\([[10,2,3]]\) binarized Galois-qudit code[1]
Description
CSS code obtained by binarizing a \([[5,1,3]]_4\) Galois-qudit CSS code in the self-dual normal basis \(\{\omega,\omega^2\}\).
A stabilizer tableau for the binarized code is [1] \begin{align} \begin{array}{cccccccccc} Z & I & Z & I & Z & I & Z & I & I & I \\ I & Z & I & Z & I & Z & I & Z & I & I \\ I & I & Z & I & Z & Z & I & Z & Z & I \\ I & I & I & Z & Z & I & Z & Z & I & Z \\ X & I & X & I & X & I & X & I & I & I \\ I & X & I & X & I & X & I & X & I & I \\ I & I & X & I & I & X & X & X & X & I \\ I & I & I & X & X & X & X & I & I & X \end{array}~. \tag*{(1)}\end{align}
Transversal and Permutation-Based Gates
The code is permutation-equivalent to its Hadamard dual: transversal physical Hadamards followed by swapping the two qubits in each binarized \(\mathbb{F}_4\) coordinate preserve the codespace and implement a logical \(H^{\otimes 2}\) up to logical \(\mathrm{SWAP}\). This is the binarized version of the permutation-assisted Hadamard available from plain self-duality of the starting \(q=4\) Galois-qudit CSS code [1].Cousins
- Binarized-and-concatenated (B&C) phantom code— The binarized \([[10,2,3]]\) code is not phantom, but concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[20,2,6]]\) B&C phantom code admitting fold-diagonal logical \(SS\) gates [1].
- \([[4,2,2]]\) Four-qubit code— Concatenating each qubit pair of the binarized \([[10,2,3]]\) code with the \([[4,2,2]]\) code yields the \([[20,2,6]]\) B&C phantom code [1,2].
- \([[20,2,6]]\) B&C phantom code— The \([[20,2,6]]\) code is obtained by concatenating each qubit pair of the \([[10,2,3]]\) binarized Galois-qudit code with the \([[4,2,2]]\) code [1].
- \([[5,1,3]]_4\) Galois-qudit CSS code— Binarizing the \([[5,1,3]]_4\) code in the self-dual normal basis \(\{\omega,\omega^2\}\) yields a \([[10,2,3]]\) qubit CSS code [1].
Primary Hierarchy
Parents
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[10,2,3]]\) binarized Galois-qudit code
References
- [1]
- J. M. Koh, A. Gong, A. C. Diaconu, D. B. Tan, A. A. Geim, M. J. Gullans, N. Y. Yao, M. D. Lukin, and S. Majidy, “Entangling logical qubits without physical operations”, (2026) arXiv:2601.20927
- [2]
- B. W. Reichardt, D. Aasen, and R. Chao, “Fire and ice: Partially fault-tolerant quantum computing with selective state filtering”, (2026) arXiv:2605.15344
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2026-05-20)
Cite as:
“\([[10,2,3]]\) binarized Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_10_2_3, arXiv:2606.11484