11institutetext:  Department of Astrophysics, Astronomy & Mechanics, Faculty
of Physics, National and Kapodistrian University of Athens, Panepistimiopolis
Zografou, Athens 15784, Greece 22institutetext: Institute of Astronomy,
Astrophysics, Space Applications and remote Sensing, National Observatory of
Athens, GR-15236 Palaia Pendeli, Greece 33institutetext: National Observatory
of Athens, GR-18100 Thessio, Athens, Greece 44institutetext: Sector of
Astrophysics, Astronomy & Mechanics, Department of Physics, Aristotle
University of Thessaloniki, Thessaloniki 54124, Greece

# Host galaxy and orientation differences between different AGN types

Anamaria Gkini 1122 Manolis Plionis 3344 Maria Chira 2244 Elias Koulouridis 22

###### Abstract

Aims. The main purpose of this study is to investigate aspects regarding the
validity of the active galactic nucleus (AGN) unification paradigm (UP). In
particular, we focus on the AGN host galaxies, which according to the UP
should show no systematic differences depending on the AGN classification.

Methods. For the purpose of this study, we used (a) the spectroscopic Sloan
Digital Sky Survey (SDSS) Data Release (DR) 14 catalogue, in order to select
and classify AGNs using emission line diagnostics, up to a redshift of
$z=0.2$, and (b) the Galaxy Zoo Project catalogue, which classifies SDSS
galaxies in two broad Hubble types: spirals and ellipticals.

Results. We find that the fraction of type 1 Seyfert nuclei (Sy1) hosted in
elliptical galaxies is significantly larger than the corresponding fraction of
any other AGN type, while there is a gradient of increasing spiral-hosts from
Sy1 to LINER, type 2 Seyferts (Sy2) and composite nuclei. These findings
cannot be interpreted within the simple unified model, but possibly by a co-
evolution scheme for supermassive black holes (SMBH) and galactic bulges.

Furthermore, for the case of spiral host galaxies we find the Sy1 population
to be strongly skewed towards face-on configurations, while the corresponding
Sy2 population range in all host galaxy orientation configurations has a
similar, but not identical, orientation distribution to star-forming (SF)
galaxies. These results also cannot be interpreted by the standard unification
paradigm, but point towards a significant contribution of the galactic disc to
the obscuration of the nuclear region. This is also consistent with the
observed preference of Sy1 nuclei to be hosted by ellipticals, that is, the
dusty disc of spiral hosts contributes to the obscuration of the broad-line
region (BLR), and thus relatively more ellipticals are expected to appear
hosting Sy1 nuclei.

###### Key Words.:

galaxies: Seyfert - galaxies: active - galaxies: nuclei - galaxies: evolution
- galaxies: bulges - galaxies: statistics

##  1 Introduction

The unification model of active galactic nucleus (AGN) explains the wide
variety of features discerned in different classes of AGN in terms of the
anisotropic geometry of the black hole’s immediate surroundings (e.g.
Antonucci, 1993; Urry & Padovani, 1995; Netzer, 2015)). It is well known that
AGNs have a central supermassive black hole (SMBH) that accretes surrounding
matter, and a toroidal structure by dust molecular gas that absorbs a fraction
of the emitted radiation. Near the SMBH, the high-speed gas clouds produce the
broad-line region (BLR), while far away from this the torus clouds that move
at low speeds produce the narrow-line region (NLR). In the simplest form of
the unification paradigm (UP), the observed differences between type 1 Seyfert
(Sy1) and type 2 Seyfert (Sy2) objects are due to orientation effects with
respect to the line of sight to the observer.

An important implication of this simple model is that, since the observed
differences between AGN types are attributed solely to orientation effects,
the host galaxies of AGN should be intrinsically the same (e.g. Netzer, 2015;
Hickox & Alexander, 2018). Thus, despite their success in explaining a range
of AGN observed features, such as the absence of broad-line features in the
spectra of Sy2 galaxies, it has become clear that the simplest models of
unification are inconsistent with observations and cannot explain aspects such
as the lack of Sy1s in isolated pairs of galaxies (González et al., 2008) and
in compact groups (Martínez et al., 2008; Bitsakis et al., 2010), the lack of
hidden BLRs in many Sy2 AGNs in polarised spectra (Tran, 2001, 2003), and the
absence of detected BLRs in low-accretion-rate AGNs (e.g. Nicastro et al.,
2003; Elitzur & Ho, 2009). Specifically, according to studies investigating
the differences between Sy1s and Sy2s (e.g. Keel, 1980; Kinney et al., 2000;
Koulouridis et al., 2006; Rigby et al., 2006; Martínez-Sansigre et al., 2006a;
Lacy et al., 2007; Lagos et al., 2011; Ramos Almeida et al., 2011; Elitzur,
2012; Villarroel et al., 2012; Koulouridis, 2014; Villarroel & Korn, 2014;
Netzer, 2015; Villarroel et al., 2017; Bornancini & García Lambas, 2018; Zou
et al., 2019; Yang et al., 2019; Bornancini & García Lambas, 2020; Malizia et
al., 2020) and the connection with their host galaxies at redshifts
$0.03<z<0.2$ (e.g. Villarroel et al., 2012; Koulouridis et al., 2013;
Villarroel & Korn, 2014), it has been shown that type 1 and type 2 Seyfert
galaxies have different optical, mid-infrared, X-ray, and morphological
properties (e.g. Sorrentino et al., 2006; Slavcheva-Mihova & Mihov, 2011;
Villarroel et al., 2017; Chen & Hwang, 2017; Bornancini & García Lambas,
2018), and they also reside in statistically different environments (e.g.
Koulouridis et al., 2013; Villarroel & Korn, 2014; Jiang et al., 2016;
Bornancini et al., 2017; Bornancini & García Lambas, 2018, 2020). In
particular, Bornancini & García Lambas (2020) showed that the ultraviolet
(UV), optical, and mid-infrared (IR) colour distribution of the different AGN
classes differ significantly, while Villarroel & Korn (2014) found that the
activity and colour between the neighbours of Sy1s and Sy2s are significantly
different, and that the spiral fraction of the host galaxies depends on the
environment of Sy1 and Sy2 galaxies in different ways. Moreover, it is found
that the neighbours of Sy2 AGN are more star-forming and bluer than Sy1 AGNs
(see also Koulouridis et al., 2013) and also that Sy2 hosts are surrounded by
a larger number of dwarf galaxies (Villarroel et al., 2012). Additionally, the
morphology of Sy1 galaxies shows no indications of close interactions, which
means either that they rarely merge (Koulouridis, 2014), or that they are
extremely short-lived AGNs (Villarroel et al., 2012). Bornancini & García
Lambas (2018) find that Sy2s have more abundant neighbours at high redshifts
($0.3<z<1.1$) and that Sy1 hosts are preferably elliptical or compact
galaxies, while Sy2 hosts present a broader Hubble-type distribution.

Many studies (e.g. Ramos Almeida et al., 2011; Elitzur, 2012; Villarroel et
al., 2017; Suh et al., 2019) reveal that differences between the two types of
AGN also lie in the star formation rate (SFR), in the structure of the torus
and the physics of the central engines. Most notably, Suh et al. (2019) and
Villarroel et al. (2017) found that the two Seyfert populations have different
luminosities with Villarroel et al. (2017) supporting that type 1 AGNs are ten
times more luminous than type 2s. Suh et al. (2019) also concluded that Sy1
and Sy2 host galaxies seem to have different SFRs, while Villarroel et al.
(2017) detected many more supernovae (SNe) in type 2 AGN hosts, which implies
differences between stellar ages and more recent star formation in Sy2 hosts.
On the other hand, Bornancini & García Lambas (2018) found no difference
between the SFR distribution of tracer galaxies in both AGN samples at larger
scales ($r_{p}<500$ kpc). In addition, in the context of differences in the
structural features of the torus, Ramos Almeida et al. (2011) found that the
Sy2 torus is broader and has more clouds than that of Sy1 and that the optical
depth of the clouds in Sy1 torus is larger than in Sy2. They suggest that the
type 1/type 2 classification depends on the torus intrinsic properties rather
than in the torus inclination (e.g. Elitzur, 2012; Audibert et al., 2017).
Previous work (e.g. Mendoza-Castrejón et al., 2015) has also claimed that the
Sy1 and Sy2 active nuclei have different torus clumpiness, while they also
found a dependence on whether the host is in a merger or isolated.

Since, observations do not fully comply with the predictions of the simple ’A
type 2 AGN is just a type 1 AGN viewed through a dust’ model, alternative or
complementary factors affecting the observed AGN types should be sought in
order to explain the AGN variety. Indeed, Koulouridis et al. (2006) and Jiang
et al. (2016), when studying the environments of Sy1 and Sy2 galaxies at low
redshifts, found that both AGN classes have similar clustering properties
(e.g. Zou et al., 2019), but at scales smaller than $100kpc$ Sy2s have
significantly more neighbours than Sy1s (e.g. Bornancini & García Lambas,
2018). These results contrast with those of Melnyk et al. (2018), who obtained
similar local- and large-scale environments for the different AGN types. Jiang
et al. (2016) also found significant differences in the infrared colour
distributions of the host galaxies of the two AGN types. Besides this, Powell
et al. (2018) reported that nearby type 2 AGNs reside in more massive halos
than type 1s, which is unlike the results of Jiang et al. (2016), who found
that their halo masses are similar.

Furthermore, some studies (e.g. Keel, 1980; Maiolino & Rieke, 1995; McLeod &
Rieke, 1995; Malkan et al., 1998; Matt, 2000; Schmitt et al., 2001; Martínez-
Sansigre et al., 2006a; Rigby et al., 2006; Lacy et al., 2007; Lagos et al.,
2011; Goulding et al., 2012; Netzer, 2015; Burtscher et al., 2016; Jiang et
al., 2016; Buchner & Bauer, 2017; Lanzuisi et al., 2017; Bornancini & García
Lambas, 2018; Zou et al., 2019; Malizia et al., 2020; Masoura et al., 2021)
claim that the host galaxy might have a non-negligible contribution to the
optical obscuration of nuclei. For example, He et al. (2018) suggest that
obscuration on both circum-nuclear ($\sim$ pc ) and galactic ($\sim$ kpc)
scales are important in shaping and orienting the AGN narrow-line regions.
Further evidence comes from high-resolution observations, in nearby type 2
AGNs, which have revealed that the nucleus is covered by a large-scale – a few
hundred pc – dust filament or diffuse dust lane (Prieto et al., 2014).
Moreover, Chen et al. (2015) found that the obscuration at X-ray and optical
bands in type 2 quasars is connected to the far-IR-emitting dust clouds
usually located on the scale of the host galaxy. Furthermore, Malizia et al.
(2020) recently published an analysis using a hard X-ray-selected AGN sample
showing that material located in the host galaxy on scales of hundreds of
parsec, while not aligned with the absorbing torus, can be extended enough to
hide the BLR of some Sy1s causing their misclassification as Sy2 objects and
giving rise to the deficiency of around 24% of Sy1s in edge-on galaxies.

The different average AGN luminosities and stellar ages of the host galaxies
of the Sy1 and Sy2 populations could be explained within the framework of an
evolutionary scenario. According to Koulouridis et al. (2006) and Krongold et
al. (2002), in some cases the interaction between gas-rich galaxies ignites
starburst activity, while large amounts of gas and dust obscure the central
nuclear region at this stage. As the starburst dies off, the remaining
molecular gas and dust forms a torus around the disc and eventually the AGN
will attenuate the obscuring medium. Namely, this model proposes an AGN
evolutionary sequence going from starburst to type 2, and finally to type 1,
Seyfert galaxies (e.g. Springel et al., 2005; Hopkins et al., 2006;
Koulouridis et al., 2013; Elitzur et al., 2014; Villarroel & Korn, 2014;
Mendoza-Castrejón et al., 2015; DiPompeo et al., 2017a; Yang et al., 2019;
Spinoglio & Fernández-Ontiveros, 2019).

