stab_18_2_5, xzzx_10_2_3, xzzx_7_1_3

Author: valbert4

codes/quantum/qubits/small_distance/small/10/stab_10_1_2.yml

@@ -10,11 +10,12 @@ logical: qubits
 name: '\([[10,1,2]]\) Vasmer-Kubica code'
 introduced: '\cite{arxiv:2112.01446}'
 
-# based on qiskit database brute force search
+# FT statement based on qiskit database brute force search, but Webster routines spit out only one gate and not all instances, so its not a proof
+#    This is the smallest code that implements a fault-tolerant logical \(T\) gate using a diagonal depth-one Clifford circuit \cite{preset:AlbertPC26}.
+
 description: |
   A stabilizer code obtained by morphing the \([[15,1,3]]\) quantum Reed-Muller code on a subset whose child code is the \([[8,3,2]]\) smallest interesting color code \cite{arxiv:2112.01446}.
-   This is the smallest code that implements a fault-tolerant logical \(T\) gate using a diagonal depth-one Clifford circuit \cite{preset:AlbertPC26}.
-   It is the smallest known stabilizer code with a fault-tolerant logical \(T\) gate, implemented via physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates \cite{arxiv:2112.01446}.
+  It is the smallest known stabilizer code with a fault-tolerant logical \(T\) gate, implemented via physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates \cite{arxiv:2112.01446}.
 
   A stabilizer tableau for the code is \cite{arxiv:2403.13732}
   \begin{align}

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

@@ -0,0 +1,64 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: xzzx_10_2_3
+physical: qubits
+logical: qubits
+
+name: '\([[10,2,3]]\) rotated toric code'
+short_name: '\([[10,2,3]]\)'
+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}.
+
+  A stabilizer tableau for this code is
+  \begin{align}
+  \begin{array}{cccccccccc}
+    Z & Z & I & Z & Z & I & I & I & I & I \\
+    I & I & Z & Z & I & Z & Z & I & I & I \\
+    I & I & I & I & Z & Z & I & Z & Z & I \\
+    Z & I & I & I & I & I & Z & Z & I & Z \\
+    X & I & X & X & I & I & I & I & I & X \\
+    I & X & X & I & X & X & I & I & I & I \\
+    I & I & I & X & X & I & X & X & I & I \\
+    I & I & I & I & I & X & X & I & X & X
+  \end{array}~.
+  \end{align}
+
+
+protection: |
+  Distance 3, correcting any single-qubit error.
+  Achieves the lower bound \(n \geq d^2+1 = 10\) for weight-four GB codes of odd distance \cite{arxiv:2203.17216}.
+
+features:
+  transversal_gates:
+    - 'Transversal Hadamard-SWAP: qubit permutation \(m \mapsto -m\) (mod 10) paired with sublattice swap
+      \(A \leftrightarrow B\), followed by Hadamard on all qubits \cite{arxiv:2509.06865}.'
+    - 'Logical Hadamard without SWAP via \(m \mapsto 3m\) (mod 10) followed by Hadamard \cite{arxiv:2509.06865}.'
+
+relations:
+  parents:
+    - code_id: twisted_xzzx
+      detail: 'This is the \(d=3\) instance of the \([[d^2+1,2,d]]\) family of twisted XZZX toric codes
+        (parameters \(a=1\), \(b=3\)), presented in its CSS form \cite{arxiv:2509.06865}.'
+    - code_id: bipartite_cyclic_cluster
+      detail: 'The \([[10,2,3]]\) rotated toric code is a \([[d^2+1,2,d]]\) BCC code for \(d=3\) \cite{arxiv:2509.06865}.'
+    - code_id: small_distance_qubit_stabilizer
+  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: 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}.'
+
+
+
+# Begin Entry Meta Information
+_meta:
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-05-26'

