# H code[1]

## Description

Family of \([[k+4,k,2]]\) CSS codes with transversal Hadmard gates; relevant to magic state distillation. 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 \(k,r\rightarrow \infty\). This matches a conjectured bound for \(\gamma\) [2].

## 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 \(10^{-7}\).

## Parent

## Zoo code information

## References

## Cite as:

“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/tree/main/codes/quantum/qubits/h_code_jones.yml.