2d + 3d stabilizer

codes/quantum/groups/topological/quantum_double.yml

@@ -44,8 +44,6 @@ relations:
       detail: 'Quantum-double Hamiltonians can be expressed in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see \cite[Sec. 3.3]{arxiv:2111.12096}.'
     - code_id: tqd
       detail: 'The anyon theory corresponding to a quantum-double code is a TQD with trivial cocycle. These models realize local topological order (LTO) \cite{arxiv:2309.13440}.'
-    - code_id: string_net
-      detail: 'String-net model reduces to the quantum-double model for group categories.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits/subsystem/topological/topological_abelian.yml → codes/quantum/properties/block/topological/topological_abelian.yml

@@ -4,8 +4,6 @@
 #######################################################
 
 code_id: topological_abelian
-physical: qudits
-logical: qudits
 
 name: 'Abelian topological code'
 #introduced: ''

codes/quantum/properties/hamiltonian/self_correct.yml

@@ -37,7 +37,7 @@ relations:
     - code_id: symmetry_protected_self_correct
       detail: 'A self-correcting quantum memory does not require symmetry for self correction.'
   cousins:
-    - code_id: translationally_invariant_stabilizer
+    - code_id: 3d_stabilizer
       detail: '3D translationally-invariant qubit stabilizer code families with constant \(k\) support logical string operators and thus cannot be self-correcting \cite{arXiv:1103.1885}. For non-constant \(k\), such families can support at most a logarithmic energy barrier \cite{arXiv:1101.1962}.'
     - code_id: higher_dimensional_surface
       detail: 'The 4D toric code is a self-correcting quantum memory \cite{arXiv:quant-ph/0110143,arXiv:0811.0033}.'

codes/quantum/properties/stabilizer/topological_stabilizer/2d_stabilizer.yml

@@ -0,0 +1,39 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: 2d_stabilizer
+# includes both Galois and modular
+
+name: '2D topological stabilizer code'
+
+description: |
+  Translationally invariant stabilizer code in two spatial dimensions.
+
+  Any prime-qudit code can be converted to several copies of the prime-qudit 2D surface code along with some trivial codes \cite{arXiv:1812.11193}.
+
+features:
+  decoders:
+    - 'Tensor-network based decoder for 2D codes subject to correlated noise \cite{arxiv:1809.10704}.'
+    - 'Standard stabilizer-based error correction can be performed even in the presence of perturbations to the codespace \cite{arxiv:2401.06300}.'
+
+relations:
+  parents:
+    - code_id: translationally_invariant_stabilizer
+  cousins:
+    - code_id: surface
+      detail: 'Translation-invariant 2D qubit topological stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit \cite{arXiv:1103.4606,arXiv:1107.2707,arXiv:1607.01387}.
+      There exists an algorithm with which one can determine the fusion and braiding rules of a 2D translationally invariant qubit code, and decompose the given code into copies of the surface code \cite{arxiv:2312.11170}.'
+    - code_id: quantum_double_abelian
+      detail: 'Translation-invariant 2D prime-qudit topological stabilizer codes are equivalent to several copies of the prime-qudit surface code via a local constant-depth Clifford circuit \cite{arxiv:1812.11193}.'
+    - code_id: holographic
+      detail: '2D topological stabilizer codes admit a bulk-boundary correspondence similar to that of holographic codes, namely, the boundary Hilbert space of the former cannot be realized via local degrees of freedom \cite{arxiv:2312.04617}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-01-27'

codes/quantum/properties/stabilizer/topological_stabilizer/3d_stabilizer.yml

@@ -0,0 +1,36 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: 3d_stabilizer
+# includes both Galois and modular
+
+name: '3D topological stabilizer code'
+
+description: |
+  Translationally invariant stabilizer code in three spatial dimensions.
+
+  Three-dimensional qubit codes can be characterized by four
+  coarse classes \cite{arXiv:1908.08049}:
+
+  1. \textit{Abelian topological phase}: Excitations are mobile in all 3 dimensions, as is typical in a topological code. Such codes are conjectured to be equivalent to a \(\mathbb{Z}_2\) gauge theory, i.e., multiple copies of the 3D surface code or its variant where the charge excitation is a fermion.
+
+  2. \textit{Foliated type-I fracton phase}: Excitations are mobile in less than 3 dimensions, but codes can be grown by \textit{foliation}, i.e., stacking copies of the 2D surface code.
+
+  3. \textit{Fractal type-I fracton phase}: Excitations are mobile in less than 3 dimensions, and codes are not foliated.
+
+  4. \textit{Type-II fracton phase}: Excitations are not mobile in any dimension and there are no string operators.
+
+
+relations:
+  parents:
+    - code_id: translationally_invariant_stabilizer
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-01-27'

codes/quantum/properties/block/topological/fracton.yml → codes/quantum/properties/stabilizer/topological_stabilizer/fracton.yml

