Nick new

k-local-review.tex

@@ -963,6 +963,55 @@ \subsection{Asymmetric Cubic Reduction}
 
 \newpage
 
+
+\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 $t_a \in \{ -1,0,1\}$.
+
+\costsec
+
+1 auxiliary variable for a three or four local term
+
+\prossec
+\begin{itemize}
+\item Very efficient in terms of resources.
+\end{itemize}
+
+\conssec
+\begin{itemize}
+\item Only reproduces the ground state manifold, not higher excited states.
+\end{itemize}
+
+\examplesec
+%\begin{eqnarray}
+%(4\,b_a+2\,\sum_{i=1}^4 b_i-6)^2\begin{cases}=0 & \mathrm{iff}\, \sum_{i=1}^4 b_i \mathrm{\, is\, odd}  \\
+%>0 & \mathrm{otherwise} \end{cases}
+%\end{eqnarray}
+\begin{eqnarray}
+-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}
+
+
+\altformsec
+%\begin{eqnarray}
+%(2\,z_a+\sum_{i=1}^4z_i)^2\begin{cases}=0 & \mathrm{iff}\, \sum_{i=1}^4 z_i=\pm 2 \\
+%>0 & \mathrm{otherwise} \end{cases}
+%\end{eqnarray}
+\begin{eqnarray}
+-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
+\begin{itemize}
+\item Discovered in: \cite{Rocchetto2016}.
+\end{itemize}
+
+\newpage
+
 \section{Methods that introduce auxiliary variables to quadratize a SINGLE positive term (Positive Term Reductions, PTR)}
 
 \subsection{Positive Term Reduction}
@@ -2182,6 +2231,54 @@ \subsection{Lower bounds for SFRs (Anthony, Boros, Crama, Gruber, 2014)}
 
 \newpage
 
+\subsection{Odd Sector Selection (Rochetto, Benjamin, Li, 2016)}
+
+\summarysec
+
+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
+
+1 auxiliary variable for a three or four local term
+
+\prossec
+\begin{itemize}
+\item Very efficient in terms of resources.
+\end{itemize}
+
+\conssec
+\begin{itemize}
+\item Only reproduces the ground state manifold, not higher excited states.
+\end{itemize}
+
+\examplesec
+%\begin{eqnarray}
+%(4\,b_a+2\,\sum_{i=1}^4 b_i-6)^2\begin{cases}=0 & \mathrm{iff}\, \sum_{i=1}^4 b_i \mathrm{\, is\, odd}  \\
+%>0 & \mathrm{otherwise} \end{cases}
+%\end{eqnarray}
+\begin{eqnarray}
+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
+%\begin{eqnarray}
+%(2\,z_a+\sum_{i=1}^4z_i)^2\begin{cases}=0 & \mathrm{iff}\, \sum_{i=1}^4 z_i=\pm 2 \\
+%>0 & \mathrm{otherwise} \end{cases}
+%\end{eqnarray}
+\begin{eqnarray}
+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
+\begin{itemize}
+\item Discovered in: \cite{Rocchetto2016}.
+\end{itemize}
+
+\newpage
+
+
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%5
@@ -2521,6 +2618,82 @@ \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
@@ -2593,6 +2766,7 @@ \subsection{Flag Based SAT Mapping\label{sub:flag_SAT}}
 % \end{itemize}
 % 
 % \newpage
+%>>>>>>> 124543eb8af7ee94a551a5f88571879c749f8386
 
 % \subsection{Multibody Operators in PAQC}
 %