Commit d42fbe48 by Aniruddha Bapat

### Structured the project sheet and started writing a warm-up section

parent b26424a5
 ... ... @@ -65,7 +65,7 @@ Recently, we worked on a project for routing quantum information, if you're interested you can read the paper at~\cite{Bapat2020}. Here, we will try to abstract away the underlying quantum operations and operate at a higher level of routing. \paragraph{The Problem:} We will consider the path graph $P_n=(V,E)$ in this project, \paragraph{The problem:} We will consider the path graph $P_n=(V,E)$ in this project, i.e., vertices $1,\dots,n$ connected as \begin{tikzpicture} \graph { 1 -- 2 -- "$\dots$" -- n }; ... ... @@ -74,23 +74,22 @@ We assign each node $i\in V$ a \emph{token}, with a destination $\pi(i)$ given b To get the token at node $i$, we define $t(i)$ as the token at node $i$ and, in an abuse of notation, let $\pi(t(i))$ be the destination of that token. The goal is to route each token to its destination. \paragraph{The Model:} We can route tokens by performing \emph{reversals}, \paragraph{The model:} We can route tokens by performing \emph{reversals}, which exchange locations of tokens. A reversal $\rev{i,j}$, for $i,j \in V$ and $i\leq j$, performs transpositions $\prod_{k=0}^{(j-i)/2} \begin{pmatrix} i+k & j-k \end{pmatrix}$ of the tokens. It is possible to implement any permutation using several reversals, e.g., one could order tokens starting from the token at node 1 by performing $\rev{1,\pi(t(1))}$, $\rev{2,\pi(t(2))}$, etc. Each reversal needs an amount of time to be implemented% \footnote{The time is the number of swap gates that can be performed in the quantum model in the same amount of time.}, namely A reversal $\rev{i,j}$ needs an amount of time% \footnote{The time is the number of swap gates that can be performed in the quantum model in the same amount of time.} c(n) \coloneqq \sqrt{{(n+1)^2} - p(n)} / 3\,, c(n) \coloneqq \sqrt{{(m+1)^2} - p(m)} / 3 where $p(n) \coloneqq n \pmod{2}$ is the parity of $n$. This is a bit of a mouthful to work with, so it may help to simply approximate this by $(n+1)/3$. to be implemented, where $m=|j-i|+1$ is the number of nodes in the reversed segment, and $p(m) \coloneqq m \pmod{2}$ is the parity of $m$. This is a bit of a mouthful to work with, so it may help to simply approximate this by $(m+1)/3$. \paragraph{Our Goals:} We are interested in routing any given permutation $\pi$ in a \emph{time} that is minimal. Note that reversals that do not overlap can be performed simultaneously, i.e. $\rev{i,j}$ and $\rev{k,l}$ with \$i
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