topological refs

codes/quantum/groups/topological/quantum_double.yml

@@ -13,13 +13,22 @@ introduced: '\cite{arxiv:quant-ph/9707021}'
 description: |
   Group-GKP stabilizer code whose codewords realize 2D modular gapped topological order defined by a finite group \(G\).
   The code's generators are few-body operators associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit of dimension \( |G| \) located at each edge of the tesselation).
+  The code dimension depends on the type of boundary used \cite{arxiv:1006.5479}.
 
   The physical Hilbert space has dimension \( |G|^E  \), where \( E \) is the number of  edges in the tessellation. The dimension of the code space is the number of orbits of the conjugation action of \( G \) on \( \text{Hom}(\pi_1(\Sigma),G) \), the set of group homomorphisms from the fundamental group of the surface \( \Sigma \) into the finite group \( G \) \cite{arxiv:1908.02829}. When \( G \) is Abelian, the formula for the dimension simplifies to \( |G|^{2g} \), where \( g \) is the genus of the surface \( \Sigma \).
 
-  The codespace is the ground-state subspace of the quantum double model Hamiltonian.
+  The codespace is the ground-state subspace of the quantum double model Hamiltonian, while local excitations are characterized by anyons.
+  Different types of anyons are labeled by irreducible representations of the group's quantum double algebra, \(D(G)\) (a.k.a. Drinfield center) \cite{arxiv:1006.5479}.
+  Not all isomorphic non-Abelian groups give rise to different quantum doubles \cite{arxiv:math/0605530}.
+
   For non-Abelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model \cite{doi:10.1007/3-540-49208-9_31,arxiv:quant-ph/0306063,doi:10.1017/CBO9780511792908}.
   The fusion space of such non-Abelian anyons has dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.
 
+
+
+
+
+
 protection: |
   Error-correcting properties established in Ref. \cite{arxiv:1908.02829}.
   The code distance is the number of edges in the shortest non contractible cycle in the tesselation or dual tesselation  \cite{arxiv:quant-ph/0110143}.

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

@@ -48,6 +48,8 @@ description: |
   \end{align}
   relates the chiral central charge (modulo 8) to the exchange statistics and quantum dimensions.
 
+  There exist functions of code states that extract the total quantum dimension \(D\) \cite{arxiv:hep-th/0510092,arxiv:cond-mat/0510613}, the topological \(S\)-matrix \cite{arxiv:1111.2342,arxiv:1407.2926}, and the chiral central charge \(c_-\) \cite{arxiv:2110.06932}.
+
 # Frohlich braiding anyons doi:10.1142/S0129055X90000107,
 # Mention [39-41] in arxiv:2211.03798
 
@@ -59,7 +61,7 @@ features:
   \end{align}
   and a generalization of the formula to the non-orientable case can be found in Ref. \cite{arxiv:1612.07792}.'
   encoders:
-    - 'The unitary circuit depth required to initialize in a general topologically ordered state using geometrically local gates on an \(L\times L\) lattice is \(\Omega(L)\) \cite{arxiv:quant-ph/0603121}, irrespective of whether the ground state admits Abelian or non-Abelian anyonic excitations.
+    - 'The unitary circuit depth required to initialize in a general topologically ordered state using geometrically local gates on an \(L\times L\) lattice is \(\Omega(L)\) \cite{arxiv:quant-ph/0603121,arxiv:quant-ph/0603114}, irrespective of whether the ground state admits Abelian or non-Abelian anyonic excitations.
     However, only a finite-depth circuit and one round of measurements is required for non-Abelian topological orders with a Lagrangian subgroup \cite{arxiv:2209.03964}.'
   general_gates:
     - 'Ising anyon braiding and fusion were studied in a phenomenological model that was the first to study error correction with non-Abelian anyons \cite{arxiv:1311.0019}.'

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

@@ -36,7 +36,7 @@ features:
     - 'A depth-\(L^2\) circuit that grows the code out of a small patch on an \(L\times L\) square lattice using CNOT gates (i.e., "local moves") \cite{arxiv:quant-ph/0110143,arxiv:0712.0348}.'
     - 'Teleportation-based state injection into the planar code \cite{arxiv:1202.1016}.'
     - 'Graph-state based adaptive circuit \cite{arxiv:quant-ph/0703143,arxiv:1105.2111}.'
-    - 'For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit \cite{arxiv:2002.00362,arxiv:2110.02020} or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates \cite{arxiv:0712.0348,arxiv:0806.4583,arxiv:1207.0253} (matching lower bounds \cite{arxiv:quant-ph/0603121,arxiv:1810.03912}).'
+    - 'For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit \cite{arxiv:2002.00362,arxiv:2110.02020} or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates \cite{arxiv:0712.0348,arxiv:0806.4583,arxiv:1207.0253} (matching lower bounds \cite{arxiv:quant-ph/0603121,arxiv:quant-ph/0603114,arxiv:1810.03912}).'
     - 'Stabilizer measurement-based circuit of linear depth \cite{arxiv:quant-ph/0110143,arxiv:1404.2495}.'
     - 'Any geometrically local unitary circuit on a lattice \(\Lambda\) that prepares a state whose energy density with respect to the surface code Hamiltonian is \(\epsilon\) must have depth of order \(\Omega(\min(\sqrt{|\Lambda|},1/\epsilon^{\frac{1-\alpha}{2}}))\) for any \(\alpha>0\) \cite{arxiv:2210.06796}.'
     - 'Single-shot state preparation \cite{arxiv:1904.01502}.'

codes/quantum/qudits/stabilizer/topological/quantum_double_abelian.yml

@@ -15,12 +15,15 @@ description: |
   The corresponding anyon theory is defined by an Abelian group.
   All such codes can be realized by a stack of modular-qudit surface codes because all Abelian groups are Kronecker products of cyclic groups.
 
+  There exists an invariant that can be computed to uniquely characterize the anyons of a state in an Abelian quantum-double topological phase \cite{arxiv:1407.2926}.
 
 protection: |
   Error-correcting properties established in Ref. \cite{arxiv:1804.03203} using operator algebra theory.
   Correcting the maximum number of correctable errors is \(NP\)-complete \cite{arxiv:2404.08552}.
 
 features:
+  encoders:
+    - 'Any geometrically local unitary circuit connecting two quantum double models whose groups are not isomorphic must have depth at linear linear in \(n\) \cite{arxiv:1407.2926}.'
   decoders:
     - 'Efficient decoder correcting below the code distance \cite{arxiv:2404.08552}.'