# The hard to soft transition of GX 339–4 as seen by Insight–HXMT

Honghui Liu11affiliation: Center for Field Theory and Particle Physics and
Department of Physics, Fudan University, 200438 Shanghai, China. †E-mail:
<EMAIL_ADDRESS>, Cosimo Bambi11affiliation: Center for Field Theory and
Particle Physics and Department of Physics, Fudan University, 200438 Shanghai,
China. †E-mail<EMAIL_ADDRESS>† †affiliationmark:  , Jiachen
Jiang22affiliation: Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge CB3 0HA, UK , Javier A. García33affiliation: Cahill Center for
Astronomy and Astrophysics, California Institute of Technology, 1216 E.
California Boulevard, Pasadena, CA 91125, USA 44affiliation: Dr. Karl Remeis-
Observatory and Erlangen Centre for Astroparticle Physics, Sternwartstr. 7,
D-96049 Bamberg, Germany , Long Ji55affiliation: School of Physics and
Astronomy, Sun Yat-Sen University, 519082 Zhuhai, China , Lingda
Kong66affiliation: Institut für Astronomie und Astrophysik, Kepler Center for
Astro and Particle Physics, Eberhard Karls, Universität, Sand 1, D-72076
Tübingen, Germany , Xiaoqin Ren77affiliation: Key Laboratory for Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
100049 Beijing, China 88affiliation: University of Chinese Academy of
Sciences, Chinese Academy of Sciences, 100049 Beijing, China , Shu
Zhang77affiliation: Key Laboratory for Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China
88affiliation: University of Chinese Academy of Sciences, Chinese Academy of
Sciences, 100049 Beijing, China , Shuangnan Zhang77affiliation: Key Laboratory
for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy
of Sciences, 100049 Beijing, China 88affiliation: University of Chinese
Academy of Sciences, Chinese Academy of Sciences, 100049 Beijing, China [

###### Abstract

We present an analysis of the relativistic reflection spectra of GX 339–4
during the hard-to-soft transition of its 2021 outburst observed by
Insight–HXMT. The strong relativistic reflection signatures in the data
suggest a high black hole spin ($a_{*}>0.86$) and an intermediate disk
inclination angle ($i\approx$ 35∘–43∘) of the system. The transition is
accompanied by an increasing temperature of the disk and a softening of the
corona emission while the inner disk radius remains stable. Assuming a
lamppost geometry, the corona height is also found to stay close to the black
hole across the state transition. If we include the Comptonization of the
reflection spectrum, the scattering fraction parameter is found to decrease
during the state transition. We also perform an analysis with a reflection
model designed for hot accretion disks of stellar mass black holes where the
surface of the innermost accretion disk is illuminated by emission from the
corona and the thermal disk below. Our results support the scenario in which
the state transition is associated with variations in the corona properties.

###### Subject headings:

accretion, accretion disks — black hole physics — X-rays: binaries

##  1\. Introduction

The outbursts of black hole X-ray binaries (XRBs) are characterized by orders
of magnitude change in the X-ray luminosity (Remillard & McClintock, 2006). It
is generally believed that there are dramatic changes in the geometry of the
accretion flow during the outburst (e.g. Zdziarski & Gierliński, 2004; Done et
al., 2007; Plant et al., 2015). If plotting on the hardness intensity diagram
(HID), the outburst often moves anticlockwise to form a q-shaped pattern (e.g.
Fender et al., 2004; Belloni et al., 2005). At the early stage of the
outburst, the source luminosity is low and its X-ray spectrum is usually
dominated by a hard power-law component, which is thought to be originated
from inverse Compton scattering of seed photons by a hot ($\sim$ 100 keV)
corona (e.g. Dove et al., 1997; Zdziarski & Gierliński, 2004). This is the so
called low hard (LH) state that is located at the lower right corner of the
HID. At this phase, the source spectrum remains hard as it gets brighter.
Moreover, a high-energy cut-off of the broadband X-ray spectrum is commonly
observed and is thought to be determined by the coronal temperature. The
energy of this cut-off is found to decease with the source luminosity (e.g.
Yamaoka et al., 2005; Miyakawa et al., 2008; Joinet et al., 2008, but see Yan
et al. 2020 for a opposite correlation at even lower luminosities).

At some point, the source makes a fast transition from the hard state to a
thermal-emission-dominated soft state through the intermediate states (e.g.
Homan & Belloni, 2005; Belloni, 2010). The transition, which can happen in a
wide range of luminosities, often completes in a few days and is associated
with softening of the source spectrum. This softening is a result of the
increasing contribution from the disk thermal emission and steepening of the
corona emission. Overall, the X-ray spectrum appears as pivoting around
certain energy (e.g. Del Santo et al., 2008). The cut-off energy tends to rise
during the hard to soft transition (e.g. Motta et al., 2009) and in some
cases, no high-energy cut-off is found up to $\sim$ 1 MeV in the soft state
(e.g. Tomsick et al., 1999; Belloni et al., 2006).

A commonly accepted scenario to explain the state transition is the truncated
disk model (e.g. Esin et al., 1997). In this scenario, the standard
geometrically thin and optically thick accretion disk (Shakura & Sunyaev,
1973) is truncated at large radius in the low hard state when the accretion
rate is low. A hot radiatively inefficient accretion flow is filled between
the truncation radius and the black hole (e.g. Narayan, 2005). As the
accretion rate increases, the disk moves closer to the black hole, which
produces stronger disk thermal emission. X-ray observations of XRBs have
provided good evidence that the disk is truncated at large radius in low
luminosities (e.g. Tomsick et al., 2009) and reaches the innermost stable
circular orbit (ISCO) in the high flux soft state (e.g. Gierliński & Done,
2004; Steiner et al., 2010). However, it remains to be known at which point
the disk exactly reaches the ISCO and what mechanism triggers the state
transition. Moreover, in some cases, the measurements of the size of inner
edge are found to be controversial and model dependent (e.g. Dziełak et al.,
2019; Mahmoud et al., 2019; Zdziarski et al., 2021).

Relativistic reflection component, namely the reflected corona emission by the
optically thick accretion disk extending close to the black hole, is a
powerful tool to probe the spacetime properties (e.g. Reynolds, 2014; Bambi,
2017; Liu et al., 2019; Bambi et al., 2021) and the geometry of the accreting
system (e.g. García et al., 2015; Wilkins & Gallo, 2015; Wang et al., 2020;
Szanecki et al., 2020). One of the main features of the relativistic
reflection component is the broad iron line around 6.4 keV. The broad line
profile is a result of skewed iron K lines due to the strong relativistic
effect near the black hole (e.g. Doppler effect, light bending and
gravitational redshift, Fabian et al., 2000; Dauser et al., 2010). This is why
this broad line feature can be used to map the innermost regions around the
accreting black hole (e.g. Fabian et al., 1989).

GX 339–4 is a classical low mass X-ray binary located at 8–12 kpc with a black
hole mass of 4–11 $M_{\odot}$ (Zdziarski et al., 2019). Strong relativistic
reflection signatures have been found in this source in the hard and soft
states (e.g. García et al., 2015; Miller et al., 2004; Liu et al., 2022).
Previous studies have found that the black hole in GX 339–4 has a very high
spin ($a_{*}\sim 0.95$, García et al., 2015; Parker et al., 2016). The
inclination angle of the accretion disk should have an intermediate value
(Fürst et al., 2015; Parker et al., 2016). The source goes into bright
outburst every a few years and thus is an ideal object for the study of the
accretion flow through different spectra states. Recently, the hard-to-soft
transition of GX 339–4 has been studied by Sridhar et al. (2020) with RXTE/PCA
data based on reflection analysis. The inner edge of the accretion disk was
found to be close to ISCO during the transition. Starting from February 2021,
GX 339–4 went into a new outburst that lasted for a few months. The hard to
soft transition of this outburst has been captured by the Chinese X-ray
satellite Hard X-ray Modulation Telescope (dubbed Insight–HXMT, Zhang et al.,
2014). Therefore, we study the Insight–HXMT data to understand the evolution
of the accretion geometry.

In this paper, we aim at studying the relativistic reflection spectra of GX
339–4 during its 2021 outburst with Insight–HXMT data. In Sec. 2, we describe
the data reduction process. The spectral fitting results are presented in Sec.
3. We discuss the results in Sec. 4.

##  2\. Observation and data reduction

Insight–HXMT monitored the 2021 outburst of GX 339–4 with observations on
nearly every day (see Fig.1). Starting from 2021 March 25, the source made a
fast transition from the hard state to the soft state in about one week. The
soft to hard transition, which should happen at a lower luminosity, was not
captured because of the high background of Insight–HXMT. The purpose of this
work is to study the evolution of the system during the state transition.
Therefore, we select Insight–HXMT observation data during the transition and
investigate their broadband energy spectra. Details about the selected
Insight–HXMT observations are listed in the appendix.

