\([[k+4,k,2]]\) H code[1] 

Description

Family of \([[k+4,k,2]]\) 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-1 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 scaling exponent \(\gamma\to 1^{+}\) 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 \(10^{-7}\).

Parents

Cousin

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
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Zoo Code ID: quantum_h

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
BibTeX:
@incollection{eczoo_quantum_h, title={\([[k+4,k,2]]\) H code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_h} }
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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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/magic/quantum_h.yml.