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

codes/classical/analog/lattice/bw/barnes_wall.yml

@@ -15,7 +15,7 @@ description: |
   Member of a family of \(2^{m+1}\)-dimensional lattices, denoted as BW\(_{2^{m+1}}\), that are the densest lattices known.
   Members include the integer square lattice \(\mathbb{Z}^2\), \(D_4\), the Gosset \(E_8\) lattice, and the \(\Lambda_{16}\) lattice, corresponding to \(m\in\{0,1,2,3\}\), respectively.
 
-  Its automorphism group is the Clifford group \cite{arxiv:math/0001038,arxiv:0712.1939,arxiv:2404.17677}.
+  Its automorphism group is the \hyperref[topic:clifford]{real Clifford group} \cite{arxiv:math/0001038,arxiv:0712.1939,arxiv:2404.17677}.
 
 protection: |
   BW lattices in dimension \(2^{m+1}\) have a nominal coding gain of \(2^{m/2}\).

codes/classical/properties/block/block.yml

@@ -19,7 +19,7 @@ description: |
 
   \begin{defterm}{Asymptotic notation}
   \label{topic:asymptotics}
-  We are often interested in how parameters of particular infinite block-code families scale with increasing block length \(n\), necessitating the use of asymptotic or Bachmann–Landau notation. The table below summarizes the notation used throughout the EC Zoo for relating functions \(f,g\) that both grow with \(n\).
+  We are often interested in how parameters of particular infinite block-code families scale with increasing block length \(n\), necessitating the use of asymptotic or Bachmann–Landau notation; see the book \cite{manual:{Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2022). Introduction to algorithms. MIT press.}}. The table below summarizes the notation used throughout the EC Zoo for relating functions \(f,g\) that both grow with \(n\).
   \begin{table}
     \begin{cells}
     \celldata<c H, c H>{relation & behavior}

codes/classical/rings/over_zq/easy/pentacode.yml

@@ -20,7 +20,7 @@ relations:
     - code_id: small_distance
   cousins:
     - code_id: best
-      detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/BF01388558}.'
+      detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/bf01388558}.'
 
 
 # Begin Entry Meta Information

codes/classical/spherical/group_orbit/sidelnikov.yml

@@ -12,10 +12,10 @@ introduced: '\cite{manual:{V. M. Sidelnikov, On a finite group of matrices and c
 
 description: |
   Slepian group-orbit code of dimension \(2^r\), approximate asympotic size \(2.38 \cdot 2^{r(r+1)/2+1}\), and distance \(1\).
-  Code is constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group \cite{arxiv:math/0001038}, onto the vector \((1,0,0,\cdots,0)\).
+  Code is constructed by applying elements of an index-two subgroup of the \hyperref[topic:clifford]{real Clifford group}, when taken as a subgroup of the orthogonal group \cite{arxiv:math/0001038}, onto the vector \((1,0,0,\cdots,0)\).
   This group is the automorphism group of BW lattice, and the resulting codes coincide with the optimal spherical codes for dimensions \(\{4,8,16\}\).
 
-  Taking the orbit under the entire real Clifford group yields spherical codes twice the points and with distance \(2-\sqrt{2}\).
+  Taking the orbit under the entire \hyperref[topic:clifford]{real Clifford group} yields spherical codes twice the points and with distance \(2-\sqrt{2}\).
 
 relations:
   parents:
@@ -24,7 +24,7 @@ relations:
       detail: 'The orbit of any point under the real Clifford subgroup is a spherical 7-design \cite{doi:10.1023/A:1018723416627}, and some are 11-designs \cite{manual:{V. M. Sidelnikov, “Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial”, Algebra i Analiz, 11:4 (1999), 183–203; St. Petersburg Math. J., 11:4 (2000), 673–686}}.'
   cousins:
     - code_id: barnes_wall
-      detail: 'The automorphism group of BW lattices is a subgroup of index 2 of a real Clifford group \cite{manual:{V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform. 33 (1997), 35–54 (1997); English translation in Problems Inform. Transmission 33 (1997), 29–44},doi:10.1109/ISIT.1997.613373} (see \cite{arxiv:math/0001038,arxiv:2404.17677} for an explanation).'
+      detail: 'The automorphism group of BW lattices is a subgroup of index 2 of a \hyperref[topic:clifford]{real Clifford group} \cite{manual:{V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform. 33 (1997), 35–54 (1997); English translation in Problems Inform. Transmission 33 (1997), 29–44},doi:10.1109/ISIT.1997.613373} (see \cite{arxiv:math/0001038,arxiv:2404.17677} for an explanation).'
 
 # Begin Entry Meta Information
 _meta:

codes/classical/spherical/polytope/disphenoidal288cell.yml

@@ -23,7 +23,7 @@ relations:
       detail: 'The disphenoidal 288-cell code forms a spherical 7-design \cite{doi:10.1109/ITW.2003.1216742}.'
   cousins:
     - code_id: sidelnikov
-      detail: 'The disphenoidal 288-cell code is a group-orbit code with the group being the real Clifford group in \(4\) dimensions.'
+      detail: 'The disphenoidal 288-cell code is a group-orbit code with the group being the \hyperref[topic:clifford]{real Clifford group} in \(4\) dimensions.'
 
 
 # Begin Entry Meta Information

codes/quantum/oscillators/coherent_state/clifford_group.yml

@@ -12,7 +12,7 @@ introduced: '\cite{arxiv:2302.11593}'
 
 description: |
   A \(((2^r,2,2-\sqrt{2},8))\) QSC for \(r \geq 2\) constructed using the real-Clifford subgroup-orbit code.
-  Logical constellations are constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group \cite{arxiv:math/0001038} to \(2\) different vectors on the complex sphere.
+  Logical constellations are constructed by applying elements of an index-two subgroup of the \hyperref[topic:clifford]{real Clifford group}, when taken as a subgroup of the orthogonal group \cite{arxiv:math/0001038} to \(2\) different vectors on the complex sphere.
   The code is known as the \textit{Witting code} for \(r=2\) because its two logical constellations form vertices of Witting polytopes.
 
 relations:

codes/quantum/oscillators/coherent_state/clifford_qsc.yml

@@ -11,12 +11,12 @@ 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.
+  QSC whose projection is onto a copy of an irreducible representation of the single-qubit \hyperref[topic:clifford]{Clifford group} \(2O\), taken as the binary octahedral subgroup of the group \(SU(2)\) of Gaussian rotations.
   Its codewords consist of non-uniform superpositions of 48 coherent states.
 
 features:
   general_gates:
-    - 'The Clifford group \(2O\) can be realized via Gaussian rotations.
+    - 'The \hyperref[topic:clifford]{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}.'
 
 

codes/quantum/properties/block/dynamic/monitored_random_circuits.yml

@@ -10,7 +10,7 @@ introduced: '\cite{doi:10.1103/PhysRevX.9.031009,doi:10.1103/PhysRevB.98.205136,
 
 description: |
   Error-correcting code arising from a monitored random circuit. Such a circuit is described by a series of intermittant random local projective Pauli measurements with random unitary time-evolution operators.
-  An important sub-family consists of \textit{Clifford monitored random circuits}, where unitaries are sampled from the Clifford group \cite{arxiv:1901.08092}.
+  An important sub-family consists of \textit{Clifford monitored random circuits}, where unitaries are sampled from the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:1901.08092}.
   When the rate of projective measurements is independently controlled by a probability parameter \(p\), there can exist two stable phases, one described by volume-law entanglement entropy and the other by area-law entanglement entropy.
   The phases and their transition can be understood from the perspective of quantum error correction, information scrambling, and channel capacities \cite{arxiv:1903.05124,arxiv:1905.05195}.
 
@@ -60,7 +60,7 @@ relations:
     - code_id: topological
       detail: 'Topological order can be generated in 2D monitored random circuits \cite{arxiv:2011.06595}.'
     - code_id: random_stabilizer
-      detail: 'An important sub-family of monitored random-circuit codes are the Clifford monitored random-circuit codes, where unitaries are sampled from the Clifford group \cite{arxiv:1901.08092}.'
+      detail: 'An important sub-family of monitored random-circuit codes are the Clifford monitored random-circuit codes, where unitaries are sampled from the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:1901.08092}.'
 
 
 # Begin Entry Meta Information

codes/quantum/properties/block/tensor_network/holographic_tensor.yml

@@ -33,7 +33,7 @@ features:
 
   transversal_gates:
     - 'There exist holographic approximate codes with arbitrary transversal gate sets for any compact Lie group \cite{arxiv:2108.11402}.
-    However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group \cite{arxiv:2103.13404}.'
+    However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:2103.13404}.'
 
   code_capacity_threshold:
     - 'The ideal holographic tensor-network code (perfect representation of AdS/CFT) should be able to protect a central bulk operator against erasures of half of the physical qubits on the boundary, in line with AdS-Rindler reconstruction \cite{arxiv:1503.06237}.'

codes/quantum/properties/hamiltonian/qltc.yml

@@ -33,7 +33,7 @@ protection: |
 
   \textit{Soundness amplification} \cite[Thm. 1.2]{arxiv:2309.05541} can be used to obtain a constant-soundness (i.e., \(R = O(1)\)) QLTC family from a CSS family with a sub-constant value, with the former's locality being at most polynomial in \(1/R\).
 
