non_stab + cws update after Leonid

codes/classical/q-ary_digits/ag/rs/reed_solomon.yml

@@ -24,6 +24,7 @@ description: |
   An RS code with length \(n=q-1\) whose points \(\alpha_i\) are \(i-1\)st powers of a primitive \(n\)th root of unity is a \textit{narrow-sense RS code}.
   In an alternative convention (not used here), the primitive-root case is called an RS code, and the general-root case is a generalized RS code.
 
+
 protection: 'Since each polynomial \(f_{\mu}\) is of degree less than \(k\), it has less than \(k\) roots; this is called the \textit{degree mantra}.
 Therefore, the polynomial can be determined from its values at \(k\) points. This means that RS codes can correct erasures on up to \(n-k\) registers. The resulting distance, \(d=n-k+1\), saturates the Singleton bound.'
 

codes/quantum/groups/group_cluster_state.yml

@@ -0,0 +1,29 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: group_cluster_state
+physical: groups
+logical: groups
+
+name: 'Group-based cluster-state code'
+introduced: '\cite{arxiv:1408.6237}'
+
+description: |
+  Code consisting of (typically one) group-based cluster states based on a finite group \(G\) \cite{arxiv:1408.6237}.
+  Such cluster states can be defined using a graph and conditional group multiplication operations.
+
+
+relations:
+  parents:
+    - code_id: group_gkp
+      detail: 'Group-based cluster states are stabilized by \(X\)-type Pauli matrices representing \(G\) and generalized \(Z\)-type operators \cite{arxiv:1408.6237}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-04-03'

codes/quantum/groups/group_gkp.yml

@@ -55,7 +55,8 @@ description: |
   \end{table}
 
 
-protection: 'Protects against generalized bit-flip errors \(g\in G\) that are inside the fundamental domain of \(G/K\). Protection against phase-flip errors determined by branching rules of irreps of \(G\) into those of \(K\), and further into those of \(H\).'
+protection: 'Protects against generalized bit-flip errors \(g\in G\) that are inside the fundamental domain of \(G/K\).
+Protection against phase-flip errors determined by branching rules of irreps of \(G\) into those of \(K\), and further into those of \(H\).'
 
 features:
   transversal_gates: 'Group-GKP codes corresponding to the \(G^{k_1} \subseteq G^{ k_2} \subset G^{n}\) group construction admit \(X\)-type transversal Pauli gates representing \(G\) \cite{arxiv:1902.07714}.'
@@ -76,8 +77,6 @@ relations:
       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.
       An \([[n,k,d]]_{\mathbb{R}}\) analog stabilizer code corresponds to the \(\mathbb{R}^{ k_1} \subseteq \mathbb{R}^{ k_2} \subset \mathbb{R}^{n}\) group construction, where \(k=k_2/k_1\).
       GKP stabilizer codes for \(n\) modes correspond to subgroups \(\mathbb{Z}^m\) for \(m<n\).'
-    - code_id: cluster_state
-      detail: 'Cluster states can be generalized to finite groups \cite{arxiv:1408.6237}.'
 
 # Begin Entry Meta Information
 _meta:

codes/quantum/qubits/majorana/kitaev_chain.yml

@@ -11,7 +11,7 @@ name: 'Kitaev chain code'
 introduced: '\cite{arXiv:cond-mat/0010440}'
 
 description: |
-  An \([[n,1,1]]_{f}\) Majorana stabilizer code forming the ground-state of the Kitaev Majorana chain (a.k.a. Kitaev Majorana wire) in its fermionic topological phase, which is equivalent to the 1D quantum Ising model in the symmetry-breaking phase via the Jordan-Wigner transformation.
+  An \([[n,1,1]]_{f}\) Majorana stabilizer code forming the ground-state of the Kitaev Majorana chain (a.k.a. Kitaev Majorana wire) in its fermionic topological phase, which is unitarily equivalent to the 1D quantum Ising model in the symmetry-breaking phase via the Jordan-Wigner transformation.
   The code is usually defined using the algebra of two anti-commuting Majorana operators called \textit{Majorana zero modes (MZMs)} or \textit{Majorana edge modes (MEMs)}.
 
   Codewords have different values of the fermionic parity.

codes/quantum/qubits/nonstabilizer/cws/cws.yml → codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/cws.yml

@@ -12,22 +12,24 @@ short_name: 'CWS'
 introduced: '\cite{arXiv:cs/0610159,arXiv:0708.1021}'
 
 description: |
-  Member of a family of codes that strictly generalizes qubit stabilizer codes.
+  A code defined using a cluster state and a set of \(Z\)-type Pauli strings defined by a binary classical code.
 
