# Precision MARS Mass Reconstruction of Abell 2744:
Synergizing the Largest Strong Lensing and Densest Weak Lensing Datasets from
JWST

Sangjun Cha Department of Astronomy, Yonsei University, 50 Yonsei-ro, Seoul
03722, Korea  Kim HyeongHan Department of Astronomy, Yonsei University, 50
Yonsei-ro, Seoul 03722, Korea  Zachary P. Scofield Department of Astronomy,
Yonsei University, 50 Yonsei-ro, Seoul 03722, Korea  Hyungjin Joo Department
of Astronomy, Yonsei University, 50 Yonsei-ro, Seoul 03722, Korea  M. James
Jee Department of Astronomy, Yonsei University, 50 Yonsei-ro, Seoul 03722,
Korea  Department of Physics and Astronomy, University of California, Davis,
One Shields Avenue, Davis, CA 95616, USA

###### Abstract

We present a new high-resolution free-form mass model of Abell 2744, combining
both weak-lensing (WL) and strong-lensing (SL) datasets from JWST. The SL
dataset comprises 286 multiple images, presenting the most extensive SL
constraint to date for a single cluster. The WL dataset, employing photo-$z$
selection, yields a source density of $\raise
0.81805pt\hbox{$\scriptstyle\sim$}350~{}\rm arcmin^{-2}$, marking the densest
WL constraint ever. The combined mass reconstruction enables the highest-
resolution mass map of Abell 2744 within the $\raise
0.81805pt\hbox{$\scriptstyle\sim$}1.8~{}$Mpc$\times$1.8 Mpc reconstruction
region to date, revealing an isosceles triangular structure with two legs of
$\raise 0.81805pt\hbox{$\scriptstyle\sim$}1$ Mpc and a base of $\raise
0.81805pt\hbox{$\scriptstyle\sim$}0.6$ Mpc. Although our algorithm MAximum-
entropy ReconStruction (MARS) is entirely blind to the cluster galaxy
distribution, the resulting mass reconstruction remarkably well traces the
brightest cluster galaxies with the five strongest mass peaks coinciding with
the five most luminous cluster galaxies within $\lesssim 2\arcsec$. We do not
detect any unusual mass peaks that are not traced by the cluster galaxies,
unlike the findings in previous studies. Our mass model shows the smallest
scatters of SL multiple images in both source ($\raise
0.81805pt\hbox{$\scriptstyle\sim$}0\farcs 05$) and image ($\raise
0.81805pt\hbox{$\scriptstyle\sim$}0\farcs 1$) planes, which are lower than the
previous studies by a factor of $\raise 0.81805pt\hbox{$\scriptstyle\sim$}4$.
Although MARS represents the mass field with an extremely large number of
$\scriptstyle\sim$300,000 free parameters, it converges to a solution within a
few hours thanks to our utilization of the deep learning technique. We make
our mass and magnification maps publicly available.

††journal: ApJ††software: Astropy (Astropy Collaboration et al., 2013, 2018,
2022), Matplotlib (Hunter, 2007), NumPy (Harris et al., 2020), PyMultiNest
(Buchner et al., 2014), PyTorch (Paszke et al., 2019), SciPy (Virtanen et al.,
2020), SEXtractor (Bertin & Arnouts, 1996), WebbPSF (Perrin et al., 2012,
2014)

##  1 Introduction

Gravitational lensing is the most powerful method for reconstructing the mass
distribution of astronomical objects without any dynamical assumptions.
Cluster mass reconstruction based on lensing is typically classified into two
types: weak lensing (WL) and strong lensing (SL). WL uses shape distortions of
background galaxies (e.g., Kaiser & Squires, 1993; Hoekstra et al., 2000; Jee
et al., 2006; Umetsu et al., 2011; Finner et al., 2017; Schrabback et al.,
2021; Kim et al., 2021; Finner et al., 2023), while SL employs multiply-lensed
images (e.g., Broadhurst et al., 2005; Zekser et al., 2006; Richard et al.,
2014; Jauzac et al., 2015; Kawamata et al., 2016; Caminha et al., 2016; Diego
et al., 2016; Limousin et al., 2016; Mahler et al., 2018; Lagattuta et al.,
2019; Zitrin, 2021; Vanzella et al., 2021; Bergamini et al., 2023a).

WL can provide measurements for a wide area, sometimes extending beyond the
virial radius of galaxy clusters. Its limitation is the intrinsic shape
noise/alignment of background galaxies and systematics from instruments. In
general, WL requires averaging of background galaxy shapes to reduce the shape
noise, which inevitably smooths out substructures. The quality and resolution
of WL mass reconstruction depend on the density of the source galaxies and the
reconstruction algorithms (Bartelmann & Schneider, 2001; Schneider, 2006).

SL enables us to obtain more precise and higher-resolution mass maps. Since SL
uses the positions of individual multiply-lensed features, it is immune to
intrinsic source shapes and instrumental systematics, providing better
performance in S/N and resolution. However, since SL features are observed
only in the vicinity of cluster centers, SL mass reconstruction techniques
alone cannot provide direct constraints beyond this so-called SL regime. SL
mass reconstruction techniques quality improves with the number of multiple
images and their redshift information availability.

To attain a cohesive understanding of the cluster mass profile from the
central region to the outskirts, one of the most effective approaches is to
combine the WL and SL signals (e.g., Bradač et al., 2005; Cacciato et al.,
2006; Jee et al., 2007b; Merten et al., 2009; Oguri et al., 2012; Giocoli et
al., 2014; Zitrin et al., 2015). However, mass reconstruction with the optimal
combination of WL and SL signals is challenging for various reasons. For the
mass models based on analytic profiles, degrees of freedom are too low to
properly account for both SL and WL signals simultaneously and sufficiently
with high fidelity (underfitting). Free-form methods may have advantages in
this regard because of their high degrees of freedom. However, free parameters
usually outnumber observables, and thus the solutions are not unique unless
carefully regularized (overfitting). Aside from these technical challenges,
the scarcity of high-quality datasets providing WL and SL has also been an
obstacle. Whereas SL data have been predominantly provided by Hubble Space
Telescope observations, wide-field imaging data for WL measurements have been
obtained mostly through ground-based observations. Thus, the severe disparity
in mass resolution between the two regimes makes it difficult to implement an
optimal smoothing scheme that preserves significant substructures while
suppressing noisy fluctuations.

In this study, we present a new high-resolution free-form mass model of Abell
2744 (A2744 hereafter) at $z=0.308$ from the central region to the outskirts
by combining unprecedentedly large WL and SL datasets, utilizing the recent
high-quality wide-field ($\raise 0.81805pt\hbox{$\scriptstyle\sim$}7\farcm
6\times 7\farcm 6$) JWST images. The SL dataset consists of 286 multiple
images, whereas the WL source density based on photo-z selection reaches
$\raise 0.81805pt\hbox{$\scriptstyle\sim$}350$ arcmin-2, which is the largest
dataset ever used for cluster mass reconstruction.

The galaxy cluster A2744, nicknamed Pandora’s cluster, was introduced as a
peculiar cluster due to its complex and puzzling substructures by Merten et
al. (2011). There is an offset between its intracluster medium (ICM) and mass
(Shan et al., 2010), resembling the “Bullet Cluster” (Clowe et al., 2006).
Also, Merten et al. (2011) reported the presence of a mass clump without a
counterpart in luminous galaxies111The authors referred to the feature as
“ghost.”. The cluster was one of the prime targets in the Hubble Frontier
Fields (HFF) program (Coe et al., 2015; Lotz et al., 2017) and has been a
subject of a number of studies encompassing lensing (e.g., Wang et al., 2015;
Jauzac et al., 2016; Medezinski et al., 2016; Sebesta et al., 2019; Finner &
Jee, 2022; Cha & Jee, 2023), X-ray (e.g., Eckert et al., 2015), and radio
observations (e.g., Venturi et al., 2013; Pearce et al., 2017; Rajpurohit et
al., 2021).

Recently, under the program Ultra-deep NIRCam and NIRSpec ObserVations before
the Epoch of Reionization (UNCOVER) (Bezanson et al., 2022; Weaver et al.,
2023), deep wide-field ($\raise 0.81805pt\hbox{$\scriptstyle\sim$}45$ arcmin2)
JWST imaging observations were carried out. Utilizing the unprecedented depth
and resolution of the new A2744 JWST data, several interesting scientific
results have been published (e.g., Furtak et al., 2023a; Morishita et al.,
2023; Castellano et al., 2023; Vulcani et al., 2023; Atek et al., 2023).
However, in the case of the lens model, only lensing models based on
parametric SL techniques have been presented. They employ the light-trace-mass
(LTM) assumption to model the masses of the cluster galaxies. The center
positions of the cluster-scale dark matter halos are allowed to move within
the prior intervals centered on the BCGs, with $\lesssim 3\arcsec$ and
$\lesssim 5\arcsec-30\arcsec$ for Furtak et al. (2023b) and Bergamini et al.
(2023b), respectively.

This study provides the first non-LTM and profile-independent mass
reconstruction using not only multiple images (SL) but also distortion (WL)
signals, which are densely distributed across the entire A2744 JWST field. We
employ the MAximum-entropy ReconStruction (MARS) algorithm (Cha & Jee, 2022,
2023). MARS is a free-form mass reconstruction method that utilizes cross-
entropy to regularize its solution, resulting in a quasi-unique solution
despite the number of free parameters greatly exceeding that of the
observables. Cha & Jee (2022) tested the fidelity of MARS with the synthetic
cluster data (Meneghetti et al., 2017) and demonstrated that MARS is one of
the best-peforming methods. The test with real HFF cluster data showed that
the image-plane scatter of the multiple images is lower than any of the
publicly available mass models (Cha & Jee, 2023). In the current study, we
extend the previous MARS algorithm of Cha & Jee (2022) to accommodate WL
constraints as well.

This paper is structured as follows. In §2, we introduce the JWST NIRCam
imaging data and reduction steps. In §3, we describe the WL analysis method
and our algorithm for the mass reconstruction combining WL and SL. We show our
results in §4. In §5, we discuss our results and conclude in §6. Unless stated
otherwise, we assume a flat $\Lambda$CDM cosmology with the dimensionless
Hubble constant parameter $h=0.7$ and the matter density
$\Omega_{M}=1-\Omega_{\Lambda}=0.3$. The plate scale at the cluster redshift
($z=0.308$) is $4.536~{}\rm kpc~{}\rm arcsec^{-1}$.

##  2 Data

###  2.1 JWST NIRCam Images

Our WL+SL analysis utilizes the publicly available NIRcam mosaic images of
A2744 processed by the UNCOVER team222https://jwst-uncover.github.io/DR1.html,
who combined the three JWST programs: 1) JWST-DD-ERS-1324 (PI: T. Treu; Treu
et al., 2022), 2) JWST-GO-2561 (PIs: I. Labbe and R. Bezanson; Bezanson et
al., 2022), and 3) JWST-DD-2756 (PI: W. Chen; Chen et al., 2022). We created
color-composite images from the 7 filter imaging data (F115W, F150W, F200W,
F277W, F356W, F410M, and F444W) and identified additional multiple images used
for SL constraints, based on their morphological properties and colors. We use
F200W to measure WL because it provides the optimal scale for the point spread
function (PSF) sampling (Finner et al., 2023). In order to obtain the PSF
model on the mosaic images, it is necessary to derive PSF models for input
frames and stack them with proper rotations and dithers. Thus, we retrieved
the input CAL files from the Mikulski Archive for Space Telescopes
(MAST)333https://archive.stsci.edu/ and found the coordinate transformation
from the detector reference frame to the mosaic reference frame for each input
CAL image. For more details on the mosaic image, we refer readers to Weaver et
al. (2023).

###  2.2 SL Data

Figure 1: Multiple image distributions in the A2744 field. Orange (cyan)
circles indicate the locations of the gold-class (silver-class) multiple
images, including the knots of extended multiple images. The color-composite
image is created using the F444W filter for red, F277W filter for green, and
F115W filter for blue. The notation from Bergamini et al. (2023b) is followed
for labeling the 5 brightest galaxies. The displayed field of view is
$400\arcsec\times 400\arcsec$. Figure 2: Newly identified multiple image
candidates from this study. We find 6 candidate SL systems around G2 (a) and
G1 (b); see Figure 1 for their locations. The cyan circles and numbers
indicate the locations and IDs of the multiple images, respectively. The
color-composite images are created using F200W (red), F150W (green), and F115W
(blue). Table 1: Multiple image catalog of A2744.

