~

codes/quantum/oscillators/stabilizer/oscillator_stabilizer.yml

@@ -17,6 +17,7 @@ description: |
   Bosonic code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting displacement operators.
   Displacements form the stabilizers of the code, and have continuous eigenvalues, in contrast with the discrete set of eigenvalues of qubit stabilizers.
   As a result, exact codewords are non-normalizable, so approximate constructions have to be considered.
+  Stabilizer groups can contain discrete or continuous subgroups and can admit logical qudit and/or oscillator logical subspaces.
 
   Stabilizer codewords encoding a finite-dimensional codespace admit a discrete infinite stabilizer group and encode quantum information in a lattice.
   Such \hyperref[code:qudits_into_oscillators]{qudit-into-oscillator} stabilizer codes are \hyperref[code:gkp]{GKP} and \hyperref[code:multimodegkp]{multimode GKP} codes.

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

@@ -4,36 +4,42 @@
 #######################################################
 
 code_id: translationally_invariant_stabilizer
-# includes both Galois and modular
+# includes Galois, modular, and oscillator
 
 name: 'Lattice stabilizer code'
-introduced: '\cite{arxiv:1101.1962,arxiv:1204.1063,doi:10.7907/GCYW-ZE58}'
+introduced: '\cite{arxiv:1101.1962,arxiv:1204.1063,doi:10.7907/GCYW-ZE58,arxiv:2411.04993}'
 # geometrically local would also cover hyperbolic, Euclidean overlaps with CSS
 
 alternative_names:
    - 'Topological stabilizer code'
 # 1D is not very topological...
 
 description: |
-  A geometrically local modular-qudit or Galois-qudit stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\).
-  On an infinite lattice, its stabilizer group is generated by few-site Pauli operators and their translations, in which case the code is called \textit{translationally invariant stabilizer code}.
+  A geometrically local modular-qudit, Galois-qudit, or bosonic stabilizer code with sites organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\).
+  On an infinite lattice, its stabilizer group is generated by few-site Pauli-type operators and their translations, in which case the code is called \textit{translationally invariant stabilizer code}.
   Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions.
   Lattice defects and boundaries between different codes can also be introduced.
 
+  \subsection{Modular- and Galois-qudit lattice stabilizer 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 lattice 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
+  A single modular- or Galois-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}).
 
   For periodic boundary conditions, this mapping can be thought of as a quantum extension of the \hyperref[topic:Cyclic-to-polynomial-correspondence]{cyclic-to-polynomial correspondence}.
   For open boundary conditions, this mapping extends the mapping used in quantum convolutional codes to multiple spatial dimensions.
   \end{defterm}
 
+  \subsection{Bosonic lattice stabilizer codes}
+
+  Bosonic lattice stabilizer codes can contain discrete or continuous subgroups and can admit logical qudit and/or oscillator logical subspaces.
+  Such codes can realize topological phases of matter that are expected not to be realizable with qudit stabilizer codes \cite{arxiv:2411.04993}. 
 
 
 #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.