The low-energy (LE), medium-energy (ME) and high-energy (HE) detectors onboard
Insight–HXMT cover the energy range of 1-250 keV (Chen et al., 2020; Cao et
al., 2020; Liu et al., 2020; Zhang et al., 2020). The energy spectra are
extracted following the HXMT Data Reduction Guide v2.04111The guide can be
found from here: http://hxmtweb.ihep.ac.cn/SoftDoc/496.jhtml. We follow the
extraction approach provided here: https://code.ihep.ac.cn/jldirac/insight-
hxmt-code-collection/-/tree/master/version2.04. and using the software HXMTDAS
v2.04. The background spectra are estimated using the scripts hebkgmap,
mebkgmap and lebkgmap (Liao et al., 2020a; Guo et al., 2020; Liao et al.,
2020b). As indicated in Tab. 2, short exposures on the same day are merged to
increase the signal to noise ratio. Before merging the data, we first check
the lightcurve and hardness ratio and confirm that there are no significant
flares or dips. In the end, we obtain energy spectra for seven epochs. The
spectra are then binned to ensure a minimal signal to noise ratio of 25 before
fitting them to physical models. We use data in 2–10 keV, 10–25 keV and 30-100
keV bands for LE, ME and HE respectively.

Figure 1.— (left): The observation log of GX 339–4 from Insight–HXMT Low
Energy (LE) instruments. The hardness is defined as the ratio between count
rates in 4–10 keV and 2–4 keV. (right): The hardness intensity diagram of GX
339–4 from Insight–HXMT. Each point represents data from one day and is marked
with the epoch number. The red color denotes the observations analyzed in this
work.

##  3\. Spectral analysis

Spectral fittings are conducted with XSPEC v12.11.1 (Arnaud, 1996). We
implement element abundances of Wilms et al. (2000) and cross-sections of
Verner et al. (1996). $\chi^{2}$ statistics are used to find the best-fit
values and uncertainties (at 90% confidence level unless otherwise specified)
of the model parameters.

###  3.1. Relativistic reflection features

As the first step, we fit the seven spectra simultaneously with a simple
absorbed continuum model which consists of a multicolor disk component
(diskbb; Mitsuda et al., 1984), a power-law component with a high energy
cutoff (cutoffpl) and a Galactic absorption model (tbabs; Wilms et al., 2000).
In XSPEC notation, the model can be written as constant*tbabs*(diskbb +
cutoffpl). The constant is required to quantify the cross-normalization
between the three instruments onboard Insight–HXMT. The column density
($N_{\rm H}$) of the tbabs is set as a free parameter and tied across all
spectra.

The ratios between the data and the best-fit models for the seven spectra are
shown in Fig. 2, from which we can identify the missing model components. We
can see that the common features in Fig. 2 are a broad excess around 6–7 keV
and a hump around 30 keV. These features are commonly seen in the X-ray
spectra of black hole XRBs (e.g. Miller et al., 2013; García et al., 2015;
Jiang et al., 2019a) and can be explained by the corona emission being
reflected by the optically thick accretion disk (e.g. Ross et al., 1999). The
position of the neutral iron K$\alpha$ fluorescent line (6.4 keV) is marked
with vertical dashed line in Fig. 2. The broad emission profile around this
line is a result of relativistically skewed iron K lines. Moreover, the
electron down scattering of high energy photons and photoelectric absorption
on the disk leads to the “Compton hump” around 30 keV (George & Fabian, 1991).

Figure 2.— Data to model ratios for an absorbed continuum model
tbabs*(diskbb+cutoffpl) for the seven epochs. The vertical line marks the
position of 6.4 keV. Data are rebinned for visual clarity.

###  3.2. Spectral fitting with reflection models

To model the relativistic reflection features, we implement the widely used
reflection model relxill v1.4.3 (García et al., 2014). The full model reads as
constant*tbabs*(diskbb+relxill+cutoffpl) (Model 1). With this model, we are
able to measure parameters of the system like the black hole spin ($a_{*}$),
the inclination angle ($i$) of the accretion disk with respect to our line of
sight and the size of the disk inner edge ($R_{\rm in}$). The seven spectra
are fitted simultaneously. The column density of the Galactic absorption
($N_{\rm H}$), the black hole spin, the inclination angle and the iron
abundance of the system should not change on the short timescale we are
seeing. Therefore, these parameters are linked across all spectra during the
spectral fitting. The reflection fraction parameter of relxill is set to $-1$
so that the model returns only the reflection component. The outer radius of
the disk $R_{\rm out}$ is fixed at 400 $R_{\rm g}$ (where $R_{\rm g}=GM/c^{2}$
is the gravitational radius, $M$ is the black hole mass).

####  3.2.1 Lamppost geometry

The emissivity profile, defined as the radial dependence of the intensity of
reflected emission, is a vital component of the reflection model. In
principle, if the corona geometry is known, the emissivity profile can be
self-consistently calculated. Therefore, we implement the relxilllp (Model
1A), which is one of the models in the relxill package. The model assumes a
lamppost geometry (Dauser et al., 2013), in which the corona is a point source
above the black hole at certain height ($h$). This model offers a good fit
with $\chi^{2}$/d.o.f=3697.6/3384. The best-fit values are shown in Tab. 1.
The spectral components and residuals of the best-fit model are shown in Fig
3, from which we do not see significant unresolved features. With this model,
we can fit with both the spin parameter and the inner disk radius free. There
is usually strong degeneracy between the two parameters (e.g. Dauser et al.,
2013). Our data can break this degeneracy because the disk extends very close
to the ISCO (Fabian et al., 2014). If we fix the spin parameter at the maximum
value the model allows ($a_{*}=0.998$), $R_{\rm in}$ is still not constrained
for Epoch 1 but is smaller than twice of the ISCO from Epoch 4 to Epoch 7
($\chi^{2}$/d.o.f=3697.6/3385). Moreover, if we set $R_{\rm in}$ at ISCO for
all epochs, the spin is measured to be $0.93\pm 0.01$
($\chi^{2}$/d.o.f=3704.0/3391).

Table 1Best-fit values with the lamppost geometry (Model 1A).

Component | Parameter | Epoch 1 | Epoch 2 | Epoch 3 | Epoch 4 | Epoch 5 | Epoch 6 | Epoch 7
---|---|---|---|---|---|---|---|---
tbabs |  $N_{\rm H}$ (1022 cm-2) | $0.99_{-0.06}^{+0.04}$
diskbb |  $T_{\rm in}$ (keV) | $0.50_{-0.03}^{+0.03}$ | $0.548_{-0.01}^{+0.014}$ | $0.656_{-0.008}^{+0.008}$ | $0.725_{-0.008}^{+0.011}$ | $0.725_{-0.009}^{+0.01}$ | $0.766_{-0.007}^{+0.006}$ | $0.757_{-0.007}^{+0.011}$
relxilllp | $a_{*}$ | $>0.95$
|  $h$ ($R_{\rm g}$) | $3.5_{-P}^{+P}$ | $3.1_{-0.5}^{+0.5}$ | $2.0_{-P}^{+4}$ | $2.20_{-0.1}^{+0.21}$ | $2.06_{-P}^{+0.18}$ | $2.35_{-0.29}^{+0.28}$ | $2.03_{-P}^{+0.24}$
| Incl | $42.0_{-1.4}^{+1.3}$
|  $A_{\rm Fe}$ (solar) | $10.0_{-0.6}^{+P}$
|  $R_{\rm in}$ (ISCO) | $70_{-P}^{+P}$ | $1.4_{-P}^{+0.7}$ | $3.2_{-1.9}^{+1.2}$ | $1.56_{-0.07}^{+0.08}$ | $1.44_{-0.09}^{+0.08}$ | $1.62_{-0.13}^{+0.12}$ | $1.6_{-0.1}^{+0.1}$
| $\log(\xi)$ | $4.29_{-0.06}^{+0.06}$ | $3.89_{-0.23}^{+0.17}$ | $4.51_{-0.11}^{+0.06}$ | $3.71_{-0.08}^{+0.09}$ | $4.06_{-0.2}^{+0.22}$ | $4.14_{-0.19}^{+0.2}$ | $3.79_{-0.12}^{+0.13}$
cutoffpl | $\Gamma$ | $1.696_{-0.013}^{+0.018}$ | $2.087_{-0.019}^{+0.028}$ | $2.260_{-0.019}^{+0.06}$ | $2.460_{-0.03}^{+0.021}$ | $2.451_{-0.019}^{+0.024}$ | $2.511_{-0.024}^{+0.028}$ | $2.55_{-0.08}^{+0.1}$
|  $E_{\rm cut}$ (keV) | $150_{-20}^{+20}$ | $160_{-20}^{+40}$ | $>400$ | $>400$ | $>330$ | $>410$ | $130_{-60}^{+50}$
cflux | (10-8 erg cm-2 s-1) |  |  |  |  |  |  |
$F_{\rm diskbb}$ |  | $0.23_{-0.03}^{+0.03}$ | $0.552_{-0.028}^{+0.019}$ | $0.935_{-0.023}^{+0.019}$ | $1.314_{-0.017}^{+0.03}$ | $1.355_{-0.025}^{+0.021}$ | $1.603_{-0.022}^{+0.019}$ | $1.85_{-0.05}^{+0.03}$
$F_{\rm relxilllp}$ |  | $0.77_{-0.06}^{+0.10}$ | $0.74_{-0.08}^{+0.08}$ | $1.43_{-0.11}^{+0.13}$ | $1.52_{-0.12}^{+0.18}$ | $1.52_{-0.12}^{+0.2}$ | $1.64_{-0.14}^{+0.19}$ | $1.9_{-0.5}^{+1.1}$
$F_{\rm cutoffpl}$ |  | $0.71_{-0.07}^{+0.05}$ | $1.28_{-0.15}^{+0.13}$ | $0.94_{-0.20}^{+0.10}$ | $2.04_{-0.21}^{+0.13}$ | $1.77_{-0.12}^{+0.1}$ | $1.90_{-0.27}^{+0.09}$ | $1.6_{-0.3}^{+0.4}$
|  $L_{\rm 0.1-100keV}/L_{\rm Edd}$ (%) | 10.9 | 16.4 | 21.1 | 31.1 | 29.6 | 32.8 | 34.1
constant | ME/LE | $1.023_{-0.014}^{+0.013}$
| HE/LE | $1.110_{-0.023}^{+0.019}$
|  $\chi^{2}$/d.o.f | 3697.6/3384

