four_qubit_permutation_invariant

codes/quantum/properties/block/symmetric/permutation_invariant.yml

@@ -15,7 +15,7 @@ description: |
 #Such a code is said to be stabilized by the symmetric group \(S_n\) on \(n\) elements, in a generalization of stabilizer codes to binary codes utilizing (nonabelian) group actions.'
 
 protection: |
-  Permutation invariant codes of distance \(d\) can protect against \(d-1\) deletion errors \cite{arxiv:2102.02494,arXiv:2102.03015}, i.e., erasures of qubits at unknown locations.
+  Permutation invariant codes of distance \(d\) can protect against \(d-1\) deletion errors \cite{arxiv:2001.08405,arxiv:2004.00814,arxiv:2102.02494,arXiv:2102.03015}, i.e., erasures of subsystems at unknown locations.
 
   Other protection depends on the code family.
   The GNU permutation-invariant family (parameterized by \(t\)) protects against arbitrary weight \(t\) qubit errors and approximately corrects spontaneous decay errors \cite{arXiv:1302.3247,doi:10.1103/PhysRevA.93.042340}.

codes/quantum/qubits/permutation_invariant/combinatorial_permutation_invariant.yml

@@ -25,6 +25,7 @@ relations:
     - code_id: qubit_permutation_invariant
 
 
+
 # Begin Entry Meta Information
 _meta:
   # Change log - most recent first

codes/quantum/qubits/permutation_invariant/four_qubit_permutation_invariant.yml

@@ -0,0 +1,44 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: four_qubit_permutation_invariant
+
+name: 'Four-qubit single-deletion code'
+introduced: '\cite{arxiv:2001.08405,arxiv:2004.00814}'
+
+description: |
+  Four-qubit permutation-invariant code that is the smallest qubit code to correct one deletion error.
+
+  In terms of \hyperref[topic:dicke]{Dicke states}, logical codewords are
+  \begin{align}
+  \begin{split}
+    |0_{L}\rangle&\propto|D_{0}^{4}\rangle+|D_{4}^{4}\rangle\\
+    |1_{L}\rangle&\propto|D_{2}^{4}\rangle~.
+  \end{split}
+  \end{align}
+
+protection: |
+  The smallest qubit code to correct one deletion error.
+
+relations:
+  parents:
+    - code_id: qubit_permutation_invariant
+    - code_id: small_distance_quantum
+      detail: 'The four-qubit single-deletion code corrects a single deletion error.'
+  cousins:
+    - code_id: binomial
+      detail: 'The four-qubit single-deletion code can be obtained from the "0-2-4" single-mode binomial code by substituting Fock states with \hyperref[topic:dicke]{Dicke states}.'
+    - code_id: stab_4_2_2
+      detail: 'A basis of codewords for the four-qubit single-deletion code consists of the \(|\overline{00}\rangle\) and \(|\overline{01}\rangle+|\overline{10}\rangle+|\overline{11}\rangle\)states of the four-qubit code.'
+    - code_id: combinatorial_permutation_invariant
+      detail: 'The combinatorial permutation-invariant code \(Q_{1,1,1,-}\) is another example of a four-qubit code correcting a single deletion error \cite[Sec. 5.1]{arxiv:2310.05358}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2023-04-18'

codes/quantum/qubits/permutation_invariant/gnu/gnu_permutation_invariant.yml

@@ -14,7 +14,7 @@ introduced: '\cite{arXiv:1302.3247,arxiv:1512.02469}'
 description: |
   Permutation-invariant code whose codewords can be expressed as superpositions of \hyperref[topic:dicke]{Dicke states} with coefficients are square-roots of the binomial distribution.
 
-  Logical codewords for codes encoding a single qubit \cite{arXiv:1302.3247} are
+  In terms of \hyperref[topic:dicke]{Dicke states}, logical codewords for codes encoding a single qubit \cite{arXiv:1302.3247} are
   \begin{align}
   |\overline{\pm}\rangle = \sum_{\ell=0}^{m} \frac{(\pm 1)^\ell}{\sqrt{2^m}} \sqrt{m \choose \ell} |D^n_{g \ell}\rangle~.
   \end{align}

codes/quantum/qubits/permutation_invariant/qubit_permutation_invariant.yml

@@ -4,6 +4,8 @@
 #######################################################
 
 code_id: qubit_permutation_invariant
+physical: qubits
+# Qudit GNU codes don't encode logical qubits
 
 name: 'Permutation-invariant qubit code'
 
@@ -16,6 +18,10 @@ description: |
   Each Dicke state in the code can be \textit{shifted} by adding a shift \(s\) to both \(n\) and \(w\).
   \end{defterm}
 
+protection: |
+  Permutation invariant qubit codes of distance \(d\) can protect against \(d-1\) deletion errors \cite{arxiv:2001.08405,arxiv:2004.00814}, i.e., erasures of subsystems at unknown locations.
+
+
 features:
   decoders:
     - 'Schur-Weyl-transform based decoder \cite{arxiv:2212.06285}.