fix 422

codes/quantum/qubits/small_distance/small/stab_4_2_2.yml

@@ -31,9 +31,11 @@ description: |
   \begin{align}
     |\overline{\pm}\rangle = \frac{1}{2}(|00\rangle \pm |11\rangle)^{\otimes 2}~.\\
   \end{align}
-  This code can be thought of as a concatenation of a two-qubit bit-flip with a two-qubit phase-flip code. The subcode \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) \cite{arxiv:quant-ph/0103042} has also been studied against amplitude-damping noise. This subcode \cite{arxiv:2006.03071} and the subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) \cite{arxiv:1912.09410}, \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \cite{arXiv:2102.06132}, and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) \cite{arXiv:2102.13071} are small planar surface codes.
+  This code can be thought of as a concatenation of a two-qubit bit-flip with a two-qubit phase-flip code.
+
+protection: |
+  Detects a single-qubit error \cite{arXiv:quant-ph/9603031} or single erasure \cite{arXiv:quant-ph/9610042}. Not able to correct arbitrary single-qubit errors because \( \lfloor \frac{d-1}{2} \rfloor =0 \). Approximately corrects a single amplitude damping error \cite{arXiv:quant-ph/9704002} (see also \cite{arxiv:quant-ph/0103042}).
 
-protection: 'Detects a single-qubit error \cite{arXiv:quant-ph/9603031} or single erasure \cite{arXiv:quant-ph/9610042}. Not able to correct arbitrary single-qubit errors because \( \lfloor \frac{d-1}{2} \rfloor =0 \). Approximately corrects a single amplitude damping error \cite{arXiv:quant-ph/9704002}.'
 
 features:
   transversal_gates: 'Transversal Pauli, Hadamard, and two-qubit \(R\) \cite{arxiv:1610.03507}.
@@ -54,8 +56,6 @@ realizations:
   - 'Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits \cite{arXiv:2102.13071}.'
   - 'The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device \cite{arxiv:2305.13581}.'
 
-notes:
-  - 'Concatenating \([[4,2,2]]\) code with surface code can generate 2D topological code with a reasonable circuit-based threshold \cite{doi:10.26421/QIC16.15-16-1}.'
 
 relations:
   parents:
@@ -68,23 +68,26 @@ relations:
     - code_id: iceberg
     - code_id: group_4_2_2
       detail: 'The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}_2\).'
+    - code_id: quantum_polar
+      detail: '\([[4,2,2]]\) code is a small quantum polar code \cite{manual:{Kyungjoo Noh, \href{https://github.com/errorcorrectionzoo/eczoo_data/files/7652763/Leung_code_as_quantum_polar_code.pdf}{Leung code as quantum polar code}, 2017.}}.'
   cousins:
     - code_id: rotated_surface
-      detail: 'Various \([[4,1,2]]\) subcodes are small rotated planar codes \cite{arxiv:2006.03071,arxiv:1912.09410,arXiv:2102.06132,arXiv:2102.13071}.'
+      detail: 'The subcodes \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) \cite{arxiv:2006.03071}, \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) \cite{arxiv:1912.09410}, \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \cite{arXiv:2102.06132}, and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) \cite{arXiv:2102.13071} of the \([[4,2,2]]\) code are small planar rotated surface codes.'
     - code_id: bacon_shor
       detail: 'The error-detecting \([[4,1,1,2]]\) Bacon-Shor code, which reduces to a subcode of the \hyperref[code:stab_4_2_2]{\([[4,2,2]]\) code} for a particular gauge configuration, has gauge operators \(\{XXII,IIXX,ZIZI,IZIZ\}\).'
     - code_id: quantum_parity
-      detail: '\([[4,1,2]]\) subcode \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) is the smallest member of the sub-family of \([[m^2,1,m]]\) QPC codes.'
+      detail: 'The \([[4,1,2]]\) subcode \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) is the smallest member of the sub-family of \([[m^2,1,m]]\) QPC codes.'
     - code_id: stab_5_1_3
       detail: 'The \([[4,2,2]]\) code can be derived from the five-qubit code using a protocol that converts an \([[n,k,d]]\) code into an \([[n-1, k+1, d-1]]\) code; see Sec. 3.5 in Gottesman \cite{arXiv:quant-ph/9705052}.'
-    - code_id: quantum_polar
-      detail: '\([[4,2,2]]\) code is a small quantum polar code \cite{manual:{Kyungjoo Noh, \href{https://github.com/errorcorrectionzoo/eczoo_data/files/7652763/Leung_code_as_quantum_polar_code.pdf}{Leung code as quantum polar code}, 2017.}}.'
     - code_id: approximate_qecc
       detail: '\([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) \cite{arXiv:quant-ph/9704002} and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) \cite{arxiv:quant-ph/0103042} approximately correct a single amplitude damping error.'
     - code_id: binomial
       detail: '\([[4,1,2]]\) subcode consisting of \(\{|\overline{00}\rangle\) and any other codeword reduces to the \(0,2,4\) binomial code when the basis labels in each codeword are written as in base-ten. Such a mapping can be generalized \cite{manual:{Linshu Li, private communication, 2018}}.'
     - code_id: heavy_hex
       detail: 'Magic states prepared using the \([[4,1,2]]\) subcode can be injected into a largest heavy-hex code \cite{arxiv:2110.04285,arxiv:2305.13581}.'
+    - code_id: quantum_concatenated
+      detail: 'Concatenating \([[4,2,2]]\) code with surface code can generate 2D topological code with a reasonable circuit-based threshold \cite{doi:10.26421/QIC16.15-16-1}.'
+
 
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