capital abelian

codes/classical/bits/reed_muller.yml

@@ -51,7 +51,7 @@ relations:
     - code_id: divisible
       detail: 'An RM\((r,m)\) code is \(2^{\left\lceil m/r\right\rceil-1}\)-divisible, according to McEliece''s theorem \cite{doi:10.1016/0097-3165(71)90066-5,doi:10.1016/0012-365X(72)90032-5}.'
     - code_id: group
-      detail: 'RM codes are group-algebra codes \cite{doi:10.1007/BF01072842,manual:{Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.}}\cite[Ex. 16.4.11]{preset:HKSalgebra}. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_2 G_m \), where \(\mathbb{F}_2=GF(2)\). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).'
+      detail: 'RM codes are group-algebra codes \cite{doi:10.1007/BF01072842,manual:{Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.}}\cite[Ex. 16.4.11]{preset:HKSalgebra}. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary Abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_2 G_m \), where \(\mathbb{F}_2=GF(2)\). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).'
   cousins:
     - code_id: bch
       detail: 'RM\(^*(r,m)\) codes are equivalent to subcodes of BCH codes of designed distance \(2^{m-r}-1\) while RM\((r,m)\) are subcodes of extended BCH codes of the same designed distance \cite[Ch. 13]{preset:MacSlo}.'

codes/classical/bits/tanner/regular_tanner/regular_binary_tanner.yml

@@ -36,7 +36,7 @@ features:
 
   encoders:
     - 'Quadratic algorithm: This technique works for all linear block codes and encodes using matrix multiplication \cite{doi:10.1145/258533.258575}.'
-    - 'Using the nonabelian Fast Fourier Transform and exploiting the symmetry of the underlying graph, an encoding algorithm that requires \(O(n^{4/3})\) has been devised in \cite{doi:10.1145/258533.258575}.'
+    - 'Using the non-Abelian Fast Fourier Transform and exploiting the symmetry of the underlying graph, an encoding algorithm that requires \(O(n^{4/3})\) has been devised in \cite{doi:10.1145/258533.258575}.'
     - 'A modified construction yields codes that may be encoded in linear time yet maintain similar performance \cite{doi:10.1109/18.556668}.'
 
   decoders:

codes/classical/groups/group_classical.yml

@@ -19,7 +19,7 @@ relations:
 
 
 #    - code_id: bits_into_bits
-#      detail: 'Group-alphabet codes whose alphabet is based on the field \(GF(2)\), taken to be an abelian group under addition, are binary codes.'
+#      detail: 'Group-alphabet codes whose alphabet is based on the field \(GF(2)\), taken to be an Abelian group under addition, are binary codes.'
 
 
 # Begin Entry Meta Information

codes/classical/q-ary_digits/cyclic/group.yml

@@ -14,7 +14,7 @@ alternative_names:
 
 description: |
   An \( [n,k]_q \) code based on a finite group \( G \) of size \(n \).
-  A group-algebra code for an abelian group is called an \textit{abelian group-algebra code}.
+  A group-algebra code for an Abelian group is called an \textit{Abelian group-algebra code}.
 
   \subsection{Group algebra}
   \label{topic:group-algebra}
@@ -36,17 +36,17 @@ description: |
   \subsection{Group-algebra code}
   A group-algebra code is a \( k \)-dimensional linear subspace of the \hyperref[topic:group-algebra]{group algebra} of \( G\) with coefficients in the field \(GF(q) = \mathbb{F}_q\) with \(q\) elements. To be precise, the code must be closed under permutations corresponding to the elements of the group \( G \); therefore, \( G \) must be a subgroup of the permutation automorphism group of the code, which is defined as the group of permutations of the physical bits that preserve the code space. This leads us to the formal definition of a group-algebra code: a group-algebra code is an ideal in the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_q G \).
 
-#protection: 'The class of abelian group-algebra codes is very general, for example including all group-algebra codes of size \(n \leq 23 \). As such it is very difficult to say anything about the distance of abelian groups codes without specializing to a particular family'
+#protection: 'The class of Abelian group-algebra codes is very general, for example including all group-algebra codes of size \(n \leq 23 \). As such it is very difficult to say anything about the distance of Abelian groups codes without specializing to a particular family'
 
 notes:
   - 'See \cite{preset:HKSalgebra}\cite[pg. 58]{doi:10.1007/978-94-011-3810-9} for introductions to group-algebra codes.'
-  - 'Not all abelian group-algebra codes are for cyclic groups (cyclic codes) or for elementary abelian \( p \) groups (e.g. Reed Muller codes \cite{doi:10.1007/BF01119999}). For example, there is a binary code with parameters \( [45,13,16] \) which is an abelian group-algebra code for the group \( G = \mathbb{Z}_3 \times \mathbb{Z}_{15} \). '
+  - 'Not all Abelian group-algebra codes are for cyclic groups (cyclic codes) or for elementary Abelian \( p \) groups (e.g. Reed Muller codes \cite{doi:10.1007/BF01119999}). For example, there is a binary code with parameters \( [45,13,16] \) which is an Abelian group-algebra code for the group \( G = \mathbb{Z}_3 \times \mathbb{Z}_{15} \). '
 