Note. Best-fit parameters for the model tbabs*(diskbb+relxilllp+cutoffpl).
Parameters with ∗ are fixed during the fit and the symbol $P$ denotes the
upper or lower boundary. Parameters that are only shown for Epoch 4 are tied
across all observations. The flux of each spectral components are calculated
in the 0.1-100 keV band using the cflux model. The absorption corrected X-ray
flux ($L_{\rm 0.1-100keV}$) is calculated with the flux command in XSPEC.

####  3.2.2 Broken power-law emissivity

The other method commonly used to deal with the emissivity profile is to fit
it phenomenologically with a broken power-law (i.e., $\epsilon\propto
1/r^{q_{\rm in}}$ for $R_{\rm in}<r<R_{\rm br}$ and $\epsilon\propto
1/r^{q_{\rm out}}$ for $R_{\rm br}<r<R_{\rm out}$ where $R_{\rm br}$ is the
breaking radius). We test both the power-law emissivity ($q_{\rm in}=q_{\rm
out}$) and the broken power-law emissivity (implemented with the model
relxill) to look for a better description of the data. In the latter case, we
fix the outer index fixed at 3 ($q_{\rm out}=3$), which is the value predicted
by Newtonian limit for a compact corona. Moreover, for the first three epochs,
the inner index ($q_{\rm in}$) and the breaking radius ($R_{\rm br}$) can not
be constrained by the data. Therefore, we fix $q_{\rm in}$ at 3 for those
data. The broken power-law emissivity provides a better statistics
($\chi^{2}/\nu=3673.1/3383$), with the $\chi^{2}$ lowered by 13 with 2 more
free parameters. The best-fit values for the broken power-law emissivity
(Model 1B) are shown in Tab. 3.

####  3.2.3 High-density reflection

The disk density ($n_{\rm e}$) of the models above is fixed at 1015 cm-3. This
value might be appropriate for massive black holes but XRBs with a stellar
mass black hole are known to have a higher density (e.g. Jiang et al., 2019a,
b). Therefore, we implement the relxillD model (assuming a broken power-law
emissivity, Model 1C) in which $\log(n_{\rm e})$ is allowed to vary between 15
and 19 (García et al., 2016). The best-fit parameters are shown in Tab. 4.
Compared to the model above, the high-density model slightly improves the
$\chi^{2}$ by 9 with 7 more degrees of freedom and provides consistent
constraints on the spin parameter and the inclination angle. Five out of the
seven observations require a higher density than 1015 cm-3. The measurements
on the ionization parameter are systematically lowered. This is expected since
a higher density leads to stronger soft X-ray emission, which is similar to
the effect of a higher ionization parameter.

####  3.2.4 The Comptonized continuum

The models tested above provide good fits to the data. However, the column
density of the tbabs component is higher than what reported in previous
spectral analysis (e.g. $7.7\pm 0.2\times 10^{21}$ cm-2 in Parker et al.,
2016, with the same versions of abundances and cross-sections). This is
recovered if we replace cutoffpl with a physically motivated Comptonization
model nthcomp (Zdziarski et al., 1996; Życki et al., 1999). The reflection
component is also changed to the corresponding flavor relxillcp (Model 2). We
link the seed photon temperature for the nthcomp model to the $T_{\rm in}$
parameter in diskbb. The best-fit parameters for this model are shown in Tab.
5. Compared with relxill, this model increases the statistics by
$\Delta\chi^{2}=64$ but lowers the column density of the Galactic absorption
to $7.3\pm 0.6\times 10^{21}$ cm-2. This is because the Comptonization model
considers cutoffs at both low and high energy ends. Therefore, the X-ray
luminosity estimated from the Comptonization model should be more realistic
than that from the model with cutoffpl. We also note that our data cannot well
constrain the coronal temperature (or the high energy cutoff).

####  3.2.5 Distant reflector

The spectra of GX 339–4 may require a distant reflection component to fit a
possible narrow-line feature (e.g. Wang-Ji et al., 2018; Sridhar et al.,
2020). To test this possibility, we add a xillver (García & Kallman, 2010;
García et al., 2013) component to Model 1A. The relativistic reflection model
assumes a broken power-law emissivity profile. The ionization parameter of
this distance reflection component is fixed at $\log(\xi)=0$. The other
parameters of the distance reflector are linked to the corresponding values in
the relativistic reflection component. Adding this component improves the
$\chi^{2}$ by only 10 with 7 more free parameters.

To test if our data quality is good enough to distinguish such a distant
reflector, we simulate 1000 spectra with a model that includes a distant
reflection component. The response files, exposure time and model parameters
(Tab. 1) are taken from our Epoch 5 which has the longest exposure. The flux
(0.1–100 keV) of the distant reflection component is set to be 2% (e.g.
Sridhar et al., 2020) of the relativistic component. Then we fit the spectra
with two models: one without the distant reflection and the other includes it.
The latter model always gives a better statistics with $\Delta\chi^{2}>50$ and
one more degrees of freedom, which indicates that the distant component can be
detected if it exists. Therefore, we conclude that, for the observation with
the longest exposure time, the distant reflection component can be detected if
its flux is larger than 2% of the relativistic component. The reason why we do
not detect the distant component in our analysis could be that its flux is
lower than the limit provided by our data quality. Similarly, in Jiang et al.
(2019a), the intermediate state of GX 339–4 requires no distance reflection.

Figure 3.— The best-fit model components for the seven epochs and the
corresponding residuals with the model tbabs*(diskbb+relxilllp+cutoffpl). The
black solid line represents the total model. The magenta, red and green dashed
lines are for the corona emission, the disk component and the reflected
component respectively.

####  3.2.6 Comptonization of the reflection component

The power-law emission is the result of inverse Compton scattering of the disk
photons by the hot electrons (but see Sridhar et al., 2021, 2022) in the
corona. Some of the reflected emission should also intercept the corona and be
Comptonized. Therefore, a self-consistent model should take the Comptonized
reflection spectrum into account (Steiner et al., 2017; Wang-Ji et al., 2018).
We consider this scenario with the Compton scattering kernel simplcut (Steiner
et al., 2017). The full model reads as tbabs*simplcut*(diskbb+relxillcp)
(Model 3). The photon index ($\Gamma$) and electron temperature ($kT_{\rm e}$)
of simplcut are linked to the corresponding parameters in relxillcp. There are
two more parameters in the Compton scattering kernel: the reflection fraction
$R_{f}$ and the scattered fraction $f_{\rm sc}$. We fix the reflection
fraction parameter at the flux ratio between the reflection and power-law
components in the 20–40 keV band derived from Model 2 and let the scattered
fraction parameter free to vary.

