Merge branch 'main' of https://github.com/errorcorrectionzoo/eczoo_data

code_extra/bib_preset.yml

@@ -265,6 +265,11 @@ HKSstc:
     flm: >-
         F. Oggier, "Spacetime Coding." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) \href{https://doi.org/10.1201/9781315147901}{DOI}
 
+HKSnetwork:
+  _ready_formatted:
+    flm: >-
+        F. R. Kschischang, "Network Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) \href{https://doi.org/10.1201/9781315147901}{DOI}
+
 HKSfountain:
   _ready_formatted:
     flm: >-

codes/classical/matrices/rank-metric/gabidulin.yml

@@ -8,10 +8,12 @@ physical: matrices
 logical: matrices
 
 name: 'Gabidulin code'
-introduced: '\cite{manual:{E. M. Gabidulin, \textit{Theory of Codes with Maximum Rank Distance}, Problemy Peredachi Informacii, Volume 21, Issue 1, \emph{3–16} (1985)},doi:10.1109/18.75248}'
+introduced: '\cite{manual:{E. M. Gabidulin, \textit{Theory of Codes with Maximum Rank Distance}, Problemy Peredachi Informacii, Volume 21, Issue 1, \emph{3–16} (1985)},doi:10.1016/0097-3165(78)90015-8,doi:10.1109/18.75248}'
 
 alternative_names:
   - 'Vector rank-metric code'
+  - 'Delsarte-Gabidulin code'
+#HKS Ch 29
 
 description: |
   A linear code over \(GF(q^N)\) that corrects errors over rank metric instead of the traditional Hamming distance. Every element \(GF(q^N)\) can be written as an \(N\)-dimensional vector with coefficients in \(GF(q)\), and the rank of a set of elements is rank of the matrix formed by their coefficients.

codes/classical/q-ary_digits/projective/subspace.yml

@@ -0,0 +1,43 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: subspace
+physical: q-ary_digits
+logical: q-ary_digits
+
+name: 'Subspace code'
+introduced: \cite{doi:10.1109/TIT.2008.926449}
+
+description: |
+  A code that is a set of subspaces of a projective space \(PG(n-1,q)\).
+
+protection: |
+  Subspace codes are quantified with respect to the subspace distance \cite{doi:10.1109/TIT.2008.926449} or injection distance \cite{arxiv:0805.3824}.'
+  
+  Generalizations of various bounds for ordinary \(q\)-ary codes have been developed for subspace codes; see \cite{preset:HKSnetwork}.
+
+features:
+  decoders:
+    - 'List decoding up to the Singleton bound \cite{doi:10.1145/2488608.2488715}.'
+
+realizations:
+  - 'Packet-based transmission over networks \cite{preset:HKSnetwork}.'
+
+relations:
+  parents:
+    - code_id: projective
+  cousins:
+    - code_id: gabidulin
+      detail: 'Gabidulin codes can be used to construct asymptotically good subspace codes \cite{doi:10.1109/TIT.2003.809567,doi:10.1109/TIT.2008.926449}.'
+    - code_id: rank_metric
+      detail: 'Subspace and rank-metric codes are closely related \cite{doi:10.1109/TIT.2008.928291}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-08-23'

codes/quantum/qubits/qubits_into_qubits.yml

@@ -102,7 +102,9 @@ features:
       See Refs. \cite{arxiv:quant-ph/0304125,arxiv:0811.0898,arxiv:1310.6813,preset:GottesmanBook} for generators, relations, and normal form. 
       The group cannot be expressed as a semidirect product of the Pauli and symplectic groups \cite{arxiv:2406.09951}.
       Restricting the group to real-valued elements yields the \textit{real Clifford group}.
-      Single-qubit Clifford gates, modulo phases, realize the \(2O\) binary octahedral subgroup of \(SU(2)\).
+      Single-qubit Clifford gates, together with Paulis, realize a group with \(192\) elements.
+      Modding out phases yields the \(48\)-element \(2O\) binary octahedral subgroup of \(SU(2)\).
+      Further modding out the Pauli group, which corresponds to the quaternion group \(Q\), yields the permutation group \(S_3\), which consists of permutations of the three non-identity single-qubit Pauli matrices.
       \end{defterm}
     - 'Computing using Clifford gates only can be efficiently simulated on a classical computer, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,manual:{E. Knill, private communication, ca. 1998.}}.
     Universal quantum computing can be achieved using Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate \cite{arxiv:quant-ph/9503016}.

codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml

@@ -38,7 +38,7 @@ protection: |
 
 features:
   transversal_gates:
-    - 'Transversal \(SH\) gates \cite[Sec. 8.2]{arxiv:quant-ph/9705052}.'
+    - 'Transversal \(SH\) and \(HS\) gates \cite[Sec. 8.2]{arxiv:quant-ph/9705052}.'
     - 'The three-block transversal gate mapping each physical \(X \to XYZ\) and each \(Z \to ZXY\) implements a logical gate \cite{arxiv:quant-ph/9702029}\cite[Exam. 2]{arxiv:quant-ph/9703048}.'
 
   fault_tolerance:

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -116,6 +116,7 @@ features:
     - 'Entanglement purification \cite{arxiv:quant-ph/0210069}.'
     - 'Reinforcement learning encoding circuits \cite{arxiv:2402.17761}.'
     - 'ZX calculus provides a canonical form of an encoding circuit \cite{arxiv:2406.12083}.'
+    - 'Automated fault-tolerant encoding circuit synthesis \cite{arxiv:2408.11894}.'
 
 # Realizing transversal gates outside of the Clifford group requires certain higher-order (i.e., non-quadratic) constraints to be satisfied on the code \cite{arxiv:1209.2426}.
 
@@ -139,6 +140,7 @@ features:
     - 'Homomorphic gadgets fault-tolerant measurement unify Steane and Shor error correction \cite{arxiv:2211.03625}.'
     - 'A fault-tolerant error-correction protocol using \(O(d\log d)\) syndrome measurements can be applied to any CSS code with distance \(d \geq \Omega(n^{\alpha})\) for any \(\alpha > 0\) \cite{arxiv:2002.05180}.'
     - 'Fault-tolerant measurement-free scheme for low-distance CSS codes \cite{arxiv:2307.13296}.'
+    - 'Automated fault-tolerant encoding circuit synthesis \cite{arxiv:2408.11894}.'
   code_capacity_threshold:
     - 'Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights \cite{arxiv:1208.2317,arxiv:1412.6172}.'
   decoders:

codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml

@@ -90,7 +90,7 @@ features:
     ML decoding \cite{arxiv:quant-ph/0110143} is \(\#P\)-hard in general for the surface code \cite{arxiv:2309.10331}.'
     - 'Union-find decoder \cite{arxiv:1709.06218} uses the \textit{union-find data structure} \cite{doi:10.1145/364099.364331,doi:10.1137/0202024,doi:10.1145/62.2160}, solving the MPE decoding problem exactly for low-weight errors under depolarizing noise. A subsequent modification utilizes the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) \cite{arxiv:2107.13589}. Belief union find is a combination of belief-propagation and union-find \cite{arxiv:2203.04948}.
     Strictly local (as opposed to partially local) union find \cite{arxiv:2305.18534} has a worst-case runtime of \hyperref[topic:asymptotics]{order} \(O(d^3)\) in the distance \(d\).'
-    - 'Modified MWPM decoders: pipeline MWPM (accounting for correlations between events) \cite{arxiv:1310.0863,arxiv:2205.09828}; modification tailored to asymmetric noise \cite{arxiv:1812.01505}; parity blossom MWPM and fusion blossom MWPM \cite{arxiv:2305.08307}, a modification utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) \cite{arxiv:2107.13589}; belief perfect matching (a combination of belief-propagation and MWPM) \cite{arxiv:2203.04948}; spanning tree matching (STM) and rapid-fire (RFire) decoders \cite{arxiv:2405.01151}; ordered decoding based on MWPM \cite{arxiv:2408.01393}.
+    - 'Modified MWPM decoders: pipeline MWPM (accounting for correlations between events) \cite{arxiv:1310.0863,arxiv:2205.09828}; modification tailored to asymmetric noise \cite{arxiv:1812.01505}; parity blossom MWPM and fusion blossom MWPM \cite{arxiv:2305.08307}, a modification utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) \cite{arxiv:2107.13589}; belief perfect matching (a combination of belief-propagation and MWPM) \cite{arxiv:2203.04948}; spanning tree matching (STM) and rapid-fire (RFire) decoders \cite{arxiv:2405.01151}; ordered decoding based on MWPM \cite{arxiv:2408.01393}; Libra decoder \cite{arxiv:2408.12135}.
     Combinining, or \textit{harmonizing}, various decoders can improve performance \cite{arxiv:2401.12434}.'
     - 'Renormalization group (RG) \cite{arxiv:0911.0581,arxiv:1304.6100,arxiv:1411.3028}; see Ref. \cite{arxiv:1310.2393} for the planar surface code.'
     - 'Linear-time ML erasure decoder \cite{arxiv:1703.01517}.'

codes/quantum/qudits_galois/stabilizer/evaluation/binary_quantum_goppa.yml

@@ -10,9 +10,6 @@ logical: galois
 name: 'Quantum Goppa code'
 introduced: '\cite{arxiv:quant-ph/0006061,arxiv:quant-ph/0501074,doi:10.1007/s11128-006-0047-9}'
 
-alternative_names:
-  - 'Quantum AG code'
-
 description: |
   A Galois-qudit CSS code constructed using two Goppa codes.
   
@@ -31,29 +28,20 @@ description: |
 protection: 'Protects against weight \(t\) errors where \( 0 < t \leq  \lfloor \frac{d^*-g-1}{2} \rfloor \) where \( d^* = \text{deg} G + 2 -2g \) and \(g\) is the genus of the function field and \(d \geq n - \lfloor \frac{deg G}{2} \rfloor\). Such codes can exceed the \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{doi:10.1007/s11128-006-0047-9}.'
 
 features:
-  rate: 'Quantum Goppa codes \cite{arxiv:quant-ph/0006061} and other quantum codes constructed from AG codes \cite{arxiv:quant-ph/0107129} can be asymptotically good. There exist three such families \cite{arxiv:2408.07764,arxiv:2408.09254,arxiv:2408.10140} that admit a diagonal transversal gate at the third level of the \term{Clifford hierarchy}.'
-
-  magic_scaling_exponent: 'By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum Goppa codes that admits a diagonal transversal gate at the third level of the \term{Clifford hierarchy} and attains a zero magic-state yield parameter, \(\gamma = 0\) \cite{arxiv:2408.07764}. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Two other such asymptotically good families exist \cite{arxiv:2408.09254,arxiv:2408.10140}, admitting a different diagonal gate at the third level of the \term{Clifford hierarchy}.'
-
+  rate: 'Quantum Goppa codes \cite{arxiv:quant-ph/0006061} can be asymptotically good.'
 
   encoders:
     - 'Encoding defined in Ref. \cite{arxiv:quant-ph/0107129} uses a technique from Ref. \cite{arxiv:quant-ph/0005008} to encode quantum stabilizer codes.'
 
