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codes/quantum/oscillators/stabilizer/hyperplane/analog_stabilizer.yml

@@ -15,7 +15,7 @@ description: |
   Oscillator-into-oscillator stabilizer code encoding \(k\) logical modes into \(n\) physical modes. An \(((n,k,d))_{\mathbb{R}}\) analog stabilizer code is denoted as \([[n,k,d]]_{\mathbb{R}}\), where \(d\) is the code's distance.
 
   Analog stabilizer codes admit continuous stabilizer group of displacements.  This group can equivalently be defined in terms of its Lie algebra. The codespace is equivalently the common \(0\)-eigenvalue eigenspace of the Lie algebra generators, which are mutually commuting linear combinations of oscillator position and momentum operators called \textit{nullifiers} \cite{arxiv:0903.3233} or \textit{annihilators}.
-  An analog stabilizer code admitting a set a set of nullifiers such that each nullifier consists of either position or momentum operators is called an \textit{analog CSS code}.
+  An analog stabilizer code admitting a set of nullifiers such that each nullifier consists of either position or momentum operators is called an \textit{analog CSS code}.
 
 protection: 'Protect against erasures of at most \(d-1\) modes, or arbitrarily large dispalcements on those modes. If an error operator does not commute with a nullifier, then that error is detectable. Protection of logical modes against small displacements cannot be done using only Gaussian resources \cite{arxiv:0811.3128,arxiv:1810.00047} (see also \cite{arxiv:quant-ph/0204052,arxiv:quant-ph/0204085}). There are no such restrictions for non-Gaussian noise \cite{arxiv:0811.3616}.'