With this model, we are assuming an equal fraction of the disk photons and the
reflected photons are scattered by the corona. The best-fit parameters are
shown in Tab. 6. We note that including the effect of Comptonization of the
disk and reflection photons does not change significantly the measurements of
parameters like $T_{\rm in}$, $\Gamma$ and $R_{\rm in}$. The measurements of
the spin and inclination parameters are also not affected. The scattered
fraction decreases from 0.12 to 0.07, which may indicate changes of the disk-
corona system during the transition.

####  3.2.7 The refhiden model

In the relxill model, the ionization balance on the disk is calculated using
only the incident corona emission. In reality, we would expect that the
thermal emission from the disk can also affect the ionization state. This is
particularly important in XRBs since the disk temperature of XRBs can be very
high ($\sim$ 1 keV) and the thermal emission can dominate the X-ray spectra.
In some cases, the disk emission has been found to return to the disk and the
reflection model assuming the primary continuum to be only a blackbody (García
et al., 2022b) fits the data well (e.g. Connors et al., 2020, 2021). However,
in the intermediate states, the strength of the disk and power-law components
are comparable (see Fig. 3 and Tab. 1). As a consequence, considering only one
of the components (disk or corona) to ionize the disk atmosphere is not
appropriate.

Therefore, we implement the refhiden model that is specifically designed for
accreting stellar mass black holes (Ross & Fabian, 2007). In this model, the
disk atmosphere is ionized by the combination of a power-law component from
above and a blackbody from below. Note that here the blackbody is assumed to
be single-temperature, which is different from the multi-temperature disk
blackbody component. The model also includes the effect of Comptonization in
the accretion disk. The full model reads as tbabs*simplcut*relconv*refhiden
(Model 4). The broadening kernel relconv (Dauser et al., 2010) is required to
account for the relativistic effects. The disk thermal emission is already
incorporated in refhiden so there is no need to add additional disk component.
The parameters in the refhiden model include: the hydrogen number density on
the disk ($H_{\rm den}$), the temperature of the blackbody from the disk
($kT_{\rm bb}$), the flux ratio between the power-law and blackbody components
and the power-law index $\Gamma$ (Illum/BB). We link $\Gamma$ of the refhiden
to the same parameter in simplcut. To better constrain the evolution of other
parameters, we fix the black hole spin at 0.998 for the fitting.

The model provides an acceptable fit ($\chi^{2}/\nu=3788.8/3386$) and the
best-fit parameters are shown in Tab. 7. The variations of spectral parameters
from this model during the state transition are shown in the right panels of
Fig. 4 and can be compared with that from Model 1A in the left panels. The
refhiden model gives a blackbody temperature ($kT_{\rm bb}$) that is
systematically lower than the $T_{\rm in}$ found in other fits with the diskbb
model. This is within expectation because the temperature in refhiden is the
effective mid-plane temperature of the disk. The disk surface temperature
would be higher after including the effect of color correction (see the
discussion in Reis et al., 2008). Moreover, diskbb assumes a multi-temperature
blackbody while a single temperature is used in refhiden. This is probably
also affecting the value of the scattering fraction ($f_{\rm sc}$) of simplcut
but the decreasing trend of this parameter along with the transition is
retained. The number density of hydrogen is higher at the early phase of the
transition (the first two epochs). One parameter of refhiden that clearly
changes through the transition is the flux ratio between the illuminating
power-law and the blackbody from the disk mid-plane.

##  4\. Discussion

Figure 4.— The evolution of spectral parameters along with X-ray hardness of
the source. The definition of the hardness is the same as in Fig. 1. Lower and
upper limits are marked with arrows. Panels in the left show the parametes
from Model 1A except the scattering fraction ($f_{\rm sc}$, from Model 3). In
the third panel in the left, the red color represents estimation of the inner
disk radius from the normalization parameter of diskbb of Model 3 and the size
of the ISCO is chosen for a black hole with $a_{*}=0.95$. The reflection
strength ($R_{\rm s}$) is defined as the ratio between the observed flux of
the reflected and the direct corona emission in the 0.1–100 keV band.
Parameters from Model 4 with the refhiden model are shown in the right panels.

###  4.1. Evolution of the disc-corona system

For the transition from the hard to the soft state, GX 339–4 gets
progressively brighter in X-ray. In Fig. 4, we show how the spectral
parameters evolve with X-ray hardness of the source. The state transition can
be characterized by decreasing of the hardness. This process is apparently
associated with the steepening of the corona emission component, with a photon
index ($\Gamma$) evolving from 1.7 to 2.5 (see the top panel of Fig. 4).
Meanwhile, the X-ray luminosity increases from 10% to 20% of the Eddington
limit assuming a black hole mass of 10 $M_{\odot}$ and a distance of 8 kpc
(see Tab. 5). There is a clear trend of the inner disk temperature increasing
from 0.5 to 0.76 keV. The contribution of the disk component (in 0.1–100 keV
band) also increases from 13% to 35%. This increase of soft cooling photons
could explain the softening of the corona emission (e.g. Del Santo et al.,
2008).

As for the coronal temperature ($kT_{\rm e}$), the data do not provide good
constraint on this parameter. Only lower limits are found for Epoch 2–7 (see
Tab. 5), while these lower limits are still higher than the measurement of
Epoch 1. This weak tendency of increasing coronal temperature during the state
transition is consistent with previous broadband spectral analysis of GX 339-4
(e.g. Motta et al., 2009; Sridhar et al., 2020) and other systems (e.g. Zhang
et al., 2022). It is different from the tendency of decreasing coronal
temperature with luminosity in the hard state (e.g. García et al., 2015) which
is thought to be a result of stronger cooling by the seed photons. What is
happening in the state transition could be that the decrease of the optical
depth (e.g. McConnell et al., 2002; Del Santo et al., 2008) causes less
scattering between the seed photons and corona electrons and then increases
the coronal temperature. Note that this increase of coronal temperature will
be limited by the run-away electron-positron pair production process in the
corona (e.g. Fabian et al., 2015).

Another parameter that is of interest is the inner radius of the accretion
disk ($R_{\rm in}$). There are already reflection-based studies that have
investigated the accretion geometry of GX 339–4 in the hard state. In García
et al. (2015), the luminosity range from 1.7% to 17% of the Eddington limit
($L_{\rm Edd}$) has been studied with data from RXTE/PCA and the $R_{\rm in}$
is found to be close to the ISCO ($<5~{}R_{\rm ISCO}$). Plant et al. (2015)
have studied similar luminosity levels with data from XMM-Newton in the timing
mode but found large truncations of the disk (see also Basak & Zdziarski,
2016). This discrepancy could be due to the complex pile-up effect in the
timing mode that can affect the iron K line (e.g. Miller et al., 2010). The
regime below 1% of the Eddington luminosity has also been studied with
reflection models (e.g. Shidatsu et al., 2011; Petrucci et al., 2014; Wang-Ji
et al., 2018; Wang et al., 2020). These studies indicate that $R_{\rm in}$
should decrease with increasing luminosity and reaches 20 $R_{\rm g}$ when the
luminosity is above 1% Eddington (see Fig. 6 of García et al., 2019).

Our analysis concerns the reflection spectra of GX 339–4 during the hard-to-
soft transition that covers the luminosity range from 10% to 20% $L_{\rm
Edd}$. The results show that $R_{\rm in}$ is not constrained in Epoch 1. This
is probably because of the low statistics of the data since Epoch 1 has the
shortest exposure among all epochs. After Epoch 2, $R_{\rm in}$ is consistent
with the ISCO in some models and still below a few times the ISCO in other
models. This indicates that the disk inner edge is relatively stable across
the state transition.222We note that the Insight–HXMT spectra of this outburst
in the bright hard state ($>$ 8% $L_{\rm Edd}$, before our Epoch 1) have also
been analyzed and the inner radius is found to be constant ($\sim 3~{}R_{\rm
ISCO}$, Ren et al. in prep). The inner disc radius can also be estimated from
the normalization parameter of the diskbb model (Kubota et al., 1998). We use
values from Model 3 since it takes into account both the transmitted and
scattered disk photons. The results are shown in Fig. 4 in red assuming a
distance of 8 kpc, a black hole mass of 10 $M_{\odot}$, an inclination of 40
deg, a color correction factor of 1.7 and a relativistic correction factor of
0.412. It again shows that the inner disk radius is stable through the
transition although the absolute values need to be taken with caution as there
are systematic uncertainties in the assumptions for mass, distance and color
correction.

Similar results have been found by Sridhar et al. (2020) who studies the
2002–2003 and 2004–2005 outbursts of GX 339–4 with data from RXTE/PCA. In
their work, by fixing the black hole spin at 0.998, the $R_{\rm in}$ is found
to reach close to the ISCO at the onset of the transition for both outbursts
even though the transition luminosity differs by a factor of 3. A stable disk
during the state transition has also been found in other systems (e.g. XTE
J1550–564, Rodriguez et al. 2003; MAXI J1820+070, Wang et al. 2021).

