~

codes/quantum/properties/block/holographic.yml

@@ -6,17 +6,19 @@
 code_id: holographic
 
 name: 'Holographic code'
+short_name: 'HQECC'
 introduced: '\cite{arxiv:1503.06237}'
 
 description: |
   Block quantum code whose features serve to model aspects of the AdS/CFT holographic duality and, more generally, quantum gravity.
 
-
 # SYK codes do not seem to be related to Lego, so parent is block quantum
 relations:
   parents:
     - code_id: qecc
   cousins:
+    - code_id: approximate_qecc
+      detail: 'Universal subspace approximate error correction is used to model black holes \cite{arxiv:1807.06041}.'
     - code_id: approximate_oaecc
       detail: 'Properties of holographic codes are often quantified in the Heisenberg picture, i.e., in terms of operator algebras \cite{arxiv:1411.7041,arxiv:1612.00017}.'
 

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

@@ -6,15 +6,14 @@
 code_id: holographic_tensor
 
 name: 'Holographic tensor-network code'
-introduced: '\cite{arxiv:2009.10329,arxiv:2109.11996}'
+introduced: '\cite{arxiv:1503.06237,arxiv:1601.01694,arXiv:1801.05289,arxiv:2009.10329,arxiv:2109.11996}'
 
 description: |
-  Quantum Lego code whose encoding isometry forms a holographic tensor network, i.e., a tensor network  tiling hyperbolic space.
+  Quantum Lego code whose encoding isometry forms a holographic tensor network, i.e., a tensor network  associated with a tiling of hyperbolic space.
   Physical qubits are associated with uncontracted tensor legs at the boundary of the tesselation, while logical qubits are associated with uncontracted legs in the bulk.
   The number of layers emanating form the central point of the tiling is the \textit{radius} of the code.
 
-  The encoding map often models radial time evolution for a fixed time slice in Anti de Sitter (AdS) space, mapping operators in the bulk of AdS, represented by logical qudits, onto operators on the boundary of the corresponding Conformal Field Theory (CFT), represented by physical qudits.
-  Encoding can often be represented by a tensor network associated with a tiling of hyperbolic space.
+  The encoding map models radial time evolution for a fixed time slice in Anti de Sitter (AdS) space, mapping operators in the bulk of AdS, represented by logical qudits, onto operators on the boundary of the corresponding Conformal Field Theory (CFT), represented by physical qudits.
   See \cite[Defn. 4.3]{arxiv:2108.11402} for a technical formulation.
 
 protection: |
@@ -37,11 +36,11 @@ features:
     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}.'
 
   code_capacity_threshold:
-    - 'The ideal holographic 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}.'
-    - 'Holographic codes are argued to have a \textit{algebraic threshold}, for which the error rate scales polynomially (as opposed to exponentially) in the thermodynamic limit \cite{arxiv:2202.04710}. Such a threshold is governed by the underlying conformal field theory describing the boundary.'
+    - '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}.'
+    - 'Holographic tensor-network codes are argued to have a \textit{algebraic threshold}, for which the error rate scales polynomially (as opposed to exponentially) in the thermodynamic limit \cite{arxiv:2202.04710}. Such a threshold is governed by the underlying conformal field theory describing the boundary.'
 
 notes:
-  - 'Review of holographic tensor network codes \cite{arXiv:2102.02619}.'
+  - 'Review of holographic tensor-network codes \cite{arXiv:2102.02619}.'
 
 
 relations:
@@ -54,8 +53,6 @@ relations:
       In other words, logical legs resulting from the conversion of codes to tensors must remain logical in the final tensor network, and the same for physical.
       Contracting logical legs is another word for gluing two logical legs together.'
   cousins:
-    - code_id: approximate_qecc
-      detail: 'Universal subspace approximate error correction is used to model black holes \cite{arxiv:1807.06041}.'
     - code_id: random_stabilizer
       detail: 'Random holographic tensor-network codes reproduce many aspects of holography \cite{arxiv:1601.01694,arXiv:1801.05289,arxiv:2105.12067}.'
     - code_id: hamiltonian

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

@@ -19,7 +19,7 @@ description: |
   The class of codes constructed using the framework depends on the choice of atomic Lego blocks.
 
