group-rep + Clifford QSC

codes/quantum/oscillators/coherent_state/cat.yml

@@ -47,12 +47,15 @@ relations:
       detail: 'Cat codes are QSCs on the one-dimensional complex sphere.'
     - code_id: bosonic_rotation
       detail: 'The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) \cite{arxiv:1901.08071}.'
-    # - code_id: coherent_constellation
-    #   detail: 'Cat-code codewords are constructed using a coherent-state constellation that forms the cyclic group \(\mathbb{Z}_{2S+2}\).'
+    - code_id: group_representation
+      detail: 'Cat codes are group representation codes with \(G\) being a cyclic group \cite{arxiv:2306.11621}.'
   cousins:
     - code_id: number_phase
       detail: 'In the limit as \(N,S \to \infty\), phase measurement in the cat code has vanishing variance, just like in a number-phase code \cite{arxiv:1901.08071}.'
 
+# - code_id: coherent_constellation
+#   detail: 'Cat-code codewords are constructed using a coherent-state constellation that forms the cyclic group \(\mathbb{Z}_{2S+2}\).'
+
 
 # Begin Entry Meta Information
 _meta:

codes/quantum/oscillators/coherent_state/clifford_qsc.yml

@@ -0,0 +1,33 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: clifford_qsc
+
+name: 'Clifford group-representation QSC'
+introduced: '\cite{arXiv:2306.11621}'
+
+description: |
+  QSC whose projection is onto a copy of an irreducible representation of the single-qubit Clifford group \(2O\), taken as the binary octahedral subgroup of the group \(SU(2)\) of Gaussian rotations.
+
+features:
+  general_gates:
+    - 'The Clifford group \(2O\) can be realized via Gaussian rotations.
+    The \(T\) and \(CZ\) gates can be realized using quartic Kerr operations \cite{arxiv:2306.11621}.'
+
+
+relations:
+  parents:
+    - code_id: qsc
+      detail: 'The Clifford group-representation QSC has non-uniform coefficients.'
+    - code_id: group_representation
+      detail: 'The Clifford group-representation QSC is a group-representation code with \(G\) being the binary octahedral subgroup of \(SU(2)\).'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-02-21'

codes/quantum/oscillators/fock_state/constant_excitation/one_hot_quantum.yml

@@ -19,13 +19,16 @@ protection: |
 features:
   general_gates:
     - 'Non-deterministic gates using linear optics and photon-number resolving detectors \cite{arxiv:2302.07357}.'
+    - 'If the code is embedded into Fock space, the group \(SU(q)\) can be realized via Gaussian rotations \cite{arxiv:2306.11621}.'
 
 
 relations:
   parents:
     - code_id: chuang-leung-yamamoto
     - code_id: constant_excitation
     - code_id: permutation_invariant
+    - code_id: group_representation
+      detail: 'One-hot quantum codes are group-representation codes with \(G = SU(q)\) \cite{arxiv:2306.11621}.'
   cousins:
     - code_id: one_hot
       detail: 'The one-hot quantum code is the quantum version of the one-hot code.'

codes/quantum/properties/block/covariant/covariant.yml

@@ -9,7 +9,8 @@ name: 'Covariant code'
 introduced: '\cite{arxiv:1709.04471}'
 
 description: |
-  A block code on \(n\) subsystems that admits a group \(G\) of transversal gates. The group has to be finite for finite-dimensional codes due to the Eastin-Knill theorem. Continuous-\(G\) covariant codes, necessarily infinite-dimensional, are relevant to error correction of quantum reference frames \cite{arxiv:1709.04471} and error-corrected parameter estimation.
+  A block code on \(n\) subsystems that admits a group \(G\) of transversal gates. The group has to be finite for finite-dimensional codes due to the Eastin-Knill theorem.
+  Continuous-\(G\) covariant codes, necessarily infinite-dimensional, are relevant to error correction of quantum reference frames \cite{arxiv:1709.04471} and error-corrected parameter estimation.
 