The corona height shows similar tendency as $R_{\rm in}$, which is not
constrained in Epoch 1 and remains close to the black hole ($<6~{}R_{\rm g}$)
in the following observations. When the primary source is close to the black
hole, the light bending effect would concentrate its radiation to the
innermost part of the accretion disk (e.g. Dauser et al., 2013). This is
indeed what we find with the broken power-law emissivity, with which we obtain
a steep inner emissivity index (see Tab. 3). We also define an empirical
reflection strength parameter ($R_{\rm s}$) which is the ratio between the
reflected and the power-law flux in the 0.1–100 keV band. This is different
from but similar to the reflection fraction parameter ($R_{\rm f}$), which is
the ratio between the corona intensity that illuminates the disk and the
corona intensity that reaches the observer (Dauser et al., 2016). Variation of
the reflection strength may indicate changes of the disk-corona system (e.g.
You et al., 2021). The lowest panel in the left of Fig. 4 shows no significant
changes of this parameter, which is consistent with the prediction of a stable
inner disk radius.

We note that relativistic reflection analysis based on the single lamppost
geometry may only be sensitive to the parts of the corona that are close to
the black hole since they dominate the reflection emissivity (see Wang et al.,
2021). In practice, the two-lamppost reflection models have been used to mimic
a vertically extended corona (e.g. Buisson et al., 2019). The upper lamppost
could be responsible for the narrow cores in the iron band that is usually
assumed to originate from a distant reflector (e.g. García et al., 2019). We
do not explore further in this aspect since our data does not require the
presence of a narrow iron line.

The scattering fraction parameter ($f_{\rm sc}$) in simplcut is related to the
geometry of the disk-corona system and also the optical depth of the corona.
From Fig. 4, we see that, except for Epoch 1, $f_{\rm sc}$ from Model 3 shows
a tentative decreasing trend through the transition although the uncertainties
are large. The trend is more evident in Model 4 (see the right panel of Fig.
4). In the fit with refhiden, we find that the flux ratio between the
illuminating power-law component and the blackbody component from the disk
(Illum/BB) also decreases by a factor of 3 from Epoch 1 to Epoch 7. Meanwhile,
the temperature of the disk mid-plane ($T_{\rm bb}$) only changes in a narrow
range (0.4–0.5 keV). Given these variation patterns and the fact that the
inner disk radius remains stable, we argue that the state transition is more
likely to be related to the corona properties instead of the inner edge of the
disk. This scenario is also supported by recent studies on the reverberation
lag of black hole transients (Wang et al., 2022), in which the lag is found to
become longer across the hard-to-soft transition and is explained as a
vertically expanding jet. Similar corona behavior has also been found in other
sources by analyzing the lag spectrum with Comptonization models (e.g. García
et al., 2021, 2022a; Méndez et al., 2022; Zhang et al., 2022; Peirano et al.,
2023).

The ionization parameter $\xi$ is defined as $\xi=L/nr^{2}$ where $L$ is the
ionizing luminosity, $n$ is the density and $r$ is the distance to the
illuminating source. We do not see strong variation on the ionization
parameter. This is within our expectation since the source luminosity changes
by only a factor of 2. The variation of the ionization parameter would be
comparable to its fitting uncertainties if the density and distance do not
change. Among the seven epochs, a high ionization parameter ($\log(\xi)>3.6$)
is always required to fit the reflection spectra. This high ionization state
is consistent with previous studies of GX 339–4 at similar luminosity levels
(e.g. Sridhar et al., 2020). With the measured ionization parameter and the
known source luminosity, we can calculate the disk density and compare it with
our measurements (e.g. Tab. 4). Using the Equ. 9 of Zdziarski & De Marco
(2020), we calculate a disk density of $n_{\rm cal}\sim 10^{22}$ cm-3 with the
averaged illuminating flux and ionization parameter in Tab. 4. This value is
higher than what we measured with the relxillD model and exceeds the upper
limit ($10^{19}$ cm-3)333Note that in the latest version of relxill, which was
released after writing of this paper, the density is allowed to vary up to
$10^{20}$ cm-3. See http://www.sternwarte.uni-
erlangen.de/~dauser/research/relxill/. allowed by the model. Similar
discrepancy has been found in other intermediate data of GX 339–4 (e.g.
Zdziarski & De Marco, 2020). It might be explained if the inner disk radius is
30 times larger than the ISCO but it would conflict with the broad iron line
we are seeing. More work in developing the reflection models would help to
better understand this problem.

###  4.2. Measurement of system parameters

By fitting the seven spectra simultaneously with relativistic reflection
models, we are able to measure the black hole spin parameter of GX 339–4. We
obtain a constraint of $a_{*}>0.95$ if assuming a lamppost geometry. When
implementing a broken power-law emissivity in relxill, we obtain the lowest
$\chi^{2}$ among all models and get a constraint of $a_{*}>0.86$ (see Fig. 5).
This measurement confirms the high spin nature of the black hole in GX 339–4
as found in previous studies with data in the hard (e.g. Reis et al., 2008;
Ludlam et al., 2015; García et al., 2015) or soft states (e.g. Parker et al.,
2016).

Previous reflection based analysis have constrained the disk inclination angle
of GX 339–4 to be 30∘–60∘ (e.g. García et al., 2015; Parker et al., 2016).
Moreover, by studying the near-infrared lightcurve of GX 339–4, Heida et al.
(2017) have measured the orbital inclination to be 37∘–78∘. A more recent work
by Zdziarski et al. (2019) updated the inclination to $\approx$ 40∘–60∘ based
on considering evolutionary models for the donor. Our measurement of the
inclination angle of the inner accretion disk ranges from 35 to 43 degree,
which is consistent with the previous studies via different methods. We also
find that the constraint on the inclination with reflection spectroscopy
slightly changes ($42.0_{-1.4}^{+1.3}$ with the lamppost geometry and
$36.6_{-1.3}^{+1.5}$ for the broken power-law emissivity) with the treatments
of the emissivity profile (see Fig. 5). These differences on parameter
measurements due to systematic uncertainties are expected for reflection
spectroscopy (e.g. Zhou et al., 2020; Cárdenas-Avendaño et al., 2020). It is a
result of the lack of our knowledge about the system. We note that these
systematic uncertainties should be considered when reporting measurements of
parameters with disk reflection models. Other sources of systematic
uncertainties include simplifications in the models (e.g. the discussion in
Tripathi et al., 2021) or instrumental effects (see the review in Bambi et
al., 2021).

We always obtain a very high iron abundance that is near ten times of the
solar value. We note that the measurement may not represent the true abundance
of the system. The super-solar iron abundance is a well known issue in the
fitting of the relativistic reflection features (e.g. Dauser et al., 2012;
Jiang et al., 2018; García et al., 2018). It has been shown that allowing the
electron density ($n_{\rm e}$) of the disk to go higher than what is assumed
in the standard reflection model ($\log(n_{\rm e}/{\rm cm}^{-3})$=15) could
help to lower the iron abundance (e.g. Tomsick et al., 2018; Jiang et al.,
2019a). However, our high-density model still gives a very high iron abundance
($A_{\rm Fe}>8.6$). Another solution to solve this high iron abundance problem
is to allow different photon indexes for the directly observed primary
emission and the reflection spectrum (e.g. Fürst et al., 2015). We test this
scenario but it does not help to lower the iron abundance (with $A_{\rm
Fe}>9.0$). We note that the corona structure may be more complex than two
uniform clouds of plasma (e.g. Uttley et al., 2011). Moreover, Ross et al.
(2002) have found that ignoring the returning radiation could also affect the
measurement of the iron abundance while more recent studies on the returning
radiation do not find strong impact (e.g. Riaz et al., 2021; Dauser et al.,
2022). There is also the possibility that the high iron abundance is real and
caused by radiative levitation of metal ions in the inner part of the
accretion disk (e.g. Reynolds et al., 2012).

Figure 5.— The $\chi^{2}$ contours for the black hole spin and the inclination
parameters for the lamppost geometry and broken power-law emissivity. The
horizontal lines represent the 1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence
levels for a single parameter of interest.

##  5\. Conclusion

In this work, we analyzed the broadband spectra (2–100 keV) of GX 339–4 during
the bright state transition of its 2021 outburst observed by Insight–HXMT.
Strong relativistic reflection features were found in the spectra, which
allowed us to study the evolution of the disk-corona system during the
transition. The main results are the following:

  * •

The hard to soft transition is associated with a stronger contribution of the
disk thermal component to the X-ray spectrum and a steepening of the power-law
emission.

  * •

The inner disk radius stays close to the ISCO during the transition. This is
even true when we consider a reflection model that assumes the ionzing
continuum to be a combination of the corona emission from above and a
blackbody emission from below.