   The individual Lego blocks and resulting quantum Lego codes can be stabilizer \cite{arxiv:2009.10329,arxiv:2109.11996} or non-stabilizer \cite{arxiv:2109.08158,arxiv:2308.05152}.
-  They need not be isometries, meaning that this class of codes generalizes block-perfect tensor-network codes.
+  They need not be isometries, meaning that this class of codes generalizes \hyperref[code:block_perfect]{planar-perfect tensor}-network codes.
   However, both the logical and physical degrees of freedom must have the same local dimension.
 
   For example, any stabilizer code can be built out of atomic blocks like the 2-site repetition code, single-site trivial stabilizer codes, and tensor products of the \(|0\rangle\) state.

codes/quantum/properties/block/tensor_network/single_tensor/ame.yml

@@ -5,25 +5,25 @@
 
 code_id: ame
 
-name: '\(((n,2,\lfloor(n+1)/2\rfloor))_q\) perfect-tensor code'
+# \(((n,q,\lfloor(n+1)/2\rfloor))_{\mathbb{Z}_q}\)
+name: 'Perfect-tensor code'
 
 alternative_names:
   - 'AME code'
 
 description: |
-  Block quantum code whose encoding isometry is a perfect tensor.
-  This code stems from an \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))_q\) \hyperref[topic:ame]{AME state}.
+  Block quantum code encoding one subsystem into \(n\) subsystems whose encoding isometry is a perfect tensor.
+  This code stems from an AME\((n,q)\) \hyperref[topic:ame]{AME state}, or equivalently, a \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))_{\mathbb{Z}_q}\) code.
 
   \begin{defterm}{Absolutely maximally entangled (AME) state}
   \label{topic:ame}
   A state on \(n\) subsystems is \(d\)\textit{-uniform} \cite{arXiv:quant-ph/0310137,arxiv:1404.3586} (a.k.a. \(d\)-maximally mixed \cite{arxiv:1211.4118}) if all reduced density matrices on up to \(d\) subsystems are maximally mixed.
-  A \(K\)-dimensional subspace of \(d-1\)-uniform states in \(q\)-dimensional qudits is equivalent to a \hyperref[topic:quantum-weight-enumerator]{pure} \(((n,K,d))_q\) code \cite{arxiv:0704.0251,arxiv:1907.07733}.
-  An AME state (a.k.a. maximally multi-partite entangled state \cite{arxiv:0710.2868,arxiv:1002.2592}) is a \(\lfloor n/2 \rfloor\)-uniform state, corresponding to a \hyperref[topic:quantum-weight-enumerator]{pure} \(((n,1,\lfloor n/2 \rfloor + 1))_q\) code.
-  The rank-\(n\) tensor formed by the basis-expansion coefficients of such states is a \textit{perfect tensor} (a.k.a. multi-unitary tensor), meaning that it is proportional to an isometry for any bipartition of its indices into a set \(A\) and a complementary set \(A^{\perp}\) such that \(|A|\leq|A^{\perp}|\).
-  A perfect-tensor code is obtained by picking one leg of this tensor to denote the logical qubit index, thus treating this tensor as an encoding isometry for the perfect-tensor code.
+  A \(K\)-dimensional subspace of \(d-1\)-uniform states of \(q\)-dimensional subsystems is equivalent to a \hyperref[topic:quantum-weight-enumerator]{pure} \(((n,K,d))_q\) code \cite{arxiv:0704.0251,arxiv:1907.07733}.
+  An AME state (a.k.a. maximally multi-partite entangled state \cite{arxiv:0710.2868,arxiv:1002.2592}) is a \(\lfloor n/2 \rfloor\)-uniform state, corresponding to a \hyperref[topic:quantum-weight-enumerator]{pure} \(((n,1,\lfloor n/2 \rfloor + 1))_{\mathbb{Z}_q}\) code.
+  The rank-\(n\) tensor formed by the encoding isometry of such codes is a \textit{perfect tensor} (a.k.a. multi-unitary tensor), meaning that it is proportional to an isometry for any bipartition of its indices into a set \(A\) and a complementary set \(A^{\perp}\) such that \(|A|\leq|A^{\perp}|\).
   \end{defterm}
 
-  Stabilizer perfect-tensor codes can be converted to \hyperref[topic:ame]{AME states} via established shortening/lengthening procedures.
+  Stabilizer Galois-qudit perfect-tensor codes can be converted to \hyperref[topic:ame]{AME states} via established shortening/lengthening procedures \cite[Table 1]{arxiv:quant-ph/0508070}\cite{arxiv:1502.05267}.
   For example, an \([[n,0,d]]\) AME state can be converted into an \([[n-1,1,d-1]]\) perfect-tensor code by tracing over one qubit \cite[Sec. 3.5]{arxiv:quant-ph/9705052}.
 
 
@@ -34,7 +34,7 @@ notes:
 relations:
   parents:
     - code_id: block_perfect
-      detail: 'A planar-perfect-tensor code whose encoding isometry is a perfect tensor is a perfect-tensor code.'
+      detail: '\hyperref[code:block_perfect]{Planar-perfect tensor}s are automatically \hyperref[code:block_perfect]{planar-perfect tensor}s.'
   cousins:
     - code_id: quantum_mds
       detail: '\hyperref[topic:ame]{AME states} for even \(n\) are examples of quantum MDS codes with no logical qubits \cite{arXiv:quant-ph/0310137,arxiv:1701.03359}.
@@ -44,7 +44,7 @@ relations:
     - code_id: orthogonal_array
       detail: 'Orthogonal arrays and \(d\)-uniform quantum states are related \cite{arXiv:1404.3586,arxiv:1708.05946}.'
     - code_id: qudit_cluster_state
-      detail: 'Since any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit \cite{arxiv:quant-ph/0111080} (see also \cite{arxiv:quant-ph/0308151,arxiv:quant-ph/0703112}), \hyperref[topic:ame]{AME states} can be understood as qudit cluster states \cite{arxiv:1306.2879}.'
+      detail: 'Since any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit \cite{arxiv:quant-ph/0111080} (see also \cite{arxiv:quant-ph/0308151,arxiv:quant-ph/0703112}), \hyperref[topic:ame]{stabilizer AME states} can be understood as modular-qudit cluster states \cite{arxiv:1306.2879}.'
 
 
 # Begin Entry Meta Information

codes/quantum/properties/block/tensor_network/single_tensor/block_perfect.yml

@@ -23,7 +23,7 @@ relations:
       # Parent should not be holographic_tensor since any encoding isometry is a radius-one holographic tensor-network code
   cousins:
     - code_id: category_quantum
-      detail: 'Several modular fusion categories can be used to define block-perfect tensors \cite{arxiv:1804.03199}.'
+      detail: 'Several modular fusion categories can be used to define \hyperref[code:block_perfect]{planar-perfect tensor}s \cite{arxiv:1804.03199}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/dynamic/floquet/hyperbolic_floquet.yml

@@ -22,7 +22,7 @@ features:
   decoders:
     - 'Syndrome structure allows for MWPM decoding.'
   threshold:
-    - '\(0.1\%\) under standard circuit-level depolarising noise \cite{arxiv:2308.03750}.'
+    - '\(0.1\%\) under standard circuit-level depolarizing noise \cite{arxiv:2308.03750}.'
     - '\(0.1\%\) under phenomenological error model including depolarizing and measurement errors for the octagonal codes \cite{arxiv:2309.10033}.'
 
 relations:

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

@@ -27,10 +27,10 @@ relations:
     - code_id: rotated_surface
     - code_id: holographic_5_1_2
       detail: 'The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code.
-      The encoding of more general SCF holographic codes is a holographic tensor network consisting of encoding isometries for the \([[5,1,2]]\) rotated surface code, which are block-perfect tensors.'
+      The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a \hyperref[code:block_perfect]{planar-perfect tensor}.'
     - code_id: block_perfect
       detail: 'The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code.
-      The encoding of more general SCF holographic codes is a holographic tensor network consisting of encoding isometries for the \([[5,1,2]]\) rotated surface code, which are block-perfect tensors.'
+      The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a \hyperref[code:block_perfect]{planar-perfect tensor}.'
     - code_id: small_distance_quantum
   cousins:
     - code_id: steane

codes/quantum/qubits/small_distance/small/stab_6_1_3.yml

@@ -28,7 +28,7 @@ relations:
     - code_id: qubit_stabilizer
     - code_id: holographic_6_1_3
       detail: 'The \([[6,1,3]]\) six-qubit stabilizer code is the smallest six-qubit-tensor holographic code.
-      The encoding of more general SCF holographic codes is a holographic tensor network consisting of encoding isometries for the \([[6,1,3]]\) six-qubit stabilizer code.'
+      The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[6,1,3]]\) six-qubit stabilizer code.'
     - code_id: small_distance_quantum
   cousins:
     - code_id: subsystem_stabilizer

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

@@ -104,12 +104,9 @@ 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.'
-    - code_id: holographic_steane
-      detail: 'The Steane code is the smallest heptagon holographic code.
-      The encoding of more general heptagon holographic codes is a holographic tensor network consisting of encoding isometries for the Steane code, which are block-perfect tensors.'
     - code_id: block_perfect
       detail: 'The Steane code is the smallest heptagon holographic code.
-      The encoding of more general heptagon holographic codes is a holographic tensor network consisting of encoding isometries for the Steane code, which are block-perfect tensors.'
+      The encoding of more general heptagon holographic codes is a holographic tensor network consisting of the encoding isometry for the Steane code, which is a \hyperref[code:block_perfect]{planar-perfect tensor}.'
   cousins:
     - code_id: quantum_cyclic
       detail: 'The Steane code is equivalent to a cyclic code via qubit permutations \cite[Exam. 1]{arxiv:1108.5490}.'
@@ -120,6 +117,11 @@ relations:
     - code_id: eastab
       detail: 'The Steane code is globally equivalent to a \([[6,1,3;1]]\) code, which is the smallest EA CSS code with that distance \cite{arxiv:0803.1495}.'
 
+# Mentioned via concatenated_steane already
+# - code_id: holographic_steane
+#   detail: 'The Steane code is the smallest heptagon holographic code.
+#   The encoding of more general heptagon holographic codes is a holographic tensor network consisting of the encoding isometry for the Steane code, which is a \hyperref[code:block_perfect]{planar-perfect tensor}.'
+
 
 #      detail: 'Steane code is the smallest member of a family of Reed-Muller-based CSS codes.'
 #      detail: 'Steane code is the smallest member of a family of quantum Hamming codes.'

codes/quantum/qubits/stabilizer/data_syndrome.yml

@@ -8,7 +8,7 @@ physical: qubits
 logical: qubits
 
 name: 'Quantum data-syndrome (QDS) code'
-short_name: 'QDS code'
+short_name: 'QDS'
 introduced: '\cite{arxiv:1409.2559,doi:10.1109/ISIT.2014.6874892,doi:10.1109/ISIT.2015.7282628,doi:10.1109/ISIT.2016.7541704,arxiv:1907.01393}'
 
 description: |

codes/quantum/qubits/stabilizer/holographic/holographic_5_1_2.yml

@@ -8,27 +8,27 @@ physical: qubits
 logical: qubits
 
 name: 'Surface-code-fragment (SCF) holographic code'
-short_name: 'SCF holographic code'
+short_name: 'SCF holographic'
 introduced: '\cite{arxiv:2008.10206}'
 
 description: |
   Holographic tensor-network code constructed out of a network of encoding isometries of the \([[5,1,2]]\) rotated surface code.
   The structure of the isometry is similar to that of the HaPPY code since both isometries are rank-six tensors.
-  In the case of the SCF holographic code, the isometry is only a block-perfect tensor (as opposed to a \hyperref[topic:ame]{perfect tensor}).
+  In the case of the SCF holographic code, the isometry is only a \hyperref[code:block_perfect]{planar-perfect tensor} (as opposed to a \hyperref[topic:ame]{perfect tensor}).
 