ID | R.A. (J2000) | Decl. (J2000) | z1 | Class2 |
---|---|---|---|---|---
1.1 | 39.967047 | -1.5769172 | 0.8041 | gold |
1.2 | 39.976273 | -1.5760558 | 0.8041 | gold |
1.3 | 39.968691 | -1.5766113 | 0.8041 | gold |
2.1 | 39.973825 | -1.584229 | 0.7251 | gold |
2.2 | 39.971003 | -1.5850422 | 0.7251 | gold |
$\cdots$ |  |  |  |  |

Note. — 1Redshifts indicated with an asterisk (*) represent the redshifts
estimated using MARS (see §3.2 for details). 2“Class” is described in §2.2.
3The complete multiple-image catalog is available as online supplementary
material.

We combine SL data from various sources in the literature and identify new
multiple images. We classify the multiple images into gold and silver classes.
Gold-class images are those with spectroscopic redshifts. Silver-class images
do not possess spectroscopic redshifts, but they either have photometric
redshifts or have been identified as multiple images by various studies (e.g.,
Jauzac et al., 2015; Kawamata et al., 2016; Mahler et al., 2018; Bergamini et
al., 2023a; Furtak et al., 2023b; Bergamini et al., 2023b).

Figure 1 shows the multiple image distributions that we compiled for the
current study. In region A, we adopt the catalog of Cha & Jee (2023), who
compiled the SL data from Jauzac et al. (2015), Kawamata et al. (2016), Mahler
et al. (2018), and Bergamini et al. (2023a). All multiple images in Bergamini
et al. (2023a) have spectroscopic redshifts. If a multiple-image system
without a spectroscopic redshift is agreed to be a valid system by the three
papers (Jauzac et al., 2015; Kawamata et al., 2016; Mahler et al., 2018), the
system is classified as a silver class. In regions B and C, we adopt 79
multiple images from Furtak et al. (2023b) and 9 multiple images, which are
knots (distinctive features such as star-forming regions in the extended
multiple images) of system 68 from Bergamini et al. (2023b). If neither
spectroscopic nor photometric redshift information is available, or if there
is a considerable disagreement among the photometric redshifts within the same
system (greater than $0.15\Delta z/(1+\overline{z_{phot}})$, where
$\overline{z_{phot}}$ indicates the mean of the photometric redshifts within
the same system), we treat the redshift of the system as a free parameter. We
also free the redshift when MARS cannot converge multiple images with its
input photometric redshift (see §3.2 for details).

In addition to the compiled catalog above, we have identified 16 new multiple-
image candidates from 6 systems in region B (Figure 2) and all of them are
classified as silver-class images. When multiple photometric redshifts are
available for a system, we adopt the mean value as the system redshift.

A total of 286 images (136 gold and 150 silver images), including 91 knots,
are utilized for mass reconstruction of A2744. We list multiple images in
Table 1, where we follow the numbering scheme of Furtak et al. (2023b) for the
multiple images in regions B and C.

###  2.3 WL Data

####  2.3.1 PSF Modeling

Figure 3: WebbPSF model corrections for ellipticity and size. We present the
result for F200W, where we measure WL signals. Left: Original complex
ellipticity components of stars measured in the A2744 NIRCam images (blue) and
residuals (black) computed by subtracting the WebbPSF model values from the
observed values. The median and standard error for the residuals are provided
in the top right corner of the plot. Right: The residual size $R$ between the
observed stars and the PSFs (blue) and the residual computed after applying a
difference kernel (black). In the top left of the plot, the median residuals
before $\left<R\right>$ and after the kernel $\left<R_{s}\right>$, including
the associated standard errors, are provided.

In order to accurately perform WL analysis, one must model and correct for the
PSF. The relatively small field of view of NIRCam provides only a small number
of stars per pointing. Thus, in some cases producing an empirical PSF model
across the detector based on the limited number of observed stars is not
feasible. Jee et al. (2007a) overcame this problem for HST by building a PSF
library from the observations of dense stellar fields such as globular
clusters and then using the observed stars in a WL field to find the matching
library. However, currently, the amount of NIRCam data for dense stellar
fields is not sufficient to build such an extensive PSF library as in Jee et
al. (2007a). Instead, in the current study, we employ the wavefront sensor
data of JWST, which provide optical path difference (OPD) maps. To utilize
these OPD maps, we use the package WebbPSF (Perrin et al., 2012, 2014).

WebbPSF provides two OPD maps: one from before and the other from after the
input (observation) date. The OPD map closest to the date of observation
cannot always be guaranteed to be the best choice since JWST’s optical
alignments are monitored and adjusted on a regular basis. For the A2744 data,
we verified that both OPD maps give similar results when compared with the PSF
in the input frame. Therefore, we decided to choose the OPD map closest to
each observation date to maintain consistency.

To verify that the PSFs are properly reconstructed by WebbPSF, we first
collect star postage-stamp images for each detector from the A2744 data. Then,
the PSF reconstruction for each star is produced using the selected OPD map
for the given observation date. These star stamps are taken from calibrated,
non-distortion corrected images with the native pixel scale of the NIRCam
short wavelength channel. The residuals between the stars and the model are
computed for the size and ellipticity. These shape parameters are measured
using the following quadrupole moments (Jee et al., 2007a):

$Q_{i,j}=\frac{\int d^{2}\theta
W(\theta)I(\theta)(\theta_{i}-\bar{\theta}_{i})(\theta_{j}-\bar{\theta}_{j})}{\int
d^{2}\theta W(\theta)I(\theta)}\;,\quad i,j\in{1,2}\;.$ (1)

$I(\theta)$ is the pixel intensity at $\theta$, $\bar{\theta}_{i,j}$ is the
center of the star, and $W(\theta)$ is a circular Gaussian weight function
utilized to suppress noise in the outer regions of the PSF. We can then define
the complex ellipticity components ($e_{1}$, $e_{2}$) and size ($R$):

$\displaystyle e_{1}+ie_{2}$
$\displaystyle=\frac{Q_{11}-Q_{22}+2iQ_{12}}{Q_{11}+Q_{22}+2(Q_{11}Q_{22}-Q^{2}_{12})^{\frac{1}{2}}}\;,$
(2) $\displaystyle R$ $\displaystyle=\sqrt{Q_{11}+Q_{22}}\;.$ (3)

For an ellipse with semi-major and -minor axes $a$ and $b$ and position angle
$\phi$ (measured counterclockwise from the reference axis), $e_{1}$ and
$e_{2}$ correspond to

$e_{1}=\frac{a-b}{a+b}\cos(2\phi)\;,\quad e_{2}=\frac{a-b}{a+b}\sin(2\phi)$
(4)

with $(a-b)/(a+b)$ referred to as the ellipticity $e$.

The left panel of Figure 3 shows the ellipticities of the observed stars and
the residuals (observed star ellipticity $-$ model PSF ellipticity). The
detector-induced ellipticity in the observed stars is corrected by the model,
as the residual points are centered around $e_{1},e_{2}=0$. The right panel
shows the residual size $R$ computed by subtracting the observed model size
from the star size. The initial result is given as the blue distribution, with
the median being $\left<R\right>=(8.22\pm 0.06)\times 10^{-2}$. The PSFs
produced by the WebbPSF package are systematically smaller than the observed
stars, and this difference is likely due to WebbPSF not sufficiently taking
into account detector effects such as interpixel capacitance or intrapixel
sensitivity variations. To remedy this issue, a difference kernel is applied
to the PSFs; we used a Gaussian kernel with an empirically determined kernel
size. Represented by the black histogram, the residuals after applying the
kernel are approximately centered at $R=0$, with the median being
$\left<R_{s}\right>=(3.35\pm 8.37)\times 10^{-4}$. Given the small magnitudes
for both the ellipticity and size residuals ($\sim 10^{-4}$), our PSF model
produced by WebbPSF and the difference kernel should be sufficient for WL
analysis.

After ensuring that the chosen model can effectively reproduce the detector
effects seen in NIRCam data, a distortion-corrected and oversampled PSF model
is required for use with the A2744 mosaic image. For each of the eight NIRCam
detectors (NRCA1-4 and NRCB1-4) and for each observation date, one hundred
evenly-spaced (distortion corrected and resampled with a pixel scale of 0.02)
31 pixel $\times$31 pixel PSFs are created using WebbPSF. A principal
component analysis (PCA) is then performed to allow for the production of a 31
pixel $\times$31 pixel PSF at any detector position. Although WebbPSF could be
used to produce a PSF at each galaxy centroid position, the high source count
and the potential need for multiple stacked and rotated PSFs per galaxy make
PCA a more practical choice given the lengthy execution time required by
WebbPSF. A more detailed description of PSF modeling with PCA can be seen in
Jee et al. (2007a). To model a PSF at each galaxy position in the mosaic
image, we identify all contributing input frames, retrieve their PSF models,
and stack the results with proper weights and rotations.

The distribution of stars across the A2744 field allows for PSF modeling with
PCA using the mosaic image in the same manner as in Finner et al. (2023),
which can be compared to the modeling method used in this study. Further
analysis and justification for the choice of PSF modeling strategy can be
found in Appendix A.

####  2.3.2 Ellipticity Measurement & Source Selection

We use a forward-modeling approach to measure the ellipticity of each galaxy
before it is convolved by the JWST PSF. The PSF model predicted at the
location of each source was convolved with an elliptical Gaussian and fit to
the galaxy using the MPFIT optimizer (Markwardt, 2009). We fixed the
background and centroid of a source to the values output by SExtractor (Bertin
& Arnouts, 1996). The free parameters are the position angle ($\phi$), semi-
major and -minor axes ($a$ and $b$), and normalization. Ideally, the average
of the galaxy ellipticity $(e_{1},e_{2})=(e\cos 2\phi,e\sin 2\phi)$ should be
an unbiased estimator of the reduced shear $g$. However, because of a number
of factors, in practice, the estimator is biased. Among them, two outstanding
contributors are “noise bias” and “model bias.” Noise bias occurs due to the
nonlinear relation between pixel noise and parameter noise whereas model bias
is caused by the fact that the galaxy model (in this case, the elliptical
Gaussian) is different from the actual galaxy profile. In addition, the
blending effect is also a significant source of shear bias. Instead of
characterizing these biases individually, we performed WL image simulations
matching the JWST quality and derived multiplicative factors of 1.11 and 1.07
for $g_{1}$ and $g_{2}$, respectively (Jee et al., 2013; Finner et al., 2023).

To select source galaxies, we utilized the photometric redshift catalog
provided by the UNCOVER team444https://jwst-uncover.github.io/DR1.html. As a
conservative measure, we selected sources with a photometric redshift greater
than 0.4 as background objects. Additionally, we imposed shape quality
criteria based on fitting status and recovered shapes. We discarded sources
whose MPFIT STATUS parameter is different from unity because this typically
indicates unstable fitting. Also, the minimum ellipticity measurement error is
set to $\delta e=0.4$. When the object size is reported to be too small, the
source is typically either point source-like or unrealistically compact. We
avoided these cases by imposing that the semi-minor axis is greater than 0.4
pixels. The mean photometric redshift of the sources is 2.5, and the source
density is $\sim 350~{}\rm arcmin^{-2}$, which is the highest among all
existing WL studies (the typical source density in HST-based WL is $\raise
0.81805pt\hbox{$\scriptstyle\sim$}100~{}\rm arcmin^{-2}$).

We note that we did not explicitly mask out the SL areas when selecting WL
sources. In general, because galaxy shapes become curved in the SL regime,
their ellipticities may underrepresent the local reduced shear. We find that
approximately 60 objects are located within $0\farcs 2$ of the multiple
images. Since they comprise only $\raise
0.81805pt\hbox{$\scriptstyle\sim$}0.6$% of the WL sources, we do not think
that the bias caused by these sources is significant. Furthermore, given the
much stronger constraining power from the SL multiple images in the SL regime,
the bias, if any, should be negligible.

##  3 Method

###  3.1 Lensing Theory

In this section, we provide a brief review of lensing theory, covering the
range from the outskirts (WL) to the central regions (SL) of galaxy clusters.
For more details, we refer readers to review papers (e.g., Bartelmann &
Schneider, 2001; Kochanek, 2006; Kneib & Natarajan, 2011; Hoekstra et al.,
2013). In the WL regime, the characteristic scale of the variation in the
distortion of the background galaxy image becomes much smaller than the galaxy
size, and thus the change in the galaxy shape is approximated by the following
matrix $\mathbf{A}$:

$\mathbf{A}=(1-\kappa)\begin{pmatrix}1-g_{1}&-g_{2}\\\
-g_{2}&1+g_{1}\end{pmatrix},$ (5)

where $\kappa$ indicates the convergence and $g_{1(2)}$ denotes the first
(second) component of the reduced shear $g=({g_{1}}^{2}+{g_{2}}^{2})^{1/2}$.
The reduced shear $g$ is computed as $g=\gamma/(1-\kappa)$, where $\gamma$ is
shear.