  * •

If assuming a lamppost geometry, the mesured corona height is always close to
the black hole.

  * •

The scattering fraction, which is the fraction of the disk photons that are
scattered by the hot corona, decreases along with the transition.

  * •

The data provide constraint on the black hole spin ($a_{*}>0.86$) and the
inclination parameter ($i\approx$ 35∘–43∘) of the system.

Acknowledgments – We thank the anonymous referee for helpful comments to
improve the manuscript. We thank Yuexin Zhang for useful discussion. This work
was supported by the National Natural Science Foundation of China (NSFC),
Grant No. 11973019 and Grant No. 12250610185, the Natural Science Foundation
of Shanghai, Grant No. 22ZR1403400, the Shanghai Municipal Education
Commission, Grant No. 2019-01-07-00-07-E00035, and Fudan University, Grant No.
JIH1512604. JJ acknowledges support from the Leverhulme Trust, the Isaac
Newton Trust and St Edmund’s College.

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##  Appendix A The HXMT observations

Table 2Insight–HXMT observations of GX 339–4 analyzed in this paper

Reference name | Date1 | obsID | Exposure (ks) | Start time | Stop time
---|---|---|---|---|---
|  |  | LE | ME | HE | MJD | MJD
Epoch 1 | 20210325 | P030402403702 | 1.86 | 3.31 | 2.30 | 59298.012 | 59298.151
|  | P030402403703 |  |  |  | 59298.151 | 59298.289
Epoch 2 | 20210326 | P030402403801 | 3.46 | 5.75 | 7.95 | 59299.602 | 59299.776
|  | P030402403802 |  |  |  | 59299.776 | 59299.909
|  | P030402403803 |  |  |  | 59299.909 | 59300.003
Epoch 3 | 20210327 | P030402403901 | 2.40 | 3.86 | 3.88 | 59300.796 | 59300.923
|  | P030402403902 |  |  |  | 59300.923 | 59301.059
Epoch 4 | 20210328 | P030402403903 | 7.25 | 12.9 | 13.2 | 59301.059 | 59301.204
|  | P030402403904 |  |  |  | 59301.204 | 59301.343
|  | P030402403905 |  |  |  | 59301.343 | 59301.498
|  | P030402403906 |  |  |  | 59301.498 | 59301.634
|  | P030402403907 |  |  |  | 59301.634 | 59301.794
|  | P030402403908 |  |  |  | 59301.794 | 59301.916
|  | P030402403909 |  |  |  | 59301.916 | 59302.052
Epoch 5 | 20210329 | P030402403910 | 7.46 | 13.3 | 13.6 | 59302.052 | 59302.189
|  | P030402403911 |  |  |  | 59302.189 | 59302.347
|  | P030402403912 |  |  |  | 59302.347 | 59302.515
|  | P030402403913 |  |  |  | 59302.515 | 59302.648
|  | P030402403914 |  |  |  | 59302.648 | 59302.787
|  | P030402403915 |  |  |  | 59302.787 | 59302.961
|  | P030402403916 |  |  |  | 59302.961 | 59303.120
Epoch 6 | 20210330 | P030402403917 | 5.63 | 11.2 | 11.4 | 59303.120 | 59303.260
|  | P030402403918 |  |  |  | 59303.260 | 59303.399
|  | P030402403919 |  |  |  | 59303.399 | 59303.551
|  | P030402403920 |  |  |  | 59303.551 | 59303.684
|  | P030402403921 |  |  |  | 59303.684 | 59303.816
|  | P030402403922 |  |  |  | 59303.816 | 59303.949
|  | P030402403923 |  |  |  | 59303.949 | 59304.081
Epoch 7 | 20210331 | P030402403924 | 4.57 | 9.95 | 10.2 | 59304.081 | 59304.214
|  | P030402403925 |  |  |  | 59304.214 | 59304.346
|  | P030402403926 |  |  |  | 59304.346 | 59304.479
|  | P030402403927 |  |  |  | 59304.479 | 59304.611
|  | P030402403928 |  |  |  | 59304.611 | 59304.743
|  | P030402403929 |  |  |  | 59304.743 | 59304.876
|  | P030402403930 |  |  |  | 59304.876 | 59305.037

Note. (1) The observation date is presented in the form of yyyymmdd.

##  Appendix B The best-fit tables

Table 3Best-fit values with a broken power-law emissivity with $q_{\rm out}=3$
(Model 1B).

Component | Parameter | Epoch 1 | Epoch 2 | Epoch 3 | Epoch 4 | Epoch 5 | Epoch 6 | Epoch 7
---|---|---|---|---|---|---|---|---
tbabs |  $N_{\rm H}$ (1022 cm-2) | $0.99_{-0.06}^{+0.05}$
diskbb |  $T_{\rm in}$ (keV) | $0.50_{-0.04}^{+0.04}$ | $0.540_{-0.012}^{+0.013}$ | $0.658_{-0.009}^{+0.009}$ | $0.718_{-0.011}^{+0.008}$ | $0.726_{-0.011}^{+0.01}$ | $0.765_{-0.008}^{+0.009}$ | $0.752_{-0.009}^{+0.011}$
relxill | $a_{*}$ | $>0.86$
| $q_{\rm in}$ | $3^{*}$ | $3^{*}$ | $3^{*}$ | $10_{-3}^{+P}$ | $10_{-4}^{+P}$ | $9_{-4}^{+P}$ | $7.4_{-1.7}^{+P}$
| $q_{\rm out}$ | $3^{*}$
|  $R_{\rm br}$ ($R_{\rm g}$) | - | - | - | $3.5_{-0.3}^{+0.5}$ | $3.6_{-0.3}^{+0.8}$ | $3.62_{-0.28}^{+1.3}$ | $4.2_{-0.9}^{+0.9}$
| Incl | $36.6_{-1.3}^{+1.5}$
|  $A_{\rm Fe}$ (solar) | $10.0_{-1.1}^{+P}$
|  $R_{\rm in}$ (ISCO) | $40_{-33}^{+P}$ | $3.6_{-1.6}^{+4.0}$ | $2.3_{-0.6}^{+1.4}$ | $1.0_{-P}^{+0.8}$ | $1.0_{-P}^{+0.6}$ | $1.1_{-P}^{+0.9}$ | $1.0_{-P}^{+0.7}$
| $\log(\xi)$ | $4.29_{-0.1}^{+0.05}$ | $4.54_{-0.07}^{+0.05}$ | $4.60_{-0.1}^{+0.07}$ | $4.02_{-0.15}^{+0.21}$ | $4.35_{-0.22}^{+0.12}$ | $4.31_{-0.16}^{+0.14}$ | $4.1_{-0.3}^{+0.3}$
cutoffpl | $\Gamma$ | $1.701_{-0.025}^{+0.029}$ | $2.054_{-0.013}^{+0.017}$ | $2.27_{-0.06}^{+0.04}$ | $2.42_{-0.04}^{+0.04}$ | $2.42_{-0.04}^{+0.03}$ | $2.504_{-0.027}^{+0.027}$ | $2.48_{-0.1}^{+0.15}$
|  $E_{\rm cut}$ (keV) | $140_{-20}^{+30}$ | $>450$ | $>400$ | $>220$ | $>330$ | $>410$ | $130_{-60}^{+70}$
cflux |  |  |  |  |  |  |  |
$F_{\rm diskbb}$ | (10-8 erg cm-2 s-1) | $0.219_{-0.024}^{+0.03}$ | $0.556_{-0.029}^{+0.020}$ | $0.93_{-0.03}^{+0.04}$ | $1.310_{-0.025}^{+0.023}$ | $1.360_{-0.023}^{+0.03}$ | $1.597_{-0.025}^{+0.022}$ | $1.85_{-0.06}^{+0.04}$
$F_{\rm relxill}$ | (10-8 erg cm-2 s-1) | $0.73_{-0.13}^{+0.16}$ | $1.26_{-0.08}^{+0.07}$ | $1.44_{-0.19}^{+0.17}$ | $1.38_{-0.12}^{+0.14}$ | $1.45_{-0.18}^{+0.11}$ | $1.62_{-0.16}^{+0.2}$ | $1.56_{-0.29}^{+0.5}$
$F_{\rm cutoffpl}$ | (10-8 erg cm-2 s-1) | $0.73_{-0.20}^{+0.07}$ | $0.57_{-0.12}^{+0.15}$ | $0.9_{-0.3}^{+0.5}$ | $1.78_{-0.24}^{+0.25}$ | $1.52_{-0.25}^{+0.22}$ | $1.80_{-0.4}^{+0.23}$ | $1.3_{-0.4}^{+0.6}$
constant | ME/LE | $1.035_{-0.014}^{+0.014}$
| HE/LE | $1.158_{-0.023}^{+0.018}$
|  $\chi^{2}$/d.o.f | 3673.1/3383

Note. Best-fit parameters for the model tbabs*(diskbb+relxill+cutoffpl).
Parameters with ∗ are fixed during the fit and the symbol $P$ denotes the
upper or lower boundary. Parameters that are only shown for Epoch 4 are tied
across all observations.