Studies in polarised light to Sy2 galaxies have shown that in many low-
luminosity AGN the dusty torus is absent, while the BLR is also not detected
(e.g. Elitzur & Shlosman, 2006; Perlman et al., 2007; Trump et al., 2011;
Koulouridis, 2014; Hernández-Ibarra et al., 2016). These results are
consistent with those of Trump et al. (2011), who support the notion that
above a specific accretion rate ($(L/L_{\rm Edd}\gtrsim 0.01)$) AGNs can be
observed as broad-line or as obscured narrow-line AGNs, while for $(L/L_{\rm
Edd}\lesssim 0.01)$ the BLR becomes non-detectable. Additionally, the
obscuring torus tends to become weaker or disappears (Elitzur et al., 2014).

The above findings could be incorporated within the evolutionary scheme. If
the accretion-rate-dependent scenario is valid, one would expect that AGNs
could lose their torus or/and their BLR at the end of the AGN duty cycle, as
the accretion rate drops below a critical value (e.g. Elitzur & Ho, 2009;
Elitzur et al., 2014; Koulouridis, 2014). This implies that firstly, AGNs can
appear as type 1, after the quenching of the star-forming activity by the AGN
feedback and the disappearance of the torus (Krongold et al., 2002) and
secondly, these type 1s will evolve to true type 2 AGNs due to the elimination
of the BLR, based, for example, on the wind-disc scenario of Elitzur & Ho
(2009).

In summary, the plethora of results of many relevant studies clearly indicate
that the viewing angle alone cannot fully account for the different AGN types.
It is within this ideology that the current study lies, investigating the
orientation properties of spiral hosts, as well as the Hubble-type
distribution of different types of AGNs.

After the presentation of the data used in Sect. 2, the main body of our
analysis is organised as follows. In Sect. 3.1, we study the morphology
frequency distribution of host galaxies for different AGN types comparing to
that of the non-active star-forming sample, after statistically matching their
respective redshift distribution in order to suppress possible evolutionary
effects. In Sect. 3.2, we study the frequency distribution of spiral host
galaxy orientations ($b/a$) for the Sy1 and Sy2 sub-samples, comparing with
that of non-active star-forming (SF) galaxies (which we use as a control
sample). We only used galaxies with high ’spirality’ probability $>0.8$ (as
defined by the Galaxy Zoo project), also statistically matching their
respective redshifts and stellar mass distributions.

##  2 Observational data

For the purposes of the current study, we used galaxy catalogues extracted
from the Sloan Digital Sky Survey (SDSS) Data Release (DR) 14 (Abolfathi et
al., 2018) in five bands (u,g,r,i,z) with a magnitude limit for the
spectroscopic sample of $m_{r}=17.77$ in the r-band, in order to have a
homogeneous magnitude cut-off over the largest possible SDSS area. Our
catalogues consist of $3.800$ Sy1, $56.846$ Sy2, $120.025$ Composite,
$107.034$ LINER, and $263.223$ SF galaxies with redshift $z<0.2$. The above
classifications are derived from the spectroscopic reanalysis by the MPA-JHU
teams of a large sub-sample of SDSS galaxies, as described in Tremonti et al.
(2004) and Brinchmann et al. (2004). The morphology characterisation of the
galaxies is based on the Galaxy Zoo Project (Lintott et al., 2008), a crowd-
sourced astronomy project that asks citizens to characterise galaxies as
spirals or ellipticals and to determine the rotation direction of spirals by
inspecting SDSS galaxy images (Raddick et al., 2007). In order to ensure the
reliable classification of our galaxies, we performed a quality cut on our
catalogue, rejecting galaxies with Ha rest-frame equivalent width (EW) $<8$ Å,
which, although it is a rather arbitrary limit, we verified to be adequately
secure via an inspection of a sufficiently large sub-sample.

The Sy1 sample (see Appendix) comprises all galaxies with a Balmer line width
of $\sigma$ greater than 500 km/sec (FWHM¿1180 km/s). We note that in the SDSS
database all such sources are catalogued with $\sigma$=500 km/sec, while this
is actually a lower limit. Visual inspection of a large number of spectra
validated that these sources are bona fide broad-line Seyferts. All spectra
with an emission line of $\sigma>$200 km/sec are also characterised as broad
line in the SDSS database. We visually reviewed all spectra in our spiral
galaxy sample that fall in this category (450 sources), and we conclude that
they contain very few that could be unambiguously classified as broad-line
AGNs. Therefore, to reduce noise in our analysis, we chose to exclude all
spectra with Balmer lines of $\sigma<$500 km/sec from our Sy1 sample.

After the above procedures, our final sample of secure objects consists of
1.378 Sy1, 7.498 Sy2, 26.544 Composite, 1.926 LINER, and 203.298 SF galaxies.

The classification between different type of narrow-line AGNs and SF galaxies
has been performed utilising the BPT diagram classification method of Baldwin
et al. (1981).

It is important to note that we will use the SF galaxies, of which the vast
majority are spirals, as our control sample since their disc orientations
cover the whole range of viewing angles with respect to the line of sight.
Additionally, they are mainly non-AGN galaxies, and even when hosting an AGN
in their centres the star formation dominates the emission. This leads to
spectra characteristic of non-AGN galaxies (Siebenmorgen et al., 2015).

Before proceeding, we felt it necessary to investigate the level of
consistency between the morphology classification of the Galaxy Zoo project
and our own assessment via an inspection of the host galaxy image. To this
end, we selected a small sub-sample of images and spectra of galaxies of the
highest ’spirality’ (probability of being a spiral) and ’ellipticity’
(probability of being an elliptical). In detail, we selected 50 nearby
galaxies, five spirals, and five ellipticals of every AGN type, with redshift
$z<0.1$ in order to ensure a high-quality image. We concluded that although
the Galaxy Zoo morphology classification is reliable for the majority of our
galaxies, we found some cases in which the assessment of the citizens
participating in the Galaxy Zoo project seems to be dubious. Our analysis
showed that we can trust the characterisation of spirals up to $z\sim 0.2$
while that of ellipticals only up to $z\sim 0.1$ since the spiral features are
more difficult to distinguish at higher redshifts, where a galaxy image with
weak spirality can be interpreted as being an elliptical.

##  3 Methodology and results

This section is organised as follows. In Sect. 3.1 we study the Hubble-type
frequency distributions of the host galaxies for the different samples: Sy1,
Sy2, LINER, Composite, or SF, while in Sect. 3.2 we compare the projected
axial-ratio (related to the orientation with respect to the line of sight)
frequency distributions for the spiral hosts of Sy1s and Sy2s.

###  3.1 Hubble-type distribution

In this section, we seek to reveal if there is any correlation between the AGN
type and the Hubble-type morphology of their host galaxies. According to the
simplest unification scheme, the different AGN classes are a result of
different viewing angles with respect to the orientation of an obscuring
torus, and thus the properties of the host galaxies should not show any
statistically significant differences. For this study, we used sub-samples of
Sy1, Sy2, LINER, Composite, and SF galaxies, derived from our SDSS catalogue
with a redshift limit of $z<0.1$, in order to obtain more robust Hubble-type
classification (as discussed in Sect. 2). For these sub-samples, we generated
the frequency distribution of the Zoo ’ellipticity’ and ’spirality’
probabilities, which within the context of the simplest unification scheme,
are expected to be statistically the same for all AGN classes (e.g. Antonucci,
1993; Urry & Padovani, 1995).

In order to be able to compare the different probability distributions,
avoiding possible evolutionary effects, it is necessary to take into account
any statistically significant differences between the redshift distributions
of the different samples, which we present in Fig.1-(upper panel). Due to the
different number of objects in the sub-samples, and in order to reveal
systematic trends among the different sub-samples, we normalised the
distributions dividing with the total number of objects in each sub-sample.
Using a random sampling procedure, we matched the normalised redshift
distributions to a common fractional distribution (Fig.1 \- lower panel).
Because of the low number of Sy1 galaxies and in order to avoid further
depleting it, we re-sampled all other activity types, so that their normalised
distributions are matched to that of the Sy1 sample.

Figure 1: Upper panel: Normalised redshift frequency distributions for the
four sub-samples of AGN types and SF galaxies, limited to $z<0.1$. The
different sub-samples are colour-coded as denoted in the key. Lower panel:
Redshift-matched normalised distribution. The shaded area corresponds to the
1$\sigma$ Poisson uncertainty.

For the redshift-matched sub-samples, in Fig. 2 we present the ellipticity
(upper panel) and spirality (lower panel) probability distributions. The two
quantities are complementary, which is to be expected since the sum of the two
Zoo probabilities should be roughly equal to unity. Moreover, we should note
that SF galaxies, with spectra dominated by young stellar populations, are by
definition spirals (Hubble, 1926), and indeed, as seen in Fig. 2, their
spirality distribution (black solid line) peaks at high-spirality
probabilities, while their respective ellipticity distributions, which are
roughly complementary, peak at zero-ellipticity probabilities. The SF galaxy
sample can thus be used as a control sample of the Zoo Project spirality and
ellipticity distributions of the various AGN host galaxies.

In Fig. 2, we see that the different AGN classes are distributed in the whole
range of probabilities indicating a wide range of Hubble-type hosts, with the
predominance of spirals. However, the normalised frequency distribution of the
different AGN types are dissimilar at a statistically significant level (as
indicated by the 1$\sigma$ Poisson uncertainty), which implies a different
Hubble-type distribution for the different AGN types, a result that
contradicts the original unification paradigm according to which there should
be no dependence of the AGN class on the host galaxy’s Hubble-type
classification (e.g. Antonucci, 1993). Interestingly, the Sy1s show a peak at
both high- and low- spirality probabilities, indicating a relatively higher
fraction of Sy1s, with respect to other AGN types, residing in elliptical
hosts.

For a more revealing comparison of the previously discussed morphology
difference of the various AGN host galaxies with respect to SF galaxies, in
Fig.3 we present the excess factor by which the fractional number of the
various AGN types exceeds that corresponding to SF galaxies, for each
spirality or ellipticity probability:

$\Delta(\rm AGN,p)=\frac{\rm N_{i}(AGN)/N_{tot}(AGN)}{\rm
N_{i}(SF)/N_{tot}(SF)}-1\;.$

Figure 2: Probability distributions of ellipticity (upper panel) and spirality
(lower panel) for the Sy1, Sy2, LINER, Composite and SF host galaxies, limited
to $z<0.1$. The different AGN and SF sub-samples are colour-coded as in Fig.1,
while the shaded area corresponds to the 1$\sigma$ Poisson uncertainty.

Upon inspecting Fig.3, it becomes evident that Sy1s show the highest relative
preference for elliptical hosts with respect to LINERs, Sy2s, and Composites,
although all AGN types, each at a different degree, appear in elliptical
hosts.

Figure 3: Values of the factor $\Delta({\rm AGN,p})$ by which the different
AGN types exceed the corresponding fractional number of SF galaxies for the
ellipticity (upper panel) and spirality ((lower panel) cases. The different
AGN and SF sub-samples are colour-coded as in Fig.1, while the shaded area
corresponds to the 1$\sigma$ Poisson uncertainty.

###  3.2 Effect of the galactic disc on the obscuration of the AGN

We wish to test the hypothesis that the AGN host galaxy contributes to the
obscuration of the AGN emission. To this end, we studied the orientation of
spiral galaxies hosting Sy1 and Sy2 nuclei, limited to $z<0.2$. We only
selected spiral hosts for this test since the orientation of spirals with
respect to the line of sight can be quantified via their projected axis ratio,
$(b/a)$. Furthermore, in order to reduce noise, we required a high Galaxy Zoo
spirality probability, that is $p>0.8$. We also used the corresponding sub-
sample of SF galaxies as a reference (or control) sample.

####  3.2.1 Matching the redshift distributions

In Figs.4-5-(upper panel) ,we present the Sy1 and Sy2 sub-sample normalised
redshift distributions and compare them with the corresponding distribution of
SF galaxies. As can be clearly seen, and also as quantified by the Kolmogorov-
Smirnov (KS) test (p-value $\sim$ $0$ and $\sim$ $10^{-4}$, respectively), the
distributions are significantly different. For $z\lesssim 0.08$, the fraction
of both the AGN populations is lower than that of the SF galaxies, while for
higher values, $z\gtrsim 0.1$, the AGN fractions are greater than that of SF
galaxies. Whether due to observational biases or evolutionary effects,
understanding such differences is out of the scope of the current work.
However, we needed to ensure that our results would not be affected by such
biases and thus followed the same re-sampling technique as in Sect. 3.1, in
order to obtain matched-redshift distributions, which are shown in Figs. 4-5
-(lower panel).

Figure 4: Upper panel: Normalised redshift distribution of spiral Sy1 and SF
galaxies. Lower panel: Normalised redshift-matched distribution of Sy1s and
the corresponding SF control sample. The Sy1 and the SF sub-samples are
colour-coded as in Fig.1, while the shaded area corresponds to the 1$\sigma$
Poisson uncertainty.

Figure 5: Upper panel: Normalised redshift distribution of spiral Sy2 and SF
galaxies. Lower panel: Normalised redshift-matched distribution of Sy2 and the
corresponding SF control sample. The Sy2 and the SF sub-samples are colour-
coded as in Fig.1, while the shaded area corresponds to the 1$\sigma$ Poisson
uncertainty.

We can now proceed to a meaningful comparison of the orientation distribution
of Sy1s and Sy2s with respect to the SF case. We expect that, according to the
simplest unification model, the orientation distributions of Sy1s and Sy2s
hosted in spiral galaxies must be identical to that of spiral SF galaxies. In
presenting our results, we again normalise the distributions by the total
number of objects in each sub-sample.

Figure 6: Distribution of the projected axial ratio for spiral galaxies
hosting Sy1s and the corresponding control sample of SF galaxies. The Sy1 and
the SF sub-samples are colour-coded as in Fig.1, while the shaded area
corresponds to the 1$\sigma$ Poisson uncertainty.

In Fig.6, we present the comparative plot of the normalised axis-ratio ($b/a$)
distributions for the spiral galaxies hosting Sy1 nuclei (purple) and for the
control sample of SF galaxies. It is evident that the two distributions are
different, which we also confirm with a KS test (p-value $\sim 10^{-8}$); the
$b/a$ distribution of Sy1 is skewed towards high $b/a$ values, indicating
inclination angles closer to face-on orientations, while the control sample
covers the full range of orientation angles. The distribution of the control
sample shows a peak at $b/a\approx 0.35$, whereas the Sy1 population peaks at
$b/a\approx 0.7$. Moreover, for $b/a<0.4$ we find that the fraction of Sy1
host galaxies decreases dramatically compared to SF galaxies. Thus, we
conclude that the type 1 Seyferts tend to be more ’face-on’ compared to SF
galaxies, and only a very small fraction of Sy1 galaxies are found to have
$b/a$ values close to the edge-on orientation.

Figure 7: Distribution of projected axial ratios for spiral galaxies hosting
type 2 AGNs and the corresponding control SF sample. The Sy2 and the SF sub-
samples are colour-coded as in Fig.1, while the shaded area corresponds to the
1$\sigma$ Poisson uncertainty.

In Fig. 7, we present the respective $b/a$ comparison plot for the Sy2 and the
star-forming subsamples, and we find that although their distributions are
significantly more similar than the corresponding of Fig.6, they are still
statistically different as confirmed by a KS test (p-value $\sim 10^{-11}$).
In detail, both the type 2 Seyferts and the SF galaxies are distributed in the
whole range of $b/a$, with the star-forming peaking at b/a$\approx 0.35$,
while the Sy2 appear to have two local maxima, at b/a$\approx 0.37$ and at
b/a$\approx 0.65$. A significantly higher fraction of Sy2 galaxies has values
of $b/a\in(0.3,0.8)$, while for values $b/a<0.3$ (close to edge-on
orientations) and $b/a>0.8$ (close to face-on orientations), the fraction of
Sy2s is lower than that of SF galaxies.

####  3.2.2 Matching stellar mass distributions

Comparing AGN samples of different intrinsic luminosity distributions may
introduce a bias to the results. To this end, one could use physical
properties that are proxies of the AGN intrinsic luminosity, as the luminosity
of [O III]5007 emission line or the stellar mass of the host galaxies.
However, since we are studying the additional absorption by the host galaxy
that may also obscure the NLR, the use of the [OIII] line can bias our
results. Therefore, we chose to match the stellar mass distributions of our
samples, derived by the MPA-JHU teams, using the methodology defined in
Kauffmann et al. (2003) and applied to photometric data as described in Salim
et al. (2007). Furthermore, this normalisation eliminates any possible bias
introduced by the excessive absorption in one of the samples because of the
uneven distribution of massive hosts.

For the already redshift-matched samples, in Fig.8 we present the normalised
stellar mass distributions of type 1 (upper panel) and type 2 (lower panel)
AGNs with the corresponding distribution of SF galaxies. It is evident, but
also confirmed by a KS test, that the distributions are statistically
different (p-value $\sim 0$). We find that both AGN classes are identified
with more massive host galaxies compared to the sample of SF galaxies, a
result which is in agreement with that of Bornancini & García Lambas (2020).
Even though the direct comparison of the Sy1 and Sy2 stellar mass
distributions is out of the scope of this work, it is noteworthy to mention
that many previous studies (e.g. Yang et al., 2017, 2019) have shown that both
AGN types have similar stellar mass hosts (see discussion in Zou et al.
(2019)), which we also confirmed by a simple inspection of the corresponding
figures.

In order to acquire more robust results and to ensure the removal of possible
luminosity biases we follow the same re-sampling technique, as in Sect. 3.1,
to obtain matched stellar-mass distributions, which are shown in Fig. 8
(dashed distributions), at the expense, however, of significantly reducing the
size of the AGN samples. This also excludes the most massive host galaxies
from our samples, which could indeed reduce the significance of the effect
(galaxy-disc obscuration of nuclear region) that we are seeking to
investigate.

We can now detail the axial-ratio (b/a) distributions for the spiral-redshift-
and stellar-mass-matched sub-samples of Sy1, Sy2, and star-forming galaxies.
In Fig.9, we present the normalised b/a comparative plot for Sy1 galaxies and
the corresponding control sample. The KS test confirmed that the two
distributions are different (p-value $\sim 10^{-3}$). Compared to Fig.6,
despite the reduction of the Sy1 sample size, both plots show similar
behaviours, confirming that the results obtained in Sect. 3.2.1 are robust.

Correspondingly, in Fig.10 we present the normalised comparative plot for Sy2
and SF galaxies, and again the resulting distributions are similar to those of
Fig.7, again confirming that the main results presented in Sect. 3.2.1 are
robust. However, in this case there is a small but distinct difference at
small b/a values with the fraction of Sy2 being lower than that of SF for
$b/a<0.2$, in contrast with Fig.7 where this transition occurs for $b/a\approx
0.3$. Thus, when matching to stellar mass (Fig.10), the fraction of Sy2 with
small b/a is not significantly different to that of SF galaxies (Fig.7).

Figure 8: Normalised stellar-mass distribution of spiral Sy1 (upper panel) and
Sy2 (lower panel) with the corresponding SF and stellar-mass-matched sample
(dashed line). The Sy1, Sy2, and SF sub-samples are colour-coded as in Fig.1,
while the shaded area corresponds to the 1$\sigma$ Poisson uncertainty. Figure
9: Distribution of the projected axial ratio for stellar-mass-matched spiral
galaxies hosting Sy1s and the corresponding control sample of SF galaxies. The
Sy1 and SF sub-samples are colour-coded as in Fig.1, while the shaded area
corresponds to the 1$\sigma$ Poisson uncertainty.  Figure 10: Distribution of
the projected axial ratio for stellar-mass-matched spiral galaxies hosting
Sy2s and the corresponding control sample of SF galaxies. The Sy2 and SF sub-
samples are colour-coded as in Fig.1, while the shaded area corresponds to the
1$\sigma$ Poisson uncertainty.

##  4 Discussion

###  4.1 Why do Sy1 nuclei favour elliptical hosts?

There are various indications for differences in the host galaxies of AGNs in
the literature. Our results in the low-redshift regime are in agreement with
the results of Bornancini & García Lambas (2018), based on higher redshift
$(0.3<z<1.1)$ type 1 and type 2 AGNs from the COSMOS2015 catalogue, who showed
that the type 1 AGN host galaxies appear more elliptical and compact than
those of type 2 AGNs that span the whole spiral to elliptical Hubble-type
range. In addition, using data from the fourth SDSS Data Release (DR4),
Sorrentino et al. (2006) found that 76% of type 1 Seyfert host galaxies are
elliptical, while the corresponding ratio of Seyfert 2 hosted in early-type
galaxies is 56.8%. Slavcheva-Mihova & Mihov (2011) found that more type 1 than
type 2 AGNs prefer elliptical hosts. Similarly, Villarroel et al. (2017) and
Chen & Hwang (2017) concluded that Sy2 nuclei reside more in spiral hosts
($\sim 30-40\%$) than Sy1 nuclei do ($\sim 20\%$), in disagreement with the
simplest unification model.

The different bulge distributions of Sy1 and Sy2 might be related to an
evolutionary sequence of AGN activity (e.g. Krongold et al., 2002; Tran, 2003;
Koulouridis et al., 2006; Villarroel et al., 2012; Koulouridis et al., 2013;
Villarroel & Korn, 2014). These results indicate a possible co-evolution
scheme between galaxies and SMBHs (e.g. Springel et al., 2005; Hopkins et al.,
2006, 2008b, 2008a; Madau & Dickinson, 2014) (see also reviews by Kormendy &
Ho, 2013; Heckman & Best, 2014).

At the initial merging phase, during the enhanced star-forming activity and
accretion, the AGN is mostly obscured because of the large amount of gas and
dust in the circum-nuclear region. However, the AGN will eventually be
revealed after the surrounding material is consumed, or expelled by radiation
pressure. In parallel to the AGN evolution, the stellar population of the
merger will also rapidly redden and the host will eventually be transformed to
a quiescent elliptical galaxy (e.g. Hopkins et al., 2008b, a). AGN feedback
plays an essential role in both AGN and galaxy transformation and probably
leads to the observed SMBH-bulge relation (e.g. Magorrian et al., 1998;
Ferrarese & Merritt, 2000; Gebhardt et al., 2000). Although, major mergers and
powerful AGNs are far less common in the local Universe, compared to redshift
$z>1$, the difference found in the current work could be partly due to the co-
evolution of the bulge and AGN activity after such events.

Alternatively, or in addition, the difference could also be due to the
obscuration of the BLR from the gas- and dust-rich disc of a spiral galaxy,
especially in edge-on systems. This possibility is discussed further in the
next section.

###  4.2 Why do Sy1 nuclei favour face-on orientations of the host galaxy?

Our analysis also showed that the orientations of the Sy1 and Sy2 spiral host
galaxies are significantly different when compared to the control sample of
the spiral SF galaxies. This is also unexpected within the simplest
unification scheme, and the interpretation of our results points towards two
possible scenarios:

  1. 1.

additional obscuration of the AGN by dust in the galactic disc near the
circum-nuclear region

  2. 2.

some level of statistical co-alignment of the plane of the torus and that of
the galactic disc.

According to the latter scenario, Sy1 galaxies, which have a ’face-on’ torus,
also have more frequently face-on host galaxy orientations. However, if that
is the case, we should correspondingly expect a higher fraction of ’edge-on’
host galaxies in the Sy2 distribution with respect to SF galaxies, but this is
not observed. Without excluding this scenario, since processes like merging
may induce a rearrangement and facilitate such a co-planarity, we do not have
a clear indication to confirm this hypothesis (see, however, Maiolino & Rieke
(1995) for a relevant discussion).