codes/quantum/qubits/small_distance/small/18/stab_18_2_5.yml

@@ -0,0 +1,70 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: stab_18_2_5
+physical: qubits
+logical: qubits
+
+name: '\([[18,2,5]]\) BCC code'
+introduced: '\cite{arxiv:2509.06865}'
+
+description: |
+  BCC code on 18 qubits encoding 2 logical qubits with distance 5, found by computer search \cite{arxiv:2509.06865}.
+
+  The code is defined by \(\mathcal{S}=\{5,11,15,17\}\) in the \hyperref[code:bipartite_cyclic_cluster]{BCC code} framework,
+  with even qubits \((0,2,\ldots,16)\) forming the \(A\)-sublattice and odd qubits \((1,3,\ldots,17)\) forming the \(B\)-sublattice.
+  The subset \(\{5,11,17\}\subset\mathcal{S}\) is closed under adding 6 (mod 18),
+  so the conjugated stabilizer \(UX_0X_6U^\dagger\) has large cancellations, yielding weight-4 stabilizers
+  \(X_{-3}X_0X_3X_6\) and its cyclic shifts by even integers (and likewise with \(X\to Z\)).
+
+  A stabilizer tableau for the code is
+  \begin{align}
+  \begin{smallmatrix}
+    X & X & X & I & I & X & I & X & I & I & I & X & I & X & I & X & I & I \\
+    I & I & X & X & X & I & I & X & I & X & I & I & I & X & I & X & I & X \\
+    I & X & I & I & X & X & X & I & I & X & I & X & I & I & I & X & I & X \\
+    I & X & I & X & I & I & X & X & X & I & I & X & I & X & I & I & I & X \\
+    I & X & I & X & I & X & I & I & X & X & X & I & I & X & I & X & I & I \\
+    I & I & I & X & I & X & I & X & I & I & X & X & X & I & I & X & I & X \\
+    I & X & I & I & I & X & I & X & I & X & I & I & X & X & X & I & I & X \\
+    I & X & I & X & I & I & I & X & I & X & I & X & I & I & X & X & X & I \\
+    I & Z & Z & Z & I & I & Z & I & Z & I & Z & I & I & I & Z & I & Z & I \\
+    Z & I & I & Z & Z & Z & I & I & Z & I & Z & I & Z & I & I & I & Z & I \\
+    Z & I & Z & I & I & Z & Z & Z & I & I & Z & I & Z & I & Z & I & I & I \\
+    I & I & Z & I & Z & I & I & Z & Z & Z & I & I & Z & I & Z & I & Z & I \\
+    Z & I & I & I & Z & I & Z & I & I & Z & Z & Z & I & I & Z & I & Z & I \\
+    Z & I & Z & I & I & I & Z & I & Z & I & I & Z & Z & Z & I & I & Z & I \\
+    Z & I & Z & I & Z & I & I & I & Z & I & Z & I & I & Z & Z & Z & I & I \\
+    I & I & Z & I & Z & I & Z & I & I & I & Z & I & Z & I & I & Z & Z & Z
+  \end{smallmatrix}~.
+  \end{align}
+
+features:
+  transversal_gates:
+    - 'Transversal Hadamard-SWAP logical gate (inherited from BCC code structure) \cite{arxiv:2509.06865}.'
+    - 'Logical Hadamard without SWAP: the code is self-dual, so transversal physical Hadamard implements logical \(H^{\otimes 2}\) \cite{arxiv:2509.06865}.'
+  decoders:
+    - 'An ancilla syndrome extraction scheme with 18 shared ancilla qubits (one per data qubit position across two code blocks)
+      detects all single bit-flip errors; a further reduction to 9 entangled ancilla qubits
+      is numerically shown to preserve fault tolerance to distance 5 \cite{arxiv:2509.06865}.'
+  fault_tolerance:
+    - 'An ancilla syndrome extraction scheme with 18 shared ancilla qubits (one per data qubit position across two code blocks)
+      detects all single bit-flip errors; a further reduction to 9 entangled ancilla qubits
+      is numerically shown to preserve fault tolerance to distance 5 \cite{arxiv:2509.06865}.'
+
+relations:
+  parents:
+    - code_id: bipartite_cyclic_cluster
+      detail: 'The \([[18,2,5]]\) BCC code is the smallest BCC code with parameters \([[n,2,5]]\) \cite{arxiv:2509.06865}.'
+    - code_id: self_dual_css
+      detail: 'The stabilizer group of the \([[18,2,5]]\) BCC code is invariant under transversal Hadamard \cite{arxiv:2509.06865}.'
+    - code_id: small_distance_qubit_stabilizer
+
+
+# Begin Entry Meta Information
+_meta:
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-05-26'