-  transversal_gates:
-    - 'There exist three asymptotically good code families \cite{arxiv:2408.07764,arxiv:2408.09254,arxiv:2408.10140} that admit a diagonal transversal gate at the third level of the \term{Clifford hierarchy}.'
-
   decoders:
     - 'Farran algorithm \cite{arxiv:math/9910151}.'
 
 relations:
   parents:
-    - code_id: galois_css
-      detail: 'Goppa codes can be realized in the CSS code construction \cite{doi:10.1007/s11128-006-0047-9}.'
+    - code_id: quantum_ag
   cousins:
     - code_id: goppa
       detail: 'Classical Goppa codes over various algebraic curves are used to construct quantum Goppa codes.'
-    - code_id: quantum_triorthogonal
-      detail: 'By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum Goppa codes that admits a diagonal transversal gate at the third level of the \term{Clifford hierarchy} and attains a zero magic-state yield parameter, \(\gamma = 0\) \cite{arxiv:2408.07764}. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Two other such asymptotically good families exist \cite{arxiv:2408.09254,arxiv:2408.10140}, admitting a different diagonal gate at the third level of the \term{Clifford hierarchy}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits_galois/stabilizer/evaluation/quantum_ag.yml

@@ -0,0 +1,46 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: quantum_ag
+physical: galois
+logical: galois
+
+name: 'Quantum AG code'
+introduced: '\cite{arxiv:quant-ph/0107129}'
+
+description: |
+  A Galois-qudit CSS code constructed using two linear AG codes.
+  
+
+features:
+  rate: 'Quantum AG codes \cite{arxiv:quant-ph/0107129} can be asymptotically good. There exist three such families \cite{arxiv:2408.07764,arxiv:2408.09254,arxiv:2408.10140} that admit a diagonal transversal gate at the third level of the \term{Clifford hierarchy}.'
+
+  magic_scaling_exponent: 'By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the \term{Clifford hierarchy} and attains a zero magic-state yield parameter, \(\gamma = 0\) \cite{arxiv:2408.07764}. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Two other such asymptotically good families exist \cite{arxiv:2408.09254,arxiv:2408.10140}, admitting a different diagonal gate at the third level of the \term{Clifford hierarchy}.'
+
+
+  encoders:
+    - 'Encoding defined in Ref. \cite{arxiv:quant-ph/0107129} uses a technique from Ref. \cite{arxiv:quant-ph/0005008} to encode quantum stabilizer codes.'
+
+  transversal_gates:
+    - 'There exist three asymptotically good code families \cite{arxiv:2408.07764,arxiv:2408.09254,arxiv:2408.10140} that admit a diagonal transversal gate at the third level of the \term{Clifford hierarchy}.'
+
+relations:
+  parents:
+    - code_id: galois_css
+      detail: 'Quantum AG codes can be realized in the CSS code construction \cite{doi:10.1007/s11128-006-0047-9}.'
+  cousins:
+    - code_id: ag
+    - code_id: quantum_triorthogonal
+      detail: 'By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the \term{Clifford hierarchy} and attains a zero magic-state yield parameter, \(\gamma = 0\) \cite{arxiv:2408.07764}. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Two other such asymptotically good families exist \cite{arxiv:2408.09254,arxiv:2408.10140}, admitting a different diagonal gate at the third level of the \term{Clifford hierarchy}.'
+    - code_id: shimura
+      detail: 'The AG codes used in an asymptotically good construction of quantum AG codes with non-Clifford transversal gates \cite{arxiv:2408.09254} are those of the TVZ codes.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-08-23'

codes/quantum/qudits_galois/stabilizer/evaluation/rs/galois_grs.yml

@@ -25,8 +25,8 @@ relations:
     - code_id: generalized_reed_solomon
     - code_id: quantum_mds
       detail: 'Some Galois-qudit GRS codes are quantum MDS \cite{arxiv:1311.3009}.'
-    - code_id: galois_css
-      detail: 'Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.'
+    - code_id: quantum_ag
+      detail: 'Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction from GRS codes, which are evaluation AG codes.'
     - code_id: stabilizer_over_gfqsq
       detail: 'Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.'
     - code_id: quantum_concatenated