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‎codes/quantum/qubits/small_distance/small/10/stab_10_2_3.yml‎

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- code_id: bc_phantom
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detail: 'The binarized \([[10,2,3]]\) code is not phantom, but concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[20,2,6]]\) B\&C phantom code admitting fold-diagonal logical \(SS\) gates \cite{arxiv:2601.20927}.'
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- code_id: stab_4_2_2
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detail: 'Concatenating each qubit pair of the binarized \([[10,2,3]]\) code with the \([[4,2,2]]\) code yields a \([[20,2,6]]\) phantom code \cite{arxiv:2601.20927}.'
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detail: 'Concatenating each qubit pair of the binarized \([[10,2,3]]\) code with the \([[4,2,2]]\) code yields the \([[20,2,6]]\) B\&C phantom code \cite{arxiv:2601.20927,arxiv:2605.15344}.'
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# Begin Entry Meta Information

‎codes/quantum/qubits/small_distance/small/20/stab_20_2_6.yml‎

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#######################################################
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## This is a code entry in the error correction zoo. ##
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## https://github.com/errorcorrectionzoo ##
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#######################################################
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code_id: stab_20_2_6
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physical: qubits
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logical: qubits
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name: '\([[20,2,6]]\) B\&C phantom code'
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short_name: '\([[20,2,6]]\)'
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introduced: '\cite{arxiv:2601.20927,arxiv:2605.15344}'
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description: |
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Self-dual CSS code on 20 physical qubits encoding two logical qubits with distance 6.
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The code is obtained by binarizing the \([[5,1,3]]_4\) code in the self-dual normal basis \(\{\omega,\omega^2\}\) to the \([[10,2,3]]\) binarized Galois-qudit code and then concatenating each qubit pair with the \([[4,2,2]]\) code \cite{arxiv:2601.20927}.
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Physical qubits are arranged in five blocks of four, where block \(j\) corresponds to the \(j\)-th \(\mathbb{F}_4\) qudit of the \([[5,1,3]]_4\) code.
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Stabilizer generators of the \([[4,2,2]]\) inner code on each block are \(ZZZZ\) and \(XXXX\); the outer stabilizers are the lifted images of the eight stabilizer generators of the \([[10,2,3]]\) code using the logical-operator map
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\(X^\omega\mapsto XXII\), \(X^{\omega^2}\mapsto XIXI\), \(Z^\omega\mapsto IZIZ\), \(Z^{\omega^2}\mapsto IIZZ\) within each block.
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A stabilizer tableau for the code is \cite{arxiv:2601.20927}
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\begin{align}
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\begin{smallmatrix}
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Z & Z & Z & Z & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I \\
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I & I & I & I & Z & Z & Z & Z & I & I & I & I & I & I & I & I & I & I & I & I \\
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I & I & I & I & I & I & I & I & Z & Z & Z & Z & I & I & I & I & I & I & I & I \\
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I & I & I & I & I & I & I & I & I & I & I & I & Z & Z & Z & Z & I & I & I & I \\
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I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & Z & Z & Z & Z \\
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I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & I & I & I \\
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I & I & Z & Z & I & I & Z & Z & I & I & Z & Z & I & I & Z & Z & I & I & I & I \\
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I & I & I & I & I & Z & I & Z & I & Z & Z & I & I & I & Z & Z & I & Z & I & Z \\
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I & I & I & I & I & I & Z & Z & I & Z & I & Z & I & Z & Z & I & I & I & Z & Z \\
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X & X & X & X & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I \\
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I & I & I & I & X & X & X & X & I & I & I & I & I & I & I & I & I & I & I & I \\
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I & I & I & I & I & I & I & I & X & X & X & X & I & I & I & I & I & I & I & I \\
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I & I & I & I & I & I & I & I & I & I & I & I & X & X & X & X & I & I & I & I \\
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I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & I & X & X & X & X \\
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X & X & I & I & X & X & I & I & X & X & I & I & X & X & I & I & I & I & I & I \\
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X & I & X & I & X & I & X & I & X & I & X & I & X & I & X & I & I & I & I & I \\
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I & I & I & I & X & X & I & I & X & I & X & I & I & X & X & I & X & X & I & I \\
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I & I & I & I & X & I & X & I & I & X & X & I & X & X & I & I & X & I & X & I
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\end{smallmatrix}~.
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\end{align}
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Rows 1--5 are the \(Z\)-type inner stabilizers of each \([[4,2,2]]\) block; rows 6--9 are the lifted outer \(Z\)-type stabilizers; rows 10--14 are the \(X\)-type inner stabilizers; rows 15--18 are the lifted outer \(X\)-type stabilizers.
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protection: |
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Detects errors on up to 5 qubits and corrects errors on up to 2 qubits.
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features:
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transversal_gates:
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- 'Logical CNOT gates in both directions between the two logical qubits are realized by qubit permutations within a code block \cite{arxiv:2601.20927,arxiv:2605.15344}.'
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- 'Fold-diagonal logical \(SS\) gates are available from the self-duality of the outer \([[5,1,3]]_4\) code and the inner \([[4,2,2]]\) layer \cite{arxiv:2601.20927}.'
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fault_tolerance:
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- 'Selective state filtering (post-selection): logical error rates of \(\approx 2\times 10^{-6}\) per round of Steane-style error correction at physical error rate \(p=10^{-3}\), without being fully fault-tolerant \cite{arxiv:2605.15344}.'
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relations:
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parents:
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- code_id: bc_phantom
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detail: 'The \([[20,2,6]]\) code is the B\&C phantom code obtained from the \([[5,1,3]]_4\) Galois-qudit CSS code \cite{arxiv:2601.20927,arxiv:2605.15344}.'
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- code_id: self_dual_css
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detail: 'The \([[20,2,6]]\) code is a self-dual CSS code obtained from the \([[5,1,3]]\) code via the BLT mapping and concatenation with \([[4,2,2]]\) \cite[Corr. 2]{arxiv:2605.15344}\cite[Corr. 1]{arxiv:1004.3791}.'
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cousins:
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- code_id: stab_5_1_3
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detail: 'The \([[20,2,6]]\) code is obtained from the \([[5,1,3]]\) five-qubit code via the BLT mapping (Lemma 1) and concatenation with the \([[4,2,2]]\) code (Corollary 2) \cite{arxiv:2605.15344}\cite[Corr. 1]{arxiv:1004.3791}.'
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- code_id: stab_4_2_2
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detail: 'The \([[20,2,6]]\) code is obtained by concatenating each qubit pair of the \([[10,2,3]]\) binarized code with the \([[4,2,2]]\) code \cite{arxiv:2601.20927,arxiv:2605.15344}.'
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- code_id: stab_10_2_3
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detail: 'The \([[20,2,6]]\) code is obtained by concatenating each qubit pair of the \([[10,2,3]]\) binarized Galois-qudit code with the \([[4,2,2]]\) code \cite{arxiv:2601.20927}.'
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# Begin Entry Meta Information
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_meta:
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changelog:
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- user_id: VictorVAlbert
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date: '2026-05-22'