codes/quantum/qubits/small_distance/small/7/xzzx_7_1_3.yml

@@ -0,0 +1,43 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: xzzx_7_1_3
+physical: qubits
+logical: qubits
+
+name: '\([[7,1,3]]\) XZZX cyclic code'
+introduced: '\cite{arxiv:1703.08179}'
+
+description: |
+  A \([[7,1,3]]\) cyclic non-CSS code whose stabilizer generators are the seven cyclic shifts of the weight-four Pauli string \(XZIZXII\),
+  any six of which are independent.
+  The non-identity support of this generator is \(XZZX\) (at positions 0, 1, 3, 4), making this a member of the twisted XZZX code family.
+  It is one of sixteen distinct indecomposable \([[7,1,3]]\) codes \cite{arxiv:0709.1780}.
+
+  A stabilizer tableau for the code is given by \cite[ID 227]{preset:qiskit}
+  \begin{align}
+  \begin{array}{ccccccc}
+    X & Z & I & Z & X & I & I \\
+    I & X & Z & I & Z & X & I \\
+    I & I & X & Z & I & Z & X \\
+    X & I & I & X & Z & I & Z \\
+    Z & X & I & I & X & Z & I \\
+    I & Z & X & I & I & X & Z
+  \end{array}~.
+  \end{align}
+
+
+relations:
+  parents:
+    - code_id: twisted_xzzx
+      detail: 'The \([[7,1,3]]\) XZZX cyclic code is a cyclic non-CSS code whose generators are the cyclic shifts of the weight-four Pauli string \(XZIZXII\), which has \(XZZX\) support.'
+    - code_id: small_distance_qubit_stabilizer
+
+
+# Begin Entry Meta Information
+_meta:
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-05-27'

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bipartite_cyclic_cluster.yml

@@ -0,0 +1,74 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: bipartite_cyclic_cluster
+physical: qubits
+logical: qubits
+
+name: 'Bipartite cyclic cluster (BCC) code'
+short_name: 'BCC'
+introduced: '\cite{arxiv:2509.06865}'
+
+description: |
+  Cyclic CSS code constructed from a bipartite cluster state with cyclic invariance,
+  emphasizing simplicity of state preparation over simplicity of stabilizers.
+
+  A BCC code encodes \(k\) logical qubits into \(n = k n_0\) physical qubits.
+  Qubits are labeled by pairs \((m,j)\) with \(m\) defined modulo \(n_0\) and \(1 \leq j \leq k\),
+  partitioned into sets \(A\) (indices \(j \in \mathcal{A}\)) and \(B\) (indices \(j \notin \mathcal{A}\)).
+  The code is defined by a bipartite graph \(G\) on the qubits, with edges only between \(A\) and \(B\),
+  that is invariant under cyclic shifts \((m,j) \to (m+1,j)\).
+  Code states are prepared by initializing \(A\)-qubits in \(|\pm\rangle\) and \(B\)-qubits in \(|0/1\rangle\),
+  then applying CNOT gates (source in \(A\), target in \(B\)) along the edges of \(G\).
+  The \(2^k\) resulting basis states confirm that \(k\) logical qubits are encoded.
+  After a Hadamard on each \(B\)-qubit, this gives a CSS code.
+
+  The \(X\)-type stabilizers arise from conjugating \(X_{m,j} X_{m+1,j}\) (for \(j \in \mathcal{A}\)) by the CNOT circuit,
+  and the \(Z\)-type stabilizers from \(Z_{m,j} Z_{m+1,j}\) (for \(j \in \mathcal{B}\)).
+
+  For \(k=2\), the graph \(G\) is parameterized by a set \(\mathcal{S}\) of odd integers modulo \(n = 2n_0\):
+  even qubit \(m\) connects to odd qubit \(m'\) iff \((m'-m) \bmod n \in \mathcal{S}\).
+
+  A related non-CSS construction, the \textit{cyclic cluster code}, drops the bipartiteness requirement on \(G\).
+  For \(d=3\), this construction applied to the \([[10,2,3]]\) BCC code yields the \([[5,1,3]]\) five-qubit perfect code \cite{arxiv:2509.06865}.
+
+protection: |
+  The distance satisfies \(d \leq |\mathcal{S}| + 1\),
+  since \(U^\dagger X_{m,j} U\) for \(j \in \mathcal{A}\) gives a logical operator of weight \(|\mathcal{S}|+1\) \cite{arxiv:2509.06865}.
+  A weight-four GB code of odd distance \(d\) satisfies \(n \geq 1 + d^2\) \cite{arxiv:2203.17216},
+  so the \([[d^2+1,2,d]]\) BCC codes are optimal among weight-four BCC codes.
+
+features:
+  encoders:
+    - 'Initialize \(A\)-qubits in \(|+\rangle\) and \(B\)-qubits in \(|0\rangle\),
+      then apply CNOT gates along edges of \(G\), followed by Hadamard on each \(B\)-qubit.'
+  transversal_gates:
+    - 'For \(k=2\), all BCC codes admit a transversal Hadamard-SWAP logical gate
+      via qubit permutation \((m,1) \mapsto (-m,2)\) followed by Hadamard on all qubits \cite{arxiv:2509.06865}.'
+    - 'The \([[d^2+1,2,d]]\) BCC codes also admit a logical Hadamard without SWAP
+      via \(m \mapsto md \pmod{n}\) followed by Hadamard \cite{arxiv:2509.06865}.'
+  fault_tolerance:
+    - 'For distance-3 BCC codes (\(|\mathcal{S}|=2\)), a single error produces at most weight-1 logical error,
+      so fault tolerance is preserved \cite{arxiv:2509.06865}.'
+
+relations:
+  parents:
+    - code_id: generalized_bicycle
+      detail: 'BCC codes are GB codes with weight-two circulant \(A\) (polynomial \(a(x)=1+x\)) \cite{arxiv:2509.06865}.'
+    - code_id: quantum_cyclic
+      detail: 'BCC codes are invariant under cyclic shifts by construction \cite{arxiv:2509.06865}.'
+  cousins:
+    - code_id: cluster_state
+      detail: 'BCC codes are obtained by applying Hadamard on the \(B\)-sublattice of a bipartite cluster state,
+        converting CZ-gate preparation into CNOT-gate preparation \cite{arxiv:2509.06865}.'
+    - code_id: twisted_xzzx
+      detail: 'The \([[d^2+1,2,d]]\) twisted XZZX toric codes (parameters \(a=1,b=d\)) are Clifford-equivalent to BCC codes for odd \(d\) \cite{arxiv:2509.06865}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-05-26'

