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pegasus.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\usepackage{amsfonts,amsmath} % amsmath added here instead to prevent clash with lyx function
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\begin_body
\begin_layout Title
Pegasus: The second connectivity graph for large-scale quantum annealing
hardware
\end_layout
\begin_layout Author
Nike Dattani
\end_layout
\begin_layout Author Email
\end_layout
\begin_layout Affiliation
Harvard-Smithsonian Center for Astrophysics
\end_layout
\begin_layout Affiliation
National Research Council of Canada
\end_layout
\begin_layout Author
\begin_inset VSpace -2mm
\end_inset
Szilard Szalay
\end_layout
\begin_layout Author Email
\end_layout
\begin_layout Affiliation
\begin_inset Note Note
status open
\begin_layout Plain Layout
Strongly Correlated Systems
\begin_inset Quotes eld
\end_inset
Lendület
\begin_inset Quotes erd
\end_inset
Research Group,
\end_layout
\end_inset
Wigner Research Centre for Physics
\begin_inset Note Note
status open
\begin_layout Plain Layout
of the Hungarian Academy of Sciences, 29-33, Konkoly-Thege Miklós út, Budapest,
H-1121, Hungary
\end_layout
\end_inset
\end_layout
\begin_layout Author
\begin_inset VSpace -2mm
\end_inset
Nicholas Chancellor
\end_layout
\begin_layout Author Email
\end_layout
\begin_layout Affiliation
Joint Quantum Centre, Durham University
\end_layout
\begin_layout Abstract
Pegasus is a graph which offers substantially increased connectivity between
the qubits of quantum annealing hardware compared to the graph Chimera.
It is the first fundamental change in the connectivity graph of quantum
annealers built by D-Wave since Chimera was introduced in 2009 and then
used in 2011 for D-Wave's first commercial quantum annealer.
In this article we describe an algorithm which defines the connectivity
of Pegasus and we provide what we believe to be the best way to graphically
visualize Pegasus in order to see which qubits couple to each other.
\begin_inset Note Note
status open
\begin_layout Plain Layout
We provide comparisons between Chimera and Pegasus
\end_layout
\end_inset
As supplemental material, we provide a wide variety of different visualizations
of Pegasus which expose different properties of the graph in different
ways.
We provide an open source code for generating the many depictions of Pegasus
that we show.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide true
sideways false
status open
\begin_layout Plain Layout
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
The Chimera graph,
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Three different depictions of the Chimera graph, with open edges to show
that the pattern repeats.
\begin_inset CommandInset label
LatexCommand label
name "fig:chimera"
\end_inset
\end_layout
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\begin_layout Plain Layout
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wide false
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\begin_inset Caption Standard
\begin_layout Plain Layout
Tilted classic
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Graphics
filename metapostfigs/fig_ChimeraStdRect5x5_crop.pdf
width 33text%
\end_inset
\end_layout
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\begin_inset Float figure
wide false
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status open
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Diamond
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\end_layout
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\begin_inset Graphics
filename metapostfigs/fig_ChimeraDmdRect5x5_crop.pdf
width 33text%
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\end_layout
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\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
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Upper triangle
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\begin_inset Graphics
filename metapostfigs/fig_ChimeraTrpRect5x5_crop.pdf
width 33text%
\end_inset
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\end_inset
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\end_inset
\end_layout
\begin_layout Standard
The 128 qubits of the first commercial quantum annealer (D-Wave One, released
in 2011) were connected by a graph called Chimera (first defined publicly
in 2009
\begin_inset CommandInset citation
LatexCommand citep
key "Neven2009"
literal "false"
\end_inset
), which is rather easy to describe: A 2D array of
\begin_inset Formula $K_{4,4}$
\end_inset
graphs, with one `side' of each
\begin_inset Formula $K_{4,4}$
\end_inset
being connected to the same corresponding side on the
\begin_inset Formula $K_{4,4}$
\end_inset
cells directly above and below it, and the other side being connected to
the same corresponding side on the
\begin_inset Formula $K_{4,4}$
\end_inset
cells to the right and left of it (see Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:chimera"
plural "false"
caps "false"
noprefix "false"
\end_inset
).
The qubits can couple to up to 6 other qubits, since each qubit couples
to 4 qubits within its
\begin_inset Formula $K_{4,4}$
\end_inset
unit cell, and to 2 qubits the
\begin_inset Formula $K_{4,4}$
\end_inset
cells above and below it (or to the left and right of it).
All commercial quantum annealers built to date follow this graph connectivity,
with just larger and larger numbers of
\begin_inset Formula $K_{4,4}$
\end_inset
cells (see Table 1).
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Chimera graphs in all commercial quantum annealers to date.
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Array of
\begin_inset Formula $K_{4,4}$
\end_inset
cells
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Total # of qubits
\end_layout
\end_inset
</cell>
</row>
<row topspace="2mm">
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
D-Wave One
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $4\times4$
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</cell>
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\begin_inset Text
\begin_layout Plain Layout
128
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</cell>
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<row>
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\begin_inset Text
\begin_layout Plain Layout
D-Wave Two
\end_layout
\end_inset
</cell>
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\begin_inset Text
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\begin_inset Formula $8\times8$
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512
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D-Wave 2X
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $12\times12$
\end_inset
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</cell>
<cell alignment="center" valignment="top" usebox="none">
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\begin_layout Plain Layout
1152
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</cell>
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\begin_inset Text
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D-Wave 2000Q
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</cell>
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\begin_inset Formula $16\times16$
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</cell>
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\begin_inset Text
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2048
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\end_inset
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\begin_layout Standard
In 2018, D-Wave announced the construction of a (not yet commercial) quantum
annealer with a greater connectivity than Chimera offers, and a program
(NetworkX) which allows users to generate certain Pegasus graphs.
\begin_inset Note Note
status open
\begin_layout Plain Layout
Our algorithm only generates edges in the inside of Pegasus, and not at
the boundary, where the definition will have to be a bit different.
\end_layout
\end_inset
However, no explicit description of the graph connectivity in Pegasus has
been published yet, so we have had to apply the process of reverse engineering
to determine it, and the following section describes the algorithm we have
established for generating Pegasus.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset VSpace -5mm
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset VSpace -25mm
\end_inset
\end_layout
\begin_layout Section
Algorithm for generating Pegasus
\end_layout
\begin_layout Standard
\begin_inset VSpace -5mm
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\end_layout
\begin_layout Standard
\begin_inset Float figure
wide true
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status open
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\begin_layout Plain Layout
The Pegasus graph.
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "62text%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "black"
backgroundcolor "none"
status open
\begin_layout Plain Layout
\begin_inset Float figure
wide false
sideways false
status open
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A patch of size
\begin_inset Formula $(X,Y,Z)=(2,2,3)$
\end_inset
cropped out of a Pegasus graph.
Grey edges
\begin_inset Newline newline
\end_inset
are part of the three layers of Chimera graphs, while black and blue edges
form the remainder of the Pegasus graph (the latter connecting different
layers).
\begin_inset CommandInset label
LatexCommand label
name "fig:pegasus"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset ERT
status open
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\backslash
setbox1=
\backslash
hbox{
\backslash
includegraphics[height=4cm]{metapostfigs/fig_ChimeraStdRect5x5_crop.pdf}}
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status open
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\backslash
vspace{-6mm}
\backslash
includegraphics[width=
\backslash
textwidth]{metapostfigs/bottomLeftCrop.pdf}
\backslash
llap{
\backslash
hspace{-9.65cm}
\backslash
fboxsep=0mm
\backslash
raisebox{7.8cm}{
\backslash
fcolorbox{gray}{white}{
\backslash
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