‎codes/quantum/qubits/small_distance/small/8/stab_8_2_3.yml‎

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introduced: '\cite{arxiv:2008.05051}'
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description: |
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A non-CSS Hermitian eight-qubit stabilizer code that has the largest automorphism group of size 1728
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A non-CSS Hermitian eight-qubit stabilizer code that has the largest automorphism group of size 1728
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among the 20 inequivalent \([[8,2,3]]\) codes \cite{arxiv:2501.17447}.
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The code has an exceptionally short fault-tolerant syndrome measurement sequence \cite{arxiv:2008.05051,arxiv:2501.17447}.
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parents:
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- code_id: stabilizer_over_gf4
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- code_id: small_distance_qubit_stabilizer
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cousins:
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- code_id: qubit_concatenated
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detail: 'Applying the BLT mapping to the \([[8,2,3]]\) Hermitian code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[32,4,6]]\) self-dual CSS code \cite[Corr. 2]{arxiv:2605.15344}.'
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- code_id: self_dual_css
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detail: 'Applying the BLT mapping to the \([[8,2,3]]\) Hermitian code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[32,4,6]]\) self-dual CSS code \cite[Corr. 2]{arxiv:2605.15344}.'
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- code_id: stab_4_2_2
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detail: 'Applying the BLT mapping to the \([[8,2,3]]\) Hermitian code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[32,4,6]]\) self-dual CSS code \cite[Corr. 2]{arxiv:2605.15344}.'
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# Begin Entry Meta Information

‎codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml‎

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\end{align}
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where \(H\) is the parity-check matrix of the classical code.
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Every Hermitian qubit stabilizer code has stabilizer generators with \textit{paired support} \cite[Def. 4]{arxiv:2605.15344}: Pauli operators \(P\) and \(Q\) have paired support if \(P\), \(Q\), and \(P \cdot Q\) all have the same support, and a set of generators has paired support if it can be partitioned into such pairs. The converse does not hold in general.
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All code automorphisms lie in the \hyperref[topic:clifford]{Clifford group} \cite[Corr. 16]{arxiv:quant-ph/9704043}.
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protection: |
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For an additive self-orthogonal code \(C \subseteq \mathbb{F}_4^n\), the resulting qubit stabilizer code has distance
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\begin{align}