The convergence $\kappa$ is given by:

$\kappa=\frac{\Sigma}{\Sigma_{c}},$ (6)

where $\Sigma$ ($\Sigma_{c}$) is the (critical) surface mass density.
$\Sigma_{c}$ can be computed as follows:

$\Sigma_{c}=\frac{c^{2}D_{s}}{4\pi GD_{d}D_{ds}},$ (7)

where $c$ is the speed of light, $D_{s(d)}$ denotes the angular diameter
distance to the source (lens), and $D_{ds}$ represents the angular diameter
distance between the lens and the source. The shear $\gamma$ is related to the
convergence $\kappa$ through the following:

$\bm{\gamma}\mathbf{(x)}=\frac{1}{\pi}\int\mathbf{D}(\mathbf{x}-\mathbf{x^{\prime}})\kappa(\mathbf{x^{\prime}})d\mathbf{x^{\prime}},$
(8)

where the kernel $\mathbf{D}$ at the position $(x_{1},x_{2})$ is defined as:

$\mathbf{D}=-\frac{1}{(x_{1}-\mathbf{i}x_{2})^{2}}.$ (9)

In the SL regime, the absolute value of the reduced shear can exceed unity
($|\bm{g}|>1$). In this case, we replace $\bm{g}$ with $1/\bm{g^{*}}$ in
equation 5, where $\bm{g^{*}}$ represents the complex conjugate of $\bm{g}$.

The relation between the observed image position $\bm{\theta}$ and the source
position $\bm{\beta}$ follows the lens equation:

$\bm{\beta}=\bm{\theta}-\bm{\alpha}(\bm{\theta}),$ (10)

where $\bm{\alpha}$ is called the deflection angle. The deflection angle
$\bm{\alpha}$ can be computed through the convolution of the convergence
$\kappa$ or the differentiation of the deflection potential $\Psi$. The MARS
algorithm utilizes the convolution to obtain the deflection angle
$\bm{\alpha}$ as follows:

$\bm{\alpha}(\bm{\theta})=\frac{1}{\pi}\int\kappa(\bm{\theta}^{\prime})\frac{\bm{\theta}-\bm{\theta}^{\prime}}{|\bm{\theta}-\bm{\theta}^{\prime}|^{2}}\bm{d^{2}{\theta}}^{\prime}.$
(11)

Since the mass outside the reconstruction field affects both deflections
(equation 11) and shears (equation 8) within the reconstruction field, we make
the field size of the model 40% larger ($\raise
0.81805pt\hbox{$\scriptstyle\sim$}2.5~{}$Mpc$\times 2.5$ Mpc) than the actual
reconstruction field ($\raise
0.81805pt\hbox{$\scriptstyle\sim$}1.8~{}$Mpc$\times 1.8$ Mpc).

###  3.2 MARS WL + SL Mass Reconstruction Algorithm

We employ the MARS algorithm (Cha & Jee, 2022, 2023) to reconstruct the mass
distribution of A2744. In our previous studies, the application of MARS was
limited to SL mass modeling. In the current study, we revised MARS so that now
it can also utilize WL signals. The new MARS minimizes the following function:

$f=m\chi^{2}_{SL}+w\chi^{2}_{WL}+rR,$ (12)

where $\chi^{2}_{SL}$ and $\chi^{2}_{WL}$ represent the $\chi^{2}$ terms for
SL and WL observables, respectively, and $R$ is the regularization term. The
weight parameters $m,w$, and $r$ determine the relative importance of the SL,
WL, and regularization terms, respectively.

Reduction of $\chi^{2}_{SL}$ decreases the scatters of the multiple images in
the source plane (i.e., positions of delensed multiple images).
$\chi^{2}_{SL}$ is defined as:

$\chi^{2}_{SL}=\sum_{i=1}^{I}\sum_{j=1}^{J}\frac{(\bm{\theta}_{i,j}-\bm{\alpha}_{i,j}(z)-\bm{\beta}_{i})^{2}}{{\sigma_{i}}^{2}},$
(13)

where

$\bm{\beta}_{i}=\frac{1}{J}\sum_{j=1}^{J}(\bm{\theta}_{i,j}-\bm{\alpha}_{i,j}(z)).$
(14)

$I$ is the total number of systems, and $J$ is the number of multiple images
from each system. As is done in Cha & Jee (2023), we treat each “knot”
(distinctive feature such as star-forming region) within a multiple image as
an individual image. We refer readers to Cha & Jee (2023) for details.

We define $\chi^{2}_{WL}$ as follows:

$\chi^{2}_{WL}=\sum_{i=1}^{I}\sum_{j=1}^{2}\frac{({g}_{i,j}-\epsilon_{i,j})^{2}}{\sigma_{s}^{2}({g})+\sigma_{m,i}^{2}},$
(15)

where ${g}_{i,j}(\epsilon_{i,j})$ indicates the $j^{th}$ component of the
expected reduced shear (observed ellipticity) evaluated at the position and
redshift of the $i^{th}$ WL source. $\sigma_{m,i}$ is the measurement error
for the $i$th WL source, and $\sigma_{s}({g})$ is the shape noise per
ellipticity component for the expected reduced shear $g$. In general, the
shape noise decreases as $g$ increases (i.e., every source galaxy becomes
stretched in a nearly identical way regardless of its intrinsic shape when $g$
approaches unity). We use the following function to model the effect:

$\sigma_{s}(g)=(1-{g}^{2})\sigma_{s}(0).$ (16)

where $\sigma_{s}(0)=0.25$ is the intrinsic shape noise in the region where
there is no shape distortion by gravitational lensing.

By adopting the maximum cross entropy, MARS regularizes the mass
reconstruction to prevent overfitting and achieve the smoothest possible
solutions unless the substructures are strongly required by the data. The
regularization term $R$ is given by:

$R=\sum\left(p-\kappa+\kappa\mathrm{ln}\frac{\kappa}{p}\right),$ (17)

where $\kappa$ and $p$ are the convergence and prior, respectively. The prior
is updated for each epoch of minimization by smoothing the convergence map
obtained from the previous iteration using a Gaussian kernel. For more
details, we refer readers to Cha & Jee (2022, 2023). The kernel sizes are
$\sigma=0.6,1.2,2.4$ pixels for the $140\times 140,280\times 280,\rm
and~{}560\times 560$ mass grids, respectively. We double the kernel size when
only WL data are used.

The WL+SL mass reconstruction run is carried out in the following steps:

  1. 1.

Perform an SL-only (i.e., set $w=0$ in equation 12) low-resolution mass
reconstruction with a $140\times 140$ mass grid, which includes 20-cell thick
stripes at the boundaries outside the reconstruction field (the actual mass
reconstruction field has a resolution of $100\times 100$). We start
minimization with a flat $\kappa=0.1$ convergence field. The minimization ends
when the multiple image positions converge in the source plane.

  2. 2.

Add the $\chi^{2}_{WL}$ term to evolve the SL-only solution from step 1 to the
WL + SL one.

  3. 3.

Increase the resolution of the grid from step 2 to $280\times 280$ (by a
factor of two) and restart the WL + SL mass reconstruction.

  4. 4.

Repeat step 3 by further increasing the resolution to $560\times 560$ (now the
marginal stripe is 80-cell thick, and the resolution within the reconstruction
field is $400\times 400$). We end the reconstruction process when the
$\chi^{2}_{WL}$ per WL component reaches unity while the multiple image
scatters in the source plane are consistent with noise.

We treat the redshift of a multiple-image system as a free parameter if
neither its spectroscopic nor photometric redshift is known. In case MARS
cannot converge a multiple-image system with its known photometric redshift,
we also free its redshift. These freed redshifts are constrained along with
the mass reconstruction, and we refer to the values as model redshifts. As
done in Cha & Jee (2023), we set a flat prior for a model redshift with
$z_{model}=[z_{cluster}+0.1,15]$, where $z_{cluster}=0.308$ is the redshift of
A2744.

Including the peripheral grid cells, the final resolution of the resulting
mass map is $560\times 560$, requiring $\sim 300,000$ free parameters. To
minimize our target function $f$ (Equation 12) with this large number of free
parameters, we utilize the Adam (adaptive moment) optimizer (Kingma & Ba,
2014). The Adam optimizer is commonly used in deep learning for optimizing
complex models with an extremely large number of free parameters. Thanks to
its efficiency, MARS converges to a solution within just a few hours.

Although our main result is the one constrained by both WL and SL data, it is
instructive to examine how the solution changes with respect to the full WL+SL
reconstruction when only one of the two datasets is used. Therefore, we repeat
the above mass reconstruction run with one dataset at a time [for the WL-only
(SL-only) reconstruction, we set $m=0~{}(w=0)$ in equation 12] and include the
comparisons in our discussion.

Hereafter, when we present our $\kappa$ field, unless stated otherwise,
$\kappa$ is scaled to $D_{ds}/D_{s}=1$. The corresponding critical surface
mass density (i.e., $\kappa=1$) is $1.777\times 10^{9}M_{\odot}~{}\rm
kpc^{-2}$.

##  4 Result

###  4.1 Projected Mass Distribution

Figure 4: Mass contours of A2744 overlaid on the color-composite images. The
yellow contours indicate the convergence $\kappa$. The upper left (upper
right) panel displays the mass contours obtained from the WL-only (SL-only)
mass map, while the lower panel presents the mass contours derived from the WL
+ SL mass map. In the WL-only mass map, the contours correspond to
$\kappa=[0.15,0.2,0.3,0.4,0.5,0.6,0.7,0.8]$. For the SL-only and WL + SL mass
maps, the contours indicate $\kappa=[0.15,0.2,0.3,0.4,0.6,0.9,1.2,1.8,2.4]$.
To mitigate pixel-scale artifacts, we apply Gaussian smoothing with a kernel
of $\sigma=2\arcsec~{}(\sigma=1\arcsec)$ to the WL-only (WL + SL) mass
contours. In the lower right panel, unsmoothed mass distributions are
displayed as a color map. The color-composite images are the same as shown in
Figure 1. Figure 5: Mass contours overlaid on the cluster galaxy number and
luminosity density maps. The left (right) panel shows the color map indicating
the smoothed (Gaussian kernel with $\sigma\sim 10\arcsec$) cluster galaxy
number density (luminosity). The black solid lines represent the mass contours
derived from our combined mass map, which are the same as shown in Figure 4.
The white dashed line presents the footprint of the JWST observations.

We present our mass reconstruction results in Figure 4. The overall mass
structure of A2744 within the current mass reconstruction field revealed by
the WL+SL datasets is isosceles triangular and characterized by the three main
subclusters: the northern (G3), northwestern (G1+G2), and southern (BCG-N +
BCG-S) mass substructures. The legs of the isosceles triangle (the distances
between BCG-S and G2 and between BCG-S and G3) are $\raise
0.81805pt\hbox{$\scriptstyle\sim$}1$ Mpc whereas the base (the distance
between G3 and G2) is $\raise 0.81805pt\hbox{$\scriptstyle\sim$}0.6$ Mpc. The
northwestern (southern) substructure is further resolved into two smaller
peaks: G1 and G2 (BCG-N and BCG-S). It is remarkable that the five strongest
mass peaks are precisely aligned with the five most luminous cluster galaxies
($\lesssim 2\arcsec$), although MARS is entirely blind to the cluster galaxy
distribution and never uses the LTM assumption.

A comparison of the WL+SL result with the WL-only (upper left) and the SL-only
(upper right) ones delivers a few important takeaway messages. First, the
centroids of the five strongest mass peaks are well-constrained by either
dataset. Although it is not surprising to see these alignments with the SL-
only result, it is unprecedented that the WL data alone can constrain the mass
centroids at this high-significance level (S/N $\gtrsim 9\sigma$). We believe
that this is enabled by the unprecedentedly high WL source density ($\raise
0.81805pt\hbox{$\scriptstyle\sim$}350$ arcmin-2). Second, the SL-only mass
reconstruction provides strong constraints only within the SL regimes, which
are confined to the $r\lesssim 0.3$ Mpc region in the southern subcluster, the
$r\lesssim 0.2$ Mpc region in the northwestern subcluster, and the $r\lesssim
0.1$ Mpc region in the northern subcluster. Outside the SL regime, the mass
density in the SL-only result is determined by the initial prior (i.e., a flat
convergence with $\kappa=0.1$ ). We need the WL dataset to put meaningful
constraints outside the SL regimes. Third, the WL dataset does not detect any
significant mass peaks other than the aforementioned five mass peaks (BCG-N,
BCG-S, G1, G2, and G3). Jauzac et al. (2016) presented WL+SL analysis and
reported identifications of eight significant substructures in A2744. Among
them, four (N, NW, S3, and core in their notation) coincide with our mass
peaks. The other four are not present in our mass reconstruction. Among these
four, one substructure (Wbis in their notation) is located near the field
boundary of the JWST footprint, and thus our result cannot be used to rule out
its presence.