   Denoting the code's encoding map as \(U\), covariance is equivalent to
   \begin{align}
@@ -54,6 +55,8 @@ relations:
   parents:
     - code_id: block_quantum
       detail: 'Covariant codes for \(n>1\) are block quantum codes.'
+    - code_id: group_representation
+      detail: 'Covariant codes are block group-representation codes \cite[Lemma 2]{arxiv:2306.11621}.'
   cousins:
     - code_id: approximate_qecc
       detail: 'Normalizable constructions of infinite-dimensional \(G\)-covariant codes for continuous \(G\) are approximately error-correcting.'

codes/quantum/properties/group_representation.yml

@@ -0,0 +1,35 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: group_representation
+
+name: 'Group-representation code'
+introduced: '\cite{arXiv:2005.10910,arxiv:2306.11621}'
+
+description: |
+  Code whose projection is onto a copy of an irreducible representation of a group \(G\) of unitary operations.
+  Error correction ability is not guaranteed, but can be searched in the multiplicity space of the irrep in case there is more than one copy present.
+
+features:
+  encodings:
+    - 'General encoding map \cite{arxiv:2306.11621}.'
+  general_gates:
+    - 'By definition, a group \(G\) of gates can be realized on the code using the unitary operations used to define the code.'
+
+
+relations:
+  parents:
+    - code_id: qecc
+# cousins:
+#   - code_id: qsc
+#     detail: 'There exist group-representation codes that are quantum spherical codes with non-uniform coefficients; such codes, by definition, admit a desired group of gates via Gaussian rotations \cite{arxiv:2306.11621}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-02-21'

codes/quantum/properties/stabilizer/stabilizer.yml

@@ -30,6 +30,8 @@ relations:
   parents:
     - code_id: commuting_projector
       detail: 'Codespace is the ground-state space of the \textit{code Hamiltonian}, which consists of an equal linear combination of stabilizer generators and which can be made into a commuting-projector Hamiltonian.'
+    - code_id: group_representation
+      detail: 'Stabilizer codes are group-representation codes since their projections are onto the trivial irrep of the stabilizer group \cite{arxiv:2306.11621}.'
 
 
 # Begin Entry Meta Information

codes/quantum/spins/single_spin/j_gross.yml

@@ -32,15 +32,15 @@ description: |
   Finally, \(|\overline{0} \rangle\) is defined as the \(+1\) eigenvalue of \(\overline{\sigma}_z\) and \(|\overline{1} \rangle = \overline{\sigma}_x |\overline{0} \rangle \).
 
 features:
-  encoders:
-      - 'Encoders applicable for Clifford codes for arbitrary groups \cite{arxiv:2306.11621}.'
   transversal_gates: 'Discrete subgroups of \(SU(2)\) can be realized transversally.'
   general_gates:
     - 'Universal computation results from being able to prepare a single logical state, perform one measurement, and the following logical gates: the phase gate (\( \overline{S} \)), the Hadamard gate (\(\overline{H}\)), the conditional phase gate (\(\overline{CZ}\)), and the square root of the phase gate (\(\overline{T}\)). Single-qubit Cliffords can be generated using \(\overline{S}\) and \(\overline{H}\), the extension to multiple-qubit Cliffords is done using \(\overline{CZ}\), and \(\overline{T}\) is to transform to non-Clifford states. Together these gates can be used to create all logical unitaries, while preparation and measurement complete universal quantum computation.'
 
 relations:
   parents:
     - code_id: single_spin
+    - code_id: group_representation
+      detail: 'Clifford codes are group-representation codes with \(G\) being a subgroup of \(SU(2)\) \cite{arxiv:2306.11621}.'
 
 
 # Begin Entry Meta Information

codes/quantum/spins/single_spin/su3_spin.yml

@@ -13,13 +13,12 @@ introduced: '\cite{arXiv:2312.00162}'
 description: |
   An extension of Clifford codes to the group \(SU(3)\), whose codespace is a projection onto a particular irrep of a subgroup of \(SU(3)\).
 
-features:
-  encoders:
-      - 'Encoders applicable for Clifford codes for arbitrary groups \cite{arxiv:2306.11621}.'
 
 relations:
   parents:
     - code_id: single_spin
+    - code_id: group_representation
+      detail: '\(SU(3)\) spin codes are group-representation codes with \(G\) being a subgroup of \(SU(2)\).'
 
 
 # Begin Entry Meta Information