-  They are usually denoted by \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \) where \(\mathcal{G}\) is a graph and \(\mathcal{C}\) is a \( (n,K,d) \) binary classical code.
-  From the graph we form the unique graph state (stabilizer state) \( |G \rangle \).
-  From the classical code we form Pauli \(Z\)-type operators \( W_i = Z^{c_{i,1}} \otimes \cdots \otimes Z^{c_{i,n}} \), where \(c_{i,j} \) is the \(j\)-th bit of the \(i\)-th classical codeword.
-  The CWS codewords are then \( | i \rangle =  W_i | G \rangle \).
+  The Galois-qudit CWS construction takes in \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \), where \(\mathcal{G}\) is a graph, and where \(\mathcal{C}\) is an \((n,K,d)\) binary code.
+  From the graph, we form the unique cluster state \( |\mathcal{G} \rangle \).
+  From the binary code, we form Pauli \(Z\)-type operators \( W_i = Z^{c_{i,1}} \otimes \cdots \otimes Z^{c_{i,n}} \), where \(c_{i,j} \) is the \(j\)-th coordinate of the \(i\)-th classical codeword.
+  The CWS codewords are then \( | i \rangle =  W_i | \mathcal{G} \rangle \).
 
-  The above definition corresponds to the \textit{standard form} of a CWS codes, and there is an alternative description that is locally Clifford-equivalent.
-  In particular, we can describe CWS codes as \( \mathcal{Q} = (S,\mathcal{W})\) where \(S\) is a stabilizer group and \( \mathcal{W} = \{ w_\ell \}_{\ell = 1}^K \) is a family of \(K\) \(n\)-qubit Pauli strings. We then form CWS codeswords as \( | i \rangle = w_i | S \rangle \), where \( | S \rangle \) is the (unique) stabilizer state of \(S\).
+  The above definition corresponds to the \textit{standard form} of CWS codes.
+  Since any stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit \cite{arxiv:quant-ph/0308151}\cite[Appx. A]{arxiv:1910.00471}, any code whose underlying state is a non-cluster stabilizer state is similarly equivalent to a CWS code.
 
   The term CWS was coined in Ref. \cite{arxiv:0708.1021}, and their approach is equivalent to another approach \cite{arXiv:cs/0610159} based on Boolean functions (see Ref. \cite{manual:{Zeng, Bei. Quantum operations and codes beyond the Stabilizer-Clifford framework. Diss. Massachusetts Institute of Technology, 2009.}}).
+  In an alternative convention (not used here), CWS codes are defined from an underlying stabilizer state that is not a necessarily a cluster state.
 
 
 protection: |
   Code distance \(\mathcal{Q} = ( \mathcal{G},\mathcal{C}) \) is upper bounded by the distance of the classical code \(\mathcal{C} \).
   The \hyperref[code:qubits_into_qubits]{diagonal distance} is upper bounded by \(\delta + 1\), where \(\delta\) is the minimum degree of \(\mathcal{G}\).
+
   Computing the distance is generally \(NP\)-complete, and is \(NP\)-hard for \hyperref[topic:degeneracy]{non-degenerate} codes \cite{arXiv:2203.04262}.
   Some bounds are provided in Ref. \cite{arxiv:1108.5490}.
 
@@ -46,15 +48,22 @@ notes:
 