We compare our mass map to the cluster member galaxy distributions in Figure
5. We select cluster member candidates whose photometric redshifts are within
the $0.28<z_{phot}<0.32$ range. We only select the objects that are brighter
than 24 mag. We applied $3\sigma$ clipping and performed a linear fit in the
color-magnitude diagram. The final member selection is made by identifying
galaxies within $1\sigma$ of the best-fit relation. The F277W-F444W color is
used because the combination clearly highlights the red sequence galaxies.
Since we already demonstrated that the five strongest mass peaks precisely
coincide with the five brightest galaxies, good degrees of mass-light
agreements are somewhat expected in this comparison. However, we note that the
number density peaks when smoothed do not always fall exactly on the mass
peaks. Although the cluster member catalog is incomplete, we believe that the
offsets are primarily due to asymmetric galaxy distributions around the
deepest potential wells. We suspect that the ongoing mergers may contribute to
the asymmetry. We also observe that there are some luminosity/number density
clumps, which do not have distinct mass counterparts. Perhaps, they are groups
with low mass-to-light ratios or concentrations of galaxies that are not
gravitationally bound and are only projected along the line-of-sight
direction.

###  4.2 Cumulative Projected Mass

Figure 6: Cumulative projected mass profiles of A2744. We present the radial
cumulative mass profiles of 5 halos (BCG-North, BCG-South, G1, G2, and G3).
Additionally, we include the cumulative profile from the center of the field
of view. The green (cyan) dashed lines indicate the cumulative mass profiles
from the WL NFW profile fitting result without (with) the $M-c$ relation. The
blue dashed lines represent the SL-only mass profiles derived from the current
study. The purple and brown dashed lines represent the mass profiles from
Furtak et al. (2023b) and Bergamini et al. (2023b), respectively. The red
solid line and shaded regions display the profiles and the 1-sigma
uncertainties of our combined (WL+SL) mass model, respectively.

We present the cumulative mass profiles of the five mass peaks in Figure 6.
Also displayed is the total mass profile from the field center (RA=3.568514,
DEC=-30.386321) roughly corresponding to the geometric center of the isosceles
triangle defined by the three mass peaks: G3, G2, and BCG-S. The WL+SL and SL
mass profiles are measured directly from the $\kappa$ map while the WL results
are derived by simultaneously fitting five NFW profiles to the WL data with
and without the mass-concentration ($M-c$) relation of Duffy et al. (2008); we
cannot directly use the convergence map obtained from the WL-only result to
estimate the mass because the $\kappa$ value in the SL regime is significantly
underestimated, which is the combined effect of the mass-sheet degeneracy and
regularization. We refer readers to Appendix B for more details of the NFW
fit.

Overall, the best-fit NFW profiles from WL yield smaller masses near the mass
peak centers and larger masses at large radii ($r\gtrsim 300$ kpc) than the
main WL+SL results obtained directly from the convergence map. This implies
that the densities at the mass peaks are significantly higher than the best-
fit NFW predictions derived from our WL data. We provide two-dimensional
comparisons on the issue in Appendix B. The SL-only results are similar to the
WL+SL ones in the SL regime ($r\lesssim 100$ kpc) but are systematically lower
at larger radii, where the lack of constraints makes the density default to
the initial prior.

The WL+SL result shows that the total projected mass within $r=200$ kpc from
BCG-North is $\sim 1.73\times 10^{14}M_{\odot}$, which is consistent with the
values in Bergamini et al. (2023b) and Furtak et al. (2023b). However, for the
other mass peaks, our mass model provides lower values than the parametric
models. The projected mass within 200 kpc from G1, G2, and G3, are $\sim
1.14\times 10^{14}M_{\odot}$, $\sim 1.15\times 10^{14}M_{\odot}$, and $\sim
8.77\times 10^{13}M_{\odot}$, respectively. The projected mass of G3 has the
largest difference from the value of Bergamini et al. (2023b). This is perhaps
because there is only one multiple-image system around G3 and thus our free-
form model cannot produce a sharp peak there. Except for G3, the mass profiles
from Furtak et al. (2023b) are similar to the profiles derived from our WL-
only best-fit NFW model, which are systematically lower than the ones from our
main (WL+SL) model. Our WL+SL model estimates the total projected mass within
the $r=1$ Mpc aperture from the field center to be $\sim 1.19\times
10^{15}M_{\odot}$.

###  4.3 Magnification

Figure 7: Magnification maps of the reconstructed lens models at the reference
redshift $z_{s}=10$. The left (right) panel shows the magnification map from
the SL-only (SL + WL) mass model. Unlike Figure 4, we do not apply the
smoothing to the magnification map from the combined mass model. Figure 8:
Magnification comparison with the literature. The black contours show the
magnifications from our combined mass map. The red and blue contours represent
the magnifications from Furtak et al. (2023b) and Bergamini et al. (2023b),
respectively. For each magnification, the inner (outer) contour indicates
$|\mu|=4~{}(|\mu|=2)$. The reference redshift is $z_{s}=10$.

Figure 7 displays the magnification maps from the SL-only and SL + WL models.
Similar to the mass map, the magnification map is characterized by the three
main critical curve loops. The overall structures of the critical curves are
in broad agreement with those reported in the literature (e.g., Furtak et al.,
2023b; Bergamini et al., 2023b). However, the resolution limitation and lack
of images around compact halos cause MARS to exhibit some lack of detail near
the cluster member galaxies.

The shapes of the critical curves in the SL-only model are similar to those in
the WL+SL model. These similarities are expected because the critical curves
are primarily constrained by the SL dataset. However, the WL+SL result
provides significantly higher and more detailed magnification in the
outskirts. This is because the lack of constraints makes the SL-only model
predict much lower and simpler mass densities (defaulting to the initial
prior) outside the SL regime.

In Figure 8, we also compare our magnification map with the results from
Furtak et al. (2023b)555https://jwst-uncover.github.io/DR1.html and Bergamini
et al. (2023b)666https://www.fe.infn.it/astro/lensing/, who kindly made their
results publicly available. Although the overall morphology is similar, the
details are significantly different among three magnification maps. The
magnification contours from Furtak et al. (2023b) and Bergamini et al. (2023b)
extend wider, predicting higher magnification values in the outskirts.
Especially, Bergamini et al. (2023b) suggests broad higher magnification
distributions around G3. We attribute this difference to our mass profile
constrained by the WL data decreasing faster than the parametric descriptions
used in Furtak et al. (2023b) and Bergamini et al. (2023b).

##  5 Discussion

###  5.1 Robustness Test of the Mass Model

A robust mass model is expected to accurately reproduce observed SL and WL
features without overfitting. Here we assess the quality of our mass model
using the following four metrics: lens-plane scatters, lens-plane image
reconstructions, per galaxy shear predictability, and tangential shear
profiles. It is important to note that satisfactory performance assessed by
these metrics is only a necessary condition, not a sufficient condition, for a
robust mass model.

####  5.1.1 Lens Plane Scatters of Multiple Images

Figure 9: Lens plane scatters between the observed and predicted locations of
multiple images. $\Delta_{x(y)}$ represents the deviation from the observation
along the x-axis (y-axis). The red (blue) dots indicate the lens plane scatter
distributions derived from the WL + SL (SL-only) mass map. $\Delta_{rms}$ is
the rms value of the total lens plane scatters (see equation 18).

One of the most common metrics used for the evaluation of SL mass models is
the difference between the observed and predicted locations of multiple images
on the image plane. We compute the rms of the position differences between the
observed and predicted multiple images from our mass model using the
following:

$\Delta_{rms}=\sqrt{\frac{1}{M}\sum_{m=1}^{M}|\bm{\theta}_{truth,m}-\bm{\theta}_{model,m}|^{2}},$
(18)

where $M$ represents the total number of multiple images, and
$\bm{\theta}_{truth,m}$ and $\bm{\theta}_{model,m}$ are the locations of the
observed and predicted multiple images for the $m^{th}$ image, respectively.

In Figure 9, we plot the distributions of lens-plane scatter. The rms value
for the SL-only (SL + WL) mass model is $\Delta_{rms}=0\farcs 06~{}(0\farcs
11)$. The SL-only mass model in this study yields an rms value slightly higher
than our previous result (Cha & Jee, 2023, $\Delta_{rms}=0\farcs 05$), where
the mass reconstruction was limited to the main cluster region (within the
single ACS pointing). The scatter increases approximately by a factor of two
when the WL data are included (from $0\farcs 05$ to $0\farcs 11$). This
increase is not surprising because the inclusion of the $\chi^{2}_{WL}$ term
in equation 12 effectively lowers the weight on the $\chi^{2}_{SL}$ term under
the same regularization. Nevertheless, we emphasize that this rms value
$(0\farcs 11)$ is still a factor of four lower than those in other JWST-based
SL-only studies; for instance, the scatters are $\Delta_{rms}=0\farcs 51$ and
$0\farcs 43$ for Furtak et al. (2023b) and Bergamini et al. (2023b),
respectively. Since the JWST image informs us of the location of the multiple
images within the accuracy of a few pixels, our relatively small scatter
($0\farcs 112$) should not be attributed to overfitting.

####  5.1.2 Lens-Plane Image Reconstruction

Figure 10: Lens-plane image reconstructions. The left panels show the observed
multiple images, with red circles indicating the selected images for
reconstruction. The yellow dashed circles represent the locations of multiple
images. Right panels present the reconstructed images in the lens plane, with
dimensions matching those of the left panels. The color-composite images are
the same as shown in Figure 2.

Although the lens-plane scatter metric (§5.1.1) provides a useful statistic to
assess the quality of the lens model, it does not inform us of the robustness
of the lens model on small scales in the neighborhood of the multiple image
positions. In particular, overfit models from free-form approaches fail to
recover lens-plane galaxy morphologies reliably because of high-frequency
noise.

Figure 10 displays the reconstructed multiple images in the lens plane from
our WL+SL mass model. We choose the four systems that exhibit highly distorted
multiple images because they are more sensitive to details in the mass
distribution. In general, it is easier to reconstruct the morphology of the
system with more resolved features since the mass reconstruction utilized
them. Sources 1 (a) and 68 (c) are the systems where we identified 4 knots in
the mass reconstruction while sources 4 (b) and 74 (d) are the ones for which
we only used their centroids. Although the images with multiple knots provide
better reconstructions, the case with a single constraint also yields good
results. We performed these tests with other multiple image systems and
verified that their reconstruction qualities are similar.

Therefore, we conclude that our lens model is stable on small scales in the
vicinity of the multiple image positions even after we require the mass model
to account for all WL features.

####  5.1.3 Shear Predictability per Galaxy

Figure 11: Distributions of per-galaxy $\chi^{2}_{WL}$. The green (blue)
histograms indicate the distributions of the per-galaxy $\chi^{2}_{WL}$
obtained from the WL-only (SL-only) mass map. The red histograms represent the
distributions of the per-galaxy $\chi^{2}_{WL}$ from the WL+SL mass map. The
vertical lines correspond to the mean per-galaxy $\chi^{2}_{WL}$ values from
each mass map, indicated by the numbers at the lower left (matching the color
of each histogram).

Having verified that our combined mass model reproduces the SL features in
terms of the multiple image positions and morphologies in §5.1.1 and §5.1.2,
here we discuss how much the predicted shears at the galaxy positions are
consistent with the observations. Since each galaxy’s ellipticity measurement
contains its intrinsic shape and measurement noise, as well as the shear, it
is important to include them in our judgment of the goodness of the fit. Since
Equation 15 is already designed to accommodate such a need, we decide to
utilize it and adopt the distribution of the normalized squared residual per
galaxy ellipticity component (hereafter we refer to it as per-galaxy
$\chi^{2}_{WL}$) as our metric.

Figure 11 displays the distributions of the per-galaxy $\chi^{2}_{WL}$
measured for the WL-only, SL-only, and WL+SL mass reconstructions. Both WL-
only and WL+SL models provide a mean per-galaxy $\chi^{2}_{WL}$ close to unity
whereas the value is $\raise 0.81805pt\hbox{$\scriptstyle\sim$}50$% higher for
the SL-only model. Since the SL-only models from Furtak et al. (2023b) and
Bergamini et al. (2023b), which use the JWST observations, are publicly
available, we retrieved the models and computed their per-galaxy
$\chi^{2}_{WL}$ with our WL data. We found that the mean per-galaxy
$\chi^{2}_{WL}$ from Furtak et al. (2023b) and Bergamini et al. (2023b) are
$\raise 0.81805pt\hbox{$\scriptstyle\sim$}40$% and $\raise
0.81805pt\hbox{$\scriptstyle\sim$}32$% higher, respectively, than our WL+SL or
WL-only result. This illustrates that our final WL+SL model robustly
reproduces the WL features as well as the SL features. Also, this serves as an
important lesson that a complete mass model within the current JWST A2744
field requires WL constraints.

