\([[8,1,2]]\) Shen-Wang-Cao code[1]
Description
A stabilizer code that admits a logical \(T\) gate via application of physical \(T\) gates and a \(CZ\)-like gate.
A stabilizer tableau for the code is given by [2; ID 980]. \begin{align} \begin{array}{cccccccc} X & X & I & I & X & X & I & I \\ I & I & X & X & X & X & I & I \\ I & I & I & I & I & I & X & X \\ Z & I & I & Z & I & Z & Z & Z \\ I & Z & I & Z & I & Z & Z & Z \\ I & I & Z & Z & I & I & I & I \\ I & I & I & I & Z & Z & I & I \end{array}~. \tag*{(1)}\end{align}
Transversal Gates
Logical \(T\) gate \(\bar{T}=T^{\otimes 6}\otimes K\) via physical \(T\) gates on the first six qubits and the two-qubit gate \(K\propto\cos(\pi/8)(I\otimes I)-i\sin(\pi/8)(Z\otimes Z)\propto\mathrm{diag}(1,e^{i\pi/4},e^{i\pi/4},1)\) on qubits 7 and 8 [1].Cousins
- XP stabilizer code— The \([[8,1,2]]\) code is an XP-regular code that can be obtained via the XP stabilizer formalism applied to the \([[15,1,3]]\) Reed-Muller code [1].
- \([[15,1,3]]\) quantum RM code— The \([[8,1,2]]\) code is an XP-regular code that can be obtained via the XP stabilizer formalism applied to the \([[15,1,3]]\) Reed-Muller code [1].
- Tensor-network code— The \([[8,1,2]]\) code is an XP-regular code that can be obtained via the XP stabilizer formalism applied to the \([[15,1,3]]\) Reed-Muller code [1].
Primary Hierarchy
Parents
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[8,1,2]]\) Shen-Wang-Cao code
References
- [1]
- R. Shen, Y. Wang, and C. Cao, “Quantum Lego and XP Stabilizer Codes”, Quantum 9, 1876 (2025) arXiv:2310.19538 DOI
- [2]
- Qiskit Community. Qiskit QEC framework. https://github.com/qiskit-community/qiskit-qec
Page edit log
- Victor V. Albert (2025-12-08) — most recent
Cite as:
“\([[8,1,2]]\) Shen-Wang-Cao code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stab_8_1_2