\([[k+4,k,2]]\) H code[1]
Description
Family of \([[k+4,k,2]]\) self-dual CSS codes (for even \(k\)) with transversal Hadamard gates that are relevant to magic state distillation. The four stablizer generators are \(X_1X_2X_3X_4\), \(Z_1Z_2Z_3Z_4\), \(X_1X_2X_5X_6...X_{k+4}\), and \(Z_1Z_2Z_5Z_6...Z_{k+4}\).'Protection
Detects weight-one Pauli errors. The \(r\)-level contatenated H code detects weight Pauli errors up to weight \(2^r-1\).Rate
The H codes are dense, i.e., the rate \(\frac{k}{k+4}\rightarrow 1\) as \(k \rightarrow \infty\). The distance is 2. However an \(r\)-level concatenation of H codes gives a distance of \(2^r\).Magic
A total of \(r\) rounds of magic-state distillation yields a magic-state yield parameter \(\gamma\to 1^{+}\) as \(k,r\rightarrow \infty\); see [2; Box 2]. This matches a conjectured bound for \(\gamma\) [3].Transversal Gates
Hadamard and \(TXT^{\dagger}\) gates, with the latter Clifford-equivalent to Hadamard, and where \(T=\exp(i\pi(I-Z)/8)\) is the \(\pi/8\)-rotation gate.Gates
The H codes can be used for high-quality and high-efficiency magic-state distillation [1]. Their associated multi-level magic states protocols have an efficency advantage over the 10-to-2 and 15-to-1 protocals for output error below \(10^{-7}\).Cousin
- \([[3k + 8, k, 2]]\) triorthogonal code— The H code \([[k+4,k,2]]\) family yields the \([[3k + 8, k, 2]]\) family of triorthogonal codes when level-lifted [4; Sec. VI.C].
Primary Hierarchy
Parents
H codes are level-two generalized divisible codes [4; Sec. VI.C].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[k+4,k,2]]\) H code
Children
The \([[k+4,k,2]]\) H code for \(k=2\) is the \(C_6\) code.
References
- [1]
- C. Jones, “Multilevel distillation of magic states for quantum computing”, Physical Review A 87, (2013) arXiv:1210.3388 DOI
- [2]
- E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
- [3]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
- [4]
- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
Page edit log
- Victor V. Albert (2022-04-28) — most recent
- Xiao Xiao (2022-04-28)
Cite as:
“\([[k+4,k,2]]\) H code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_h