‎codes/quantum/qubits/stabilizer/qubit_stabilizer.yml‎

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detail: 'Qubit stabilizer states can be interpreted as states that are preparable using the Euclidean path integral in 3D Chern-Simons theory, defined on manifolds that are toy models of AdS/CFT wormholes \cite{arxiv:1611.01516,arxiv:2510.15067}.'
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- code_id: topological_abelian
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detail: 'Qubit stabilizer states can be interpreted as states that are preparable using the Euclidean path integral in 3D Chern-Simons theory, defined on manifolds that are toy models of AdS/CFT wormholes \cite{arxiv:1611.01516,arxiv:2510.15067}.'
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- code_id: galois_css
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detail: 'Any \([[n,k,d]]\) qubit stabilizer code can be mapped to an \([[n,k,d]]_4\) Galois-qudit CSS code via the BLT mapping \cite[Lemma 1]{arxiv:2605.15344}\cite[Lemma 1]{arxiv:1004.3791}: each stabilizer \(P = \bigotimes_j P_j\) generates an \(XX\)-type stabilizer via \(\mathcal{D}_X\) (\(I\mapsto II,\,X\mapsto XI,\,Z\mapsto IX,\,Y\mapsto XX\)) and a \(ZZ\)-type stabilizer via \(\mathcal{D}_Z\) (\(I\mapsto II,\,X\mapsto IZ,\,Z\mapsto ZI,\,Y\mapsto ZZ\)).'
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# verify...
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# - code_id: reed_muller

‎codes/quantum/qubits/stabilizer/self_dual_css.yml‎

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- code_id: dual
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detail: 'Self-dual CSS codes arise from dual-containing (equivalently, self-orthogonal a.k.a. weakly self-dual) binary linear codes.'
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- code_id: qubit_stabilizer
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detail: 'Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[4n,2k,2d]]\) self-dual CSS code via an intermediate tetron Majorana stabilizer code \cite[Corr. 1]{arxiv:1004.3791}\cite{arxiv:1703.07847}, which preserves geometric locality of a code up to a constant factor.'
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detail: 'Any \([[n,k,d]]\) qubit stabilizer code maps to a \([[4n,2k,2d]]\) self-dual CSS code by applying the BLT mapping and concatenating each qubit pair with the \([[4,2,2]]\) code \cite[Corr. 2]{arxiv:2605.15344}\cite[Corr. 1]{arxiv:1004.3791}. The BLT mapping proceeds by first concatenating each qubit with the \hyperref[code:tetron]{tetron code} to obtain an intermediate \([[2n,k,2d]]_{f}\) Majorana stabilizer code.'
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- code_id: tetron
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detail: 'Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[4n,2k,2d]]\) self-dual CSS code via an intermediate tetron Majorana stabilizer code \cite[Corr. 1]{arxiv:1004.3791}\cite{arxiv:1703.07847}, which preserves geometric locality of a code up to a constant factor.'
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detail: 'Any \([[n,k,d]]\) qubit stabilizer code maps to a \([[4n,2k,2d]]\) self-dual CSS code by applying the BLT mapping and concatenating each qubit pair with the \([[4,2,2]]\) code \cite[Corr. 2]{arxiv:2605.15344}\cite[Corr. 1]{arxiv:1004.3791}. The BLT mapping proceeds by first concatenating each qubit with the \hyperref[code:tetron]{tetron code} to obtain an intermediate \([[2n,k,2d]]_{f}\) Majorana stabilizer code.'
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- code_id: stab_4_2_2
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detail: 'Any \([[n,k,d]]\) qubit stabilizer code maps to a \([[4n,2k,2d]]\) self-dual CSS code by applying the BLT mapping and concatenating each qubit pair with the \([[4,2,2]]\) code \cite[Corr. 2]{arxiv:2605.15344}\cite[Corr. 1]{arxiv:1004.3791}. The BLT mapping proceeds by first concatenating each qubit with the \hyperref[code:tetron]{tetron code} to obtain an intermediate \([[2n,k,2d]]_{f}\) Majorana stabilizer code.'
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- code_id: quantum_reed_muller
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detail: 'The \([[2^m,{m \choose r}, 2^{\min(r,m-r)}]]\) quantum RM family contains a self-dual sub-family for \(m=2r\), which admits logical Clifford group gates via permutations, transversal gates, and fold-transversal gates \cite{arxiv:2410.23263,arxiv:2602.09788}.'
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- code_id: quantum_divisible

‎codes/quantum/qudits_galois/small/css_5_1_3.yml‎

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detail: 'The \([[5,1,3]]_4\) code saturates the quantum Singleton bound.'
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- code_id: small_distance_quantum
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cousins:
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- code_id: stab_5_1_3
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detail: 'The \([[5,1,3]]_4\) Galois-qudit CSS code is the image of the \([[5,1,3]]\) five-qubit code under the BLT mapping \cite[Lemma 1]{arxiv:2605.15344}\cite[Lemma 1]{arxiv:1004.3791}.'
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- code_id: shortened_hexacode
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detail: 'The \([[5,1,3]]_4\) code is obtained from the shortened hexacode \cite{arxiv:2601.20927}.'
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- code_id: stab_10_2_3
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detail: 'Binarizing the \([[5,1,3]]_4\) code in the self-dual normal basis \(\{\omega,\omega^2\}\) yields a \([[10,2,3]]\) qubit CSS code \cite{arxiv:2601.20927}.'
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- code_id: bc_phantom
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detail: 'Binarizing this code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[20,2,6]]\) B\&C phantom code admitting fold-diagonal logical \(SS\) gates \cite{arxiv:2601.20927}.'
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# Begin Entry Meta Information

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