####  5.1.4 Reduced Tangential Shear Profile

Figure 12: Radial reduced tangential shears comparison. We compute the radial
tangential shears from the center of the field of view. The dots indicate the
radial reduced tangential shears. The errorbars and shaded regions show the
standard uncertainties. The black, red, blue, and green samples are obtained
from the observations, combined model, SL-only model, and WL-only model,
respectively.

The reduced tangential shear is a measure of how much the shapes of background
galaxies are tangentially aligned with respect to a reference point. Here we
adopt the center of the mosaic image as our reference because it maximizes the
radius where the measurement is derived from a complete circle. The reduced
tangential shear $g_{t}$ is evaluated via the following equation:

$g_{t}=-g_{1}\cos 2\phi-g_{2}\sin 2\phi,$ (19)

where $g_{1(2)}$ denotes the first (second) component of the reduced shear,
and $\phi$ represents the position angle of the object measured
counterclockwise from the reference axis. The amplitude of the reduced
tangential shear is given by [$\bar{\kappa}(<r)-\kappa(r)]/[1-\kappa(r)]$ and
is sensitive to the overall shape of the radial mass profile. Therefore, it is
possible that a mass model that performs well in the above per-galaxy
$\chi^{2}_{WL}$ test performs poorly in this tangential shear test, and vice
versa.

Figure 12 displays the comparison of observed tangential shears with the model
prediction. We remind the reader that the radial behavior of the current
tangential shear is different from those of the typical cases in the
literature because of two reasons. First, the reference point is at the center
of the mass reconstruction field, which is near the geometric center of the
three substructures of A2744. Second, the mass distribution of A2744 is by and
large trimodal. The relatively low projected mass density near the field
center makes the amplitude of the tangential shears remain low until the
radius reaches $r\sim 600$ kpc, which is approximately the mean distance from
the field center to the three substructures.

Some noticeable deviations from the observation are present in the SL-only and
WL-only mass models. The SL-only mass model predicts that the reduced
tangential shears are initially more negative at $r\lesssim 400$ kpc, reach
levels similar to the observation at $500~{}\mbox{kpc}\lesssim r\lesssim
700~{}\mbox{kpc}$, and become significantly lower at larger radii ($r\gtrsim
800$ kpc). This pattern is attributed to the absence of the multiple image
constraints at $r\lesssim 400$ kpc or $r\gtrsim 800$ kpc, where our SL-only
mass reconstruction tends to default to the initial prior. The WL-only mass
model predicts the tangential shears well up to $r\lesssim 500$ kpc. However,
beyond this, its prediction is systematically lower than the observation. This
is due to the fact that our maximum entropy regularization makes it difficult
for the WL-only mass reconstruction to reach high $\kappa$ values in the SL
regime since sharper mass peaks lower entropy. The WL+SL mass model provides
the best predictions, which match the observed values across the entire range.

The current tangential shear test illustrates that it is important to
incorporate both SL and WL datasets when a high-fidelity wide-field mass
reconstruction is desired in a massive cluster. As observed, WL-only mass
reconstruction is biased because the WL data alone cannot adequately inform us
of the mass distribution around the extremely high-density regions. On the
other hand, SL-only reconstruction is only robust in the SL regime, where
multiple images are densely distributed. Although this issue can be somewhat
mitigated by assuming that the mass profile follows some analytic
descriptions, in post-merger clusters, such as A2744, the assumption may
diminish our ability to learn how the mass profiles are affected by the sub-
halo collisions.

###  5.2 Comparison with Previous Studies and Merging Scenarios

Figure 13: Mass, uncertainty, and signal-to-noise maps of the WL-only mass
model. The dashed black lines indicate the footprint of the JWST observations.
All three maps are smoothed with a Gaussian kernel of $\sigma=2\arcsec$. (a)
WL-only mass map. The contour labels show the convergence $\kappa$. (b)
Uncertainty of the WL-only mass model derived from 1000 bootstrap
realizations. (c) S/N map of the WL-only mass model. The contour levels are
$3\sigma,~{}6\sigma,~{}9\sigma,~{}12\sigma$, and $15\sigma$. The red arrows
indicate the mass bridges.  Figure 14: WL + SL Mass reconstruction of A2744.
The blue intensity region represents the mass distributions from the WL + SL
mass map. The red intensity region corresponds to X-ray surface brightness
(OBSID: 7915, 8477 and 8557; PI: J. Kempner). The green region displays radio
continuum from radio observations (Venturi et al., 2013; Paul et al., 2019, ;
GMRT data at 325MHz). The color composite image is created using the
Subaru/Suprime-Cam observations, with the z band for red, R band for green,
and B band for blue (Finner & Jee, 2022). The areas enclosed with white solid
lines indicate the four known radio relics (Pearce et al., 2017; Rajpurohit et
al., 2021). The orange solid lines show the expected merger axes from the mass
bridges. The field of view is $13\farcm 46\times 13\farcm 46$.

Due to its rich and puzzling substructures, A2744 was introduced with its
nickname “Pandora’s cluster” (Merten et al., 2011). One of the notable
features highlighted in Merten et al. (2011) was the “ghost” clump with no
apparent correlation with the cluster galaxies. In addition, the authors
reported large offsets between BCGs and mass peaks, which were supported by
Medezinski et al. (2016). However, other studies (e.g., Jauzac et al., 2016;
Furtak et al., 2023b; Bergamini et al., 2023b) based on SL-only or WL + SL
datasets found neither significant offsets between BCGs and mass peaks nor the
ghost clump. A caveat in the latter studies is that the mass models are
reconstructed based on the LTM assumption, and thus the possibility of mass
peaks with considerable departure from the galaxies is not extensively
explored. The current study is the first free-form mass reconstruction of
A2744 based on WL+SL with no LTM assumption. Our JWST result supports neither
the existence of the ghost clump nor the mass-galaxy offsets, although our
analysis is completely blind to the locations of the cluster galaxies in
A2744.

Intracluster stars and globular clusters have been suggested as visible
tracers of dark matter (e.g., Jee, 2010; Alonso Asensio et al., 2020; Yoo et
al., 2022; Montes & Trujillo, 2022; Diego et al., 2023) if their formation
occurs at high redshifts (Ko & Jee, 2018; Joo & Jee, 2023; Werner et al.,
2023). Recently, Harris & Reina-Campos (2023) identified more than 10,000
intracluster globular clusters in A2744 with the same JWST imaging data that
we used in the current study. The spatial distribution of the intracluster
globular clusters in A2744 closely follows the lensing-based mass map
presented in the current and other studies (Furtak et al., 2023b; Bergamini et
al., 2023b).

The scrutiny of our mass map hints at the existence of two “mass bridges”: one
between NW and S, and the other between N and NW. These density enhancements
are mainly constrained by the WL dataset, which provides unprecedentedly high
source density ($\raise 0.81805pt\hbox{$\scriptstyle\sim$}350$ arcmin-2). To
estimate the significance of these mass bridges, we compute an uncertainty of
the WL-only mass map (Figure 13). We reconstruct 1000 mass maps from
bootstraps of the original WL catalog and measure the standard deviation
(Figure 13b), which we adopt as the uncertainty map. The S/N map (Figure 13c)
is obtained by dividing the WL-only mass map by this uncertainty map. The
significance of the contours outlining these mass bridges is $\raise
0.81805pt\hbox{$\scriptstyle\sim$}6.0~{}\sigma$ according to the S/N map.
Since A2744 is one of the most massive clusters known to date, the mass
density within the current JWST field should be significantly positive and
thus the true background level can be estimated only from studies with much
larger fields. Nevertheless, as a conservative measure, we also evaluated the
significance of the mass bridges with respect to the background level
estimated within the reconstruction field. When adopting the outermost contour
level of the mass map (Figure 13a) as the baseline, we find that the
significance decreases to $\raise
0.81805pt\hbox{$\scriptstyle\sim$}4.0~{}\sigma$, which implies that the mass
bridge features are still high in this conservative measure. The mass bridges
may be interpreted as arising from the mergers since numerical simulations
show that mass bridges develop between the two clusters after the core
passage.

Interestingly, the orientations and locations of the two largest radio relics
in A2744 are consistent with the hypothesized merger axes inferred by the two
mass bridges (Figure 14). The brighter relic (R1) is located $\raise
0.81805pt\hbox{$\scriptstyle\sim$}1$ Mpc away from the northern clump and is
on the hypothesized merger axis connecting NW and N. Also, the orientation of
the relic is perpendicular to the merger axis. The fainter relic (R2), $\raise
0.81805pt\hbox{$\scriptstyle\sim$}0.5$ Mpc southeast of the southern (main)
clump, is also on the axis connecting NW and S, and again its orientation is
orthogonal to the axis.

However, despite the above intriguing possibility, complete reconstruction of
the merging scenario of A2744 is still challenging. First of all, A2744
consists of at least five massive substructures, which provide too many
degrees of freedom in plausible merging scenario reconstruction. Furthermore,
the X-ray morphology of A2744 is complex and contains many substructures
significantly dissociated from the galaxy distributions. Although similar
degrees of gas-galaxy offsets have been observed in other binary mergers
(e.g., Clowe et al., 2006; Paraficz et al., 2016), the A2744 complexity cannot
be explained by such binary encounters.

##  6 Conclusion

Leveraging 286 multiple images and a WL source density of $\sim 350~{}\rm
arcmin^{-2}$, we have presented a new WL + SL mass model of A2744 using the
MARS free-form algorithm. For the WL analysis, we carefully modeled the PSF
and measured the ellipticities of the background sources, whose selection is
based on photometric redshifts. For the SL constraints, we compiled multiple
images from the literature and also identified new multiple-image candidates.

Our WL+SL mass reconstruction provides the highest-resolution mass map of
Abell 2744 within the $\raise 0.81805pt\hbox{$\scriptstyle\sim$}1.8$
Mpc$\times 1.8$ Mpc field to date, revealing the giant isosceles triangular
structure characterized by two legs of $\raise
0.81805pt\hbox{$\scriptstyle\sim$}1$ Mpc and a base of $\raise
0.81805pt\hbox{$\scriptstyle\sim$}0.6$ Mpc. While our algorithm MARS remains
entirely unaware of the distribution of cluster galaxies, the resulting mass
reconstruction remarkably traces the brightest cluster galaxies, with the five
strongest mass peaks coinciding with the five most luminous cluster galaxies.
The most remarkable features of our mass reconstruction include the two mass
bridges: one connecting N and NW and the other connecting NW and S. These 4
$\sigma$ features are interpreted as arising from the on-going mergers because
the merger axes defined by the features are consistent with the positions and
morphologies of the two brightest radio relics in A2744.

We support the robustness of our mass model with various tests involving lens-
plane position scatters, lens-plane morphology reconstruction, per-galaxy
$\chi^{2}_{WL}$ statistics, and tangential shear profiles. We demonstrate that
the WL+SL mass model performs well against all these tests while the
performance is not satisfactory when only one of the two (WL and SL) datasets
is used for the model construction. The comparison of the current result with
those in previous studies shows some important differences in mass profiles
and magnifications in both SL and WL regimes. We attribute them to both the
utilization of the WL data outside the SL regime and the free-form MARS
algorithm.

The present study demonstrates that with the advent of JWST era, the cluster
mass reconstruction combining WL and SL has now emerged as a critical and also
practical channel to robustly measure the mass distributions of the massive
clusters from their cores to the outskirts. This will enhance our
comprehension of cluster physics, dark matter properties, and reionization-
epoch galaxies.

We express our gratitude to Kyle Finner for providing the excellent
Subaru/Suprime-Cam color image and contributing to the discussion of the
merging scenario of A2744. Our gratitude goes to Pietro Bergamini for sharing
maps from the lens model. We also extend our thanks to the UNCOVER team for
making their photometric redshift catalog and reduced JWST images publicly
available. This work is based on observations created with NASA/ESA/CSA JWST
and downloaded from the Mikulski Archive for Space Telescope (MAST) at the
Space Telescope Science Institute (STScI). The GLASS-JWST NIRCam observations
analyzed can be accessed via https://doi.org/10.17909/mrt6-wm89 (catalog
doi:10.17909/mrt6-wm89). The JWST UNCOVER NIRCam observations analyzed can be
accessed via https://doi.org/10.17909/zn4s-0243 (catalog
doi:10.17909/zn4s-0243). The JWST DDT program 2756 NIRCam observations
analyzed can be accessed via https://doi.org/10.17909/te6f-cg91 (catalog
doi:10.17909/te6f-cg91). M. J. Jee acknowledges support for the current
research from the National Research Foundation (NRF) of Korea under the
programs 2022R1A2C1003130 and RS-2023-00219959.