 relations:
   parents:
-    - code_id: qsqc
-      detail: 'Restricting the underlying classical code to be symplectic self-dual reduces the \(((n,2^k L,d))\) QSQC construction to the \(((n,L,d))\) CWS construction \cite[Sec. IV.E]{arxiv:2311.07265}.'
+    - code_id: non_stabilizer
+      detail: 'Any CWS code can be written as a USt whose (\(K=1\)) stabilizer code is the cluster state and whose coset representatives are constructed from the binary classical code.
+      Conversely, USts are equivalent to CWS codes via a single-qubit Clifford circuit as follows \cite{arxiv:0907.2038,manual:{Li, Yunfan. Codeword Stabilized Quantum Codes and Their Error Correction. Diss. UC Riverside, 2010.}}\cite[Sec. 10.4]{doi:10.1017/CBO9781139034807.012}.
+      The set of coset representatives of any USt can be extended to a larger set iterating over the underlying stabilizer code such that all codewords can be obtained from a single stabilizer state.
+      Then, one can apply a single-qubit Clifford transformation to map said stabilizer state into a cluster state.'
     - code_id: qudit_cws
       detail: 'Modular-qudit CWS codes reduce to CWS codes for \(q=2\).'
     - code_id: galois_cws
       detail: 'Galois-qudit CWS codes reduce to CWS codes for \(q=2\).'
+    - code_id: qsqc
+      detail: 'Restricting the underlying classical code to be symplectic self-dual reduces the \(((n,2^k L,d))\) QSQC construction to the \(((n,L,d))\) CWS construction \cite[Sec. IV.E]{arxiv:2311.07265}.'
   cousins:
+    - code_id: cluster_state
+      detail: 'A single cluster-state codeword is used to construct a CWS code.'
     - code_id: movassagh_ouyang
-      detail: 'The Movassagh-Ouyang codes overlap the CWS codes but neither family is contained in the other.'
+      detail: 'The Movassagh-Ouyang codes overlap the CWS codes but neither family is contained in the other \cite{arXiv:2012.01453}.'
     - code_id: spacetime
       detail: 'CWS codes have been considered in the context of spacetime replication of quantum data \cite{arxiv:1210.0913,arXiv:1601.02544}, while STCs are designed to replicate classical data.'
 
@@ -63,6 +72,8 @@ relations:
 _meta:
   # Change log - most recent first
   changelog:
+    - user_id: VictorVAlbert
+      date: '2024-03-28'
     - user_id: VictorVAlbert
       date: '2022-04-21'
     - user_id: VictorVAlbert

codes/quantum/qubits/nonstabilizer/cws/qubit_5_6_2.yml → codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/qubit_5_6_2.yml

@@ -23,8 +23,6 @@ relations:
     - code_id: arvind
       detail: 'The \(((5,6,2))\) code is the \(((n,1+n(q-1),2))_q\) union stabilizer code for \(n=5\) and \(q=2\) \cite{arxiv:quant-ph/0210097}.'
     - code_id: quantum_cyclic
-    - code_id: non_stabilizer
-      detail: 'The six-qubit CWS code is a union stabilizer code \cite{arXiv:quant-ph/9703002,arxiv:quant-ph/0210097}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/nonstabilizer/cws/non_stabilizer.yml → codes/quantum/qubits/nonstabilizer/union_stabilizer/non_stabilizer.yml

@@ -7,7 +7,8 @@ code_id: non_stabilizer
 physical: qubits
 logical: qubits
 
-name: 'Union stabilizer code'
+name: 'Union stabilizer (USt) code'
+short_name: 'USt'
 introduced: '\cite{arXiv:quant-ph/9703002,arXiv:quant-ph/9703016,arxiv:quant-ph/9710031,arxiv:quant-ph/0210097,arxiv:0801.2144}'
 # First ref discusses unions of general codes
 
@@ -18,6 +19,13 @@ alternative_names:
 description: |
   A qubit code whose codespace consists of a direct sum of a qubit stabilizer codespace and one or more of that stabilizer code's error spaces.
 
+  Given a subset \(T\) of coset representatives of \(\mathsf{N}(\mathsf{S})/\mathsf{S}\) of a stabilizer code \([[n,k]]\) with codespace \(\mathsf{C}\) and stabilizer group \(\mathsf{S}\), one can construct the USt with codespace \cite[Def. 10.1]{doi:10.1017/CBO9781139034807.012}
+  \begin{align}
+    \mathsf{C}_{\text{USt}}=\bigoplus_{t\in T}t\mathsf{C}~.
+  \end{align}
+  The parameters of the USt are \(((n,2^k |T|))\), where \(|T|\) is the number of chosen coset representatives.
+  A USt is \textit{CSS-like} when the underlying stabilizer code is CSS, so the coset representatives from the two classical codes underlying the CSS code.
+
   Union stabilizer codes constructed in Ref. \cite{arxiv:quant-ph/0210097} include the \(((33, 155, 3))\) and \(((15, 8, 3))\) codes.
 
