~

Author: valbert4

codes/quantum/qubits/small_distance/small/10/xzzx_10_2_3.yml

@@ -14,7 +14,7 @@ introduced: '\cite{arxiv:1202.0928}\cite[Exam. 3]{arxiv:1212.6703}'
 description: |
   Rotated toric code that is the CSS form of the twisted XZZX toric code with parameters \(a=1\), \(b=3\) \cite{arxiv:1108.5490,arxiv:2101.09349},
   related to the XZZX form by Hadamard on the \(B\)-sublattice.
-  It is also the \hyperref[topic:symplectic-doubling]{symplectic double} (a.k.a. genus-one double cover) of the \([[5,1,3]]\) five-qubit perfect code \cite{arxiv:1212.6703,arxiv:2406.09951} and a BCC code \cite{arxiv:2509.06865}.
+  It is also the \hyperref[topic:symplectic-doubling]{symplectic double} (a.k.a. genus-one double cover) of the \([[5,1,3]]\) five-qubit perfect code \cite{arxiv:1212.6703,arxiv:2406.09951}, the \hyperref[topic:symplectic-doubling]{symplectic double} of the \([[5,1,2]]\) rotated surface code \cite{arxiv:2406.09951}, and a BCC code \cite{arxiv:2509.06865}.
 
   A stabilizer tableau for this code is
   \begin{align}
@@ -52,6 +52,8 @@ relations:
   cousins:
     - code_id: stab_5_1_3
       detail: 'The \([[10,2,3]]\) rotated toric code is the \hyperref[topic:symplectic-doubling]{symplectic double} (a.k.a. genus-one double cover) of the five-qubit perfect code \cite[Exam. 3]{arxiv:1212.6703}\cite{arxiv:2406.09951}. A non-CSS cyclic cluster code related to the \([[10,2,3]]\) rotated toric code yields the \([[5,1,3]]\) five-qubit perfect code for \(d=3\) \cite{arxiv:2509.06865}.'
+    - code_id: stab_5_1_2
+      detail: 'The \([[10,2,3]]\) rotated toric code is the \hyperref[topic:symplectic-doubling]{symplectic double} (a.k.a. genus-one double cover) of the \([[5,1,2]]\) rotated surface code \cite{arxiv:2406.09951}.'
     - code_id: bipartite_cyclic_cluster
       detail: 'A non-CSS cyclic cluster code related to the \([[10,2,3]]\) rotated toric code yields the \([[5,1,3]]\) five-qubit perfect code for \(d=3\) \cite{arxiv:2509.06865}.'
 

codes/quantum/qubits/small_distance/small/4/stab_4_1_2/stab_4_1_2.yml

@@ -50,8 +50,8 @@ realizations:
 
 relations:
   parents:
-    - code_id: triangular_color
-      detail: 'The \([[4,1,2]]\) twist-defect code is the smallest triangular color code with \(x\)-, \(y\)-, and \(z\)-type Pauli boundaries \cite[Fig. 7]{arxiv:1806.02820}.'
+    - code_id: twist_defect_color
+      detail: 'The \([[4,1,2]]\) twist-defect code is the smallest triangular color code with \(x\)-, \(y\)-, and \(z\)-type Pauli boundaries, which make the code non-CSS \cite[Fig. 7]{arxiv:1806.02820}.'
     - code_id: twist_defect_surface
       detail: 'The \([[4,1,2]]\) twist-defect code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere \cite{arxiv:2406.09951} via a single-qubit \hyperref[topic:clifford]{Clifford circuit}.'
     - code_id: small_distance_qubit_stabilizer

codes/quantum/qubits/small_distance/small/5/stab_5_1_2/stab_5_1_2.yml

@@ -9,6 +9,13 @@ logical: qubits
 
 name: '\([[5,1,2]]\) rotated surface code'
 
+alternative_names:
+  - '\([[5,1,2]]\) morphed Steane code'
+  - '\([[5,1,2]]\) rotated toric code'
+  - '\([[5,1,2]]\) genon code'
+  - '\([[5,1,2]]\) holographic code'
+  - '\([[5,1,2]]\) planar-perfect code'
+
 # based on qiskit database brute force search
 description: |
   A rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. This is the smallest code that implements a fault-tolerant logical \(S\) gate using a diagonal depth-one Clifford circuit \cite{preset:AlbertPC26}.
@@ -34,6 +41,18 @@ description: |
     \label{figure:512-operators}
   \end{figure}
 
+  A non-CSS genon-code form of the same stabilizer group, local Clifford equivalent to the above via Hadamard on the four corner qubits, is \cite{arxiv:2406.09951}
+  \begin{align}
+  \begin{array}{ccccc}
+    X & X & I & Z & I \\
+    I & I & X & Z & X \\
+    I & Z & Z & X & I \\
+    Z & I & I & X & Z
+  \end{array}~.
+  \end{align}
+  The missing external stabilizer \(YYYIY\) (product of all four generators) forms the back face when the code is viewed as a genon code on a sphere.
+
+
 
 features:
   general_gates:

codes/quantum/qubits/stabilizer/topological/surface/twist_defect/twisted_xzzx.yml

@@ -27,8 +27,8 @@ description: |
 protection: |
   A family of \([[a^2+b^2,k,d]]\) cyclic codes exists for all \(b > a \geq 1\) such that \(\text{gcd}(a,b)=1\) \cite[Thm. 3.9]{arxiv:2101.09349}.
   Here, \(k=1\) (\(k=2\)) and \(d=a+b\) (\(d=\max(a,b)\)) for odd \(n\) (even \(n\)).
-  The subfamily \([[d^2+1,2,d]]\) (i.e., \(a=1,b=d\) for odd \(d\)) includes \([[10,2,3]]\), \([[26,2,5]]\), \([[50,2,7]]\), \ldots \cite[Exam. 1]{arxiv:1202.0928}.
-  The subfamily \([[t^2+(t+1)^2,1,2t+1]]\) (i.e., \(a=t,b=t+1\)) includes \([[5,1,3]]\), \([[13,1,5]]\), \([[25,1,7]]\), \ldots \cite[Exam. 4]{arxiv:1202.0928}.
+  The subfamily \([[d^2+1,2,d]]\) (i.e., \(a=1,b=d\) for odd \(d\)) includes \([[10,2,3]]\), \([[26,2,5]]\), \([[50,2,7]]\), \(\ldots\) \cite[Exam. 1]{arxiv:1202.0928}.
+  The subfamily \([[t^2+(t+1)^2,1,2t+1]]\) (i.e., \(a=t,b=t+1\)) includes \([[5,1,3]]\), \([[13,1,5]]\), \([[25,1,7]]\), \(\ldots\) \cite[Exam. 4]{arxiv:1202.0928}.
   Small instances are tabulated in Ref. \cite{arxiv:2406.09951} as genus-one genon codes.
   Other types of distances have been considered for this code \cite{arxiv:2203.16486}.