topological refs

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

@@ -16,6 +16,9 @@ protection: |
   SPT codes typically do not offer protection against generic errors, but can protect against noise that respects the underlying symmetry.
 
 
+notes:
+  - 'Review on generalized (i.e., non-tensor-product) symmetries \cite{arxiv:2204.03045}.'
+
 relations:
   parents:
     - code_id: topological

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

@@ -93,6 +93,10 @@ protection: |
   This condition implies that any operator supported solely on \(A\) cannot distinguish the global projector from the local one \cite{arxiv:1001.4363,arxiv:2405.19412}.
   \end{defterm}
 
+  A notion of topological order generalizing both the \hyperref[topic:cleaning-lemma]{cleaning lemma} and the \hyperref[topic:tqo]{TQO conditions} is \textit{homogeneous topological order} \cite{arxiv:2009.13551}.
+  Related topological order definitions include equivalence under course-graining (i.e., renormalization group) \cite{arxiv:1406.5090,arxiv:1407.8203}.
+  See \cite[Sec. 4]{arxiv:2009.13551} for a discussion.
+
 
 features:
   rate: 'The logical dimension \(K\) of 2D topological codes described by unitary modular fusion categories depends on the type of manifold \(\Sigma^2\) that is tesselated to form the many-body system.
@@ -109,7 +113,7 @@ features:
 
 
 notes:
-  - 'Ref. \cite[Appx. F]{arxiv:cond-mat/0506438}\cite{doi:10.7907/5NDZ-W890,arxiv:0707.1889,doi:10.1017/9781009212717,arxiv:1508.02595,arxiv:1610.03911,arxiv:2205.05565} for introductions to topological phases.'
+  - 'Ref. \cite[Appx. F]{arxiv:cond-mat/0506438}\cite{doi:10.7907/5NDZ-W890,arxiv:0707.1889,arxiv:1508.02595,arxiv:1610.03911,doi:10.1017/9781316226308,arxiv:2205.05565} for introductions to topological phases.'
   - 'See \href{https://anyonwiki.github.io/}{AnyonWiki} for lists of categories relevant to anyons.'
 
 
@@ -125,6 +129,7 @@ relations:
       detail: 'There exist necessary and sufficient conditions for a family of cluster states to exhibit the TQO-1 property \cite{arxiv:2112.02502}.'
 
 
+
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   # Change log - most recent first

codes/quantum/properties/hamiltonian/commuting_projector.yml

@@ -12,7 +12,7 @@ description: |
   Hamiltonian-based code whose Hamiltonian terms can be expressed as orthogonal projectors (i.e., Hermitian operators with eigenvalues 0 or 1) that commute with each other.
 
 protection: |
-  Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TO conditions, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}.
+  Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the \hyperref[topic:tqo]{TQO conditions}, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}.
   This notion can be extended to semi-hyperbolic manifolds \cite{arxiv:2405.19412}.
 
   2D topological order on qubit manifolds requires weight-four Hamiltonian terms, i.e., it cannot be stabilized via weight-two or weight-three terms on nearly Euclidean geometries of qubits or qutrits \cite{arxiv:quant-ph/0308021,arxiv:1102.0770,arxiv:1803.02213}.
@@ -22,9 +22,11 @@ protection: |
 relations:
   parents:
     - code_id: hamiltonian
+      detail: 'Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the \hyperref[topic:tqo]{TQO conditions}, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}.
+      This notion can be extended to semi-hyperbolic manifolds \cite{arxiv:2405.19412}.'
   cousins:
     - code_id: topological
-      detail: 'Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TO conditions, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}.
+      detail: 'Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the \hyperref[topic:tqo]{TQO conditions}, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}.
       This notion can be extended to semi-hyperbolic manifolds \cite{arxiv:2405.19412}.'
 
 

codes/quantum/properties/hamiltonian/frustration_free.yml

@@ -11,6 +11,9 @@ name: 'Frustration-free Hamiltonian code'
 description: |
   Hamiltonian-based code whose Hamiltonian is frustration free, i.e., whose ground states minimize the energy of each term.
 
+protection: |
+   Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the \textit{local topological quantum order} (LTQO) condition (cf. the \hyperref[topic:tqo]{TQO conditions}), meaning that a notion of a phase can be defined \cite{arxiv:1109.1588,arxiv:2110.11194}.
+
 features:
   encoders:
     - 'Lindbladian-based dissipative encoding can be constructed for a codespace that is the ground-state subspace of a frustration-free Hamiltonian \cite{arxiv:0809.0613,arxiv:1112.4860,arxiv:0803.1447,arxiv:1802.00010}.'
@@ -19,10 +22,14 @@ features:
 relations:
   parents:
     - code_id: hamiltonian
+    - code_id: topological
+      detail: 'Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order condition (cf. the \hyperref[topic:tqo]{TQO conditions}), meaning that a notion of a phase can be defined \cite{arxiv:1109.1588,arxiv:2110.11194}.'
   cousins:
     - code_id: commuting_projector
       detail: 'Frustration-free Hamiltonians can contain non-commuting projectors; an example is the AKLT model \cite{doi:10.1007/978-3-662-06390-3_18}.
       On the other hand, commuting-projector Hamiltonians can be frustrated; an example is the 1D classical Ising model on a circle for odd \(n\).'
+    - code_id: topological
+      detail: 'Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order condition (cf. the \hyperref[topic:tqo]{TQO conditions}), meaning that a notion of a phase can be defined \cite{arxiv:1109.1588,arxiv:2110.11194}.'
 
