binary dihedral non-additive

codes/quantum/qubits/nonstabilizer/permutation_invariant/binary_dihedral_permutation_invariant0.yml

@@ -0,0 +1,48 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: binary_dihedral_permutation_invariant
+physical: qubits
+logical: qubits
+
+name: 'Binary dihedral permutation-invariant code'
+introduced: '\cite{arxiv:2310.17652}'
+
+description: |
+  Multi-qubit code designed to realize gates from the binary dihedral group transversally.
+  Can also be interpreted as a single-spin code.
+  The codespace projection is a projection onto an irrep of the \textit{binary dihedral group} \( \mathsf{BD}_{2N} = \{\omega I, X, P\} \) of order \(8N\), where \( \omega \) is a \( 2N \) root of unity, and \( P = \text{diag} ( 1, \omega^2) \).
+
+  The construction includes three families.
+  The first family has parameters \(((2m+3,2,3))\) for \(m\) not a power of two, realizing binary dihedral transversal gates that are not possible to realize in any qubit stabilizer code \cite[Prop. 1]{arxiv:2310.17652}.
+  The second family is the case of \(m\) being a power of two, corresponding to \(((2^{r-1}+3,2,3))\) codes, each realizing a member of the Clifford hierarchy transversally.
+  The third family consists of \(((n,2,d))\) codes with \(n = \frac{1}{4}(3d^2+6d-7+2(d\text{ mod }8))\), realizing \(S\) and \(T\) gates transversally.
+
+features:
+  transversal_gates: 'Binary dihedral group gates can be realized transversally, which include subsgroups of any level of the Clifford hierarchy.'
+
+
+relations:
+  parents:
+    - code_id: qubits_into_qubits
+    - code_id: j_gross
+      detail: 'Binary dihedral permutation-invariant codes can be interpreted as Clifford single-spin codes.'
+  cousins:
+    - code_id: small_distance
+      detail: 'The first and second families of binary dihedral permutation-invariant codes have distance three.'
+    - code_id: xp_stabilizer
+      detail: 'Binary dihedral permutation invariant codewords form error spaces of XP stabilizer codes.'
+    - code_id: diagonal_clifford
+      detail: 'The \(((2^{r-1}+3,2,3))\) family of binary dihedral permutation-invariant codes realizes the same transversal gates as the \([[2^r-1, 1, 3]]\) quantum Reed-Muller codes, but require fewer qubits in almost all cases.'
+    - code_id: quantum_triorthogonal
+      detail: 'The \(((27,2,5))\) binary dihedral permutation-invariant code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[49,1,5]]\) triorthogonal code.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2023-11-20'

codes/quantum/qubits/nonstabilizer/icosahedral_permutation_invariant.yml → codes/quantum/qubits/nonstabilizer/permutation_invariant/icosahedral_permutation_invariant.yml

@@ -12,7 +12,7 @@ introduced: '\cite{arxiv:quant-ph/0304153,arXiv:2005.10910,arxiv:2305.07023}'
 
 description: |
   Seven-qubit code designed to realize gates from the binary icosahedral group transversally.
-  Can also be interpreted as a spin-\(7/2\) spin code.
+  Can also be interpreted as a spin-\(7/2\) single-spin code.
   The codespace projection is a projection onto an irrep of the binary icosahedral group \(2I\).
 
   In terms of Dicke states, the unnormalized logical states of one version \cite{arxiv:2305.07023} of this code are

codes/quantum/qubits/nonstabilizer/xp_stabilizer.yml

@@ -11,7 +11,9 @@ name: 'XP stabilizer code'
 introduced: '\cite{arXiv:2203.00103}'
 
 description: |
-  The XP Stabilizer formalism is a generalization of the XS and Pauli stabilizer formalisms, with stabilizer generators taken from the group \( \{\omega I, X, P\}^{\otimes n} \). Here, \( \omega \) is a \( 2N \) root of unity, and \( P = \text{diag} ( 1, \omega^2) \). The codespace is a \(+1\) eigenspace of a set of XP stabilizer generators, which need not commute to define a valid codespace.
+  The XP Stabilizer formalism is a generalization of the XS and Pauli stabilizer formalisms, with stabilizer generators taken from the group \( \mathsf{BD}_{2N}^{\otimes n} = \{\omega I, X, P\}^{\otimes n} \), which is the tensor product of the binary dihedral group of order \(8N\).
+  Here, \( \omega \) is a \( 2N \) root of unity, and \( P = \text{diag} ( 1, \omega^2) \).
+  The codespace is a \(+1\) eigenspace of a set of XP stabilizer generators, which need not commute to define a valid codespace.
 
   XP stabilizer codes are classified into XP-regular and XP-non-regular, where the former can be mapped to a CSS code with similar logical operator structure.
 

codes/quantum/spins/single_spin.yml

@@ -14,6 +14,7 @@ description: |
 
 protection: |
   Noise models can be categorized as those that cause the state to leave the maximally symmetric subspace and those that do not. The former include single-spin errors akin to qubit Pauli noise. The latter include collective rotations or decays.
+  A particular error basis of interest consists of the spherical tensors \cite{arixv:2304.08611}.
 
 features:
   transversal_gates: 'When the physical Hilbert space is thought of a collective spin, logical gates for spin codes have the form \(U^{\otimes N}\), where \(U\) is a local rotation on the physical system.'
@@ -23,7 +24,8 @@ relations:
   parents:
     - code_id: spins_into_spins
     - code_id: permutation_invariant
-      detail: 'Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the permutation-invariant subspace of a set of spins.'
+      detail: 'Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the permutation-invariant subspace of a set of spins.
+      A single-spin code correcting spherical tensors can be mapped into a permutation-invariant code with an analogous distance \cite[Thm. 1]{arxiv:2310.17652}.'
     - code_id: single_subsystem
     - code_id: qecc_finite
 

codes/quantum/spins/transversal/j_gross.yml

@@ -40,7 +40,7 @@ relations:
     - code_id: single_spin
   cousins:
     - code_id: qubits_into_qubits
-      detail: 'Certain Clifford codes yield qubit codes with non-trivial distance when the single spin is treated as a collective spin of several qubits.'
+      detail: 'Certain Clifford codes yield permutation-invariant qubit codes with non-trivial distance \cite{arxiv:2304.08611} when the single spin is treated as a collective spin of several qubits.'
 
 
 # Begin Entry Meta Information