 relations:
   parents:
     - code_id: q-ary_linear
       detail: 'A linear code is a group-algebra code for a group \(G\) if and only if \(G\) is isomorphic to a regular subgroup of the code''s permutation automorphism group \cite{doi:10.1007/s10623-008-9261-z}\cite[Thm. 16.4.7]{preset:HKSalgebra}.'
-#Note that we have an isomorphism of \( \mathbb{F} \) algebras \( \mathbb{F}\mathbb{Z}_n \cong \mathbb{F}[x]/\langle x^n-1\rangle \) by taking  \( x \) to be the generator of the cyclic group. Thus we can see how cyclic codes are an example of an abelian group-algebra code.'
+#Note that we have an isomorphism of \( \mathbb{F} \) algebras \( \mathbb{F}\mathbb{Z}_n \cong \mathbb{F}[x]/\langle x^n-1\rangle \) by taking  \( x \) to be the generator of the cyclic group. Thus we can see how cyclic codes are an example of an Abelian group-algebra code.'
 
 
 # Begin Entry Meta Information

codes/classical/q-ary_digits/cyclic/q-ary_cyclic.yml

@@ -50,7 +50,7 @@ relations:
     - code_id: q-ary_linear
     - code_id: cyclic
     - code_id: group
-      detail: 'A length-\(n\) cyclic \(q\)-ary linear code is an abelian group-algebra code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).'
+      detail: 'A length-\(n\) cyclic \(q\)-ary linear code is an Abelian group-algebra code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).'
   cousins:
     - code_id: q-ary_ltc
       detail: 'Cyclic linear codes cannot be \(c^3\)-LTCs \cite{doi:10.1109/TIT.2005.851735}. Codeword symmetries are in general an obstruction to achieving such LTCs \cite{doi:10.1007/978-3-642-16367-8_12}.'

codes/classical/q-ary_digits/q-ary_additive.yml

@@ -16,7 +16,7 @@ relations:
   parents:
     - code_id: q-ary_digits_into_q-ary_digits
     - code_id: group_linear
-      detail: 'Additive \(q\)-ary codes are linear over \(G=GF(q)\) since Galois fields are abelian groups under addition.'
+      detail: 'Additive \(q\)-ary codes are linear over \(G=GF(q)\) since Galois fields are Abelian groups under addition.'
 
 
 # Begin Entry Meta Information

codes/classical/rings/rings_into_rings.yml

@@ -15,7 +15,7 @@ relations:
     - code_id: block
     - code_id: ecc_finite
     - code_id: group_classical
-      detail: 'A ring \(R\) is an abelian group under addition.'
+      detail: 'A ring \(R\) is an Abelian group under addition.'
 
 
 # Begin Entry Meta Information

codes/classical/rings/rings_linear.yml

@@ -17,7 +17,7 @@ relations:
   parents:
     - code_id: rings_into_rings
     - code_id: group_linear
-      detail: '\(R\)-linear codes are linear over \(G=R\) since rings are abelian groups under addition.'
+      detail: '\(R\)-linear codes are linear over \(G=R\) since rings are Abelian groups under addition.'
 
 
 # Begin Entry Meta Information

codes/classical/spherical/numerical/tlsc.yml

@@ -26,7 +26,7 @@ features:
 relations:
   parents:
     - code_id: slepian_group
-      detail: 'Polyphase codewords can be implemented by acting on the all-ones initial vector by diagonal orthogonal matrices whose entries are the codeword components \cite[Ch. 8]{preset:EricZin}. TLSC codes are generalizations of polyphase codes to other initial vectors and are examples of abelian Slepian-group codes.'
+      detail: 'Polyphase codewords can be implemented by acting on the all-ones initial vector by diagonal orthogonal matrices whose entries are the codeword components \cite[Ch. 8]{preset:EricZin}. TLSC codes are generalizations of polyphase codes to other initial vectors and are examples of Abelian Slepian-group codes.'
 