Table 4Best-fit values with relxillD.

Component | Parameter | Epoch 1 | Epoch 2 | Epoch 3 | Epoch 4 | Epoch 5 | Epoch 6 | Epoch 7
---|---|---|---|---|---|---|---|---
tbabs |  $N_{\rm H}$ (1022 cm-2) | $0.98_{-0.04}^{+0.04}$
diskbb |  $T_{\rm in}$ (keV) | $0.48_{-0.04}^{+0.04}$ | $0.561_{-0.014}^{+0.012}$ | $0.659_{-0.008}^{+0.008}$ | $0.717_{-0.01}^{+0.008}$ | $0.726_{-0.007}^{+0.006}$ | $0.762_{-0.007}^{+0.008}$ | $0.756_{-0.006}^{+0.008}$
relxill | $a_{*}$ | $>0.87$
| $q_{\rm in}$ | $3^{*}$ | $3^{*}$ | $3^{*}$ | $7.9_{-4}^{+P}$ | $7.3_{-2.2}^{+P}$ | $7.2_{-2.7}^{+P}$ | $6.5_{-1.4}^{+P}$
| $q_{\rm out}$ | $3^{*}$
|  $R_{\rm br}$ ($R_{\rm g}$) | - | - | - | $3.7_{-0.4}^{+0.7}$ | $3.8_{-0.6}^{+1.0}$ | $3.8_{-0.6}^{+1.2}$ | $4.2_{-0.8}^{+1.1}$
| Incl | $35.3_{-1.5}^{+1.5}$
| $A_{\rm Fe}$ | $>8.6$
|  $R_{\rm in}$ (ISCO) | $11_{-6}^{+P}$ | $1.5_{-P}^{+0.8}$ | $1.5_{-P}^{+0.9}$ | $1.69_{-0.23}^{+0.22}$ | $1.7_{-0.4}^{+0.3}$ | $1.77_{-0.26}^{+0.29}$ | $1.73_{-0.18}^{+0.25}$
| $\log(\xi)$ | $3.97_{-0.19}^{+0.17}$ | $3.69_{-0.14}^{+0.17}$ | $4.2_{-0.1}^{+0.1}$ | $3.49_{-0.15}^{+0.13}$ | $4.00_{-0.16}^{+0.22}$ | $4.0_{-0.4}^{+0.2}$ | $4.01_{-0.4}^{+0.27}$
| $\log(n_{\rm e})$ | $16_{-P}^{+P}$ | $19.0_{-0.6}^{+P}$ | $19.0_{-0.6}^{+P}$ | $18.9_{-0.9}^{+P}$ | $17.9_{-1.3}^{+0.4}$ | $18.2_{-0.5}^{+0.5}$ | $15.0_{-P}^{+2.1}$
cutoffpl | $\Gamma$ | $1.724_{-0.019}^{+0.026}$ | $2.042_{-0.011}^{+0.024}$ | $2.166_{-0.022}^{+0.04}$ | $2.369_{-0.029}^{+0.019}$ | $2.387_{-0.024}^{+0.023}$ | $2.43_{-0.04}^{+0.04}$ | $2.54_{-0.06}^{+0.09}$
|  $E_{\rm cut}$ (keV) | $100_{-14}^{+18}$ | $130_{-20}^{+30}$ | $>270$ | $>320$ | $>310$ | $>390$ | $170_{-90}^{+270}$
cflux |  |  |  |  |  |  |  |
$F_{\rm diskbb}$ | (10-8 erg cm-2 s-1) | $0.22_{-0.03}^{+0.04}$ | $0.556_{-0.029}^{+0.027}$ | $0.965_{-0.017}^{+0.017}$ | $1.346_{-0.012}^{+0.028}$ | $1.390_{-0.017}^{+0.027}$ | $1.634_{-0.012}^{+0.029}$ | $1.82_{-0.03}^{+0.05}$
$F_{\rm relxillD}$ | (10-8 erg cm-2 s-1) | $0.55_{-0.08}^{+0.09}$ | $0.69_{-0.1}^{+0.04}$ | $1.2_{-0.1}^{+0.1}$ | $1.43_{-0.3}^{+0.11}$ | $1.28_{-0.11}^{+0.13}$ | $1.44_{-0.18}^{+0.3}$ | $1.73_{-0.22}^{+0.21}$
$F_{\rm cutoffpl}$ | (10-8 erg cm-2 s-1) | $1.00_{-0.05}^{+0.09}$ | $1.21_{-0.06}^{+0.11}$ | $0.75_{-0.19}^{+0.07}$ | $1.54_{-0.14}^{+0.1}$ | $1.53_{-0.11}^{+0.08}$ | $1.51_{-0.3}^{+0.11}$ | $1.6_{-0.6}^{+0.6}$
constant | ME/LE | $1.025_{-0.014}^{+0.016}$
| HE/LE | $1.088_{-0.021}^{+0.022}$
|  $\chi^{2}$/d.o.f | 3664.5/3376

Note. Best-fit parameters for the model tbabs*(diskbb+relxillD+cutoffpl).
Parameters with ∗ are fixed during the fit and the symbol $P$ denotes the
upper or lower boundary. Parameters that are only shown for Epoch 4 are tied
across all observations.

Table 5Best-fit values with relxillcp (Model 2).

Component | Parameter | Epoch 1 | Epoch 2 | Epoch 3 | Epoch 4 | Epoch 5 | Epoch 6 | Epoch 7
---|---|---|---|---|---|---|---|---
tbabs |  $N_{\rm H}$ (1022 cm-2) | $0.73_{-0.06}^{+0.06}$
diskbb |  $T_{\rm in}$ (keV) | $0.48_{-0.04}^{+0.04}$ | $0.567_{-0.016}^{+0.016}$ | $0.676_{-0.01}^{+0.009}$ | $0.729_{-0.009}^{+0.008}$ | $0.739_{-0.009}^{+0.008}$ | $0.775_{-0.009}^{+0.008}$ | $0.768_{-0.009}^{+0.008}$
relxillcp | $a_{*}$ | $>0.85$
| $q_{\rm in}$ | $3^{*}$ | $3^{*}$ | $3^{*}$ | $10_{-3}^{+P}$ | $8.0_{-2.2}^{+P}$ | $9_{-4}^{+P}$ | $7.3_{-2.0}^{+P}$
| $q_{\rm out}$ | $3^{*}$
|  $R_{\rm br}$ ($R_{\rm g}$) | - | - | - | $3.42_{-0.12}^{+0.6}$ | $3.9_{-0.7}^{+0.9}$ | $3.6_{-0.3}^{+2.1}$ | $4.1_{-0.9}^{+1.2}$
| Incl | $37.1_{-1.3}^{+1.4}$
| $A_{\rm Fe}$ | $10.0_{-0.3}^{+P}$
|  $R_{\rm in}$ (ISCO) | $>12$ | $2.3_{-0.8}^{+1.8}$ | $2.5_{-1.1}^{+1.8}$ | $1.1_{-P}^{+0.7}$ | $1.1_{-P}^{+0.4}$ | $1.18_{-0.12}^{+0.5}$ | $1.1_{-P}^{+0.6}$
| $\log(\xi)$ | $4.56_{-0.11}^{+0.05}$ | $4.46_{-0.1}^{+0.13}$ | $4.66_{-0.07}^{+P}$ | $3.99_{-0.15}^{+0.12}$ | $4.09_{-0.18}^{+0.24}$ | $4.27_{-0.24}^{+0.12}$ | $3.93_{-0.3}^{+0.22}$
nthcomp | $\Gamma$ | $1.773_{-0.024}^{+0.04}$ | $2.092_{-0.022}^{+0.019}$ | $2.215_{-0.027}^{+0.03}$ | $2.339_{-0.016}^{+0.02}$ | $2.360_{-0.023}^{+0.022}$ | $2.391_{-0.018}^{+0.024}$ | $2.42_{-0.08}^{+0.12}$
|  $kT_{\rm e}$ (keV) | $64_{-21}^{+30}$ | $160_{-70}^{+600}$ | $>270$ | $>400$ | $>310$ | $>410$ | $65_{-30}^{+600}$
cflux |  |  |  |  |  |  |  |
$F_{\rm diskbb}$ | (10-8 erg cm-2 s-1) | $0.183_{-0.024}^{+0.03}$ | $0.506_{-0.022}^{+0.024}$ | $0.938_{-0.012}^{+0.025}$ | $1.370_{-0.022}^{+0.022}$ | $1.421_{-0.022}^{+0.024}$ | $1.665_{-0.022}^{+0.023}$ | $1.86_{-0.03}^{+0.03}$
$F_{\rm relxillcp}$ | (10-8 erg cm-2 s-1) | $0.94_{-0.25}^{+0.15}$ | $0.85_{-0.15}^{+0.12}$ | $1.10_{-0.14}^{+0.12}$ | $1.04_{-0.07}^{+0.09}$ | $1.05_{-0.07}^{+0.09}$ | $1.09_{-0.09}^{+0.11}$ | $1.11_{-0.19}^{+0.6}$
$F_{\rm nthcomp}$ | (10-8 erg cm-2 s-1) | $0.50_{-0.15}^{+0.24}$ | $0.67_{-0.08}^{+0.11}$ | $0.46_{-0.08}^{+0.08}$ | $0.528_{-0.027}^{+0.022}$ | $0.50_{-0.03}^{+0.03}$ | $0.43_{-0.03}^{+0.03}$ | $0.34_{-0.06}^{+0.04}$
|  $L_{\rm 0.1-100keV}/L_{\rm Edd}$ (%) | 10.4 | 12.9 | 15.9 | 18.8 | 19.0 | 20.3 | 21.1
constant | ME/LE | $1.022_{-0.012}^{+0.014}$
| HE/LE | $1.147_{-0.025}^{+0.034}$
|  $\chi^{2}$/d.o.f | 3737.3/3383