Our results favour the scenario where additional obscuration, caused by the
host galaxy, might affect the classification of Seyfert types, regardless of
the inclination of the torus, as also claimed by Lagos et al. (2011). This
result is similar to the one found by Keel (1980), who was the first to
discover a deficiency of Sy1 galaxies in edge-on hosts (see also Lawrence &
Elvis, 1982). Recent works support the idea that galactic-scale dust is
responsible for additional obscuration (e.g. Burtscher et al., 2016). Buchner
& Bauer (2017) showed that a large fraction (up to $40\%$) of the obscuration
observed in AGNs, at least in the Compton-thin regime, is not due to the
nuclear torus, but to the galaxy-scale gas in the host, while DiPompeo et al.
(2017b) estimated the fraction of IR-selected type 2 AGNs that are obscured by
dust outside the torus to be $\sim 25\%$. Also, using data from the
intermediate Palomar Transient Factory, the SDSS DR7, and Galaxy Zoo,
Villarroel et al. (2017) found that for the majority of Sy2 galaxies (up to
$90\%$), the obscuration must stem from larger scales: host galaxy obscuration
and/or a large-scale environment in which the host galaxies reside.

Furthermore, having studied a sample of nearby Compton-thick AGNs, Goulding et
al. (2012) concluded that the dust of the host galaxy, and not necessarily the
compact torus, is the dominant obscurer of the central engine. Using Spitzer
data of six $0.3<z<0.8$ type 2 quazars, Lacy et al. (2007) found a
contribution of the extinction towards the nucleus from an extended star-
forming disc on scales of kiloparsecs, in addition to, or instead of, the
traditional dusty torus. Moreover, also using high-redshift type 2 quasars
from Spitzer and VLA data of the Spitzer First Look Survey, Martínez-Sansigre
et al. (2006b) concluded that the nuclear region could be effectively obscured
by dust on large scales, away from the torus. Additionaly, using X-ray-
selected AGNs with spectroscopic redshifts in the Chandra Deep Field South
(CDFS), Rigby et al. (2006) argued that part of the column density that
obscures the soft X-rays may come from the galactic disc. Last but not least,
Malizia et al. (2020) used the hard X-ray-selected sample of AGNs detected by
INTEGRAL/IBIS to show that material located in the host galaxy on scales of
hundreds of parsecs and not aligned with the absorbing torus can sufficiently
hide the BLR of some type 1 AGNs, causing their classification as type 2
objects and giving rise to the deficiency of type 1 in edge-on galaxies.

Although the deficiency of edge-on Sy1 galaxies is unexpected in the simplest
unification model, it does not necessarily contradict it. It might be assumed
that in the case of edge-on Seyfert galaxies the gas and dust along the host
galaxy disc can act in the same way as the torus, blocking the direct view of
the BLR, thus leading to a classification as a Sy2 galaxy (e.g. Schmitt et
al., 2001).

###  4.3 Why there is a deficit of Sy2 nuclei in edge-on and face-on spiral
hosts with respect to SF galaxies?

Regarding the small but significant difference in the $b/a$ distributions of
the spiral Sy2 and SF galaxies (Fig.7), at low $b/a$ values a possible
explanation could be that in extreme edge-on orientations the dust of the
galactic disc can obscure not only the BLR but also the NLR region. In an
early study, McLeod & Rieke (1995) used samples of optically and soft-X-ray-
selected Seyferts and found a bias against inclined spiral hosts, while hard-
X-ray-selected samples were found unbiased. In addition, Malkan et al. (1998)
argued that in Sy2 galaxies irregular structures at large distances can
provide sufficient absorbing column density for the nuclear source. Similar
results were presented in Rigby et al. (2006), where they attributed the
appearance of X-ray-selected AGNs as optically ’dull’ to galactic absorption.

On the other hand, there is a hypothesis that low luminosity Seyferts may be
diluted by high-luminosity host galaxies, in which the continuum can hide the
AGN lines. This may be the case for high-$z$ AGNs, where the source fully
falls within the spectroscopic fiber or slit, as demonstrated by Moran &
Filippenko (2002) and later supported by Trump et al. (2009) for a sample of
high-$z$ dull AGNs in the COSMOS survey. However, our sample is limited to
$z<0.2,$ and furthermore, relevant studies showed that there is no significant
dilution in dull local AGN samples (La Franca et al., 2002; Hornschemeier et
al., 2005). In addition, Rigby et al. (2006) argued against dilution also in
high-$z$ AGNs. Therefore, we do not consider this scenario as a possible
explanation for our results. A co-alignment of the torus with the disc cannot
explain the deficit of Sy2 galaxies in edge-on systems either when compared
with the control SF sample.

Finally, we note that the smaller (higher) fraction of Sy2 (Sy1) with respect
to SF galaxies at high b/a values strengthens the hypothesis of a host galaxy
contribution to the obscuration of the BLR. Specifically, due to the fact that
in face-on galaxies the gas and the dust of the disc does not intervene
between the observer and the active nuclei, the only obscurer of the BLR is
the torus, which apparently in some cases is not sufficient to hide the BLR,
giving rise to the deficiency of type 2 in face-on galaxies (e.g. Lacy et al.,
2007).

In the context of matching the samples of Sy1, Sy2, and control SF galaxies to
stellar mass, the lack of ’hidden’ Sy2 at small b/a values compared to the
unmatched case can be explained, as we point out in Sect. 3.2.2, by the fact
that the matching procedure cuts the most massive Sy2 and Sy1 galaxies, which
are also the most abundant. Taking into account that more massive spiral
galaxies contain more dust (e.g. Whitaker et al., 2017), by excluding them
from our samples we effectively exclude those host galaxies that can obscure
both the AGN BLR and the NLR at extreme edge-on configurations. This result
further supports the scenario of a host galaxy contribution to the obscuration
of the AGN activity.

##  5 Conclusions

The main purpose of the current work was to test aspects of the simplest
unification paradigm by searching for differences in the properties of the
host galaxies of various AGN types. For our purposes, we used (a) the SDSS
DR14 spectroscopic galaxy catalogue, selecting sub-samples of Sy1 and Sy2,
LINER, Composite and SF galaxies, limited to $z<0.2,$ and (b) the results of
the Galaxy Zoo project regarding the Hubble-type morphology of these galaxies.

Our main results are listed below:

  1. 1.

We find statistically significant differences -quantified by a KS two-sample
test- of the various types of AGN host-galaxy Hubble types, with the most
significant result being that the fraction of Sy1 galaxies hosted by
ellipticals is higher than that of any other AGN class. These results can be
interpreted within a possible co-evolution scenario between galaxies and SMBS.

  2. 2.

We also find that the orientation distributions, as revealed by the disc axis
ratio ($b/a$) of the Sy1 and Sy2 spiral populations show statistically
significant differences with respect to the control sample of star-forming
galaxies (which by definition should cover all possible orientation
configurations), which is in conflict with the predictions of the simple
unified model. These differences hint towards an effect by which the dusty
galactic disc has a significant contribution to the obscuration of the broad-
line and partially also of the narrow-line nuclear region. This could also
interpret our previous result regarding the host-galaxy Hubble types of type 1
AGN. Indeed, the fact that we detect more Sy1 than any other AGN type in
elliptical hosts (which as is well known are gas and dust deficient) can be
explained if the amount of galactic dust and gas contributes to the
obscuration of the nuclear region and in particular the BLR.

These findings are at odds with the expectations of the simplest formulation
of the unification paradigm, implying that beyond the orientation of the
torus, at least two other factors play an important role in the AGN
classification: (a) the orientation of the host galaxy, and (b) evolution.

###### Acknowledgements.

Authors thank the anonymous referee for useful comments which helped us
improve the presentation of this work. This paper has made use of the data
from the SDSS projects. Funding for the Sloan Digital Sky Survey IV has been
provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy
Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High-Performance Computing at the
University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by
the Astrophysical Research Consortium for the Participating Institutions of
the SDSS Collaboration including the Brazilian Participation QUASAR BLACK HOLE
MASSES 25 Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofisica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the Korean
Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut
für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatario Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autonoma de Mexico,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.

## References

  * Abolfathi et al. (2018) Abolfathi, B., Aguado, D. S., Aguilar, G., et al. 2018, ApJS, 235, 42
  * Antonucci (1993) Antonucci, R. 1993, ARA&A, 31, 473
  * Audibert et al. (2017) Audibert, A., Riffel, R., Sales, D. A., Pastoriza, M. G., & Ruschel-Dutra, D. 2017, MNRAS, 464, 2139
  * Baldwin et al. (1981) Baldwin, J. A., Phillips, M. M., & Terlevich, R. 1981, PASP, 93, 5
  * Bitsakis et al. (2010) Bitsakis, T., Charmandaris, V., Le Floc’h, E., et al. 2010, A&A, 517, A75
  * Bornancini & García Lambas (2018) Bornancini, C. & García Lambas, D. 2018, MNRAS, 479, 2308
  * Bornancini & García Lambas (2020) Bornancini, C. & García Lambas, D. 2020, MNRAS, 494, 1189
  * Bornancini et al. (2017) Bornancini, C. G., Taormina, M. S., & Lambas, D. G. 2017, A&A, 605, A10
  * Brinchmann et al. (2004) Brinchmann, J., Charlot, S., Heckman, T. M., et al. 2004, arXiv e-prints, astro
  * Buchner & Bauer (2017) Buchner, J. & Bauer, F. E. 2017, MNRAS, 465, 4348
  * Burtscher et al. (2016) Burtscher, L., Davies, R. I., Graciá-Carpio, J., et al. 2016, A&A, 586, A28
  * Chen et al. (2015) Chen, C.-T. J., Hickox, R. C., Alberts, S., et al. 2015, ApJ, 802, 50
  * Chen & Hwang (2017) Chen, Y.-C. & Hwang, C.-Y. 2017, Astrophysics and Space Science, 362
  * DiPompeo et al. (2017a) DiPompeo, M. A., Hickox, R. C., Eftekharzadeh, S., & Myers, A. D. 2017a, MNRAS, 469, 4630
  * DiPompeo et al. (2017b) DiPompeo, M. A., Hickox, R. C., Myers, A. D., & Geach, J. E. 2017b, MNRAS, 464, 3526
  * Elitzur (2012) Elitzur, M. 2012, ApJ, 747, L33
  * Elitzur & Ho (2009) Elitzur, M. & Ho, L. C. 2009, ApJ, 701, L91
  * Elitzur et al. (2014) Elitzur, M., Ho, L. C., & Trump, J. R. 2014, MNRAS, 438, 3340
  * Elitzur & Shlosman (2006) Elitzur, M. & Shlosman, I. 2006, ApJ, 648, L101
  * Ferrarese & Merritt (2000) Ferrarese, L. & Merritt, D. 2000, ApJ, 539, L9
  * Gebhardt et al. (2000) Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13
  * González et al. (2008) González, J. J., Krongold, Y., Dultzin, D., et al. 2008, in Revista Mexicana de Astronomia y Astrofisica Conference Series, Vol. 32, Revista Mexicana de Astronomia y Astrofisica Conference Series, 170–172
  * Goulding et al. (2012) Goulding, A. D., Alexander, D. M., Bauer, F. E., et al. 2012, ApJ, 755, 5
  * He et al. (2018) He, Z., Sun, A.-L., Zakamska, N. L., et al. 2018, MNRAS, 478, 3614
  * Heckman & Best (2014) Heckman, T. M. & Best, P. N. 2014, ARA&A, 52, 589
  * Hernández-Ibarra et al. (2016) Hernández-Ibarra, F. J., Krongold, Y., Dultzin, D., et al. 2016, MNRAS, 459, 291
  * Hickox & Alexander (2018) Hickox, R. C. & Alexander, D. M. 2018, ARA&A, 56, 625
  * Hopkins et al. (2008a) Hopkins, P. F., Cox, T. J., Kereš, D., & Hernquist, L. 2008a, ApJS, 175, 390
  * Hopkins et al. (2006) Hopkins, P. F., Hernquist, L., Cox, T. J., et al. 2006, ApJS, 163, 1
  * Hopkins et al. (2008b) Hopkins, P. F., Hernquist, L., Cox, T. J., & Kereš, D. 2008b, ApJS, 175, 356
  * Hornschemeier et al. (2005) Hornschemeier, A. E., Heckman, T. M., Ptak, A. F., Tremonti, C. A., & Colbert, E. J. M. 2005, AJ, 129, 86
  * Hubble (1926) Hubble, E. 1926, Contributions from the Mount Wilson Observatory / Carnegie Institution of Washington, 324, 1
  * Jiang et al. (2016) Jiang, N., Wang, H., Mo, H., et al. 2016, ApJ, 832, 111
  * Kauffmann et al. (2003) Kauffmann, G., Heckman, T. M., White, S. D. M., et al. 2003, MNRAS, 341, 54
  * Keel (1980) Keel, W. C. 1980, AJ, 85, 198
  * Kinney et al. (2000) Kinney, A. L., Schmitt, H. R., Clarke, C. J., et al. 2000, The Astrophysical Journal, 537, 152–177
  * Kormendy & Ho (2013) Kormendy, J. & Ho, L. C. 2013, ARA&A, 51, 511
  * Koulouridis (2014) Koulouridis, E. 2014, A&A, 570, A72
  * Koulouridis et al. (2013) Koulouridis, E., Plionis, M., Chavushyan, V., et al. 2013, A&A, 552, A135
  * Koulouridis et al. (2006) Koulouridis, E., Plionis, M., Chavushyan, V., et al. 2006, ApJ, 639, 37
  * Krongold et al. (2002) Krongold, Y., Dultzin-Hacyan, D., & Marziani, P. 2002, ApJ, 572, 169
  * La Franca et al. (2002) La Franca, F., Fiore, F., Vignali, C., et al. 2002, ApJ, 570, 100
  * Lacy et al. (2007) Lacy, M., Sajina, A., Petric, A. O., et al. 2007, ApJ, 669, L61
  * Lagos et al. (2011) Lagos, C. D. P., Padilla, N. D., Strauss, M. A., Cora, S. A., & Hao, L. 2011, MNRAS, 414, 2148
  * Lanzuisi et al. (2017) Lanzuisi, G., Delvecchio, I., Berta, S., et al. 2017, A&A, 602, A123
  * Lawrence & Elvis (1982) Lawrence, A. & Elvis, M. 1982, ApJ, 256, 410
  * Lintott et al. (2008) Lintott, C. J., Schawinski, K., Slosar, A., et al. 2008, MNRAS, 389, 1179
  * Madau & Dickinson (2014) Madau, P. & Dickinson, M. 2014, Annual Review of Astronomy and Astrophysics, 52, 415–486
  * Magorrian et al. (1998) Magorrian, J., Tremaine, S., Richstone, D., et al. 1998, AJ, 115, 2285
  * Maiolino & Rieke (1995) Maiolino, R. & Rieke, G. H. 1995, ApJ, 454, 95
  * Malizia et al. (2020) Malizia, A., Bassani, L., Stephen, J. B., Bazzano, A., & Ubertini, P. 2020, A&A, 639, A5
  * Malkan et al. (1998) Malkan, M. A., Gorjian, V., & Tam, R. 1998, ApJS, 117, 25
  * Martínez et al. (2008) Martínez, M. A., del Olmo, A., Coziol, R., & Focardi, P. 2008, ApJ, 678, L9
  * Martínez-Sansigre et al. (2006a) Martínez-Sansigre, A., Rawlings, S., Lacy, M., et al. 2006a, Astronomische Nachrichten, 327, 266
  * Martínez-Sansigre et al. (2006b) Martínez-Sansigre, A., Rawlings, S., Lacy, M., et al. 2006b, Astronomische Nachrichten, 327, 266
  * Masoura et al. (2021) Masoura, V. A., Mountrichas, G., Georgantopoulos, I., & Plionis, M. 2021, arXiv e-prints, arXiv:2101.00724
  * Matt (2000) Matt, G. 2000, A&A, 355, L31
  * McLeod & Rieke (1995) McLeod, K. K. & Rieke, G. H. 1995, ApJ, 441, 96
  * Melnyk et al. (2018) Melnyk, O., Elyiv, A., Smolčić, V., et al. 2018, A&A, 620, A6
  * Mendoza-Castrejón et al. (2015) Mendoza-Castrejón, S., Dultzin, D., Krongold, Y., González, J. J., & Elitzur, M. 2015, MNRAS, 447, 2437
  * Moran & Filippenko (2002) Moran, E. C. & Filippenko, A. V. 2002, in American Astronomical Society Meeting Abstracts, Vol. 200, American Astronomical Society Meeting Abstracts #200, 17.04
  * Netzer (2015) Netzer, H. 2015, ARA&A, 53, 365
  * Nicastro et al. (2003) Nicastro, F., Martocchia, A., & Matt, G. 2003, ApJ, 589, L13
  * Perlman et al. (2007) Perlman, E. S., Mason, R. E., Packham, C., et al. 2007, ApJ, 663, 808
  * Powell et al. (2018) Powell, M. C., Cappelluti, N., Urry, C. M., et al. 2018, The Astrophysical Journal, 858, 110
  * Prieto et al. (2014) Prieto, M. A., Mezcua, M., Fernández-Ontiveros, J. A., & Schartmann, M. 2014, MNRAS, 442, 2145
  * Raddick et al. (2007) Raddick, J., Lintott, C. J., Schawinski, K., et al. 2007, in American Astronomical Society Meeting Abstracts, Vol. 211, 94.03
  * Ramos Almeida et al. (2011) Ramos Almeida, C., Levenson, N. A., Alonso-Herrero, A., et al. 2011, The Astrophysical Journal, 731, 92
  * Rigby et al. (2006) Rigby, J. R., Rieke, G. H., Donley, J. L., Alonso-Herrero, A., & Pérez-González, P. G. 2006, ApJ, 645, 115
  * Salim et al. (2007) Salim, S., Rich, R. M., Charlot, S., et al. 2007, ApJS, 173, 267
  * Schmitt et al. (2001) Schmitt, H. R., Antonucci, R. R. J., Ulvestad, J. S., et al. 2001, ApJ, 555, 663
  * Siebenmorgen et al. (2015) Siebenmorgen, R., Heymann, F., & Efstathiou, A. 2015, A&A, 583, A120
  * Slavcheva-Mihova & Mihov (2011) Slavcheva-Mihova, L. & Mihov, B. 2011, A&A, 526, A43
  * Sorrentino et al. (2006) Sorrentino, G., Radovich, M., & Rifatto, A. 2006, A&A, 451, 809
  * Spinoglio & Fernández-Ontiveros (2019) Spinoglio, L. & Fernández-Ontiveros, J. A. 2019, arXiv e-prints, arXiv:1911.12176
  * Springel et al. (2005) Springel, V., Di Matteo, T., & Hernquist, L. 2005, MNRAS, 361, 776
  * Suh et al. (2019) Suh, H., Civano, F., Hasinger, G., et al. 2019, The Astrophysical Journal, 872, 168
  * Tran (2001) Tran, H. D. 2001, ApJ, 554, L19
  * Tran (2003) Tran, H. D. 2003, ApJ, 583, 632
  * Tremonti et al. (2004) Tremonti, C. A., Heckman, T. M., Kauffmann, G., et al. 2004, ApJ, 613, 898
  * Trump et al. (2011) Trump, J. R., Impey, C. D., & Kelly, B. C. 2011, in American Astronomical Society Meeting Abstracts, Vol. 217, American Astronomical Society Meeting Abstracts #217, 430.08
  * Trump et al. (2009) Trump, J. R., Impey, C. D., Taniguchi, Y., et al. 2009, ApJ, 706, 797
  * Urry & Padovani (1995) Urry, C. M. & Padovani, P. 1995, PASP, 107, 803
  * Villarroel et al. (2012) Villarroel, B., Korn, A., & Matsuoka, Y. 2012, arXiv e-prints, arXiv:1211.0528
  * Villarroel & Korn (2014) Villarroel, B. & Korn, A. J. 2014, Nature Physics, 10, 417
  * Villarroel et al. (2017) Villarroel, B., Nyholm, A., Karlsson, T., et al. 2017, ApJ, 837, 110
  * Whitaker et al. (2017) Whitaker, K. E., Pope, A., Cybulski, R., et al. 2017, ApJ, 850, 208
  * Yang et al. (2019) Yang, G., Brandt, W. N., Alexander, D. M., et al. 2019, MNRAS, 485, 3721
  * Yang et al. (2017) Yang, G. et al. 2017, Astrophys. J., 842, 72
  * Zou et al. (2019) Zou, F., Yang, G., Brandt, W. N., & Xue, Y. 2019, arXiv e-prints, arXiv:1904.13286

##  Appendix A Example of Sy1 and Sy2 spectra

The authors, in order to demonstrate that the final Sy1 sample consists of
well-selected type 1 AGNs, present ten random Sy1 optical spectra (11),
limited to $z<0.2$, from the SDSS database. On the other hand, they
comparatively present a sample of ten random Sy2 optical spectra (12) with
redshifts up to $z=0.2$.

Figure 11: Sample of ten random optical spectra of type 1 Seyfert AGNs. The
identified emission lines are indicated. These spectra were randomly picked
from different redshift ranges up to $0.2$. credits to the SDSS.

Figure 12: Sample of ten random optical spectra of type 2 Seyfert AGNs. The
identified emission lines are indicated. These spectra were randomly picked
from different redshift ranges up to $0.2$. credits to the SDSS.


# Qubit-efficient Variational Quantum Algorithms for Image Segmentation

Supreeth Mysore Venkatesh1,2, Antonio Macaluso2, Marlon Nuske3,
Matthias Klusch2, Andreas Dengel1,3
1University of Kaiserslautern-Landau (RPTU), Kaiserslautern, Germany
2German Research Center for Artificial Intelligence (DFKI), Saarbruecken,
Germany
3German Research Center for Artificial Intelligence (DFKI), Kaiserslautern,
Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quantum computing is expected to transform a range of computational tasks
beyond the reach of classical algorithms. In this work, we examine the
application of variational quantum algorithms (VQAs) for unsupervised image
segmentation to partition images into separate semantic regions. Specifically,
we formulate the task as a graph cut optimization problem and employ two
established qubit-efficient VQAs, which we refer to as Parametric Gate
Encoding (PGE) and Ancilla Basis Encoding (ABE), to find the optimal
segmentation mask. In addition, we propose Adaptive Cost Encoding (ACE), a new
approach that leverages the same circuit architecture as ABE but adopts a
problem-dependent cost function. We benchmark PGE, ABE and ACE on
synthetically generated images, focusing on quality and trainability. ACE
shows consistently faster convergence in training the parameterized quantum
circuits in comparison to PGE and ABE. Furthermore, we provide a theoretical
analysis of the scalability of these approaches against the Quantum
Approximate Optimization Algorithm (QAOA), showing a significant cutback in
the quantum resources, especially in the number of qubits that logarithmically
depends on the number of pixels. The results validate the strengths of ACE,
while concurrently highlighting its inherent limitations and challenges. This
paves way for further research in quantum-enhanced computer vision.

###### Index Terms:

Quantum algorithms, variational circuits, image segmentation, combinatorial
optimization

##  I Introduction

Variational Quantum Algorithms (VQAs) aim to leverage NISQ devices for
practical uses through a hybrid quantum-classical approach [1]. VQAs address
classical problems, which lack polynomial-time solutions but have verifiable
answers, showcasing their potential for real-world applications. One
application area that highlights the potential of VQAs is in solving Quadratic
Unconstrained Binary Optimization (QUBO) problems, an NP-Hard class of
optimization problems with applications in various domains, including
logistics, finance, and machine learning [2]. The Quantum Approximate
Optimization Algorithm (QAOA) represents a notable example of VQAs applied to
such challenges [3]. While classical computers can handle QUBO for instances
with several thousand variables, running QAOA on similar tasks requires
thousands of qubits, a requirement that appears out of reach given the
constraints of near-term quantum technology. The scalability of QAOA is
inherently limited by the linear relationship between the number of binary
variables in QUBO problems and the number of logical qubits required,
underscoring the pressing need to explore more efficient encoding strategies.
These strategies are designed to mitigate the hardware limitations of current
quantum devices while simultaneously unlocking new possibilities for
efficiently addressing complex tasks that classical computing capabilities
cannot solve.