 
 # Begin Entry Meta Information

codes/quantum/categories/groupoid_surface.yml

@@ -11,7 +11,7 @@ name: 'Groupoid toric code'
 introduced: '\cite{arxiv:2212.01021}'
 
 description: |
-  Extension of the Kitaev surface code from abelian groups to groupoids, i.e., multi-fusion categories in which every morphism is an isomorphism \cite{doi:10.1112/blms/19.2.113}.
+  Extension of the Kitaev surface code from Abelian groups to groupoids, i.e., multi-fusion categories in which every morphism is an isomorphism \cite{doi:10.1112/blms/19.2.113}.
   Some models admit fracton-like features such as extensive ground-state degeneracy and excitations with restricted mobility.
   The robustness of these features has not yet been established.
 

codes/quantum/categories/string_net.yml

@@ -31,7 +31,7 @@ features:
     - 'Gates can be implemented through topological operations corresponding to elements of the mapping class group, which is generated by Dehn-twists along non-contractible cycles for triangulations of toroidal \cite{arXiv:1806.02358,arxiv:1806.06078} and hyperbolic \cite{arXiv:1901.11029} manifolds. Whether or not a gate set is universal depends on the choice of input category; in some cases such as the Ising category, gates can be complemented by topological charge measurements to obtain a universal gate set.'
     - 'Alternatively, one could encode the logical quantum information into the degenerate fusion space of a number of computational anyons. In this case, a universal logical gate set can be implemented through the braiding of the computational anyons \cite{arXiv:quant-ph/0001108,arXiv:math/0103200,arxiv:1002.2816}, e.g., for the case of the \hyperref[code:fibonacci]{Fibonacci} input category.'
   decoders:
-    - 'Fusing nonabelian anyons cannot be done in one step \cite{arxiv:hep-th/0110205}.'
+    - 'Fusing non-Abelian anyons cannot be done in one step \cite{arxiv:hep-th/0110205}.'
     - 'Syndrome measurement circuits analyzed in Ref. \cite{arXiv:1206.6048}.'
     - 'Clustering decoder \cite{arxiv:1607.02159}.'
 

codes/quantum/groups/group_gkp.yml

@@ -62,7 +62,7 @@ relations:
   cousins:
     - code_id: css
       detail: 'Group GKP codes are stabilized by \(X\)-type Pauli matrices representing \(H\) and all \(Z\)-type operators that are constant on \(K\).
-      However, the \(Z\)-type operators are not unitary for nonabelian groups.'
+      However, the \(Z\)-type operators are not unitary for non-Abelian groups.'
     - code_id: oscillator_stabilizer
       detail: 'The group-GKP construction encompasses all bosonic CSS codes.
       A single-mode qubit GKP code corresponds to the \(2\mathbb{Z}\subset\mathbb{Z}\subset\mathbb{R}\) group construction, and multimode GKP codes can be similarly described.

codes/quantum/groups/topological/generalized_color.yml

@@ -11,7 +11,7 @@ name: 'Generalized color code'
 introduced: '\cite{arXiv:1408.6238}'
 
 description: |
-  Member of a family of nonabelian topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)).
+  Member of a family of non-Abelian topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)).
 
 
 relations:
@@ -22,7 +22,7 @@ relations:
     - code_id: group_gkp
       detail: 'Generalized color-code Hamiltonians should be expressable in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see \cite[Sec. 3.3]{arxiv:2111.12096}.'
     - code_id: quantum_double
-      detail: 'Generalized color code for group \(G\) on the 4.8.8 lattice is equivalent to a \(G\) quantum double model and another \(G/[G,G]\) quantum double model defined using the abelianization of \(G\).'
+      detail: 'Generalized color code for group \(G\) on the 4.8.8 lattice is equivalent to a \(G\) quantum double model and another \(G/[G,G]\) quantum double model defined using the Abelianization of \(G\).'
 
 
 # Begin Entry Meta Information

codes/quantum/groups/topological/quantum_double.yml

@@ -14,11 +14,11 @@ description: |
   Group-GKP stabilizer code whose codewords realize 2D modular gapped topological order defined by a finite group \(G\).
   The code's generators are few-body operators associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit of dimension \( |G| \) located at each edge of the tesselation).
 
-  The physical Hilbert space has dimension \( |G|^E  \), where \( E \) is the number of  edges in the tessellation. The dimension of the code space is the number of orbits of the conjugation action of \( G \) on \( \text{Hom}(\pi_1(\Sigma),G) \), the set of group homomorphisms from the fundamental group of the surface \( \Sigma \) into the finite group \( G \) \cite{arXiv:1908.02829}. When \( G \) is abelian, the formula for the dimension simplifies to \( |G|^{2g} \), where \( g \) is the genus of the surface \( \Sigma \).
+  The physical Hilbert space has dimension \( |G|^E  \), where \( E \) is the number of  edges in the tessellation. The dimension of the code space is the number of orbits of the conjugation action of \( G \) on \( \text{Hom}(\pi_1(\Sigma),G) \), the set of group homomorphisms from the fundamental group of the surface \( \Sigma \) into the finite group \( G \) \cite{arXiv:1908.02829}. When \( G \) is Abelian, the formula for the dimension simplifies to \( |G|^{2g} \), where \( g \) is the genus of the surface \( \Sigma \).
 