##  Appendix A PSF Modeling

Two different PSF modeling methods were tested when analyzing the A2744 data.
The first is described in Section 2.3.1, while the other process closely
follows that from Finner et al. (2023). This second, empirically obtained PSF
model more accurately reproduces the observed PSF ellipticity at the locations
of stars in the field, as seen by the smaller magnitude and more randomly
oriented ellipticity residuals in Figure 15. However, this modeling technique
makes the assumption that the mosaic PSF is continuous despite being the
result of combining numerous exposures from various observation dates.
Additionally, the empirically obtained PSF model is subject to the possibility
of overfitting or underfitting the data based on the choice of polynomial
order.

Figure 15: Whisker plots showing ellipticity direction and magnitude at mosaic
star positions for the two different PSF models. The upper (lower) three
plots, from left to right, show star ellipticities, WebbPSF model (empirically
obtained PSF model) ellipticities, and the residual ellipticities from
subtracting the model ellipticities from the star ellipticities. Figure 16:
Mass contours of A2744 overlaid on the mass maps from the different PSF
models. The upper (lower) row shows the WL-only (WL+SL) mass maps. The black
solid lines represent the mass contours derived from our result mass maps,
which are the same as shown in Figure 4. The white dashed line presents the
footprint of the JWST observations. Figure 17: Cumulative projected mass
profile of A2744 from the center of the field of view (Same as in Figure 6).
The black (red) solid line and shaded regions indicate the cumulative mass
profile and the 1-sigma uncertainties of the combined mass model using WebbPSF
(empirically obtained PSF).

The WebbPSF model generally produces larger magnitude residual ellipticities
at mosaic star positions, which in some cases maintain the observed alignment.
However, given that the A2744 data was collected on different occasions,
modeling the PSF for each input exposure may be more robust in terms of
reproducing PSF characteristics away from observed star locations.
Additionally, we find that there are no significant differences in the
resulting mass maps utilizing the two different PSF models for WL catalogs.
Figure 16 presents WL-only and WL+SL mass maps using WL catalogs derived from
the two PSF models. While there are some small differences in details, the
overall mass distributions are highly consistent. The presence of mass bridges
is also identified when we use the WL catalog from the empirically obtained
PSF model. Furthermore, elongations of mass distributions remain regardless of
which PSF model is used.

Similar to the mass distributions, the total mass measured when using the
different models remains consistent. We plot cumulative projected mass
profiles of WL+SL mass maps from the center of the field of view in Figure 17.
The cumulative masses derived from the two PSF models show agreement within
1-sigma uncertainties. The total projected mass within the $r=1$ Mpc aperture
from the field center to be $\sim 1.19\times 10^{15}M_{\odot}$ and $\sim
1.10\times 10^{15}M_{\odot}$ from WebbPSF and the empirically obtained PSF
model, respectively. Given the assumption of continuity made with the
empirically obtained PSF model, the ambiguous choice of polynomial order, and
the similarity in the final WL results between the two models, the WebbPSF
model was chosen for the final WL analysis.

##  Appendix B Substructure Properties

Table 2: NFW profile fitting results with $M-c$ relation

|  | WL |  |  |  | WL + SL |
---|---|---|---|---|---|---|---
Component | $M_{200}$ | $c_{200}$ | $R_{200}$ |  | $M_{200}$ | $c_{200}$ | $R_{200}$
| ($10^{14}M_{\odot}$) |  | (kpc) |  | ($10^{14}M_{\odot}$) |  | (kpc)
BCG-North | $4.16\pm 0.30$ | $3.313^{+0.018}_{-0.021}$ | $1385^{+35}_{-30}$ |  | $2.084\pm 0.032$ | $3.5104\pm 0.0045$ | $1100.6\pm 5.6$
BCG-South | $2.12^{+0.20}_{-0.26}$ | $3.508\pm 0.033$ | $1105\pm 42$ |  | $1.947\pm 0.029$ | $3.5304\pm 0.0044$ | $1076.2\pm 5.3$
G1 | $0.95^{+0.12}_{-0.14}$ | $3.752^{+0.039}_{-0.046}$ | $846\pm 38$ |  | $1.052\pm 0.022$ | $3.7179\pm 0.0066$ | $876.3\pm 6.2$
G2 | $2.51^{+0.18}_{-0.21}$ | $3.456\pm 0.023$ | $1171\pm 31$ |  | $1.794\pm 0.025$ | $3.5548\pm 0.0042$ | $1047.2\pm 4.9$
G3 | $2.66\pm 0.16$ | $3.440\pm 0.018$ | $1193\pm 25$ |  | $1.392\pm 0.016$ | $3.6313\pm 0.0035$ | $962.3\pm 3.7$
Total1 | $12.40^{+0.29}_{-0.34}$ | - | - |  | $8.268\pm 0.016$ | - | -

Note. — 1The total mass is computed by summation of the virial mass from 5
fitted halos. 2For the WL only result, we use the reduced shear to fit NFW
profiles. In case of the WL + SL, we use the 2-dimensional $\kappa$
distributions to fit NFW profiles (see §B for more details).

Table 3: Comparison of the WL NFW fitting results with the main mass model

|  | WL |  |  |  | WL + SL |
---|---|---|---|---|---|---|---
Component | $r_{s}$ | $c_{200}$ | $M_{200}$ |  | $r_{s}$ | $c_{200}$ | $M_{200}$
| (kpc) |  | ($10^{14}M_{\odot}$) |  | (kpc) |  | ($10^{14}M_{\odot}$)
BCG-North | $521^{+80}_{-100}$ | $2.87^{+0.30}_{-0.38}$ | $4.96^{+0.89}_{-1.2}$ |  | $116.71^{+0.51}_{-0.74}$ | $9.996^{+0.043}_{-0.020}$ | $2.451\pm 0.028$
BCG-South | $349^{+60}_{-100}$ | $3.41^{+0.50}_{-0.66}$ | $2.37^{+0.46}_{-0.77}$ |  | $106.27^{+0.68}_{-1.2}$ | $9.915^{+0.085}_{-0.037}$ | $1.827\pm 0.024$
G1 | $222^{+40}_{-100}$ | $3.62^{+0.88}_{-1.0}$ | $0.63^{+0.11}_{-0.25}$ |  | $104.4^{+1.8}_{-3.7}$ | $8.01^{+0.22}_{-0.14}$ | $0.912^{+0.017}_{-0.021}$
G2 | $637^{+90}_{-100}$ | $2.25^{+0.22}_{-0.27}$ | $4.41^{+0.71}_{-0.80}$ |  | $134.4\pm 2.3$ | $8.13\pm 0.12$ | $2.038\pm 0.024$
G3 | $786\pm 100$ | $1.89^{+0.15}_{-0.24}$ | $4.92\pm 0.60$ |  | $179.8\pm 2.8$ | $6.025\pm 0.083$ | $1.985\pm 0.018$
Total1 | - | - | $17.3\pm 1.0$ |  | - | - | $9.214\pm 0.017$

Note. — 1The total mass is computed by summation of the virial mass from 5
fitted halos. 2For the WL only result, we use the reduced shear to fit NFW
profiles. In case of the WL + SL, we use the 2-dimensional $\kappa$
distributions to fit NFW profiles (see §B for more details).

Figure 18: WL-only NFW fitted results with $M-c$ relation of Duffy et al.
(2008). Unit of mass and virial radius are $10^{14}M_{\odot}$ and kpc,
respectively. Figure 19: WL + SL NFW fitted results with $M-c$ relation of
Duffy et al. (2008). Unit of mass and virial radius are same as Figure 18.
Figure 20: WL-only NFW fitted results without $M-c$ relation. Unit of mass and
scale radius are $10^{14}M_{\odot}$ and kpc, respectively. Figure 21: WL + SL
NFW fitted results without $M-c$ relation. Unit of mass and scale radius are
same as Figure 20. Figure 22: Residual mass maps between the NFW fitting and
the main WL+SL mass maps. (a) Two-dimensional mass models created from the
best-fit NFW parameters. (b) Subtraction of the main WL+SL mass map from the
models shown in (a). The first and second (third and fourth) rows display the
results obtained with (without) the $M-c$ relation of Duffy et al. (2008). The
black dashed lines indicate the footprints of the JWST observations. The black
solid lines indicate regions where the residual is zero. See text for
discussions.

The cumulative mass profiles that we present in §4.2 cannot be used to isolate
the mass properties of individual substructures because the measured
convergence is the result of the superposition of multiple halo profiles.
Here, we determine the substructure properties by simultaneously fitting five
NFW profiles. A2744 is a complex system with a number of merging features
including sophisticated X-ray morphologies and radio relics. Hence, this NFW
fitting is not expected to produce results devoid of bias, as mergers are
likely to cause substantial deviations from NFW descriptions in cluster mass
profiles (e.g., Lee et al., 2023). Nevertheless, simultaneous multi-halo
fitting can significantly diminish the impact of neighboring substructures.

We perform simultaneous NFW fitting in two approaches. The first method is to
fit five NFW profiles to our WL shape catalogs. In the second approach, we fit
five NFW profiles to the convergence map obtained from the WL+SL mass
reconstruction. In Tables 2 and 3, we display the results with and without the
$M-c$ relation of Duffy et al. (2008), respectively. We also present the
posterior distributions of all free parameters in Figures 18, 19, 20, and 21.
When we assume the $M-c$ relation of Duffy et al. (2008), both the WL and
WL+SL results show similar concentration values. However, in the results
obtained without the $M-c$ relation, the WL+SL model gives considerably larger
concentration values, which are attributed to the availability of the SL
constraints near the mass peaks. The merging cluster simulations of Lee et al.
(2023) demonstrate that the concentrations tend to increase because mass
infalls occur in post-collision systems. However, since WL does not densely
sample the signals near the halo centers, the concentrations are biased low,
which in turn leads to an overestimation of the cluster mass in WL analysis.
The extent of this overestimation depends on the state of the merger, with the
factor potentially reaching as high as $2-3$. The systematic differences in
both concentration and mass shown in Table 3 are consistent with the
predictions of Lee et al. (2023).

In order to visualize the two-dimensional difference between the NFW fitting
results and the main WL+SL mass map, we present the residual (subtraction of
the WL+SL mass map from the NFW fitting result) mass maps in Figure 22.
Although the details differ between the two results with and without the $M-c$
relations, similar trends are present in the residual maps. The shear-based
models produced with the best-fit NFW parameters (first and third rows) yield
lower mass densities around the five mass peaks while they predict higher
values in the outskirts. This illustrates that if we attempt to estimate the
total mass of A2744 using the extrapolation of the shear-based NFW fitting
results, the procedure will result in severe overestimation. The residuals
created with the convergence-based models (second and fourth rows) are
somewhat more complex and show larger azimuthal variations.

## References

  * Alonso Asensio et al. (2020) Alonso Asensio, I., Dalla Vecchia, C., Bahé, Y. M., Barnes, D. J., & Kay, S. T. 2020, MNRAS, 494, 1859
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167
  * Atek et al. (2023) Atek, H., Chemerynska, I., Wang, B., et al. 2023, MNRAS, 524, 5486
  * Bartelmann & Schneider (2001) Bartelmann, M., & Schneider, P. 2001, Phys. Rep., 340, 291
  * Bergamini et al. (2023a) Bergamini, P., Acebron, A., Grillo, C., et al. 2023a, A&A, 670, A60
  * Bergamini et al. (2023b) —. 2023b, ApJ, 952, 84
  * Bertin & Arnouts (1996) Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393
  * Bezanson et al. (2022) Bezanson, R., Labbe, I., Whitaker, K. E., et al. 2022, arXiv e-prints, arXiv:2212.04026
  * Bradač et al. (2005) Bradač, M., Schneider, P., Lombardi, M., & Erben, T. 2005, A&A, 437, 39
  * Broadhurst et al. (2005) Broadhurst, T., Benítez, N., Coe, D., et al. 2005, ApJ, 621, 53
  * Buchner et al. (2014) Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, A&A, 564, A125
  * Cacciato et al. (2006) Cacciato, M., Bartelmann, M., Meneghetti, M., & Moscardini, L. 2006, A&A, 458, 349
  * Caminha et al. (2016) Caminha, G. B., Grillo, C., Rosati, P., et al. 2016, A&A, 587, A80
  * Castellano et al. (2023) Castellano, M., Fontana, A., Treu, T., et al. 2023, ApJ, 948, L14
  * Cha & Jee (2022) Cha, S., & Jee, M. J. 2022, ApJ, 931, 127
  * Cha & Jee (2023) —. 2023, ApJ, 951, 140
  * Chen et al. (2022) Chen, W., Kelly, P., Castellano, M., et al. 2022, Transient Name Server AstroNote, 257, 1
  * Clowe et al. (2006) Clowe, D., Bradač, M., Gonzalez, A. H., et al. 2006, ApJ, 648, L109
  * Coe et al. (2015) Coe, D., Bradley, L., & Zitrin, A. 2015, ApJ, 800, 84
  * Diego et al. (2016) Diego, J. M., Broadhurst, T., Wong, J., et al. 2016, MNRAS, 459, 3447
  * Diego et al. (2023) Diego, J. M., Pascale, M., Frye, B., et al. 2023, arXiv e-prints, arXiv:2301.03629
  * Duffy et al. (2008) Duffy, A. R., Schaye, J., Kay, S. T., & Dalla Vecchia, C. 2008, MNRAS, 390, L64
  * Eckert et al. (2015) Eckert, D., Jauzac, M., Shan, H., et al. 2015, Nature, 528, 105
  * Finner et al. (2023) Finner, K., Faisst, A., Chary, R.-R., & Jee, M. J. 2023, ApJ, 953, 102
  * Finner & Jee (2022) Finner, K., & Jee, M. 2022, in American Astronomical Society Meeting Abstracts, Vol. 54, American Astronomical Society Meeting #240, 214.05D
  * Finner et al. (2017) Finner, K., Jee, M. J., Golovich, N., et al. 2017, ApJ, 851, 46
  * Furtak et al. (2023a) Furtak, L. J., Zitrin, A., Plat, A., et al. 2023a, ApJ, 952, 142
  * Furtak et al. (2023b) Furtak, L. J., Zitrin, A., Weaver, J. R., et al. 2023b, MNRAS, 523, 4568
  * Giocoli et al. (2014) Giocoli, C., Meneghetti, M., Metcalf, R. B., Ettori, S., & Moscardini, L. 2014, MNRAS, 440, 1899
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357
  * Harris & Reina-Campos (2023) Harris, W. E., & Reina-Campos, M. 2023, arXiv e-prints, arXiv:2307.14412
  * Hoekstra et al. (2013) Hoekstra, H., Bartelmann, M., Dahle, H., et al. 2013, Space Sci. Rev., 177, 75
  * Hoekstra et al. (2000) Hoekstra, H., Franx, M., & Kuijken, K. 2000, ApJ, 532, 88
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90
  * Jauzac et al. (2015) Jauzac, M., Richard, J., Jullo, E., et al. 2015, MNRAS, 452, 1437
  * Jauzac et al. (2016) Jauzac, M., Eckert, D., Schwinn, J., et al. 2016, MNRAS, 463, 3876
  * Jee (2010) Jee, M. J. 2010, ApJ, 717, 420
  * Jee et al. (2007a) Jee, M. J., Blakeslee, J. P., Sirianni, M., et al. 2007a, PASP, 119, 1403
  * Jee et al. (2013) Jee, M. J., Tyson, J. A., Schneider, M. D., et al. 2013, ApJ, 765, 74
  * Jee et al. (2006) Jee, M. J., White, R. L., Ford, H. C., et al. 2006, ApJ, 642, 720
  * Jee et al. (2007b) Jee, M. J., Ford, H. C., Illingworth, G. D., et al. 2007b, ApJ, 661, 728
  * Joo & Jee (2023) Joo, H., & Jee, M. J. 2023, Nature, 613, 37
  * Kaiser & Squires (1993) Kaiser, N., & Squires, G. 1993, ApJ, 404, 441
  * Kawamata et al. (2016) Kawamata, R., Oguri, M., Ishigaki, M., Shimasaku, K., & Ouchi, M. 2016, ApJ, 819, 114
  * Kim et al. (2021) Kim, J., Jee, M. J., Hughes, J. P., et al. 2021, ApJ, 923, 101
  * Kingma & Ba (2014) Kingma, D. P., & Ba, J. 2014, arXiv e-prints, arXiv:1412.6980
  * Kneib & Natarajan (2011) Kneib, J.-P., & Natarajan, P. 2011, A&A Rev., 19, 47
  * Ko & Jee (2018) Ko, J., & Jee, M. J. 2018, ApJ, 862, 95
  * Kochanek (2006) Kochanek, C. S. 2006, Strong Gravitational Lensing (Berlin, Heidelberg: Springer Berlin Heidelberg), 91–268
  * Lagattuta et al. (2019) Lagattuta, D. J., Richard, J., Bauer, F. E., et al. 2019, MNRAS, 485, 3738
  * Lee et al. (2023) Lee, W., Cha, S., Jee, M. J., et al. 2023, ApJ, 945, 71
  * Limousin et al. (2016) Limousin, M., Richard, J., Jullo, E., et al. 2016, A&A, 588, A99
  * Lotz et al. (2017) Lotz, J. M., Koekemoer, A., Coe, D., et al. 2017, ApJ, 837, 97
  * Mahler et al. (2018) Mahler, G., Richard, J., Clément, B., et al. 2018, MNRAS, 473, 663
  * Markwardt (2009) Markwardt, C. B. 2009, in Astronomical Society of the Pacific Conference Series, Vol. 411, Astronomical Data Analysis Software and Systems XVIII, ed. D. A. Bohlender, D. Durand, & P. Dowler, 251
  * Medezinski et al. (2016) Medezinski, E., Umetsu, K., Okabe, N., et al. 2016, ApJ, 817, 24
  * Meneghetti et al. (2017) Meneghetti, M., Natarajan, P., Coe, D., et al. 2017, MNRAS, 472, 3177
  * Merten et al. (2009) Merten, J., Cacciato, M., Meneghetti, M., Mignone, C., & Bartelmann, M. 2009, A&A, 500, 681
  * Merten et al. (2011) Merten, J., Coe, D., Dupke, R., et al. 2011, MNRAS, 417, 333
  * Montes & Trujillo (2022) Montes, M., & Trujillo, I. 2022, ApJ, 940, L51
  * Morishita et al. (2023) Morishita, T., Roberts-Borsani, G., Treu, T., et al. 2023, ApJ, 947, L24
  * Oguri et al. (2012) Oguri, M., Bayliss, M. B., Dahle, H., et al. 2012, MNRAS, 420, 3213
  * Paraficz et al. (2016) Paraficz, D., Kneib, J. P., Richard, J., et al. 2016, A&A, 594, A121
  * Paszke et al. (2019) Paszke, A., Gross, S., Massa, F., et al. 2019, in Advances in Neural Information Processing Systems, ed. H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, & R. Garnett, Vol. 32 (Curran Associates, Inc.)
  * Paul et al. (2019) Paul, S., Salunkhe, S., Datta, A., & Intema, H. T. 2019, MNRAS, 489, 446
  * Pearce et al. (2017) Pearce, C. J. J., van Weeren, R. J., Andrade-Santos, F., et al. 2017, ApJ, 845, 81
  * Perrin et al. (2014) Perrin, M. D., Sivaramakrishnan, A., Lajoie, C.-P., et al. 2014, in Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave, ed. J. M. O. Jr., M. Clampin, G. G. Fazio, & H. A. MacEwen, Vol. 9143, International Society for Optics and Photonics (SPIE), 91433X
  * Perrin et al. (2012) Perrin, M. D., Soummer, R., Elliott, E. M., Lallo, M. D., & Sivaramakrishnan, A. 2012, in Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, ed. M. C. Clampin, G. G. Fazio, H. A. MacEwen, & J. M. O. Jr., Vol. 8442, International Society for Optics and Photonics (SPIE), 84423D
  * Rajpurohit et al. (2021) Rajpurohit, K., Vazza, F., van Weeren, R. J., et al. 2021, A&A, 654, A41
  * Richard et al. (2014) Richard, J., Jauzac, M., Limousin, M., et al. 2014, MNRAS, 444, 268
  * Schneider (2006) Schneider, P. 2006, Weak Gravitational Lensing (Berlin, Heidelberg: Springer Berlin Heidelberg), 269–451
  * Schrabback et al. (2021) Schrabback, T., Bocquet, S., Sommer, M., et al. 2021, MNRAS, 505, 3923
  * Sebesta et al. (2019) Sebesta, K., Williams, L. L. R., Liesenborgs, J., Medezinski, E., & Okabe, N. 2019, MNRAS, 488, 3251
  * Shan et al. (2010) Shan, H., Qin, B., Fort, B., et al. 2010, MNRAS, 406, 1134
  * Treu et al. (2022) Treu, T., Roberts-Borsani, G., Bradac, M., et al. 2022, ApJ, 935, 110
  * Umetsu et al. (2011) Umetsu, K., Broadhurst, T., Zitrin, A., Medezinski, E., & Hsu, L.-Y. 2011, ApJ, 729, 127
  * Vanzella et al. (2021) Vanzella, E., Caminha, G. B., Rosati, P., et al. 2021, A&A, 646, A57
  * Venturi et al. (2013) Venturi, T., Giacintucci, S., Dallacasa, D., et al. 2013, A&A, 551, A24
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * Vulcani et al. (2023) Vulcani, B., Treu, T., Calabrò, A., et al. 2023, ApJ, 948, L15
  * Wang et al. (2015) Wang, X., Hoag, A., Huang, K. H., et al. 2015, ApJ, 811, 29
  * Weaver et al. (2023) Weaver, J. R., Cutler, S. E., Pan, R., et al. 2023, arXiv e-prints, arXiv:2301.02671
  * Werner et al. (2023) Werner, S. V., Hatch, N. A., Matharu, J., et al. 2023, MNRAS, 523, 91
  * Yoo et al. (2022) Yoo, J., Ko, J., Sabiu, C. G., et al. 2022, ApJS, 261, 28
  * Zekser et al. (2006) Zekser, K. C., White, R. L., Broadhurst, T. J., et al. 2006, ApJ, 640, 639
  * Zitrin (2021) Zitrin, A. 2021, ApJ, 919, 54
  * Zitrin et al. (2015) Zitrin, A., Fabris, A., Merten, J., et al. 2015, ApJ, 801, 44


HAMMER: Boosting Fidelity of Noisy Quantum Circuits by Exploiting Hamming Behavior ...]HAMMER: Boosting Fidelity of Noisy Quantum Circuits
by Exploiting Hamming Behavior of Erroneous Outcomes

Swamit Tannu^*

University of Wisconsin–Madison

Poulami Das

Georgia Institute of Technology

Ramin Ayanzadeh

Georgia Institute of Technology

Moinuddin Qureshi

Georgia Institute of Technology

<concept_desc>Hardware Quantum technologies</concept_desc>

<concept_desc>Computer systems organization Quantum computing</concept_desc>

[500]Computer systems organization Quantum computing

We thank Matthew Harrigan from the Google Quantum AI Group for providing us the QAOA datasets used for some of the  evaluations in this paper.
Poulami Das was funded by the Microsoft Research PhD Fellowship. Ramin Ayanzadeh was supported by the NSF Computing Innovation Fellows (CI-Fellows) program.
This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

#1     #1#1#1     #1     #1     #1     #1#1          #1#1

authorpersonGoogle Quantum AI.
yearAccessed: June 19, 2021.
titleQuantum Computer Datasheet.

[Alam, Ash-Saki, and GhoshAlam
et al2020a]
authorpersonMahabubul Alam, personAbdullah
Ash-Saki, and personSwaroop Ghosh.
Accelerating quantum approximate optimization
algorithm using machine learning. In booktitleDATE. IEEE,

[Alam, Ash-Saki, and GhoshAlam
et al2020b]
authorpersonMahabubul Alam, personAbdullah
Ash-Saki, and personSwaroop Ghosh.
Circuit Compilation Methodologies for Quantum
Approximate Optimization Algorithm. In booktitleMICRO-53.
IEEE, pages215–228.

[Arute, Arya, Babbush, Bacon, Bardin, Barends,
Boixo, Broughton, Buckley, Buell, et alArute et al2020]
authorpersonFrank Arute, personKunal Arya,
personRyan Babbush, personDave Bacon,
personJoseph C Bardin, personRami Barends,
personSergio Boixo, personMichael Broughton,
personBob B Buckley, personDavid A Buell,
et al year2020.
Quantum approximate optimization of non-planar
graph problems on a planar superconducting processor.
journalarXiv preprint arXiv:2004.04197

[Barron and WoodBarron and Wood2020]
authorpersonGeorge S Barron and
personChristopher J Wood. year2020.
Measurement error mitigation for variational
quantum algorithms.
journalarXiv preprint arXiv:2010.08520

[Biamonte, Wittek, Pancotti, Rebentrost, Wiebe,
and LloydBiamonte et al2017]
authorpersonJacob Biamonte, personPeter
Wittek, personNicola Pancotti, personPatrick
Rebentrost, personNathan Wiebe, and personSeth
Lloyd. year2017.
Quantum machine learning.
journalNature volume549,
number7671 (year2017), pages195–202.

[Bollobás and BélaBollobás and
authorpersonBéla Bollobás and
personBollobás Béla. year2001.
booktitleRandom graphs.
Number 73. publisherCambridge university press.

[Bravyi, Sheldon, Kandala, Mckay, and
GambettaBravyi et al2020]
authorpersonSergey Bravyi, personSarah
Sheldon, personAbhinav Kandala, personDavid C Mckay,
and personJay M Gambetta. year2020.
Mitigating measurement errors in multi-qubit
journalarXiv preprint arXiv:2006.14044

authorpersonFrederic Chong.
Closing the gap between quantum algorithms and
hardware through software-enabled vertical integration and co-design. In
booktitleAPS March Meeting Abstracts,
Vol. volume2018. pagesC39–004.

authorpersonInternational Business Machines
Corporation. year2017a.
titleQuantum Software Development Kit for writing quantum
computing experiments, programs, and applications.
[Online; accessed 28-AUGUST-2020].

authorpersonInternational Business Machines
Corporation. year2017b.
titleUniversal Quantum Computer Development at IBM:.
[Online; accessed 3-April-2017].

[Cross, Bishop, Sheldon, Nation, and
GambettaCross et al2019]
authorpersonAndrew W Cross, personLev S
Bishop, personSarah Sheldon, personPaul D Nation,
and personJay M Gambetta. year2019.
Validating quantum computers using randomized model
journalPhysical Review A volume100,
number3 (year2019), pages032328.

[Das, Tannu, Dangwal, and QureshiDas
et al2021b]
authorpersonPoulami Das, personSwamit
Tannu, personSiddharth Dangwal, and personMoinuddin
Qureshi. year2021b.
ADAPT: Mitigating Idling Errors in Qubits via
Adaptive Dynamical Decoupling. In booktitleMICRO-54.

[Das, Tannu, and QureshiDas
et al2021a]
authorpersonPoulami Das, personSwamit
Tannu, and personMoinuddin Qureshi.
JigSaw: Boosting Fidelity of NISQ Programs via
Measurement Subsetting. In booktitleMICRO-54.

[Das, Tannu, Nair, and QureshiDas
et al2019]
authorpersonPoulami Das, personSwamit S.
Tannu, personPrashant J. Nair, and personMoinuddin
Qureshi. year2019.
A Case for Multi-Programming Quantum Computers. In
booktitleMICRO-52 (Columbus, OH, USA)
(seriesMICRO '52). publisherAssociation for
Computing Machinery, addressNew York, NY, USA,

[Farhi, Goldstone, and GutmannFarhi
et al2014]
authorpersonEdward Farhi, personJeffrey
Goldstone, and personSam Gutmann.
A quantum approximate optimization algorithm.
journalarXiv preprint:1411.4028

[Funcke, Hartung, Jansen, Kühn, Stornati,
and WangFuncke et al2020]
authorpersonLena Funcke, personTobias
Hartung, personKarl Jansen, personStefan Kühn,
personPaolo Stornati, and personXiaoyang Wang.
Measurement Error Mitigation in Quantum Computers
Through Classical Bit-Flip Correction.
journalarXiv preprint arXiv:2007.03663

[Gokhale, Ding, Propson, Winkler, Leung, Shi,
Schuster, Hoffmann, and ChongGokhale et al2019]
authorpersonPranav Gokhale, personYongshan
Ding, personThomas Propson, personChristopher
Winkler, personNelson Leung, personYunong Shi,
personDavid I Schuster, personHenry Hoffmann, and
personFrederic T Chong. year2019.
Partial Compilation of Variational Algorithms for
Noisy Intermediate-Scale Quantum Machines. In
booktitleMICRO-52. ACM, pages266–278.

[Gokhale, Javadi-Abhari, Earnest, Shi, and
ChongGokhale et al2020]
authorpersonPranav Gokhale, personAli
Javadi-Abhari, personNathan Earnest, personYunong
Shi, and personFrederic T Chong.
Optimized quantum compilation for near-term
algorithms with openpulse. In booktitleMICRO-53. IEEE,

[Harrigan, Sung, Neeley, Satzinger, Arute,
Arya, Atalaya, Bardin, Barends, Boixo, et alHarrigan
et al2021]
authorpersonMatthew P Harrigan,
personKevin J Sung, personMatthew Neeley,
personKevin J Satzinger, personFrank Arute,
personKunal Arya, personJuan Atalaya,
personJoseph C Bardin, personRami Barends,
personSergio Boixo, et al
Quantum approximate optimization of non-planar
graph problems on a planar superconducting processor.
journalNature Physics volume17,
number3 (year2021), pages332–336.

authorpersonIBM. year2019.
titleMeasurement Error Mitigation.
[Online; accessed 26-July-2020].

authorpersonIBM. year2021.
titleIBM’s roadmap for scaling quantum technology.

[Kwon and BaeKwon and Bae2021]
authorpersonHyeokjea Kwon and
personJoonwoo Bae. year2021.
A Hybrid Quantum-Classical Approach to Mitigating
Measurement Errors in Quantum Algorithms.
journalIEEE Trans. Comput. volume70,
number9 (year2021), pages1401–1411.

[Li, Ding, and XieLi et al2019]
authorpersonGushu Li, personYufei Ding,
and personYuan Xie. year2019.
Tackling the qubit mapping problem for NISQ-era
quantum devices. In booktitleASPLOS-24.

[McClean, Romero, Babbush, and
Aspuru-GuzikMcClean et al2016]
authorpersonJarrod R McClean, personJonathan
Romero, personRyan Babbush, and personAlán
Aspuru-Guzik. year2016.
The theory of variational hybrid quantum-classical
journalNew Journal of Physics
volume18, number2 (datefeb
year2016), pages023023.

[Murali, Baker, Abhari, Chong, and
MartonosiMurali et al2019a]
authorpersonPrakash Murali, personJonathan M
Baker, personAli Javadi Abhari, personFrederic T
Chong, and personMargaret Martonosi.
Noise-Adaptive Compiler Mappings for Noisy
Intermediate-Scale Quantum Computers.
journalarXiv preprint arXiv:1901.11054

[Murali, Baker, Javadi-Abhari, Chong, and
MartonosiMurali et al2019b]
authorpersonPrakash Murali, personJonathan M
Baker, personAli Javadi-Abhari, personFrederic T
Chong, and personMargaret Martonosi.
Noise-adaptive compiler mappings for noisy
intermediate-scale quantum computers. In
booktitleASPLOS-24. pages1015–1029.

[Murali, McKay, Martonosi, and
Javadi-AbhariMurali et al2020]
authorpersonPrakash Murali, personDavid C
McKay, personMargaret Martonosi, and personAli
Javadi-Abhari. year2020.
Software Mitigation of Crosstalk on Noisy
Intermediate-Scale Quantum Computers.
journalarXiv preprint arXiv:2001.02826

[Nishio, Pan, Satoh, Amano, and
Van MeterNishio et al2019]
authorpersonShin Nishio, personYulu Pan,
personTakahiko Satoh, personHideharu Amano, and
personRodney Van Meter. year2019.
Extracting Success from IBM's 20-Qubit Machines
Using Error-Aware Compilation.
journalarXiv preprint arXiv:1903.10963

[of Sciences Engineering and
Medicineof Sciences Engineering and Medicine2019]
authorpersonNational Academies of Sciences Engineering
and personMedicine. year2019.
booktitleQuantum Computing: Progress and
publisherThe National Academies Press,
addressWashington, DC.

[Orus, Mugel, and LizasoOrus
et al2019]
authorpersonRoman Orus, personSamuel Mugel,
and personEnrique Lizaso. year2019.
Quantum computing for finance: overview and
journalReviews in Physics (year2019).

[Patel, Li, Roy, and TiwariPatel
et al2020]
authorpersonTirthak Patel, personBaolin Li,
personRohan Basu Roy, and personDevesh Tiwari.
$\{$UREQA$\}$: Leveraging Operation-Aware Error
Rates for Effective Quantum Circuit Mapping on NISQ-Era Quantum Computers.
In booktitle2020 $\{$USENIX$\}$ Annual Technical Conference
($\{$USENIX$\}$$\{$ATC$\}$ 20). pages705–711.

[Patel and TiwariPatel and Tiwari2020a]
authorpersonTirthak Patel and personDevesh
Tiwari. year2020a.
DisQ: A Novel Quantum Output State Classification
Method on IBM Quantum Computers Using Openpulse. In
booktitleICCAD-39. publisherACM, Article
articleno139, numpages9 pages.

[Patel and TiwariPatel and Tiwari2020b]
authorpersonTirthak Patel and personDevesh
Tiwari. year2020b.
VERITAS: Accurately Estimating the Correct Output
on Noisy Intermediate-Scale Quantum Computers. In
booktitleSC20. pages1–16.

[Patel and TiwariPatel and Tiwari2021]
authorpersonTirthak Patel and personDevesh
Tiwari. year2021.
Qraft: reverse your Quantum circuit and know the
correct program output. In booktitleASPLOS-26.

authorpersonJohn Preskill.
Quantum Computing in the NISQ era and beyond.
journalarXiv preprint arXiv:1801.00862

[Quantum and CollaboratorsQuantum and
authorpersonGoogle AI Quantum and
personCollaborators. year2020.
titleSycamore QAOA experimental data.
[Online; accessed 26-July-2021].

[Shi, Gokhale, Murali, Baker, Duckering, Ding,
Brown, Chamberland, Javadi-Abhari, Cross, et alShi
et al2020]
authorpersonYunong Shi, personPranav
Gokhale, personPrakash Murali, personJonathan M
Baker, personCasey Duckering, personYongshan Ding,
personNatalie C Brown, personChristopher Chamberland,
personAli Javadi-Abhari, personAndrew W Cross,
et al year2020.
Resource-Efficient Quantum Computing by Breaking
journalProc. IEEE (year2020).

[Shi, Leung, Gokhale, Rossi, Schuster,
Hoffmann, and ChongShi et al2019]
authorpersonYunong Shi, personNelson Leung,
personPranav Gokhale, personZane Rossi,
personDavid I Schuster, personHenry Hoffmann, and
personFrederic T Chong. year2019.
Optimized compilation of aggregated instructions
for realistic quantum computers. In booktitleASPLOS-24.

[Smith, Ravi, Murali, Baker, Earnest,
Javadi-Abhari, and ChongSmith et al2021]
authorpersonKaitlin N Smith,
personGokul Subramanian Ravi, personPrakash Murali,
personJonathan M Baker, personNathan Earnest,
personAli Javadi-Abhari, and personFrederic T
Chong. year2021.
Error Mitigation in Quantum Computers through
Instruction Scheduling.
journalarXiv preprint arXiv:2105.01760

[Tang, Tomesh, Suchara, Larson, and
MartonosiTang et al2021]
authorpersonWei Tang, personTeague Tomesh,
personMartin Suchara, personJeffrey Larson, and
personMargaret Martonosi. year2021.
booktitleCutQC: Using Small Quantum Computers for
Large Quantum Circuit Evaluations.
publisherACM, pages473–486.

[Tannu and QureshiTannu and Qureshi2019a]
authorpersonSwamit S. Tannu and
personMoinuddin Qureshi. year2019a.
Ensemble of Diverse Mappings: Improving Reliability
of Quantum Computers by Orchestrating Dissimilar Mistakes. In
booktitleMICRO-52 (Columbus, OH, USA)
(seriesMICRO '52). publisherACM,

[Tannu and QureshiTannu and Qureshi2019b]
authorpersonSwamit S. Tannu and
personMoinuddin K. Qureshi. year2019b.
Mitigating Measurement Errors in Quantum Computers
by Exploiting State-Dependent Bias. In booktitleMICRO-52.
publisherACM, pages279–290.

[Tannu and QureshiTannu and Qureshi2019c]
authorpersonSwamit S Tannu and
personMoinuddin K Qureshi. year2019c.
Not all qubits are created equal: a case for
variability-aware policies for NISQ-era quantum computers. In
booktitleProceedings of the Twenty-Fourth International
Conference on Architectural Support for Programming Languages and Operating
Systems. pages987–999.

[Zulehner, Paler, and WilleZulehner
et al2018]
authorpersonAlwin Zulehner, personAlexandru
Paler, and personRobert Wille.
Efficient mapping of quantum circuits to the IBM QX
architectures. In booktitle2018 Design, Automation Test in
Europe Conference Exhibition (DATE). pages1135–1138.