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Andrew Childs
reversalsort
Commits
78e91c37
Commit
78e91c37
authored
May 15, 2020
by
Eddie Schoute
Browse files
write problem statement
parent
c04e083e
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docs/problemstatement.tex
View file @
78e91c37
\documentclass
{
article
}
\documentclass
[11pt]
{
scrartcl
}
\usepackage
[T1]
{
fontenc
}
% Bibliography
\usepackage
[
%
...
...
@@ 11,13 +12,88 @@
]
{
biblatex
}
\bibliography
{
bibliography
}
\usepackage
{
amssymb,amsmath,amsthm
}
\title
{
Sorting with Weighted Reversals
}
% Math commands
\usepackage
{
mathtools
}
\DeclareMathOperator
{
\revop
}{
R
}
\DeclarePairedDelimiterXPP\rev
[1]
{
\revop
}{
(
}{
)
}{}{
#1
}
\usepackage
{
tikz
}
\usetikzlibrary
[graphs]
%Link colors
\usepackage
{
color
}
\usepackage
{
xcolor
}
\definecolor
{
darkred
}{
rgb
}{
0.4,0.15,0.15
}
\definecolor
{
darkblue
}{
rgb
}{
0.15,0.15,0.4
}
\definecolor
{
mediumblue
}{
rgb
}{
0,0,0.5
}
\definecolor
{
mycomment
}{
rgb
}{
0.3,0.7,0.8
}
\definecolor
{
mygray
}{
rgb
}{
0.5,0.5,0.5
}
\definecolor
{
lightgray
}{
rgb
}{
0.95,0.95,0.95
}
\definecolor
{
mymauve
}{
rgb
}{
0.58,0,0.82
}
\usepackage
{
hyperref
}
\hypersetup
{
%
breaklinks=true,
% bookmarks=true,
colorlinks,
linkcolor=
{
darkblue
}
,
citecolor=
{
darkblue
}
,
urlcolor=
{
mediumblue
}
,
% pdfauthor = {Aniruddha Bapat, Eddie Schoute, Alexey V. Gorshkov and Andrew M. Childs},
% pdftitle ={Nearly optimal timeindependent reversal of a spin chain},
pdfpagemode=UseOutlines,
pdfborder=
{
0 0 0
}
}
% References, load last.
\usepackage
[nameinlink,capitalise]
{
cleveref
}
\crefname
{
figure
}{
Figure
}{
Figures
}
\crefname
{
equation
}{}{}
% Do not print "Eq."
\Crefname
{
equation
}{
Equation
}{
Equations
}
\author
{
Aniruddha Bapat, Andrew Childs, Alexey Gorshkov, Eddie Schoute
}
\title
{
Routing with Weighted Reversals
}
\begin{document}
\maketitle
Hello World~
\cite
{
Bapat2020
}
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,
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.
The goal is to route each token to its destination.
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
%
\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<j<k<l
$
simultaneously,
thus only incurring the time cost once for two disjoint reversals.
\printbibliography
%
\end{document}
\ No newline at end of file
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