   The codespace is the ground-state subspace of the quantum double model Hamiltonian.
-  For nonabelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model \cite{doi:10.1007/3-540-49208-9_31,arXiv:quant-ph/0306063,doi:10.1017/CBO9780511792908}.
-  The fusion space of such nonabelian anyons has dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.
+  For non-Abelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model \cite{doi:10.1007/3-540-49208-9_31,arXiv:quant-ph/0306063,doi:10.1017/CBO9780511792908}.
+  The fusion space of such non-Abelian anyons has dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.
 
 protection: |
   Error-correcting properties established in Ref. \cite{arxiv:1908.02829}.
@@ -29,7 +29,7 @@ features:
     - 'A depth-\(L^2\) circuit that grows the code out of a small patch on an \(L\times L\) square lattice using CMULT gates (i.e., "local moves") \cite{arxiv:0712.0348,arxiv:1101.0527}.'
     - 'For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit \cite{arXiv:0901.1345,arXiv:1101.0527} or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates \cite{arXiv:0712.0348,arxiv:0806.4583}.'
     - 'For any solvable group \(G\), ground-state preparation and anyon-pair creation can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout \cite{arXiv:2112.01519,arXiv:2205.01933} (see Ref. \cite{arXiv:2112.03061} for specific dihedral groups).
-    Anyon-pair creation requires an adaptive circuit for any nonabelian \(G\) \cite{arXiv:2205.01933}.'
+    Anyon-pair creation requires an adaptive circuit for any non-Abelian \(G\) \cite{arXiv:2205.01933}.'
   general_gates:
     - 'Universal topological quantum computation possible for certain groups \cite{arxiv:quant-ph/0306063,arxiv:0901.1345}.'
   decoders:

codes/quantum/oscillators/fock_state/matrix_qm.yml

@@ -11,7 +11,7 @@ name: 'Matrix-model code'
 introduced: '\cite{arxiv:2211.08448}'
 
 description: |
-  Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a nonabelian bosonic gauge theory with a large gauge group.
+  Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group.
   The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry.
 
   A simple matrix-model code consists of two spatially separated bosons with codewords
@@ -38,7 +38,7 @@ relations:
       detail: 'Matrix-model codewords for simple codes are eigenstates of a matrix-model Hamiltonian.'
   cousins:
     - code_id: holographic
-      detail: 'Matrix-model codes are motivated by the Ads/CFT correspondence because it is manifest in continuous nonabelian gauge theories with large gauge groups \cite{arxiv:2211.08448}.'
+      detail: 'Matrix-model codes are motivated by the Ads/CFT correspondence because it is manifest in continuous non-Abelian gauge theories with large gauge groups \cite{arxiv:2211.08448}.'
     - code_id: self_correct
       detail: 'Matrix-model codes are similar to self-correcting memories in the sense that memory time becomes infinite in the thermodynamic limit, but with corrections being polynomial in \(N\).'
 

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

@@ -3,7 +3,7 @@
 ##       https://github.com/errorcorrectionzoo       ##
 #######################################################
 
-code_id: nonabelian_covariant_erasure
+code_id: non-Abelian_covariant_erasure
 
 name: '\(U(d)\)-covariant approximate erasure code'
 introduced: '\cite{arxiv:2007.09154}'

codes/quantum/properties/block/topological/topological.yml

@@ -58,8 +58,8 @@ features:
   \end{align}
   and a generalization of the formula to the non-orientable case can be found in Ref. \cite{arxiv:1612.07792}.'
   encoders:
-    - 'The unitary circuit depth required to initialize in a general topologically ordered state using geometrically local gates on an \(L\times L\) lattice is \(\Omega(L)\) \cite{arXiv:quant-ph/0603121}, irrespective of whether the ground state admits Abelian or nonabelian anyonic excitations.
-    However, only a finite-depth circuit and one round of measurements is required for nonabelian topological orders with a Lagrangian subgroup \cite{arxiv:2209.03964}.'
+    - 'The unitary circuit depth required to initialize in a general topologically ordered state using geometrically local gates on an \(L\times L\) lattice is \(\Omega(L)\) \cite{arXiv:quant-ph/0603121}, irrespective of whether the ground state admits Abelian or non-Abelian anyonic excitations.
+    However, only a finite-depth circuit and one round of measurements is required for non-Abelian topological orders with a Lagrangian subgroup \cite{arxiv:2209.03964}.'
 
 
 notes: