bare_7_1_3

codes/classical/bits/bits_into_bits.yml

@@ -15,6 +15,7 @@ description: |
   The coordinate permutation group \(S_n\) of order \(n!\) is formed by \(n\)-dimensional matrices with a 1 in each row and column \cite[Ch. 8]{preset:MacSlo}\cite[Ch. 3]{doi:10.1007/978-1-4757-6568-7}.
   The group of isometries of Hamming space is the hyperoctahedral group \(\mathbb{Z}_2\wr S_n=\mathbb{Z}_2^n\rtimes S_n\), i.e., the permutation group together with the group formed by the action of binary space on itself (under addition).
   Two binary codes are \textit{equivalent} if the codewords of one code can be mapped into those of the other under a hyperoctahedral group element \cite[Def. 1.8.8]{preset:HKSbasics}\cite{preset:HKSclass}.
+  Determining equivalence of two codes can be done by putting each in a canonical form and mapping to a graph isomorphism problem \cite{doi:10.1007/3-540-28991-7}.
 
 protection: |
   A binary code \(C\) \textit{corrects} \(t\) errors in the Hamming distance if

codes/quantum/qubits/qubits_into_qubits.yml

@@ -17,6 +17,8 @@ description: |
   Encodes \(K\)-dimensional Hilbert space into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space.
   Usually denoted as \(((n,K))\) or \(((n,K,d))\), where \(d\) is the code's distance.
 
+  The qubit codes are \textit{equivalent} if the codespace of one code can be mapped into that of the other under a tensor product of single-qubit unitary operations and a qubit permutation.
+
 protection: |
   An \(((n,K,d))\) code corrects erasure errors on up to \(d-1\) qubits.
   The number of correctable errors is often called the \textit{decoding radius}, and it is upper bounded by half of the code distance.

codes/quantum/qubits/small_distance/small/bare_7_1_3.yml

@@ -0,0 +1,33 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: bare_7_1_3
+physical: qubits
+logical: qubits
+
+name: '\([[7,1,3]]\) bare code'
+introduced: '\cite{arxiv:1702.01155}'
+
+description: |
+  A \([[7,1,3]]\) code that admits fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement.
+
+  It is one of sixteen distinct \([[7,1,3]]\) codes \cite{arxiv:0709.1780}.
+
+features:
+  decoders:
+    - 'Fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement.'
+
+relations:
+  parents:
+    - code_id: qubit_stabilizer
+    - code_id: small_distance_quantum
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-09-25'

codes/quantum/qubits/small_distance/small/twist_defect_7_1_3.yml

@@ -10,6 +10,9 @@ logical: qubits
 name: '\([[7,1,3]]\) twist-defect surface code'
 introduced: '\cite{arxiv:1612.04795}'
 
+alternative_names:
+  - '\([[7,1,3]]\) triangle code'
+
 description: |
   A \([[7,1,3]]\) code (different from the Steane code) that is a small example of a twist-defect surface code.
 

codes/quantum/qubits/stabilizer/qubit_stabilizer.yml

@@ -40,6 +40,10 @@ description: |
       \label{table:stabilizer-groups}
     \end{table}
 
+  Two qubit stabilizer codes codes are \textit{equivalent} if the codespace of one code can be mapped into that of the other under a tensor product of elements of the \hyperref[topic:clifford]{single-qubit Clifford group} and a qubit permutation.
+  Equivalence under single-qubit Clifford operations is not the same as the equivalence under a tensor product of arbitrary single-qubit unitary operations \cite{arxiv:0709.1266}.
+  A qubit stabiilzer code is \textit{decomposable} if there exists a permutation that maps the stabilizer group into a tensor product of two stabilizer groups acting on disjoint sets of qubits.
+
   \begin{defterm}{Symplectic representation}
   \label{topic:binary-symplectic-representation}
   In the symplectic representation, the single-qubit identity, \(X\), \(Y\), or \(Z\) Pauli matrices represented using two bits as \((0|0)\), \((1|0)\), \((1|1)\), and \((0|1)\), respectively.
@@ -65,6 +69,7 @@ description: |
 
   The sets of \(GF(4)\)-represented vectors for all generators yield a trace-Hermitian self-orthogonal additive quaternary code.
   This classical code corresponds to the stabilizer group \(\mathsf{S}\) while its trace-Hermitian dual corresponds to the normalizer \(\mathsf{N(S)}\).
+  In the case of stabilizer states, the correspondence is between such states and trace-Hermitian self-dual quaternary codes; such codes, and therefore such states, have been classified up to equivalence for \(n \leq 12\) \cite{arxiv:quant-ph/0503236,arxiv:math/0504522}.
 
   Alternative representations include the \textit{decoupling representation}, in which Pauli strings are represented as vectors over \(GF(2)\) using three bits \cite{arxiv:2305.17505}.