that are relevant to magic-state distillation. In the synthillation framework, these distance-two codes realize batches of logical \(CCZ\) gates using physical \(T\) gates followed by a Clifford correction.Protection
Detects single-qubit errors.Rate
Encoding rate is \(3k/(6k+2)\), approaching \(1/2\) as \(k\to\infty\).Magic
A total of \(r\) rounds of magic-state distillation yields a magic-state yield parameter \(\gamma\to 1^{+}\) as \(k,r\rightarrow \infty\). This matches the Bravyi-Haah conjectured lower bound \(\gamma \geq 1\) for concatenated triorthogonal-matrix protocols [2; Sec. VI].Transversal Gates
Quasi-transversal \(CCZ^{\otimes k}\) gates [1].Cousin
- Triorthogonal code— Campbell-Howard codes arise from the generalized-triorthogonal/quasitransversal \(G\)-matrix framework, which extends Bravyi-Haah triorthogonal matrices by allowing odd pair and triple overlaps entirely within the logical-row block \(K\) [1; Appx. D].
Primary Hierarchy
Parents
The family has distance \(2\).
\([[6k+2,3k,2]]\) Campbell-Howard code
Children
The \([[8,3,2]]\) code is the \(k=1\) member of the \([[6k+2,3k,2]]\) Campbell-Howard family with a quasi-transversal logical \(CCZ\) gate [1].
References
- [1]
- E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
- [2]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
Page edit log
- Victor V. Albert (2024-07-23) — most recent
- Earl T. Campbell (2024-07-23)
Cite as:
“\([[6k+2,3k,2]]\) Campbell-Howard code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/campbell_howard