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

parent 78e91c37
 \documentclass[11pt]{scrartcl} \documentclass[10pt]{scrartcl} \usepackage[T1]{fontenc} % Bibliography ... ... @@ -13,7 +13,7 @@ \bibliography{bibliography} \usepackage{amssymb,amsmath,amsthm} \usepackage{fullpage} % Math commands \usepackage{mathtools} \DeclareMathOperator{\revop}{R} ... ... @@ -64,36 +64,44 @@ 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. 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 }; \end{tikzpicture}, which helps simplify the problem significantly. We assign each node $i\in V$ a \emph{token}, with a destination $\pi(i)$ given by a permutation $\pi \colon V \to V$. 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. 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. The goal is to route each token to its destination. 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. We are interested in doing this in a \emph{time} that is minimal because, in fact, each reversal needs an amount of time to be implemented% 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 \begin{equation} c(n) \coloneqq \sqrt{{(n+1)^2} - p(n)} / 3\,, \end{equation} 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 $n/3$. Additionally, we allow ourselves to perform reversals $\rev{i,j}$ and $\rev{k,l}$ with \$i
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