 
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codes/quantum/properties/hamiltonian/hamiltonian.yml

@@ -31,7 +31,7 @@ description: |
 
   Hamiltonians realizing different phases cannot be adiabatically deformed into one another without a closing of the energy gap between the ground and excited states.
   Such adiabatic deformations naively would be generated by non-local Hamiltonians.
-  However, Hastings and Wen \cite{arXiv:cond-mat/0503554} (see also \cite{arXiv:quant-ph/0601019}) showed that adiabatic evolution can in fact be generated by a quasi-local operator; such evolution is often called \textit{quasi-adiabatic evolution}, \textit{quasi-adiabatic continuation}, or \textit{spectral flow}.
+  However, Hastings and Wen \cite{arXiv:cond-mat/0503554} (see also \cite{arXiv:quant-ph/0601019,arxiv:2205.10460}) showed that adiabatic evolution can in fact be generated by a quasi-local operator; such evolution is often called \textit{quasi-adiabatic evolution}, \textit{quasi-adiabatic continuation}, or \textit{spectral flow}.
 
   The unitary operation generated by a quasi-local Hamiltonian can be simulated by a quantum circuit, with the time of evolution determining the depth of the circuit.
   States in two different phases \textit{cannot} be deformed into one another via such a circuit \cite{arxiv:1004.3835}.
@@ -46,7 +46,8 @@ protection: |
   A no-go theorem states that open-boundary MPS that form a degenerate ground-state space of a gapped local Hamiltonian yield codes with distance that is only constant in the number of qubits \(n\), so MPS excitation ansatze have to be used to achieve a distance scaling nontrivially with \(n\) \cite{arxiv:1902.02115} (see also Ref. \cite{arxiv:1407.3413}).
 
 notes:
-  - 'Reviews of quantum phases of matter \cite{arxiv:1508.02595,doi:10.1017/9781009212717}'
+  - 'Reviews of various quantum phases of matter and many-body systems \cite{doi:10.1007/978-3-662-02520-8,arxiv:1311.2717,arxiv:1508.02595,arxiv:1203.4565,doi:10.1017/9781009212717,doi:10.1007/978-3-030-41265-4,doi:10.1017/CBO9781139020916,doi:10.1017/9781316480649}.'
+  - 'Book on rigorous results on stability of non-topological phases \cite{doi:10.1017/CBO9780511819681}.'
 
 
 relations:

codes/quantum/properties/qecc_finite.yml

@@ -35,7 +35,7 @@ protection: |
   \end{align}
   where the \textit{QEC matrix} elements \(c_{ij}\) are arbitrary complex numbers.
   \end{defterm}
-  The Knill-Laflamme conditions can alternatively be expressed in terms of the \hyperref[topic:complementary-channel]{complementary channel}, or in an information-theoretic way via a data processing inequality \cite{arxiv:quant-ph/9604022}\cite[Eq. (29)]{arxiv:quant-ph/9604034}.
+  The Knill-Laflamme conditions can alternatively be expressed in terms of the \hyperref[topic:complementary-channel]{complementary channel}, or in an information-theoretic way via a data processing inequality \cite{arxiv:quant-ph/9604022,arxiv:quant-ph/9702031}\cite[Eq. (29)]{arxiv:quant-ph/9604034}.
   They have been extended to sequences of multiple errors and rounds of correction \cite{arxiv:2405.17567}.
 
   \begin{defterm}{Degeneracy}

codes/quantum/quantum_into_quantum.yml

@@ -13,10 +13,9 @@ description: |
   Code designed for transmission of quantum and, optionally, classical information through a quantum channel for the purposes of robust storage, communication, or sensing. 
   Transmission can be performed with side information or entanglement.
 
-# relations:
-# parents:
-#   - code_id: eacq
 
+notes:
+  - 'States of block quantum codes can be classified in terms of the complexity of their underlying encoding circuit; see \href{https://complexityzoo.net/Complexity_Zoo_Exhibit}{Complexity Zoo exhibit}.'
 
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codes/quantum/qubits/stabilizer/topological/surface/higher_d/3d_surface.yml

@@ -67,7 +67,8 @@ relations:
     - code_id: higher_dimensional_surface
     - code_id: 3d_stabilizer
     - code_id: topological_abelian
-      detail: 'The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with bosonic charge and loop excitations (BcBl).'
+      detail: 'The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with bosonic charge and loop excitations (BcBl).
+      The welded surface code does not satisfy homogeneous topological order \cite{arxiv:2009.13551}.'
     - code_id: xyz_product
       detail: 'The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes \cite[Exam. A.1]{arxiv:2311.01328}, but done in a different way than the Chamon code; see \cite[Sec. 3.4]{arxiv:2011.09746}.'
   cousins: