refs

codes/quantum/oscillators/coherent_state/cat.yml

@@ -14,6 +14,7 @@ description: |
   Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator which utilizes a constellation of \(q(S+1)\) coherent states distributed equidistantly around a circle in phase space of radius \(\alpha\).
 
   Codewords for a qubit code (\(q=2\)) consist of a coherent state \(|\alpha\rangle\) projected onto a subspace of Fock state number modulo \(2(S+1)\). The logical state \(|\overline{0}\rangle\) is in the \(\{|0\rangle , |2(S+1)\rangle , |4(S+1)\rangle \cdots \}\) Fock-state subspace, while \(|\overline{1}\rangle\) is in the \(\{|(S+1)\rangle, |3(S+1)\rangle , |5(S+1)\rangle , |7(S+1)\rangle \cdots \}\) subspace.
+  These projected coherent states make up generalized cat states \cite{doi:10.1016/0031-8914(74)90215-8,doi:10.1007/BF02581033}. 
 
 protection: 'Due to the spacing between sets of Fock states, the distance between two distinct logical states is \(d=S+1\). Hence, this code is able to detect \(S\)-photon loss error.'
 #In addition, the codewords have \(Z_{S+1}\) rotational symmetry in phase space, and rotation errors up to angle \(\frac{\pi}{S+1}\) are correctable.'

codes/quantum/oscillators/coherent_state/two-legged-cat.yml

@@ -13,7 +13,7 @@ introduced: '\cite{arxiv:quant-ph/9809037}'
 description: |
   Code whose codespace is spanned by two coherent states \(\left|\pm\alpha\right\rangle\) for nonzero complex \(\alpha\).
 
-  An orthonormal basis for the codespace consists of the bosonic \textit{cat states}
+  An orthonormal basis for the codespace consists of the bosonic \textit{cat states} \cite{doi:10.1016/0031-8914(74)90215-8}
   \begin{align}
     |\overline{\pm}\rangle=\frac{\left|\alpha\right\rangle \pm\left|-\alpha\right\rangle }{\sqrt{2\left(1\pm e^{-2|\alpha|^{2}}\right)}}
   \end{align}

codes/quantum/oscillators/fock_state/constant_excitation/two-mode_binomial.yml

@@ -26,7 +26,7 @@ description: |
     |\overline{\mu}\rangle=\frac{1}{2^{J}}\sum_{m=0}^{2J}\left(-1\right)^{\mu m}\sqrt{{2J \choose m}}\left|2J-(S+1)m,(S+1)m\right\rangle~,
   \end{align}
   with spacing \(S\) and dephasing error parameter \(N\) such that \(J = \frac{1}{2}(N+1)(S+1)\) \cite{arxiv:1602.00008}.
-  The \(S=0\) version can be obtained by applying a \(50:50\) beamsplitter to the highest-weight Fock states \(|2J,0\rangle\) and \(|0,2J\rangle\) \cite{arxiv:1512.07605}.
+  The \(S=0\) version can be obtained by applying a \(50:50\) beamsplitter to the highest-weight Fock states \(|2J,0\rangle\) and \(|0,2J\rangle\) \cite{arxiv:1512.07605}; in this case, codewords are two-mode binomial coherent states \cite{doi:10.1088/0305-4470/4/3/009,doi:10.1103/PhysRevA.6.2211,arxiv:2104.10581}.
 
 
 relations:

codes/quantum/oscillators/oscillators.yml

@@ -63,7 +63,7 @@ features:
     Analogues of (non-Pauli) Clifford-group transformations are the \textit{Gaussian unitary transformations} (a.k.a. symplectic, Bogoliubov-Valatin, or linear canonical transformations) \cite{doi:10.1063/1.1665805,arxiv:1110.3234,manual:{Wagner, M. Unitary transformations in solid state physics. Netherlands.}}, which are unitaries generated by quadratic polynomials in positions and momenta.
     The Gaussian unitary transformation group permutes displacement operators amognst themselves, and, up to any phases, is equivalent to the symplectic group \(Sp(2n,\mathbb{R})\).'
     - 'Computing using Gaussian states and Gaussian unitaries only can be efficiently simulated on a classical computer \cite{arxiv:1210.1783,arxiv:1208.3660}; this remains true even if superpositions of Gaussian states are considered \cite{arxiv:2010.14363,arxiv:2403.19059}.
-    A cubic phase gate is required to make a universal gate set on the oscillator \cite{arxiv:quant-ph/0410100}.
+    A cubic phase gate is required to make a universal gate set on the oscillator \cite{arxiv:quant-ph/0410100}; other gates are possible, but higher-order squeezing is not well-defined \cite{doi:10.1103/PhysRevD.29.1107}.
     More generally, controllability has been proven when the normalizable state space is restricted to Shwartz space, the space of states with bounded moments of position and momentum \cite{arxiv:quant-ph/0505063}.'
     - 'Measurements can be performed by homodyne and generalized homodyne measurements \cite{arxiv:quant-ph/0511044}.'
     - 'The number-phase interpretation allows for the mapping of rotor Clifford gates into the oscillator, some of which become non-unitary (e.g., conditional occupation number addition) \cite{arxiv:2311.07679}.'

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -117,7 +117,8 @@ features:
 # Realizing transversal gates outside of the Clifford group requires certain higher-order (i.e., non-quadratic) constraints to be satisfied on the code \cite{arxiv:1209.2426}.
 
   transversal_gates:
-    - 'All CSS codes admit transversal Pauli and CNOT gates \cite{arxiv:quant-ph/9605011}.
+    - 'Transversal Pauli gates since qubit CSS codes are qubit stabilizer codes.'
+    - 'Transversal CNOT gates iff a code is CSS \cite{arxiv:quant-ph/9605011,preset:GottesmanBook}.
     Self-dual CSS codes admit a transversal Hadamard gate.'
     - '\textit{Fold-transversal} \cite{arxiv:2202.06647} Clifford gates are transversal gates combined with qubit permutations.
     Some of these can be obtained from automorphism groups of the underlying classical codes \cite[Thms. 2-3]{arxiv:1302.1035}.'

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

@@ -12,9 +12,12 @@ introduced: '\cite{arxiv:quant-ph/0605138,arxiv:cond-mat/0607736}'
 # not a lattice code because can be defined on general graph
 
 description: |
-  Member of a family of qubit CSS codes defined on a \(D\)-dimensional graph which satisfies two properties: (1) the graph is a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex, and (2) the graph is \(D+1\)-colorable.
+  Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs.
+
+  One family is defined on a \(D\)-dimensional graph which satisfies two properties: (1) the graph is a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex, and (2) the graph is \(D+1\)-colorable.
   Qubits are placed on the \(D\)-simplices and generators are supported on suitable simplices \cite{arxiv:1311.0277,arxiv:1410.0069,doi:10.7907/059V-MG69}.
-  Admissible graphs can be obtained via a fattening procedure \cite{arxiv:cond-mat/0607736}; see also a construction based on the  more general quantum pin codes \cite{arxiv:1906.11394}.
+  Admissible graphs can be obtained via a fattening procedure \cite{arxiv:cond-mat/0607736}.
+  See also a construction based on the more general quantum pin codes \cite{arxiv:1906.11394}.
 
 
 # For 2-dimensional color code, the lattice must be such that it is 3-valent and has 3-colorable faces, such as a honeycomb lattice.