@@ -6,18 +6,16 @@
 code_id: fracton
 # includes Galois and modular
 
-name: 'Fracton code'
+name: 'Fracton stabilizer code'
 #introduced: '\cite{arXiv:quant-ph/9705052}'
 
-description: 'A code whose codewords make up the ground-state space of a fracton-phase Hamiltonian.'
+description: 'A 3D translationally invariant stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase.'
 
-protection: ''
-# Can expound on normalizer etc
 
 relations:
   parents:
-    - code_id: qldpc
-      detail: 'Fracton codes admit geometrically local stabilizer generators on a cubic lattice.'
+    - code_id: 3d_stabilizer
+      detail: '3D stabilizer codes can realize fracton orders. Conversely, fracton codes need not be translationally invariant, and can realize multiple phases on one lattice.'
   cousins:
     - code_id: topological
       detail: 'Fracton phases can be understood as topological defect networks, meaning that they can be described in the language of topological quantum field theory with defects \cite{arXiv:2002.05166,arxiv:2112.14717}.'

codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_stabilizer.yml

@@ -0,0 +1,61 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: translationally_invariant_stabilizer
+# includes both Galois and modular
+
+name: 'Lattice stabilizer code'
+introduced: '\cite{arXiv:1101.1962,arXiv:1204.1063,doi:10.7907/GCYW-ZE58}'
+# geometrically local would also cover hyperbolic, Euclidean overlaps with CSS
+
+description: |
+  A geometrically local qubit, modular-qudit, or Galois-qudit stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\).
+  If the stabilizer group is generated by site-local Pauli operators and their translations, then the code is called \textit{translationally invariant stabilizer code}.
+  Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional codes.
+  Lattice defects and boundaries between different codes can also be introduced.
+  It is possible to formulate a thermodynamic limit for lattice codes, with the 1D lattice version reducing to quantum convolutional codes.
+
+  Translationally-invariant prime-qudit (\(q=p\)) stabilizer codes with \(m\) qudits per unit cell have been classified in dimensions \(D\in\{1,2\}\) in the thermodynamic limit, up to equivalence under local constant-depth Clifford circuits.
+  Any 1D (2D) code can be converted to several copies of the 1D repetition code (prime-qudit 2D surface code) along with some trivial codes \cite{arXiv:1607.01387} (\cite{arXiv:1812.11193}).
+  See 3D topological stabilizer code entry for the 3D classification.
+
+  \begin{defterm}{Pauli-to-polynomial mapping}
+  \label{topic:quantum-polynomial-mapping}
+  A single-qudit Pauli operator can be specified by the lattice coordinate of the site and the symplectic vector
+  representation of the Pauli operator within the site.
+  In an extension of the sympletic representation, each lattice coordinate can be represented by a Laurent monomial of \(D\) formal variables. For example, when \(D=2\) and \(m=1\), the product of an \(X\) acting on the qubit at lattice coordinate \((-1,2)\) and a \(Z\) acting on the qubit at \((1,0)\) can be represented by the vector \( (x^{-1} y^2 | x) \). The multiplicative group of finitely supported Pauli operators modulo phase factors on the lattice of dimension \(D\) with \(m\) prime-dimensional qubits per site is isomorphic to the additive group of Laurent polynomial column vectors of length \(2m\) in \(D\) formal variables (see Ref. \cite{arXiv:1607.01387} and Sec. IV of Ref. \cite{arXiv:1812.01625}).
+  \end{defterm}
+
+
+#The code is specified by a stabilizer group that is generated by site-local Pauli operators and is translationally invariant with respect to the lattice.
+#As an example with \(p = 2\) and \(D = 3\), \(P(1, 2, 3; e_1)P(-1, 2, 4; e_{q+2})\) is the Pauli operator that applies \(X\) to
+#  qubit \(1\) of lattice point \((1, 2, 3)\) and applies \(Z\) to qubit \(2\) of lattice point \((-1, 2, 4)\). This operator can be
+#  equivalently expressed by the Laurent polynomial \(x_1x_2^2x_3^3e_1 + x_1^{-1}x_2^2x_3^4e_{q+2}\).
+
+features:
+  decoders:
+    - 'Clustering decoder \cite{doi:10.7907/AHMQ-EG82,arXiv:1112.3252}.'
+    - 'Quantum neural-network (QNN) decoder \cite{arxiv:2401.06300}.'
+
+relations:
+  parents:
+    - code_id: qldpc
+      detail: 'Lattice stabilizer codes are geometrically local.'
+    - code_id: quantum_quasi_cyclic
+      detail: 'Lattice stabilizer codes are invariant under translations by a lattice unit cell.'
+
+
+    # - code_id: qudit_stabilizer
+    #   detail: 'Modular-qudit stabilizer codes can be thought of as lattice stabilizer codes for dimension \(D = 0\), with the lattice consisting of a single site with some number of qudits.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2022-05-15'
+    - user_id: TonyLau
+      date: '2022-04-02'