 
@@ -31,8 +39,9 @@ notes:
 
 relations:
   parents:
-    - code_id: cws
-      detail: 'Union stabilizer codes can be defined as CWS codes \cite{arxiv:0907.2038,manual:{Li, Yunfan. Codeword Stabilized Quantum Codes and Their Error Correction. Diss. UC Riverside, 2010.}}\cite[Sec. 10.4]{doi:10.1017/CBO9781139034807.012}.'
+    - code_id: qubits_into_qubits
+    - code_id: qudit_non_stabilizer
+      detail: 'Modular-qudit union stabilizer codes reduce to union stabilizer codes for \(q=2\).'
     - code_id: galois_non_stabilizer
       detail: 'Galois-qudit union stabilizer codes reduce to union stabilizer codes for \(q=2\).'
   cousins:

codes/quantum/qubits/nonstabilizer/cws/quantum_goethals_preparata.yml → codes/quantum/qubits/nonstabilizer/union_stabilizer/quantum_goethals_preparata.yml

@@ -11,7 +11,7 @@ name: '\(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code'
 introduced: '\cite{arXiv:0801.2144,arxiv:0801.2150}'
 
 description: |
-  Member of a family of \(((2^m,2^{2^m−5m+1},8))\) CWS codes constructed using the classical Goethals and Preparata codes.
+  Member of a family of \(((2^m,2^{2^m−5m+1},8))\) CSS-like union stabilizer codes constructed using the classical Goethals and Preparata codes.
 
   They can be viewed as union stabilizer codes constructed from the codespace of a \([[2^m,2^m-7m+3,8]]\) code (which itself is obtained from the Steane enlargement construction) and the coset representatives used to obtain the Goethals and Preparata codes \cite[Thm. 10.3]{doi:10.1017/CBO9781139034807.012}.
 

codes/quantum/qubits/small_distance/small/stab_5_1_3.yml

@@ -93,7 +93,7 @@ relations:
     - code_id: majorana_stab
       detail: 'The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators \cite{manual:{Aleksander Kubica, private communication, 2019}}.'
     - code_id: qubits_into_qubits
-      detail: 'Every \(((5,2,3))\) code is equivalent to the five-qubit code \cite[Corr. 10]{arxiv:quant-ph/9704043}.'
+      detail: 'Every \(((5,2,3))\) code is single-qubit-Clifford-equivalent to the five-qubit code \cite[Corr. 10]{arxiv:quant-ph/9704043}.'
     - code_id: quantum_concatenated
       detail: 'The concatenated five-qubit code has a \hyperref[topic:measurement-threshold]{measurement threshold} of one \cite{arxiv:2402.00145}.'
     - code_id: cws

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

@@ -93,8 +93,9 @@ relations:
       detail: 'The Steane code is a group-representation code with \(G\) being the \(2O\) subgroup of \(SU(2)\) \cite{arxiv:2306.11621}.'
     - code_id: concatenated_steane
       detail: 'The concatenated Steane code at level \(m=1\) is the Steane code.'
+  cousins:
     - code_id: quantum_cyclic
-      detail: 'The Steade code is equivalent to a cyclic code via qubit permutations \cite[Exam. 1]{arxiv:1108.5490}.'
+      detail: 'The Steane code is equivalent to a cyclic code via qubit permutations \cite[Exam. 1]{arxiv:1108.5490}.'
 
 
 #      detail: 'Steane code is the smallest member of a family of Reed-Muller-based CSS codes.'

codes/quantum/qubits/stabilizer/fermion_into_qubit/bksf.yml

@@ -29,7 +29,7 @@ relations:
     - code_id: mlsc
       detail: 'The BKSF code can be thought of as a particular MLSC \cite{arxiv:1812.08190}.'
     - code_id: 2d_bosonization
-      detail: 'The BKSF code is equivalent to a distance-two 2D bosonization code for a specific edge ordering \cite{arxiv:2201.05153}.'
+      detail: 'The BKSF code is a distance-two 2D bosonization code for a specific edge ordering \cite{arxiv:2201.05153}.'
 # cousins:
 #   - code_id: triangular_color
 #     detail: 'The square-lattice BKSF encoding \cite{arxiv:1810.05274} has the excitation structure as one of the toric codes underlying the 2D color code \cite{manual:{Compact fermion to qubit mappings for quantum simulation}}.'

codes/quantum/qubits/stabilizer/fracton/checkerboard.yml

@@ -12,7 +12,7 @@ introduced: '\cite{arXiv:1505.02576}'
 
 description: |
   A foliated type-I fracton code defined on a cubic lattice that admits weight-eight  \(X\)- and \(Z\)-type stabilizer generators on the eight vertices of each cube in the lattice.
-  
+
   Variants include the twisted checkerboard model \cite{arxiv:1805.06899}.
 
 features:
@@ -26,10 +26,10 @@ relations:
   parents:
     - code_id: qubit_css
     - code_id: fracton
-      detail: 'The checkerboard model is equivalent to two copies of the X-cube model via a local unitary \cite{arxiv:1806.08633}. Hence, it is a foliated type-I fracton code.'
+      detail: 'The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary \cite{arxiv:1806.08633}. Hence, it is a foliated type-I fracton code.'
   cousins:
     - code_id: xcube
-      detail: 'The checkerboard model is equivalent to two copies of the X-cube model via a local unitary \cite{arxiv:1806.08633}.'
+      detail: 'The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary \cite{arxiv:1806.08633}.'
 