 
 features:
   code_capacity_threshold:
-    - '\(7.1\%\) and \(8.2\%\) for even and odd raddi reduced-rate codes, respectively, under depolarising using the integer optimization decoder \cite{arxiv:2008.10206}.'
+    - '\(7.1\%\) and \(8.2\%\) for even and odd raddi reduced-rate codes, respectively, under depolarizing using the integer optimization decoder \cite{arxiv:2008.10206}.'
 
 relations:
   parents:
     - code_id: qubit_css
     - code_id: holographic_tensor
-      detail: 'The encoding of the heptagon holographic code is a holographic tensor network consisting of encoding isometries for the \([[5,1,2]]\) rotated surface code, which are block-perfect tensors.'
+      detail: 'The encoding of the heptagon holographic code is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a \hyperref[code:block_perfect]{planar-perfect tensor}.'
   cousins:
     - code_id: block_perfect
-      detail: 'The encoding of the heptagon holographic code is a holographic tensor network consisting of encoding isometries for the \([[5,1,2]]\) rotated surface code, which are block-perfect tensors.'
+      detail: 'The encoding of the heptagon holographic code is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a \hyperref[code:block_perfect]{planar-perfect tensor}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/stabilizer/holographic/holographic_6_1_3.yml

@@ -12,18 +12,18 @@ introduced: '\cite{arxiv:2009.10329}'
 
 description: |
   Holographic tensor-network code constructed out of a network of encoding isometries of the \([[6,1,3]]\) six-qubit stabilizer code.
-  The structure of the isometry is similar to that of the heptagon holographic code since both isometries are rank-six tensors, but the isometry in this case is neither a \hyperref[topic:ame]{perfect tensor} nor a block-perfect tensor.
+  The structure of the isometry is similar to that of the heptagon holographic code since both isometries are rank-six tensors, but the isometry in this case is neither a \hyperref[topic:ame]{perfect tensor} nor a \hyperref[code:block_perfect]{planar-perfect tensor}.
 
 
 features:
   code_capacity_threshold:
-    - '\(18.8\%\) under depolarising using tensor-network decoder \cite{arxiv:2009.10329}.'
+    - '\(18.8\%\) under depolarizing noise using tensor-network decoder \cite{arxiv:2009.10329}.'
 
 relations:
   parents:
     - code_id: qubit_stabilizer
     - code_id: holographic_tensor
-      detail: 'The encoding of the six-qubit-tensor holographic code is a holographic tensor network consisting of encoding isometries for the \([[6,1,3]]\) six-qubit stabilizer code.'
+      detail: 'The encoding of the six-qubit-tensor holographic code is a holographic tensor network consisting of the encoding isometry for the \([[6,1,3]]\) six-qubit stabilizer code.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/stabilizer/holographic/holographic_hyperinvariant.yml

@@ -12,7 +12,7 @@ short_name: 'HTN'
 introduced: '\cite{arxiv:2304.02732}'
 
 description: |
-  Holographic tensor-network error-detecting code constructed out of a hyperinvariant tensor network \cite{arxiv:1704.04229}, a MERA-like network admitting a hyperbolic geometry.
+  Holographic tensor-network error-detecting code constructed out of a hyperinvariant tensor network \cite{arxiv:1704.04229}, i.e., a MERA-like network admitting a hyperbolic geometry.
   The network is defined using two layers A and B, with constituent tensors satisfying isometry conditions (a.k.a. multitensor constraints).
 
   This code produces boundary correlation functions that align with those expected from conformal field theory (CFT) boundary states.