codes/quantum/qubits/dynamic_gen/da/da.yml

@@ -45,6 +45,8 @@ relations:
       detail: 'One of the instantaneous stabilizer codes of the 2D DA color code are stacks of toric/surface codes'
     - code_id: topological_abelian
       detail: 'Useful measurement sequences of DA Floquet codes can be extracted from topological quantum field theory \cite{arXiv:2307.10353}.'
+    - code_id: translationally_invariant_stabilizer
+      detail: 'DA codes are defined on 2D and 3D lattices, but they are not conventional stabilizer codes in that they use code switching for error correction and gates.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/mbqc/rbh.yml

@@ -41,6 +41,7 @@ features:
 relations:
   parents:
     - code_id: cluster_state
+    - code_id: 3d_stabilizer
     - code_id: walker_wang
       detail: 'The Walker-Wang model code reduces to the RBH cluster-state code when the input category \(\mathcal{C}\) is that of the surface code \cite[Sec. V.A]{arxiv:2011.04693}.'
   cousins:

codes/quantum/qubits/stabilizer/qldpc/sc_qldpc/sc_qldpc.yml

@@ -174,7 +174,7 @@ relations:
   parents:
     - code_id: qubit_stabilizer
     - code_id: translationally_invariant_stabilizer
-      detail: 'Stabilizer generator matrices of SC-QLDPC codes on infinite-length chains or grids define a class of translationally-invariant stabilizer codes.'
+      detail: 'Stabilizer generator matrices of SC-QLDPC codes on infinite-length chains or grids define a class of lattice stabilizer codes.'
   cousins:
     - code_id: sc_ldpc
       detail: 'SC-QLDPC code stabilizer-generator matrices have similar block form as the parity-check matrices of SC-LDPC codes.'

codes/quantum/qubits/stabilizer/quantum_parity.yml

@@ -36,7 +36,6 @@ notes:
 relations:
   parents:
     - code_id: generalized_shor
-    - code_id: translationally_invariant_stabilizer
     - code_id: quantum_concatenated
       detail: 'A QPC is a concatenation of a phase-flip repetition code with a bit-flip repetition code.'
   cousins:
@@ -45,6 +44,8 @@ relations:
     - code_id: majorana_stab
       detail: 'QPCs for \(m_1=m_2\) can be conveniantly expressed in terms of mutually commuting Majorana operators \cite{arxiv:quant-ph/0003137}.'
 
+#     - code_id: translationally_invariant_stabilizer
+# Doesn't seem to be since stab gens can be arbitrarily large
 
 # Begin Entry Meta Information
 _meta:

codes/quantum/qubits/stabilizer/topological/color/3d_color.yml

@@ -25,6 +25,7 @@ features:
 relations:
   parents:
     - code_id: color
+    - code_id: 3d_stabilizer
     - code_id: topological_abelian
     - code_id: quantum_triorthogonal
   cousins:

codes/quantum/qubits/stabilizer/topological/convolutional/quantum_convolutional.yml

@@ -32,7 +32,8 @@ relations:
   parents:
     - code_id: qubit_stabilizer
     - code_id: translationally_invariant_stabilizer
-      detail: 'Quantum convolutional codes are translationally-invariant stabilizer codes on an semi-infinite or infinite lattice in one dimension \cite{arxiv:1305.6973}. Some notions may be extendable to non-stabilizer codes \cite[Sec. 4]{arxiv:quant-ph/0401134}.'
+      detail: 'Quantum convolutional codes are lattice stabilizer codes on an semi-infinite or infinite lattice in one dimension \cite{arxiv:1305.6973}. Some notions may be extendable to non-stabilizer codes \cite[Sec. 4]{arxiv:quant-ph/0401134}.
+      Any prime-qudit code can be converted using a constant-depth Clifford circuit to several copies of the 1D repetition code along with some trivial codes \cite{arXiv:1607.01387}.'
 
 
 # Begin Entry Meta Information