Figure 1: The figure depicts an architecture for segmenting an image.
Generating a graph representation of the image, the distinct semantic regions
in the image are identified by finding the minimum cut in the graph that can
be formulated as Quadratic Unconstrained Binary Optimization (QUBO). The QUBO
problem is subsequently solved using a VQA by optimizing the circuit
parameters that incidentally explore an exponentially large solution space to
locate the global optimum, which can be decoded as the segmentation mask.

In this work, we adapt two existing variational quantum algorithms, which we
refer to as Parametric Gate Encoding (PGE) [4] and Ancilla Basis Encoding
(ABE) [5], to address the task of image segmentation formulated as a
combinatorial optimization problem [6]. PGE encodes the segmentation solution
in the parameterized gates exerted in the quantum ansatz, while ABE [5] uses
the probability distribution of the qubits’ basis states for encoding the
solution [4]. Furthermore, we propose an enhancement of ABE, termed Ancilla
Cost Encoding (ACE) that uses a problem-specific cost function and
demonstrates an improved performance. Importantly, these algorithms require
exponentially fewer qubits than QAOA and its variants, scaling logarithmically
with the number of image pixels. This represents a significant improvement in
scalability for image segmentation tasks using VQAs. We begin by reformulating
the image segmentation problem as a graph partition problem, subsequently
framed as a QUBO. The experiments are focused on analyzing the solution
quality and the efficiency of various classical optimizers for these
algorithms. We also provide a theoretical analysis of essential quantum
resource requirements as a function of the number of pixels of the input
image. For a well-rounded study, we also discuss the specific limitations of
each approach. Our findings open new research directions for leveraging
quantum computing in computer vision, particularly in the optimization of
quantum resources for complex computational tasks.

##  II Related Works

Supervised deep learning models have set benchmarks in image segmentation by
leveraging precisely annotated data. However, the collection of such labeled
datasets faces hurdles, including significant annotation time and costs, the
necessity of specialized knowledge, and issues scaling to vast datasets, not
to mention the inconsistencies arising from variable interpretations by
different annotators [7]. These limitations highlight the importance of
unsupervised segmentation methods, especially valuable for leveraging
extensive unlabeled data pools [8]. Nonetheless, the computational demand of
some unsupervised techniques, like graph-based segmentation [9], presents a
significant limitation, making these tasks ideal targets for quantum computing
to promise more efficient solutions.

A wide range of research has explored the use of quantum algorithms for
computer vision tasks, primarily focusing on image classification [10, 11, 12,
13] and matching [14, 15]. Additionally, quantum-inspired classical techniques
have been applied to segmentation [16] and edge-detection [17, 18], blending
quantum concepts with traditional computing infrastructure. In particular,
quantum annealers and hybrid quantum-classical approaches have demonstrated
promise for image segmentation, converting these challenges into a QUBO
problem [19, 20, 6]. Despite their innovative methods, they face limitations
in scalability and are constrained by the current capabilities of quantum
annealers for handling high-resolution images.

In contrast, our work focuses on leveraging the versatility of VQAs,
particularly suited for NISQ devices, to overcome the scalability limitations
in existing quantum and hybrid approaches. Unlike methods requiring extensive
quantum resources or fault tolerance, we adapt established encoding strategies
to harness the potential of quantum circuits in enhancing computer vision
techniques, specifically image segmentation, through efficient quantum
resource utilization.

##  III Preliminaries

The segmentation process involves representing an input image as a lattice
graph with nodes corresponding to pixels and edges representing pixel
similarity. The approach is particularly interesting as it can capture both
spatial and spectral information in the image [9]. Segmentation seeks to
partition this graph’s vertices into mutually exclusive subsets, such that the
total weight of the edges connecting vertices from one subset to another is
minimized, defined as:

$\textsc{MinimumCut}(G)=\arg\min_{A,\overline{A}}\sum_{i\in
A,j\in\overline{A}}w(v_{i},v_{j})$ (1)

where $G$ denotes the graph and $w(v_{i},v_{j})$ is the weight of the edge
connecting the nodes $v_{i}$ and $v_{j}$. Given the NP-hard nature of the min-
cut problem for arbitrary weights, classical solutions become computationally
challenging, motivating the reformulation into a QUBO problem to exploit the
power of quantum computers [21].

#### Example

Let’s consider an image of size $2\times 2$, which is converted to a grid
graph with a one-to-one mapping of the pixels to the vertices. The edge
weights are assigned as the similarity measure between the neighboring pixels.
Finding the minimum cut in this grid graph will eventually give the
segmentation of the image.

Figure 2: Transformation of a $2\times 2$ pixel image into a grid graph, where
edge weights indicate pixel similarity. The red dashed line depicts the edges
that are cut to partition the graph.

Focusing on the capabilities of existing NISQ technology, we target near-
optimal solutions as they are often tolerable for downstream computer vision
tasks. With $n$ vertices, the binary encoding of each vertex in graph $G$
facilitates the representation of potential solutions as vector
$\in\\{0,1\\}^{n}$ delineating the vertices into two disjoint subsets, with
edges intersecting these subsets identified as cuts. Such an encoding allows
to reformulate the minimum cut as a QUBO without loss of generality as[22,
23]:

$\displaystyle\vec{x}^{*}$ $\displaystyle=\arg\min_{\vec{x}}\sum_{1\leq
i<j\leq n}x_{v_{i}}(1-x_{v_{j}})w(v_{i},v_{j})$ (2)
$\displaystyle=\arg\min_{\vec{x}}\ {\vec{x}}^{T}Q{\vec{x}}$

where $\vec{x}^{*}$ represents the optimal solution for the minimum cut of
$G$. The construction of the $Q\in\mathcal{R}^{n\times n}$ matrix is derived
from the coefficients of terms in the quadratic expression of the binary
variables $x_{v_{i}}$ for $i\in\\{1,2,...,n\\}$ obtained by substituting the
edge weights $w(v_{i},v_{j})$ and simplifying.

QAOA, a special case of the Variational Quantum Eigensolver (VQE) [24], is
designed for QUBO problems, implementing a parameterized quantum circuit to
approximate the problem’s ground state. The binary encoding strategy employed
for solving the image segmentation problem using QAOA leads to scalability
challenges of representing high-resolution images with individual qubits for
each pixel. In contrast to the problem-specific ansatz of QAOA, the methods we
explore use a problem-agnostic ansatz. This gives rise to a need for efficient
encoding strategies within the VQA framework to address image segmentation,
requiring fewer qubits and offering scalable solutions for practical
applications.

##  IV Methods

In this section, we will explain the two methods and describe a new approach
to obtain the segmentation of an image employing innovative encoding
strategies for solving the QUBO problem.

###  IV-A Parametric Gate Encoding (PGE)

Given a grid graph $G$ constructed from the input image, the Laplacian matrix
$L_{G}$ provides a compact representation of the graph’s structure, capturing
information about the connectivity and degree of each vertex, and is given by
$L_{G}=D_{G}-A_{G}$, where $D_{G}$ is the degree matrix, a diagonal matrix
with vertex degrees $d(v_{i})$ for $v_{i}$ on the diagonal, and $A_{G}$ is the
adjacency matrix, a square matrix where the element $A_{G}[i,j]$ is the weight
of the edge between the vertices $v_{i}$ and $v_{j}$. If the dimension of the
matrix $L_{G}$ is not a power of $2$, without loss of generality, we pad the
matrix with zeroes to the nearest power of $2$ to ensure that the matrix
dimensions are compatible with the quantum register, as quantum states are
represented in a Hilbert space of dimension $n$ for $\log_{2}(n)$ qubits.
Since the Laplacian is a symmetric matrix consisting of real-valued edge
weights for undirected weighted graphs, it is also Hermitian and thus serves
as a suitable quantum observable [4].

The PGE method constructs a parameterized quantum circuit with
$n^{\prime}=\lceil\log_{2}(n)\rceil$ qubits, initializing with Hadamard gates
for equal superposition and applying a diagonal gate as the ansatz which can
be expressed as [4]:

$\hat{U}(\vec{\Theta})=\begin{pmatrix}\exp(i\pi f(\theta_{1}))&0&\cdots&0\\\
0&\exp(i\pi f(\theta_{2}))&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\exp(i\pi f(\theta_{2^{n^{\prime}}}))\end{pmatrix}$

(3)

where $\vec{\Theta}=(\theta_{1},\theta_{2},...\theta_{2^{n^{\prime}}})$, and
$\theta_{i}\in[0,2\pi]$ for $i\in\\{1,2,...,2^{n^{\prime}}\\}$ and $f$ is
piecewise function defined as:

$f(\theta_{i})=x_{v_{i}}=\begin{cases}0&0\leq\theta<\pi\\\
1&\pi\leq\theta<2\pi\\\ \end{cases}\forall i\in\\{1,2,...,2^{n^{\prime}}\\}$
(4)

with $n^{\prime}-2$ two-qubit CNOT gates and $n^{\prime}$ single-qubit
parametric gates, where each parametric gate in the variational ansatz encodes
a binary decision for the segmentation mask. With $L_{G}$ as the observable,
the measurement operation is performed, and the energy of the system is
evaluated as:

$C(\vec{\Theta})=\frac{2^{n^{\prime}}}{2}\langle\psi(\vec{\Theta})|L_{G}|\psi(\vec{\Theta})\rangle$
(5)

Since $L_{G}$ is also a Hermitian matrix, it is decomposed into a linear
combination of tensor products of Pauli matrices, enabling the evaluation of
the quantum observable corresponding to the graph structure. The parameters
$\vec{\Theta}$ are adjusted iteratively using a classical optimizer to
minimize the cost function (Eq. 5). The optimal parameters
$\vec{\Theta}=\\{\theta_{1},\theta_{2},...\theta_{2^{n^{\prime}}}\\}$ are
decoded to binary values (Eq. 4), obtaining the minimum cut and eventually the
binary segmentation mask of the image.

Figure 3: Circuit schematic of ABE/ACE for finding the segmentation mask of
the example image in Fig. 2. illustrating the encoding of the solution of a
QUBO problem of size $4$ into a quantum state facilitated by 3 qubits:
$q_{1},q_{2}$ are the register qubits and $q_{3}$ is the ancilla qubit. The
circuit initialization employs Hadamard gates to induce superposition across
all qubits, followed by two layers of hardware-efficient ansatz with
entanglement and rotation gates parameterized by $\vec{\Theta}$. Executing and
measuring the circuit obtains a probability distribution over the basis
states. For the basis states whose register qubits are the same encodes the
value of the binary variable. For example, with the register qubit basis
states $|\phi_{1}\rangle_{r}=|00\rangle$, we obtain the value $x_{v_{1}}=1$ as
the probability of the ancilla qubit in state $|1\rangle>|0\rangle$ (Eq. 7).

###  IV-B Ancilla Basis Encoding (ABE)

For an image with $n$ pixels to be segmented, the QUBO problem formulated will
also have $n$ variables. The ABE [5] strategy uses only $\log_{2}(n)+1$ number
of qubits, where the $\log(n)$ qubits are called the register qubits and the
additional qubit is termed as the ancilla qubit. Using a hardware efficient
ansatz where each layers consists of $\log_{2}(n)$ CNOT gates between adjacent
qubits and $\log_{2}(n)+1$ parametric $R_{y}(\theta)$ gates for each qubit,
the quantum state is represented as:

$|\psi(\vec{\Theta})\rangle=\sum_{i=1}^{n}\beta_{i}(\vec{\Theta})(a_{i}(\vec{\Theta})|0\rangle_{a}+b_{i}(\vec{\Theta})|1\rangle_{a})\otimes|\phi_{i}\rangle_{r}$
(6)

where $|\phi_{i}\rangle_{r}$ are the computational basis states of the
register qubits, $|0\rangle_{a}$ and $|1\rangle_{a}$ are the states of the
ancilla qubit, $a_{i}$ and $b_{i}$ are the amplitudes for the ancilla qubit,
and $\beta_{i}(\vec{\Theta})$ are coefficients dependent on variational
parameters $\vec{\Theta}$. The expectation measurements obtain a probability
distribution over the basis states of the qubits in the circuit and the
solution is decoded as:

$x_{v_{i}}=\begin{cases}0&|a_{i}|^{2}>|b_{i}|^{2}\\\ 1&otherwise\\\
\end{cases}\vspace*{-6pt}$ (7)

where $|a_{i}|^{2}+|b_{i}|^{2}=1$.

