css_12_1_3

Author: valbert4

codes/quantum/properties/hamiltonian/constant_excitation.yml

@@ -35,6 +35,9 @@ protection: |
 features:
   rate: 'Fock-state CE codes can be used in a protocol that achieves the two-way quantum capacity of the \hyperref[topic:ad]{AD} Gaussian channel \cite{arxiv:2203.13924}. For every \(K,t \geq 2\), there are explicitly constructible \(K\)-dimensional Fock-state CE codes with \(q=N=(K-1)t(t+1)\) modes, total excitation \(N\), and distance \(t+1\); there also exist families with logical dimension \(K = o(2^N)\) and distance of \hyperref[topic:asymptotics]{order} \(o(N/\log N)\) \cite{arxiv:2509.20545}.'
 
+  fault_tolerance:
+    - 'Fault-tolerant QEC framework for CE CSS codes using modified Shor and Steane syndrome extraction, where weight-\(2w\) stabilizers are measured using \(w\)-CE cat states and zero-controlled NOT (\(\mathrm{C}_0 X\)) gates replace standard CNOT gates to preserve the constant-excitation structure \cite{arxiv:2507.10395}.'
+
 
 relations:
   parents:
@@ -48,8 +51,11 @@ relations:
         Qubit CE codes are protected from coherent noise in the form of transversal \(Z\)-rotations because such rotations act identically on all codewords \cite{doi:10.1109/ISIT45174.2021.9518206,arxiv:2011.00197}.
         In the case of qubit CSS codes, all codes oblivious to such rotations are CE codes \cite{doi:10.1109/ISIT45174.2021.9518206,arxiv:2011.00197}.
         Any \([[n,k,d]]\) CSS code can be made into an \([[mn,k,>d]]\) CE code \cite{doi:10.1109/ISIT45174.2021.9518206}.
+        Concatenating the dual-rail code with an inner \([[n,k,d]]\) qubit stabilizer code yields a degenerate \([[2n,k,d]]\) constant-excitation stabilizer code that avoids coherent phase errors and is equivalent to a Pauli-rotated repetition-concatenated stabilizer code \cite{arxiv:2010.00538}. CSS structure is preserved when the original code is CSS \cite{arxiv:2507.10395}.
     - code_id: stab_5_1_3
-      detail: 'The five-qubit code can be concatenated with a particular decoherence-free subspace (DFS) \cite{arxiv:quant-ph/9807004,arxiv:quant-ph/9902041,arxiv:quant-ph/9908064,arxiv:quant-ph/0007013} to yield a 20-qubit CE code \cite{arxiv:quant-ph/9809081,arxiv:quant-ph/9907096}.'
+      detail: 'The five-qubit code can be concatenated with a particular decoherence-free subspace (DFS) \cite{arxiv:quant-ph/9807004,arxiv:quant-ph/9902041,arxiv:quant-ph/9908064,arxiv:quant-ph/0007013} to yield a 20-qubit CE code \cite{arxiv:quant-ph/9809081,arxiv:quant-ph/9907096}. Dual-rail concatenation of the five-qubit code yields a \([[10,1,3]]\) CE stabilizer code \cite{arxiv:2507.10395}.'
+    - code_id: qubit_stabilizer
+      detail: 'Concatenating the dual-rail code with an inner \([[n,k,d]]\) qubit stabilizer code yields a degenerate \([[2n,k,d]]\) constant-excitation stabilizer code that avoids coherent phase errors and is equivalent to a Pauli-rotated repetition-concatenated stabilizer code \cite{arxiv:2010.00538}. CSS structure is preserved when the original code is CSS \cite{arxiv:2507.10395}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/small_distance/small/12/css_12_1_3.yml