 
 _meta:

codes/quantum/qubits/stabilizer/mbqc/cluster_state.yml

@@ -14,17 +14,19 @@ alternative_names:
   - 'Graph-state code'
 
 description: |
-  Code consisting of cluster states \cite{arxiv:quant-ph/0004051}, which are stabilizer states defined on a graph. There is one stabilizer generator \(S_v\) per graph vertex \(v\) of the form
+  Code consisting of (typically one) cluster states \cite{arxiv:quant-ph/0004051}.
+  Often used in measurement-based quantum computation (MBQC), which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension.
+  This is done by encoding the computation into the features of the cluster state''s graph.
+
+  Cluster states are stabilizer states defined on a graph.
+  There is one stabilizer generator \(S_v\) per graph vertex \(v\) of the form
   \begin{align}
     S_v = X_{v} \prod_{w\in N(v)} Z_w~,
   \end{align}
   where the neighborhood \(N(v)\) is the set of vertices which share an edge with \(v\).
 
-  Cluster-state codewords are used in measurement-based quantum computation (MBQC), which substitutes the temporal dimension necessary for decoding with a spatial dimension.
-  This is done by encoding the computation into the topological features of the cluster state''s graph.
-
   An MBQC scheme can be constructed out of any qubit CSS code (via \textit{foliation} \cite{arxiv:1607.02579}) or qubit stabilizer code \cite{arxiv:1811.11780}.
-  The original MBQC scheme \cite{arxiv:quant-ph/0510135,arxiv:quant-ph/0610082} uses the RBH cluster state on the bcc lattice (equivalently, a cubic lattice with qubits on edges and faces).
+  The original MBQC scheme \cite{arxiv:quant-ph/0510135,arxiv:quant-ph/0610082} uses the RBH cluster state on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces).
 
 protection: |
   Protection is related to the stabilizer code underlying the cluster state.
@@ -84,9 +86,11 @@ relations:
   parents:
     - code_id: qubit_stabilizer
       detail: 'Cluster states are particular qubit stabilizer states defined on a graph.
-      Any qubit stabilizer code is locally equivalent to a graph code \cite{arxiv:quant-ph/0111080} (see also \cite{arxiv:quant-ph/0703112}).
-      As a corollary, any qubit stabilizer state is locally equivalent to a cluster state \cite{arxiv:quant-ph/0308151}\cite[Appx. A]{arxiv:1910.00471}.
+      Any qubit stabilizer code is equivalent to a graph code via a single-qubit Clifford circuit \cite{arxiv:quant-ph/0111080} (see also \cite{arxiv:quant-ph/0703112}).
+      As a corollary, any qubit stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit \cite{arxiv:quant-ph/0308151}\cite[Appx. A]{arxiv:1910.00471}.
       Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol \cite{arxiv:1811.11780}.'
+    - code_id: group_cluster_state
+      detail: 'Group-based cluster-state codes reduce to cluster-state codes for \(G=\mathbb{Z}_2\).'
   cousins:
     - code_id: fusion
       detail: 'FBQC and MBQC are both computational models in which computation is done by measuring resource states (which are qubit stabilizer states).

codes/quantum/qubits/stabilizer/mbqc/rbh.yml

@@ -13,7 +13,7 @@ introduced: '\cite{arxiv:quant-ph/0407255,arxiv:quant-ph/0510135,arxiv:quant-ph/
 
 description: |
   Also called an \textit{RHG (Raussendorf-Harrington-Goyal) cluster-state code}.
-  A three-dimensional cluster-state code defined on the bcc lattice (equivalently, a cubic lattice with qubits on edges and faces).
+  A three-dimensional cluster-state code defined on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces).
 
   The MBQC version of the code is defined as the unique ground state of a certain code Hamiltonian. This state is the resource state used in the first MBQC scheme \cite{arxiv:quant-ph/0510135,arxiv:quant-ph/0610082}.
   It encodes the temporal gate operations on the surface code into a third spatial dimension.