The probability of finding the optimal solution increases as the number of
measurements tends to $\infty$ [5]. While it may appear impractical, for
relatively small quantum circuits, the number of measurement operations only
needs to be large enough to adequately approximate the optimal probability
distribution; it does not necessarily need to be $\infty$. Additionally, well-
established VQAs like QAOA are guaranteed to find the optimal solution only as
the depth of the corresponding quantum circuit tends to infinity. According to
the ABE approach, the quantum circuit is typically initiated by a layer of
Hadamard gates to generate a uniform superposition, followed by a hardware-
efficient ansatz applied to evolve the quantum state. For the parameters
$\vec{\Theta}$, the cost is given by:

$C(\vec{\Theta})=\sum_{i,j=1}^{n}Q_{ij}\frac{\langle\hat{P}^{1}_{i}\rangle_{\vec{\Theta}}\langle\hat{P}^{1}_{j}\rangle_{\vec{\Theta}}}{\langle\hat{P}_{i}\rangle_{\vec{\Theta}}\langle\hat{P}_{j}\rangle_{\vec{\Theta}}}(1-\delta_{ij})+\sum_{i=1}^{n}Q_{ii}\frac{\langle\hat{P}^{1}_{i}\rangle_{\vec{\Theta}}}{\langle\hat{P}_{i}\rangle_{\vec{\Theta}}}$
(8)

where $\hat{P}_{i}$ are projectors over the basis states of the register
qubits $|\phi_{i}\rangle_{r}$, irrespective of the state of the ancilla qubit,
$\delta_{ij}=1$ when $i=j$ and $Q$ is the matrix representing the QUBO
problem. The objective is to find a binary vector $\vec{x}^{*}$ that minimizes
$\vec{x}^{\top}Q\vec{x}$. The expectation values $\langle\hat{P}_{i}\rangle$
embody the probabilities of observing the qubit states that encode the binary
variables $x_{v_{i}}$ of the QUBO solution vector $\vec{x}$. The projectors
$P_{i}$ and $\hat{P}^{1}_{i}$ are operators that map the quantum state onto
the respective basis states of the register qubits, with $P_{i}$ aligning with
the state that encodes $x_{v_{i}}=1$. $\hat{P}^{1}_{i}$ modifies this to
include the state of the ancilla qubit [5]. Finally, an optimizer running on a
classical computer optimizes the variational parameters $\vec{\Theta}$ to
minimize $C(\vec{\Theta})$ (Eq. 8).

###  IV-C Adaptive Cost Encoding (ACE)

We propose a modified cost function using the same encoding strategy as ABE.
During each iteration of the optimization, once we measure the circuit
multiple times, we get a probability distribution that can be decoded into a
binary vector $\vec{x}$ that represents a possible solution. Instead of using
the cost function in Eq. 8, we can make use of the min-cut problem’s cost
function i.e.,

$C(\vec{x})=\sum_{1\leq i<j\leq
n}x_{v_{i}}(1-x_{v_{j}})w(v_{i},v_{j})\vspace*{-6pt}$ (9)

where  $\vec{x}=x_{v_{1}},x_{v_{2}},...,x_{v_{n}}$ such that
$x_{v_{i}}\in\\{0,1\\}$ given by Eq. 7, and $w(v_{i},v_{j})\ \forall\
i,j\in\\{1,2,...,n\\}$ is the edge weight between $v_{i}$ and $v_{j}$. In case
of ABE, given two sets of parameters, say $\vec{\Theta}_{1}$ and
$\vec{\Theta}_{2}$, obtaining the binary vectors $\vec{x}_{1}$ and
$\vec{x}_{2}$ respectively (Eq. 6 and 7), if
$C(\vec{\Theta}_{1})<C(\vec{\Theta}_{2})$ (Eq. 8) does not always guarantee
that $C(\vec{x_{1}})<C(\vec{x_{2}})$ (Eq. 9) which implies that the optimal
circuit parameters may not correspond to the minimum cut. Whereas, ACE
guarantees that the optimization process obtains the solution of the original
problem, i.e., the optimal circuit parameters $\vec{\Theta}$ implies the
corresponding $\vec{x}$ minimizes the minimum cut cost function (Eq. 9).
Moreover, the QUBO formulation of the original problem also becomes obsolete
as the cost function directly involves the weights of the edges in the graph
and not the QUBO coefficients. In the following section, through our
experiments, we demonstrate ACE to have improved trainability, a strong
tendency to find better solutions and a more consistent behavior using
different optimizers with our proposed problem-specific cost function compared
to ABE.

Furthermore, the adaptive cost function can be extended to other combinatorial
optimization problems where the problem formulation expresses the solution as
a binary vector and uses the original cost function. This approach can be
crucial for tasks where the optimization problem includes constraints and
reformulating the problem as a QUBO involves integrating the constraints as
penalty terms into the cost function with an appropriate penalty coefficient,
eventually not depicting the original problem [25]. Even an optimal penalty
coefficient should be set, while a very low value can lead to the
consideration of an invalid binary vector as a solution, or a high value can
cause any valid solution to be considered optimal. Whereas, ACE allows finding
the optimal solution of the original problem without modifying the cost
function or the constraints, by setting a high or low cost for binary vectors
that do not obey the constraints depending on whether the problem is
minimization or maximization respectively.

Figure 4: The figure illustrates the performance of PGE, ABE and ACE for
finding the minimum cut on grid graphs of size $2\times 2$ and $4\times 4$
with the layers in the ansatz of the quantum circuit ranging from $1$ to $5$
in terms of the solution quality and the efficiency of the optimization
process. The plots show the average and standard deviation of the relative
errors in the obtained solution’s cost as a measure of quality and the number
of iterations to quantify the trainability using COBYLA, Powell, and SLSQP
optimizers.

#### Example

Considering again the $2\times 2$ image from the previously discussed example
(Fig. 2).

For PGE, the Laplacian $L_{G}$ is of size $4\times 4$ and the encoding scheme
uses a quantum circuit with just $2$ qubits. The circuit consists of an
initial layer of Hadamard gates, followed by the ansatz
$\hat{U}(\vec{\Theta})$ (Eq. 3) which is a diagonal gate of size $4\times 4$.
With $L_{G}$ as the observable, optimizing the cost function and obtaining the
optimal circuit parameters
$\theta^{*}_{1},\theta^{*}_{2},\theta^{*}_{3},\theta^{*}_{4}$, we can decode
the binary mask $x^{*}_{v_{1}},x^{*}_{v_{2}},x^{*}_{v_{3}},x^{*}_{v_{4}}$ (Eq.
4). While this approach is efficient in terms of the number of qubits, the
main drawback is the number of parameters which implies an overhead in the
classical optimization process.

Figure 3 represents the quantum circuit and the encoding strategy for solving
this example. In the cost function (Eq. 8) of ABE, the term
$\langle\hat{P}_{i}\rangle_{\vec{\Theta}}$ is the probability of the basis
state $(|0\rangle_{a}+|1\rangle_{a})\otimes|\phi_{i}\rangle_{r}$, and the term
$\langle\hat{P}^{1}_{i}\rangle_{\vec{\Theta}}$ is the probability of the state
$|1\rangle_{a}\otimes|\phi_{i}\rangle_{r}$, where $|0\rangle_{a}$ and
$|1\rangle_{a}$ are the basis states of the ancilla qubit. Updating the
parameters, a unitary evolution is performed over the state $|\psi_{0}\rangle$
which produces a new state
$|\psi_{1}(\vec{\Theta})\rangle=\hat{U}_{1}(\vec{\Theta})|\psi_{0}\rangle$.
The optimization iteratively adjusts $\vec{\Theta}$, and each iteration
involves running the quantum circuit and evaluating the cost function. A
classical optimizer is used to find the optimal parameters
$\vec{\Theta}_{\text{opt}}$ that minimize the cost function $C(\vec{\Theta})$.
Decoding the final quantum state
$|\psi_{1}(\vec{\Theta}_{\text{opt}})\rangle$, we obtain a binary vector
$\vec{x}$ which represents the solution of the QUBO problem, which is the
binary segmentation mask of the given image. Whereas in the case of ACE, at
each iteration of the optimizer, the measurement is decoded to get the binary
vector $\vec{x}$, and the value of the cost function in Eq. 9 is minimized by
optimizing the circuit parameters $\vec{\Theta}$.

##  V Experiments

###  V-A Experimental Settings

PGE, ABE, ACE algorithms are implemented in Qiskit framework, and the
experiments are executed on a classical computer using Qasm simulator.
Inspired from the existing literature [6], the synthetic dataset consist of
square grid graphs by sampling edge weights from a uniform distribution in the
range [-1,1], and the experiments were executed for sizes $2\times 2$ i.e., 4
pixels, and $4\times 4$ i.e., 16 pixels, as all the approaches discussed
previously behave similarly for images of sizes from $\log_{2}(n+1)$ pixels up
to size $n$. For the reproducibility of results and fair comparison, we have
executed the experiments using $5$ fixed seeds for generating the problem
instances and the initial values for the parameters of the variations
circuits. We evaluated the local optimizers from the Scipy library, detailing
the results for COBYLA, Powell, and SLSQP. For ABE and ACE, we increased the
number of layers up to a maximum of $5$, allowing additional layers to
introduce more parameters for optimization, causing significant overhead for
the classical optimizer. To ensure the solution quality improves with an
increased number of measurements [5], we conducted our experiments with
$65,536$ measurements for each optimization iteration.

We benchmarked the solution quality and optimization efficiency of PGE, ABE,
and ACE against the results from Gurobi, known for its effectiveness in
solving combinatorial optimization problems. Solution quality was evaluated by
calculating the relative error of the minimum cut value from the VQAs relative
to Gurobi, defined as $\text{Relative
Error}=\frac{\left|C_{\text{VQA}}-C_{\text{gurobi}}\right|}{\left|C_{\text{gurobi}}\right|}$.
We measured optimization efficiency by the number of iterations needed for the
cost function, which directly correlates with the frequency of quantum circuit
executions. PGE uses a constant-depth quantum circuit and shows no variance
across different layers of the ansatz. The software for these experiments was
developed using Python 3.10.

###  V-B Results

Figure 4 illustrates the experiments for synthetically generated problems
fixing $5$ seeds with layers $\in[1,5]$ using different classical optimizers.
We see that using our modified cost function in ACE parameterized by the
binary vector decoded from the measurement of the quantum circuit helps the
optimizer decide upon the optimal solution much quicker than ABE that uses a
cost function parameterized directly by the probability distribution obtained
from the measurement process. Moreover, the training is more consistent for
ACE as we observe minimal variations in the number of iterations and the
quality across $5$ seeds. The number of iterations is consistently lower for
ACE as the different costs explored during the optimization are discrete as
the minimum cut is a combinatorial optimization problem, lowering the
possibilities to explore in contrast to ABE where the optimization landscape
is not only non-convex but also infinitely many values are explored. However,
certain optimizers, such as SLSQP, demonstrate limitations when faced with a
higher number of layers, struggling with the increased overhead from
optimizing a larger parameter set and thus proving unsuitable for scenarios
demanding extensive parameter optimization. Although, PGE can be seen
performing better with Powell optimizer, the number of parameters optimized is
exponential to that of qubits. Therefore, we extend our experiments using a
more robust classical optimizer to prove the efficacy of ACE in terms of the
number of parameters to optimize.