@@ -0,0 +1,60 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: css_12_1_3
+physical: qubits
+logical: qubits
+
+name: '\([[12,1,3]]\) CE CSS code'
+introduced: '\cite{arxiv:2507.10395}'
+
+description: |
+  Twelve-qubit constant-excitation (CE) CSS code that encodes one logical qubit with distance three.
+  It is the smallest CE CSS code that corrects a single-qubit error \cite{arxiv:2507.10395}.
+  Codewords lie in a fixed Hamming-weight subspace, making the code immune to coherent noise in the form of transversal \(Z\)-rotations.
+
+  One stabilizer tableau for the code is, up to Pauli frame \cite{arxiv:2507.10395},
+  \begin{align}
+  \begin{smallmatrix}
+    X & X & X & X & I & I & I & I & I & I & I & I \\
+    I & I & X & X & X & X & I & I & I & I & I & I \\
+    I & I & I & I & I & I & X & X & X & X & I & I \\
+    I & I & I & I & I & I & I & I & X & X & X & X \\
+    Z & I & Z & I & Z & I & Z & I & Z & I & Z & I \\
+    Z & Z & I & I & I & I & I & I & I & I & I & I \\
+    I & I & Z & Z & I & I & I & I & I & I & I & I \\
+    I & I & I & I & Z & Z & I & I & I & I & I & I \\
+    I & I & I & I & I & I & Z & Z & I & I & I & I \\
+    I & I & I & I & I & I & I & I & Z & Z & I & I \\
+    I & I & I & I & I & I & I & I & I & I & Z & Z
+  \end{smallmatrix}~.
+  \end{align}
+
+protection: |
+  Corrects any single-qubit error (distance \(d=3\)).
+  Protects from collective coherent noise in the form of transversal \(Z\)-rotations, since all codewords lie in the same Hamming-weight subspace \cite{doi:10.1109/ISIT45174.2021.9518206,arxiv:2011.00197}.
+
+features:
+  fault_tolerance:
+    - 'Fault-tolerant syndrome extraction using modified Shor and Steane methods adapted for CE codes: weight-\(2w\) stabilizers are measured using \(w\)-CE cat states, and zero-controlled NOT (\(\mathrm{C}_0 X\)) gates replace standard CNOT gates to preserve the constant-excitation structure \cite{arxiv:2507.10395}.'
+
+  threshold:
+    - 'Pseudo-threshold of \(\sim 9.28 \times 10^{-4}\) (circuit-level stochastic noise) and \(\sim 5.98 \times 10^{-4}\) (with coherent corrections, \(\gamma = 0.01\)) under collective coherent noise \cite{arxiv:2507.10395}.'
+
+
+relations:
+  parents:
+    - code_id: qubit_css
+    - code_id: constant_excitation
+    - code_id: qubit_concatenated
+      detail: 'This code is obtained by dual-rail concatenation of the \([[6,1,2]]\) CSS code \cite[ID 50]{preset:qiskit}.'
+    - 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/14/phantom_14_3_3.yml

@@ -7,11 +7,12 @@ code_id: phantom_14_3_3
 physical: qubits
 logical: qubits
 
-name: '\([[14,3,3]]\) CSS phantom code'
-introduced: '\cite{arxiv:2601.20927}'
+name: '\([[14,3,3]]\) CE phantom code'
+introduced: '\cite{arxiv:2507.10395,arxiv:2601.20927}'
 
 description: |
   CSS phantom code obtained by concatenating the \([[7,3,(d_X=3,d_Z=2)]]\) punctured hypercube code with the two-qubit phase-flip repetition code.
+  The code is equivalent to the \([[14,3,3]]\) constant-excitation (CE) CSS code obtained by applying dual-rail concatenation to the \([[7,3,2]]\) punctured hypercube code, up to single-qubit Clifford gates, a physical-qubit permutation, and a Pauli frame \cite{arxiv:2507.10395}.
 
   One stabilizer tableau for the code is
   \begin{align}
@@ -37,16 +38,23 @@ features:
     - 'The code is phantom, so every ordered-pair in-block logical CNOT gate between its three logical qubits can be implemented by a physical-qubit permutation \cite{arxiv:2601.20927}.'
     - 'The Hadamard-dual code admits fold-diagonal logical \(S_iS_j\) and \(CZ_{ij}\) gates \cite{arxiv:2601.20927}.'
 
+  fault_tolerance:
+    - 'In the locally Clifford-equivalent CE CSS frame, fault-tolerant syndrome extraction can use modified Shor and Steane methods adapted for CE codes: weight-\(2w\) stabilizers are measured using \(w\)-CE cat states, and zero-controlled NOT (\(\mathrm{C}_0 X\)) gates replace standard CNOT gates to preserve the constant-excitation structure \cite{arxiv:2507.10395}.'
+
 relations:
   parents:
     - code_id: phantom
       detail: 'This \([[14,3,3]]\) code is a CSS phantom code obtained from the punctured hypercube code and the two-qubit phase-flip repetition code \cite{arxiv:2601.20927}.'
+    - code_id: constant_excitation
+      detail: 'This code is single-qubit Clifford equivalent to the \([[14,3,3]]\) CE CSS code obtained by dual-rail concatenation of the \([[7,3,2]]\) punctured hypercube code \cite{arxiv:2507.10395}.'
     - code_id: qubit_concatenated
       detail: 'This code is a concatenation of the \([[7,3,2]]\) punctured hypercube code with the two-qubit phase-flip repetition code \cite{arxiv:2601.20927}.'
     - code_id: small_distance_qubit_stabilizer
   cousins:
     - code_id: xz_7_3_2
-      detail: 'Concatenating the \([[7,3,(d_X=3,d_Z=2)]]\) punctured hypercube code with the two-qubit phase-flip repetition code yields this \([[14,3,(d_X=3,d_Z=4)]]\) CSS phantom code \cite{arxiv:2601.20927}.'
+      detail: 'Concatenating the \([[7,3,(d_X=3,d_Z=2)]]\) punctured hypercube code with the two-qubit phase-flip repetition code yields this \([[14,3,(d_X=3,d_Z=4)]]\) CSS phantom code \cite{arxiv:2601.20927}. Dual-rail concatenation of the same punctured hypercube code yields a single-qubit Clifford-equivalent CE CSS frame \cite{arxiv:2507.10395}.'
+    - code_id: steane
+      detail: 'Dual-rail concatenation of the \([[7,1,3]]\) Steane code yields a \([[14,1,3]]\) CE CSS code, from which the locally Clifford-equivalent \([[14,3,3]]\) CE CSS frame is obtained by removing two independent \(Z\)-type stabilizer generators \cite{arxiv:2507.10395}.'
     - code_id: quantum_repetition
       detail: 'The inner code in the construction is the two-qubit phase-flip repetition code \cite{arxiv:2601.20927}.'
 

codes/quantum/qubits/stabilizer/qubit_stabilizer.yml

@@ -323,8 +323,6 @@ relations:
       detail: 'Stabilizer states on \(n\) qubits form complex projective 3-designs, but not 4-designs, on \(\mathbb{C}P^{2^n}\) \cite{arxiv:1510.02767}. The \hyperref[topic:clifford]{Clifford group} is a unitary 2-design \cite{arxiv:quant-ph/0103098} and a 3-design \cite[Thm. 1.6(B)]{arxiv:math/0502080}\cite[pg. 191]{doi:10.1007/3-540-30731-1}\cite{arxiv:1510.02619,arxiv:1510.02769} on \(U(2^n)\).'
     - code_id: iceberg
       detail: 'The \([[2m,2m-2,2]]\) code for \(2m\) being a multiple of four obstructs the Clifford group from being a 4-design \cite{arxiv:1609.08172}.'
-    - code_id: constant_excitation
-      detail: 'Concatenating the dual-rail code with an inner \([[n,k,d]]\) qubit stabilizer code yields a degenerate \([[2n,k,d]]\) constant-excitation stabilizer code that avoids coherent phase errors and is equivalent to a Pauli-rotated repetition-concatenated stabilizer code \cite{arxiv:2010.00538}.'
     - code_id: ampdamp
       detail: 'Concatenating the dual-rail code with an inner \([[n,k,d]]\) qubit stabilizer code yields a degenerate \([[2n,k,d]]\) constant-excitation stabilizer code that corrects \(d-1\) \hyperref[topic:ad]{AD} errors \cite{arxiv:1001.2356,arxiv:2010.00538}.'
     - code_id: two-legged-cat