-  AEL distance amplification can be used to convert an \([[n^{\prime},k,d,w]]\) soundness-\(R\) CSS LTC family into an \([[n=n^{\prime}+O(1),k,d=O(n)]]\) family with \(w\) and \(R\) differing by a factor at most polynomial in \(w\) and \(n/d\) \cite[Thm. 1.3]{arxiv:2309.05541}.
+  AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} can be used to convert an \([[n^{\prime},k,d,w]]\) soundness-\(R\) CSS LTC family into an \([[n=n^{\prime}+O(1),k,d=O(n)]]\) family with \(w\) and \(R\) differing by a factor at most polynomial in \(w\) and \(n/d\) \cite[Thm. 1.3]{arxiv:2309.05541}.
 
 notes:
   - 'It was shown in Ref. \cite{doi:10.1109/FOCS.2017.46} that existence of a QLTC with constant parameters would implies resolution of the \textit{No low-energy trivial states} (NLTS) conjecture \cite{arxiv:1301.1363} (see also \cite{arxiv:2311.09503}).

codes/quantum/properties/stabilizer/qldpc/good_qldpc.yml

@@ -31,7 +31,7 @@ description: |
 
 
 features:
-  rate: 'The codes'' rate and distance are both separated from zero as block length goes to infinity. AEL distance amplification can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound \cite[Corr. 5.3]{arxiv:2212.09935}.'
+  rate: 'The codes'' rate and distance are both separated from zero as block length goes to infinity. AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound \cite[Corr. 5.3]{arxiv:2212.09935}.'
 
 relations:
   parents:
@@ -40,7 +40,7 @@ relations:
     - code_id: translationally_invariant_stabilizer
       detail: 'Chain complexes describing some good QLDPC codes can be ''lifted'' into higher-dimensional manifolds admitting some notion of geometric locality \cite{arxiv:2012.02249,arxiv:2309.16104}. Applying this procedure to good QLDPC codes yiels \([[n,n^{1-2/D},n^{1-1/D}]]\) lattice stabilizer codes in \(D\) spatial dimensions that saturate the \hyperref[topic:bpt-bound]{BPT bound}, up to corrections poly-logarithmic in \(n\) \cite{arxiv:2303.06755}.'
     - code_id: quantum_mds
-      detail: 'AEL distance amplification can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound \cite[Corr. 5.3]{arxiv:2212.09935}.'
+      detail: 'AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound \cite[Corr. 5.3]{arxiv:2212.09935}.'
 
 
 # Begin Entry Meta Information

codes/quantum/properties/stabilizer/qldpc/quantum_locally_recoverable.yml

@@ -21,12 +21,12 @@ protection: |
   \end{align}
   implying that locality restricts the distance of the code.
   Random QLRCs with qudit dimension \(q = 2^{O(r)}\) achieve a relative distance that is \hyperref[topic:asymptotics]{order} \(O(1/r)\) below the bound \cite[Prop. 5]{arxiv:2311.08653}.
-  Codes constructed with the help of AEL distance amplification admit a gap of \hyperref[topic:asymptotics]{order} \(O(1/r^{1/4})\) \cite[Prop. 6]{arxiv:2311.08653}.
+  Codes constructed with the help of AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} admit a gap of \hyperref[topic:asymptotics]{order} \(O(1/r^{1/4})\) \cite[Prop. 6]{arxiv:2311.08653}.
 
 
 features:
   decoders:
-    - 'Codes constructed with the help of AEL distance amplification admit efficient decoders \cite{arxiv:2311.08653}.'
+    - 'Codes constructed with the help of AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} admit efficient decoders \cite{arxiv:2311.08653}.'
 