Given the non-convex and non-differentiable nature of the problem, classical
optimizers focused on local optimization may not always secure the global
optimal solution. Although inefficient in terms of runtime, using a
metaheuristic optimizer like Differential Evolution could be more effective as
the cost function lacks smoothness. This approach helps illustrate that even a
single layer of ACE, with just a logarithmic number of parameters equal to the
number of qubits, can capably represent the solution to the original problem.

Seed | Size | Iterations | Execution Time (s) | Obtained Value | Exact Value
---|---|---|---|---|---
111 | 4 | 3463 | 236.13 | -1.13 | -1.13
16 | 14601 | 1320.92 | -1.54 | -1.54
222 | 4 | 4966 | 362.44 | -0.62 | -0.62
16 | 75225 | 3557.49 | -3.97 | -3.97
333 | 4 | 5439 | 379.46 | -1.16 | -1.16
16 | 40287 | 3477.47 | -3.57 | -3.57
444 | 4 | 9174 | 622.83 | -1.20 | -1.20
16 | 75201 | 3797.97 | -2.19 | -2.19
555 | 4 | 3737 | 268.69 | -1.86 | -1.86
16 | 75201 | 4202.49 | -2.56 | -2.56

TABLE I: Table details performance metrics for ACE with a single layer,
optimized using a differential evolution strategy. It includes reproducibility
seeds, image sizes denoted by pixel count, total optimizer iterations,
computational runtimes in seconds, and minimum cut values obtained by ACE
alongside those calculated through an exhaustive search for validation.

The results in Table I reveal that ACE can solve a combinatorial optimization
problem consisting of $n$ variables by optimizing only $\log_{2}(n)+1$
parameters. From a classical perspective, treating the quantum circuit as a
black box, ACE allows to reformulate an NP-Hard problem of size $n$ as a
multivariate optimization problem of size $O(\log_{2}(n))$ leveraging the
power of quantum computing. In contrast, PGE requires to optimize the same
number of real-valued parameters and also uses more number of gates.

###  V-C Theoretical Analysis

Here we perform an analysis on the scaling of QAOA, PGE, ABE and ACE based on
the metrics typically considered for evaluating the scalability of VQAs. In
particular, we provide the complexities of the discussed NISQ algorithms for
image segmentation in terms of qubit complexity which is a major bottleneck
for present-day applicability, the number of two-qubit entanglement gates is
vital to understanding the error rates due to decoherence, the number of
parametric gates is crucial for assessing the overhead on the classical
optimizer, and the circuit depth which is the longest sequence of quantum
gates applied from the input till the measurement operation which provides an
estimate of circuit execution time.

Method |  Qubit complexity |  Entanglement gates |  Parametric gates |  Circuit depth
---|---|---|---|---
QAOA |  $O(n)$ |  $O(n^{2})$ |  $O(Ln)$ |  $O(Ln^{2})$
PGE |  $\log_{2}(n)$ |  $n-1$ |  $n$ |  $n$
ABE/ACE |  $\log_{2}(n)+1$ |  $L(\log_{2}(n))$ |  $L(\log_{2}(n)+1)$ |  $L(\log_{2}(n)+1)$

TABLE II: Scalability analysis of VQAs for solving QUBO problems in terms of
quantum resources.

Table II provides a theoretical scalability analysis, where $n$ is the number
of binary variables in QUBO and $L$ is the number of layers in the ansatz of
the quantum circuit. Acknowledging the qubit limitations in emerging quantum
technologies, the scalability of the methods introduced here, excluding QAOA,
is notably superior. They potentially allow for the segmentation of a
$1-$megapixel image, equivalent to $2^{20}$ pixels, utilizing merely $20$
qubits.

##  VI Discussion and Conclusion

The segmentation methods explored in this paper demonstrate substantial
scalability in terms of quantum resources compared to well-known variational
quantum algorithms like QAOA. For PGE, there is no increase in circuit depth,
and the ansatz remains constant regardless of the number of edges in the
graph, only depending on the number of vertices. This implies that while the
number of entanglement gates, remains fewer than the number of variables in
the QUBO, the overall resource efficiency is maintained. The number of
parameters, which corresponds directly to the number of pixels, represents the
primary challenge for the classical optimizer due to its exponential
relationship with the number of qubits. The consistency in circuit design
across different graph connectivities suggests that PGE could benefit
significantly from more intricate connections in the image-derived graph. Such
enhancements could potentially improve the spatial information consideration
for each pixel, enriching segmentation quality without adding computational
overhead.

The methods ABE and ACE similarly display a significant reduction in quantum
resource requirements for addressing NP-Hard segmentation problems, aligning
with the conservation goals typical of near-term quantum technologies. These
methods achieve scalability primarily through their linear increase in
parameters with the number of layers, which enhances their applicability
without exponential increases in resource demands. However, the number of
circuit measurements scales exponentially with the number of qubits allowing
us to measure the ancilla qubit state for all possible register basis states
to obtain a complete binary vector solution. Experimentally, ACE has proven to
achieve optimal solutions with increased layers and parameters, provided that
a robust classical optimizer is used. This highlights the potential for
optimizing the synergy between quantum circuit design and classical
optimization techniques, advancing applications of quantum technology.
Additionally, a deeper exploration of the optimization landscape could unveil
methods allowing classical optimizers to find global optima more effectively.

In summary, the techniques covered here demonstrate the potential of new
encoding schemes, allowing vital applications like image segmentation to
benefit from emerging quantum technologies, even with the current limitations
of hardware from the NISQ era. These developments highlight the crucial role
that novel quantum computational techniques play in overcoming classical
computational challenges and open up promising directions for future research
into quantum-enhanced computer vision.

## References

  * [1] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, “Variational quantum algorithms,” _Nature Reviews Physics_ , vol. 3, no. 9, pp. 625–644, Sep 2021. [Online]. Available: https://doi.org/10.1038/s42254-021-00348-9
  * [2] F. Glover, G. Kochenberger, and Y. Du, “Applications and computational advances for solving the qubo model,” in _The Quadratic Unconstrained Binary Optimization Problem: Theory, Algorithms, and Applications_. Springer, 2022, pp. 39–56.
  * [3] E. Farhi, J. Goldstone, and S. Gutmann, “A quantum approximate optimization algorithm,” 2014.
  * [4] M. J. Rančić, “Noisy intermediate-scale quantum computing algorithm for solving an $n$-vertex maxcut problem with log($n$) qubits,” _Phys. Rev. Res._ , vol. 5, p. L012021, Feb 2023. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevResearch.5.L012021
  * [5] B. Tan, M.-A. Lemonde, S. Thanasilp, J. Tangpanitanon, and D. G. Angelakis, “Qubit-efficient encoding schemes for binary optimisation problems,” _Quantum_ , vol. 5, p. 454, May 2021. [Online]. Available: https://doi.org/10.22331/q-2021-05-04-454
  * [6] S. M. Venkatesh, A. Macaluso, M. Nuske, M. Klusch, and A. Dengel, “Q-seg: Quantum annealing-based unsupervised image segmentation,” 2023.
  * [7] O. Ronneberger, P.Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in _Medical Image Computing and Computer-Assisted Intervention (MICCAI)_ , ser. LNCS, vol. 9351. Springer, 2015, pp. 234–241, (available on arXiv:1505.04597 [cs.CV]). [Online]. Available: http://lmb.informatik.uni-freiburg.de/Publications/2015/RFB15a
  * [8] Y. Wang, J. Cheng, Y. Chen, S. Shao, L. Zhu, Z. Wu, T. Liu, and H. Zhu, “FVP: Fourier visual prompting for source-free unsupervised domain adaptation of medical image segmentation,” _IEEE Transactions on Medical Imaging_ , pp. 1–1, 2023. [Online]. Available: https://ieeexplore.ieee.org/document/10223270
  * [9] F. Yi and I. Moon, “Image segmentation: A survey of graph-cut methods,” in _2012 international conference on systems and informatics (ICSAI2012)_. IEEE, 2012, pp. 1936–1941.
  * [10] H. Neven, G. Rose, and W. G. Macready, “Image recognition with an adiabatic quantum computer i. mapping to quadratic unconstrained binary optimization,” _arXiv preprint arXiv:0804.4457_ , 2008.
  * [11] H. Neven, V. S. Denchev, G. Rose, and W. G. Macready, “Training a binary classifier with the quantum adiabatic algorithm,” _arXiv preprint arXiv:0811.0416_ , 2008.
  * [12] H. Neven, V. S. Denchev, M. Drew-Brook, J. Zhang, W. G. Macready, and G. Rose, “Nips 2009 demonstration: Binary classification using hardware implementation of quantum annealing,” _Quantum_ , vol. 4, p. 1, 2009.
  * [13] E. Grant, M. Benedetti, S. Cao, A. Hallam, J. Lockhart, V. Stojevic, A. G. Green, and S. Severini, “Hierarchical quantum classifiers,” _npj Quantum Information_ , vol. 4, no. 1, p. 65, 2018.
  * [14] N. Jiang, Y. Dang, and J. Wang, “Quantum image matching,” _Quantum Information Processing_ , vol. 15, no. 9, pp. 3543–3572, Sep 2016. [Online]. Available: https://doi.org/10.1007/s11128-016-1364-2
  * [15] H. Bhatia, E. Tretschk, Z. Lähner, M. Seelbach Benkner, M. Möller, C. Theobalt, and V. Golyanik, “Ccuantumm: Cycle-consistent quantum-hybrid matching of multiple shapes,” in _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2023.
  * [16] Ç. Aytekin, S. Kiranyaz, and M. Gabbouj, “Quantum mechanics in computer vision: Automatic object extraction,” in _2013 IEEE International Conference on Image Processing_. IEEE, 2013, pp. 2489–2493.
  * [17] S. Yuan, X. Mao, L. Chen, and Y. Xue, “Quantum digital image processing algorithms based on quantum measurement,” _Optik_ , vol. 124, no. 23, pp. 6386–6390, 2013.
  * [18] X. Fu, M. Ding, Y. Sun, and S. Chen, “A new quantum edge detection algorithm for medical images,” in _MIPPR 2009: Medical Imaging, Parallel Processing of Images, and Optimization Techniques_ , vol. 7497. SPIE, 2009, pp. 547–553.
  * [19] L. Tse, P. Mountney, P. Klein, and S. Severini, “Graph cut segmentation methods revisited with a quantum algorithm,” _arXiv preprint arXiv:1812.03050_ , 2018.
  * [20] T. Presles, C. Enderli, G. Burel, and E. H. Baghious, “Synthetic aperture radar image segmentation with quantum annealing,” _IET Radar, Sonar & Navigation_, vol. n/a, no. n/a, p. 1–13, 2024. [Online]. Available: https://ietresearch.onlinelibrary.wiley.com/doi/abs/10.1049/rsn2.12523
  * [21] S. M. Venkatesh, A. Macaluso, and M. Klusch, “Gcs-q: Quantum graph coalition structure generation,” in _Computational Science – ICCS 2023_ , J. Mikyška, C. de Mulatier, M. Paszynski, V. V. Krzhizhanovskaya, J. J. Dongarra, and P. M. Sloot, Eds. Cham: Springer Nature Switzerland, 2023, pp. 138–152.
  * [22] A. Lucas, “Ising formulations of many np problems,” _Frontiers in Physics_ , vol. 2, p. 5, 2014, edited by Jacob Biamonte, ISI Foundation, Italy.
  * [23] F. Glover, G. Kochenberger, and Y. Du, “A tutorial on formulating and using qubo models,” _arXiv preprint arXiv:1811.11538_ , 2018.
  * [24] S. Gupta and V. Kekatos, “A quantum approach for stochastic constrained binary optimization,” in _ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , June 2023, pp. 1–5.
  * [25] S. M. Venkatesh, A. Macaluso, and M. Klusch, “Bilp-q: Quantum coalition structure generation,” in _Proceedings of the 19th ACM International Conference on Computing Frontiers_ , ser. CF ’22. New York, NY, USA: Association for Computing Machinery, 2022, p. 189–192. [Online]. Available: https://doi.org/10.1145/3528416.3530235