ref

codes/quantum/oscillators/stabilizer/lattice/multimodegkp.yml

@@ -19,6 +19,8 @@ description: |
 
   The space of all single-mode GKP codes is the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed \cite{arXiv:2407.03270}.
 
+  GKP states have negative Wigner functions, but the Zak-Gross Wigner function represents them positively \cite{arxiv:2407.18394}.
+
 protection: 'The level of protection against displacement errors is quantified by the Euclidean code distance \(\Delta=\min_{x\in {\mathcal{L}}^{\perp}\setminus {\mathcal{L}}} \|x\|_2\) \cite{arxiv:2109.14645}.'
 
 features:
@@ -32,7 +34,7 @@ features:
     - 'Dissipative stabilization of finite-energy GKP states using stabilizers conjugated by \textit{cooling} (\cite{arxiv:1310.7596}, Appx. B) or \textit{damping} operator, i.e., a damped exponential of the total occupation number \cite{arxiv:2009.07941,arxiv:2201.12337}.'
 
   general_gates:
-    - 'Gaussian operations and homodyne measurements on GKP states are classically simulable, and there is a sufficient condition for an additional element to achieve universal quantum computation \cite{arxiv:2309.07820}.'
+    - 'Gaussian operations and homodyne measurements on GKP states are classically simulable, and there is a sufficient condition for an additional element to achieve universal quantum computation \cite{arxiv:2309.07820}. There is an algorithm for GKP circuit simulation whose runtime scales with the amount of negativity of the Zak-Gross Wigner function \cite{arxiv:2412.13136}.'
     - 'There is a relation between magic (i.e., how far away a state is from being a stabilizer state) and non-Gaussianity for GKP codewords \cite{arxiv:2109.13018,arxiv:2406.06418}. In particular, implementing a non-Clifford logical gate requires a higher degree of non-Gaussianity than that expressed by ideal non-normalizable GKP states \cite{arxiv:2406.06418}.'
     - 'By applying GKP error correction to Gaussian input states, computational universality can be achieved without additional non-Gaussian elements  \cite{arxiv:1903.00012}. This procedure can be alternatively desscribed as performing heterodyne detection on one half of a GKP encoded Bell pair.'
     - 'Logical shadow tomography protocol \cite{arxiv:2411.00235}.'