codes/quantum/qubits/stabilizer/topological/color/non-css/xyz_color.yml

@@ -11,7 +11,10 @@ name: 'XYZ color code'
 introduced: '\cite{arxiv:2203.16534}'
 
 description: |
-  Non-CSS variant of the 6.6.6 color code whose generators are \(XZXZXZ\) and \(ZYZYZY\) Pauli strings associated to each hexagonal in the hexagonal (6.6.6) tiling. A further variation called the \textit{domain wall color code} admits generators of the form \(XXXZZZ\) and \(ZZZXXX\) \cite{arxiv:2307.00054}.
+  A variant of the 6.6.6 color code whose generators are \(XZXZXZ\) and \(ZYZYZY\) Pauli strings associated to each hexagonal in the hexagonal (6.6.6) tiling. 
+  A further variation called the \textit{domain wall color code} admits generators of the form \(XXXZZZ\) and \(ZZZXXX\) \cite{arxiv:2307.00054}.
+  While such codes are equivalent to CSS color codes with the same distance, other properties like noise-bias performance can differ significantly.
+
 
 features:
   decoders:

…ed_surface/rotated-surface-operators.svg …ed_surface/rotated-surface-operators.svgcodes/quantum/qubits/stabilizer/topological/surface/rotated_surface/rotated_surface/rotated-surface-operators.svg renamed to codes/quantum/qubits/stabilizer/topological/surface/2d_surface/rotated_surface/rotated-surface-operators.svg codes/quantum/qubits/stabilizer/topological/surface/rotated_surface/rotated_surface/rotated-surface-operators.svg renamed to codes/quantum/qubits/stabilizer/topological/surface/2d_surface/rotated_surface/rotated-surface-operators.svg

@@ -13,7 +13,10 @@ introduced: '\cite{arxiv:1708.08474}'
 alternative_names:
   - 'Tailored surface code (TSC)'
 
-description: 'Non-CSS derivative of the surface code whose generators are \(XXXX\)  and \(YYYY\), obtained by mapping \(Z \to Y\) in the surface code.'
+description: |
+  A variant of the surface code whose generators are \(XXXX\)  and \(YYYY\), obtained by mapping \(Z \to Y\) in the surface code.
+  While this code is equivalent to a CSS surface code with the same distance, other properties like noise-bias performance can differ significantly.
+
 
 protection: As a stabilizer code, \([[n=O(d^2), k=O(1), d]]\).
 
@@ -26,8 +29,8 @@ features:
 
 relations:
   parents:
-    - code_id: clifford-deformed_surface
-      detail: XY code is obtained from the surface code by applying \(H\sqrt{Z}H\) to all qubits, thereby exchanging \(Z\leftrightarrow Y\).
+    - code_id: surface
+      detail: 'The XY surface code is obtained from the surface code by applying \(H\sqrt{Z}H\) to all qubits, thereby exchanging \(Z\leftrightarrow Y\). While it is equivalent to a CSS surface code with the same distance, but other properties like noise-bias performance can differ significantly.'
   cousins:
     - code_id: heavy_hex
       detail: 'XY surface code can be adapted for a heavy-hexagonal point set \cite{arxiv:2211.14038}.'