Merge branch 'main' of https://github.com/Gerwinlab/eczoo_data_Erwin

codes/classical/bits/easy/Checksum.yml → codes/classical/bits/easy/checksum.yml

@@ -7,7 +7,7 @@ code_id: checksum
 physical: bits
 logical: bits
 
-name: 'Checksum code'
+name: 'checksum code'
 #introduced: ''
 
 description: |

codes/classical/spherical/polytope/polygon/polygon.yml

@@ -11,7 +11,6 @@ name: 'Polygon code'
 
 description: |
   Spherical \((1,q,4\sin^2 \frac{\pi}{q})\) code for any \(q\geq1\) whose codewords are the vertices of a \(q\)-gon. Special cases include the line segment (\(q=2\)), triangle (\(q=3\)), square (\(q=4\)), pentagon (\(q=5\)), and hexagon (\(q=6\)).
-  The code forms a tight spherical \(q-1\) design.
   \begin{figure}
     \includegraphics{polygon.svg}
     \caption{\(q\)-gon code for \(q=5\). Each codeword is a vertex of the \(5\)-gon.}
@@ -23,6 +22,8 @@ relations:
   parents:
     - code_id: polytope
     - code_id: sharp_config
+    - code_id: spherical_design
+      detail: 'A \(q\)-gon a tight spherical \(q-1\) design.'
   cousins:
     - code_id: cat
       detail: 'The \(q(S+1)\)-component cat coherent-state constellation forms the vertices of a \(q(S+1)\)-gon.'

codes/classical/spherical/sharp_config/cgs_spherical.yml

@@ -20,6 +20,7 @@ protection: 'CGS isotropic subspace codes saturate the Levenshtein bound \cite[p
 
 relations:
   parents:
+    - code_id: spherical_design
     - code_id: sharp_config
       detail: 'CGS isotropic subspace codes are the only known spherical sharp configrations not derived from regular polytopes or lattices \cite{arxiv:math/0607446}.'
   cousins:

codes/quantum/properties/approximate_qecc.yml

@@ -75,7 +75,7 @@ protection: |
 
   In addition to the necessary and sufficient error correction conditions,
   there exist sufficient conditions for AQECCs.
-  Given a noise channel \(\mathcal{U}(\rho)=\sum_{n} A_n \rho A_n^{\dagger}\) where \(\for all{n}\), \(A_n\) is a Kraus operator, and code projector \(\Pi \), express the following using polar decomposition, \(A_n \Pi =U_n \sqrt{\Pi A_n^{\dagger}A_n \Pi }\), and let \(p_n\) and \(p_n\lambda_n\) be the largest and smallest eigenvalues for \(\Pi A_n^{\dagger}A_n \Pi \).
+  Given a noise channel \(\mathcal{U}(\rho)=\sum_{n} A_n \rho A_n^{\dagger}\) where \(\forall{n}\), \(A_n\) is a Kraus operator, and code projector \(\Pi \), express the following using polar decomposition, \(A_n \Pi =U_n \sqrt{\Pi A_n^{\dagger}A_n \Pi }\), and let \(p_n\) and \(p_n\lambda_n\) be the largest and smallest eigenvalues for \(\Pi A_n^{\dagger}A_n \Pi \).
   Then, we are guaranteed that if
   \begin{align}\Pi U_m^{\dagger}U_n \Pi =\delta_{mn} \Pi  \land p_n(1-\lambda_n)\le O(f(\epsilon))\end{align}
   we have a fidelity \(F \geq 1-O(f(\epsilon))\) after recovery~\cite{arXiv:quant-ph/9704002}.
@@ -91,7 +91,7 @@ protection: |
   Generalizing the notion of quantum information transmission and capacity of \hyperref[code:qecc_finite]{(exact) QECCs}, one can achieve an \(\alpha\)-bit transmission rate \(r\) quantum channel \(\mathcal{E}\) iff,
   for sufficiently large \(d\) and \(n\), and for all \(\epsilon>0\), the channel \(\mathcal{E}^{\otimes n}\)
   is able to transmit
-  \begin{align}\lceil \frac{n r}{\log(d)} \rceil \mathrm{\(\alpha\)-dits}\end{align}
+  \begin{align}\lceil \frac{n r}{\log(d)} \rceil\quad \textup{\(\alpha\)-dits}\end{align}
   with total error \(\epsilon\) across those \(\alpha\)-dits.
   The \(\alpha\)-bit capacity \(Q\) of \(\mathcal{E}\)
   is defined as the supremum of achievable transmission rates~\cite{arXiv:1706.09434}.