modifications to equations in both RBL sections

k-local-review.tex

@@ -968,7 +968,7 @@ \subsection{Even Sector Selection (Rochetto, Benjamin, Li, 2016)}
 
 \summarysec
 
-Extra binary or ternary variable is added to ensure that the energy of the even parity sector is zero and the energy of the odd sector is higher. Requires introduction of a ternary variable $T \in \{ -1,0,1\}$.
+Extra binary or ternary variable is added to ensure that the energy of the even parity sector is zero and the energy of the odd sector is higher. Requires introduction of a ternary variable $t_a \in \{ -1,0,1\}$.
 
 \costsec
 
@@ -990,7 +990,9 @@ \subsection{Even Sector Selection (Rochetto, Benjamin, Li, 2016)}
 %>0 & \mathrm{otherwise} \end{cases}
 %\end{eqnarray}
 \begin{eqnarray}
--z_1z_2z_3z_4\rightarrow 16\,T+8\,T\sum_{i=1}^4 b_i+4\,\sum_{i=1}^4\sum_{j>i}^4b_i\,b_j+16
+-z_1z_2z_3z_4=-16\,b_1b_2b_3b_4+8\,(b_1b_2b_3+b_1b_2b_4+b_1b_3b_4+b_2b_3b_4)-\nonumber \\
+4\,(b_1b_2+b_1b_3+b_1b_4+b_2b_3+b_2b_4+b_3b_4)+2\,(b_1+b_2+b_3+b_4)-1 \nonumber \\
+\rightarrow 16\,t_a^2+8\,t_a\sum_{i=1}^4 b_i+8\,\sum_{i=1}^4\sum_{j>i}^4b_i\,b_j+16
 \end{eqnarray}
 
 
@@ -1000,7 +1002,7 @@ \subsection{Even Sector Selection (Rochetto, Benjamin, Li, 2016)}
 %>0 & \mathrm{otherwise} \end{cases}
 %\end{eqnarray}
 \begin{eqnarray}
--z_1z_2z_3z_4\rightarrow (4T+\sum_{i=1}^4 z_i)^2
+-z_1z_2z_3z_4\rightarrow 16\,t_a^2+4\,t_a\,\sum_{i=1}^4 z_i+2\,\sum_{i=1}^4\sum_{j>i}^4z_i\,z_j+4
 \end{eqnarray}
 
 \refsec
@@ -2233,7 +2235,7 @@ \subsection{Odd Sector Selection (Rochetto, Benjamin, Li, 2016)}
 
 \summarysec
 
-Extra binary or ternary variable is added to ensure that the energy of the odd parity sector is zero and the energy of the even sector is higher.
+Extra binary variable is added to ensure that the energy of the odd parity sector is zero and the energy of the even sector is higher.
 
 \costsec
 
@@ -2255,7 +2257,9 @@ \subsection{Odd Sector Selection (Rochetto, Benjamin, Li, 2016)}
 %>0 & \mathrm{otherwise} \end{cases}
 %\end{eqnarray}
 \begin{eqnarray}
-z_1z_2z_3z_4\rightarrow (4\,b_a+2\,\sum_{i=1}^4 b_i-6)^2
+z_1z_2z_3z_4=16\,b_1b_2b_3b_4-8\,(b_1b_2b_3+b_1b_2b_4+b_1b_3b_4+b_2b_3b_4)+\nonumber \\
+4\,(b_1b_2+b_1b_3+b_1b_4+b_2b_3+b_2b_4+b_3b_4)-2\,(b_1+b_2+b_3+b_4)+1 \nonumber \\
+\rightarrow 16\,b_a+8\,b_a\,\sum_{i=1}^4 b_i+8\,\sum_{i=1}^4\sum_{i<j}^4 b_i\,b_j+24\,b_a+12\,\sum_{i=1}^4 b_i+32
 \end{eqnarray}
 
 \altformsec
@@ -2264,7 +2268,7 @@ \subsection{Odd Sector Selection (Rochetto, Benjamin, Li, 2016)}
 %>0 & \mathrm{otherwise} \end{cases}
 %\end{eqnarray}
 \begin{eqnarray}
-z_1z_2z_3z_4\rightarrow (2\,z_a+\sum_{i=1}^4z_i)^2
+z_1z_2z_3z_4\rightarrow 2\,z_a\,\sum_{i=1}^4z_i+2\,\sum_{i=1}^4\sum_{i<j}^4 z_i\,z_j+4
 \end{eqnarray}
 
 \refsec