 relations:
   parents:

codes/quantum/qubits/dynamic/random/haar_random.yml

@@ -31,9 +31,9 @@ relations:
     - code_id: random_circuit
   cousins:
     - code_id: local_haar_random
-      detail: 'Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the Clifford group forms a unitary 2-design and produces stabilizer codes.'
+      detail: 'Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the \hyperref[topic:clifford]{Clifford group} forms a unitary 2-design and produces stabilizer codes.'
     - code_id: t-designs
-      detail: 'Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the Clifford group forms a unitary 2-design and produces stabilizer codes.'
+      detail: 'Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the \hyperref[topic:clifford]{Clifford group} forms a unitary 2-design and produces stabilizer codes.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/qubits_into_qubits.yml

@@ -92,8 +92,16 @@ features:
   transversal_gates:
     - 'A qubit code is \(U\)-\textit{quasi-transversal} if it can realize the logical gate \(U\) in the third level of the \term{Clifford hierarchy} using the physical gate \(C T^{\otimes n}\), where \(C\) is some Clifford gate \cite[Def. 4]{arxiv:1606.01904}.'
   general_gates:
-    - 'The normalizer of the \hyperref[topic:complementary-channel]{Pauli group} is the Clifford group; see Ref. \cite{arxiv:1310.6813} for generators, relations, and normal form. The Clifford group permutes Pauli operators amongst themselves, and, up to any phases, is equivalent to the symplectic group \(Sp(2n,\mathbb{Z}_2)\).
-    The combined Pauli and Clifford group cannot be expressed as a semidirect product of those two constituents \cite{arxiv:2406.09951}.'
+    - |
+      \begin{defterm}{Clifford group}
+      \label{topic:clifford}
+      The Clifford group is the normalizer of the \hyperref[topic:pauli]{Pauli group}. 
+      The group consists of the Pauli group as well as elements that permute Pauli operators amongst themselves.
+      Up to any phases and Pauli strings, the group is equivalent to the symplectic group \(Sp(2n,\mathbb{Z}_2)\). 
+      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}.
+      \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}.
     More generally, the \textit{Solovay-Kitaev} theorem \cite{doi:10.1070/rm1997v052n06abeh002155,doi:10.1090/gsm/047} states that any subset of gates the generates a dense subgroup of the full \(n\)-qubit gate group can be used to construct any gate to arbitrary accuracy (see \cite{arxiv:quant-ph/0505030}\cite[Appx. 3]{doi:10.1017/cbo9780511976667.019}). The task of approximating a desired gate by Clifford gates and a fixed set of non-Clifford gates is called \textit{gate compilation} or \textit{circuit synthesis}.'
@@ -106,7 +114,7 @@ features:
       \begin{align}
         C_k = \{ U | U P U^{\dagger} \in C_{k-1} \}~,
       \end{align}
-      where \(P\) is any Pauli matrix, and \(C_1\) is the \hyperref[topic:complementary-channel]{Pauli group}.
+      where \(P\) is any Pauli matrix, and \(C_1\) is the \hyperref[topic:pauli]{Pauli group}.
       \end{defterm}'
     - 'Arbitrary \(n\)-qubit circuits can be implemented fault-tolerantly in a 3D architecture using \(O(n^{3/2}\log^3 n)\) qubits, and in a 2D architecture using only \(O(n^2 \log^3 n)\) qubits \cite{arxiv:2402.13863}.'
   decoders:

codes/quantum/qubits/small_distance/iceberg.yml

@@ -22,7 +22,7 @@ description: |
   Admits a basis such that each codeword is a superposition of a computational basis state labeled by an even-weight bitstring \(b\) and a state labeled by the negation of \(b\).
   Its all-zero logical state is a conventional GHz state.
 
-  All of its automorphisms lie in the Clifford group \cite[Thm. 13]{arxiv:quant-ph/9704043}.
+  All of its automorphisms lie in the \hyperref[topic:clifford]{Clifford group} \cite[Thm. 13]{arxiv:quant-ph/9704043}.
 
 protection: 'Detects a single-qubit error.'
 

codes/quantum/qubits/small_distance/small/stab_5_1_2.yml

@@ -17,9 +17,9 @@ description: |
 
 features:
   general_gates:
-    - 'Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs \cite{arxiv:2112.01446}.'
+    - 'Fault-tolerant implementation of the \hyperref[topic:clifford]{Clifford group} based on transversal gates and SWAPs \cite{arxiv:2112.01446}.'
   fault_tolerance:
-    - 'Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs \cite{arxiv:2112.01446}.'
+    - 'Fault-tolerant implementation of the \hyperref[topic:clifford]{Clifford group} based on transversal gates and SWAPs \cite{arxiv:2112.01446}.'
 
 
 relations: