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@@ -1344,6 +1344,82 @@ \subsection{Flag Based SAT Mapping\label{sub:flag_SAT}}
 
 \newpage
 
+
+\subsection{Chained Three Body Parity Operators}
+
+\summarysec
+
+Goal: Terms of the form  $z_{a_{k}}=-z_iz_j$ can be chained together to make large product terms consisting of $z$. 
+%Recall that products of $b$ correspond to parity check or ${\sc xor}$ operations, while products of $b$ correspond to logical ${\sc and}$ operations.
+The penalty term $P(z_{a_k},z_i,z_j)= \mp z_{a_k}z_iz_j$ will lead to an energy penalty unless $z_{a_k}=\pm z_iz_j$. The 3-local term can be made from gadgets.
+
+%This method was originally only used to reproduce the ground state of high locality terms, but states of the "wrong" parity ($q_k=\mp z_iz_j$) will all have the same energy as well, so it reproduces the full spectrum. 
+ 
+
+\costsec
+\begin{itemize}
+\item The best known gadget for a 3-local Ising term uses one auxilliary qubit. Based on this gadget an $n\ge4$ body Ising term can be made using $3+2(n-4)$ auxilliary qubits. 
+\end{itemize}
+
+\prossec
+\begin{itemize}
+\item Natural transmon implementation \cite{Leib2016}.
+\item Chain like structure means that long range connectivity not required.
+\end{itemize}
+
+\conssec
+\begin{itemize}
+\item Reproduces energy spectrum, but not with the same degeneracy as the original Hamiltonian.
+\item Not very symmetric.
+\end{itemize}
+
+\vspace{-1mm}
+
+\examplesec
+\vspace{-4mm}
+
+\begin{align}
+\begin{gathered}
+H_\textrm{5-local} = z_1z_2z_3z_4z_5 =32\,b_1b_2b_3b_4b_5\nonumber \\
+-16\,(b_1b_2b_3b_4+b_1b_2b_3b_5+b_1b_2b_4b_5+b_1b_3b_4b_5+b_2b_3b_4b_5) \nonumber \\
++8\,(b_1b_2b_3+b_1b_2b_4+b_1b_2b_5+b_1b_3b_4+b_1b_3b_5+b_1b_4b_5+b_2b_3b_4+b_2b_3b_4+b_2b_4b_5+b_3b_4b_5)\nonumber \\
+-4\,(b_1b_2+b_1b_3+b_1b_4+b_1b_5+b_2b_3+b_2b_4+b_2b_5+b_3b_4+b_3b_5+b_4b_5)\nonumber \\
++2(b_1+b_2+b_3+b_4+b_5)-1
+\end{gathered}
+\end{align}
+
+The full spectrum of $\pm z_1z_2z_3=\pm(8\,b_1b_2b_3-4\,b_1b_2-4\,b_2b_3+2\,b_1+2\,b_2+2\,b_3-1) $ is reproduced by:
+
+%\begin{equation}
+%P_\pm(z_1,z_2,z_3;\lambda)=\lambda\left(z_1z_2+z_2z_3+z_3z_1+2z_a(z_1+z_2+z_3)\right)\mp (z_1+z_2+z_3 +2z_a).
+%\end{equation}
+
+\begin{align}
+P_\pm(b_1,b_2,b_3)=\left(4\,(b_1b_2+b_2b_3+b_3b_1)+8\,b_a(b_1+b_2+b_3)-12\,b_a-4\,(b_1+b_2+b_3)+3\right)\nonumber \\
+\mp (2\,b_1+2\,b_2+2\,b_3 +4\,b_a-5).
+\end{align}
+
+This can be written more compactly in terms of $z$:
+
+\begin{equation}
+P_\pm(z_1,z_2,z_3)=\left(z_1z_2+z_2z_3+z_3z_1+2z_a(z_1+z_2+z_3)\right)\mp (z_1+z_2+z_3 +2z_a).
+\end{equation}
+
+Using three copies of this 3-local gadget as a building block, and using two additional auxilliary variables, the spectrum of the 5-local term $z_1z_2z_3z_4z_5$ can be reproduced by the following Hamiltonian
+
+\begin{equation}
+H_{2-\rm{local}}=P_-(b_1,b_2,b_{a_1})+P_-(b_{a_1},b_3,b_{a_2})+P_-(b_{a_2},b_4,b_5).
+\end{equation}
+
+\refsec
+\begin{itemize}
+\item Proposal with transmon implementation: \cite{Leib2016}.
+\item Three local gadget independently discovered: \cite{Chancellor2016}.
+\item Extension relating to stabilizers: \cite{Rocchetto2016}.
+\end{itemize}
+
+\newpage
+%=======
 % \subsection{Chained Three Body Parity Operators}
 % 
 % \summarysec
@@ -1416,6 +1492,7 @@ \subsection{Flag Based SAT Mapping\label{sub:flag_SAT}}
 % \end{itemize}
 % 
 % \newpage
+%>>>>>>> 124543eb8af7ee94a551a5f88571879c749f8386
 
 % \subsection{Multibody Operators in PAQC}
 %