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\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code[1; Ch. 7]

Description

Member of CSS code family constructed with a punctured self-dual RM \([2^r-1,2^{r-1},\sqrt{2}^{r-1}-1]\) code and its even subcode for \(r \geq 2\).

Transversal Gates

All single-qubit Clifford gates.

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
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Quantum rainbow code
Quantum pin codeStabilizer Hamiltonian-based Qubit QECC Quantum
Quantum Reed-Muller codeCSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Quantum Reed-Muller code
The \([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller codes are special cases of the \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) family for \(m \to 2r-1\), \(w \to 0\), and \(r \to r-1\).
\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code
Children
\([[7,1,3]]\) Steane code

References

[1]
J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
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Zoo Code ID: single_qubit_clifford

Cite as:
“\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/single_qubit_clifford
BibTeX:
@incollection{eczoo_single_qubit_clifford, title={\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/single_qubit_clifford} }
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Permanent link:
https://errorcorrectionzoo.org/c/single_qubit_clifford

Cite as:

“\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/single_qubit_clifford

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