Note. Best-fit parameters for the model tbabs*(diskbb+relxillcp+cutoffpl).
Parameters with ∗ are fixed during the fit and the symbol $P$ denotes the
upper or lower boundary. Parameters that are only shown for Epoch 4 are tied
across all observations.

Table 6Best-fit values with simplcut (Model 3).

Component | Parameter | Epoch 1 | Epoch 2 | Epoch 3 | Epoch 4 | Epoch 5 | Epoch 6 | Epoch 7
---|---|---|---|---|---|---|---|---
tbabs |  $N_{\rm H}$ (1022 cm-2) | $0.75_{-0.06}^{+0.04}$
diskbb |  $T_{\rm in}$ (keV) | $0.50_{-0.04}^{+0.04}$ | $0.575_{-0.017}^{+0.018}$ | $0.682_{-0.009}^{+0.008}$ | $0.735_{-0.007}^{+0.007}$ | $0.743_{-0.01}^{+0.007}$ | $0.777_{-0.007}^{+0.007}$ | $0.769_{-0.009}^{+0.008}$
relxillcp | $a_{*}$ | $0.92_{-0.07}^{+0.05}$
| $q_{\rm in}$ | $3^{*}$ | $3^{*}$ | $3^{*}$ | $9.7_{-3}^{+P}$ | $7.8_{-2.3}^{+P}$ | $8.9_{-5}^{+P}$ | $7.0_{-2.0}^{+P}$
| $q_{\rm out}$ | $3^{*}$
|  $R_{\rm br}$ ($R_{\rm g}$) | - | - | - | $3.32_{-0.15}^{+0.5}$ | $3.8_{-0.4}^{+1}$ | $3.45_{-0.28}^{+2.6}$ | $4.0_{-0.7}^{+1.5}$
| Incl | $37.3_{-1.1}^{+1.4}$
| $A_{\rm Fe}$ | $10.0_{-0.3}^{+P}$
|  $R_{\rm in}$ (ISCO) | $50.0_{-39}^{+P}$ | $3.0_{-0.9}^{+1.9}$ | $3.1_{-0.9}^{+1.0}$ | $1.4_{-P}^{+0.5}$ | $1.41_{-0.30}^{+0.25}$ | $1.47_{-0.29}^{+0.24}$ | $1.4_{-0.3}^{+0.5}$
| $\log(\xi)$ | $4.62_{-0.04}^{+0.03}$ | $4.59_{-0.15}^{+0.09}$ | $4.70_{-0.19}^{+P}$ | $3.99_{-0.16}^{+0.12}$ | $4.13_{-0.22}^{+0.22}$ | $4.28_{-0.24}^{+0.13}$ | $3.94_{-0.27}^{+0.6}$
| $\Gamma$ | $1.771_{-0.012}^{+0.008}$ | $2.125_{-0.022}^{+0.024}$ | $2.243_{-0.029}^{+0.04}$ | $2.356_{-0.018}^{+0.03}$ | $2.370_{-0.022}^{+0.019}$ | $2.397_{-0.016}^{+0.019}$ | $2.43_{-0.07}^{+0.11}$
|  $kT_{\rm e}$ (keV) | $90_{-20}^{+35}$ | $>80$ | $>240$ | $>280$ | $>200$ | $>270$ | $60_{-30}^{+230}$
simplcut | $f_{\rm sc}$ | $0.121_{-0.011}^{+0.004}$ | $0.175_{-0.028}^{+0.06}$ | $0.127_{-0.023}^{+0.04}$ | $0.119_{-0.006}^{+0.007}$ | $0.114_{-0.012}^{+0.01}$ | $0.0947_{-0.008}^{+0.0025}$ | $0.076_{-0.015}^{+0.01}$
cflux |  |  |  |  |  |  |  |
$F_{\rm diskbb}$ | (10-8 erg cm-2 s-1) | $0.215_{-0.028}^{+0.03}$ | $0.60_{-0.03}^{+0.06}$ | $1.066_{-0.024}^{+0.06}$ | $1.547_{-0.029}^{+0.022}$ | $1.60_{-0.04}^{+0.05}$ | $1.838_{-0.029}^{+0.021}$ | $2.009_{-0.04}^{+0.025}$
$F_{\rm relxillcp}$ | (10-8 erg cm-2 s-1) | $1.278_{-0.014}^{+0.017}$ | $1.17_{-0.11}^{+0.1}$ | $1.34_{-0.21}^{+0.05}$ | $1.21_{-0.09}^{+0.11}$ | $1.18_{-0.09}^{+0.09}$ | $1.20_{-0.09}^{+0.08}$ | $1.2_{-0.4}^{+0.7}$
|  $L_{\rm 0.1-100keV}/L_{\rm Edd}$ (%) | 10.5 | 13.7 | 16.8 | 19.7 | 19.7 | 21.0 | 21.7
constant | ME/LE | $1.018_{-0.007}^{+0.017}$
| HE/LE | $1.141_{-0.027}^{+0.020}$
|  $\chi^{2}$/d.o.f | 3734.8/3383

Note. Best-fit parameters for the model tbabs*simplcut(diskbb+relxillcp) with
a broken power-law emissivity. Parameters with ∗ are fixed during the fit and
the symbol $P$ denotes the upper or lower boundary. Parameters that are only
shown for Epoch 4 are tied across all observations. The model flux is before
Comptonization by simplcut. The absorption corrected X-ray flux is calculated
with the flux command in XSPEC.

Table 7Best-fit values with refhiden (Model 4).

Component | Parameter | Epoch 1 | Epoch 2 | Epoch 3 | Epoch 4 | Epoch 5 | Epoch 6 | Epoch 7
---|---|---|---|---|---|---|---|---
tbabs |  $N_{\rm H}$ (1022 cm-2) | $0.36_{-0.06}^{+0.09}$
relconv | $a_{*}$ | $0.998^{*}$
| $q_{\rm in}$ | $2.0_{-2.2}^{+2.0}$ | $2.17_{-0.16}^{+0.21}$ | $1.7_{-0.6}^{+0.5}$ | $2.35_{-0.14}^{+0.14}$ | $3.3_{-0.6}^{+0.7}$ | $2.7_{-0.3}^{+0.5}$ | $2.5_{-0.3}^{+0.8}$
| $q_{\rm out}$ | $=q_{\rm in}$
| Incl | $33.2_{-1.4}^{+2.4}$
|  $R_{\rm in}$ (ISCO) | $9_{-P}^{+P}$ | $1.5_{-P}^{+5}$ | $1.0_{-P}^{+20}$ | $1.6_{-P}^{+2.8}$ | $5.1_{-1.7}^{+2.5}$ | $3.8_{-1.8}^{+4}$ | $2.6_{-1.5}^{+2.9}$
refhiden |  $kT_{\rm bb}$ (keV) | $0.487_{-0.008}^{+0.013}$ | $0.400_{-P}^{+0.007}$ | $0.400_{-P}^{+0.005}$ | $0.441_{-0.004}^{+0.004}$ | $0.4403_{-0.0027}^{+0.004}$ | $0.459_{-0.004}^{+0.004}$ | $0.468_{-0.004}^{+0.005}$
| $H_{\rm den}~{}(10^{20}~{}{\rm cm}^{-3})$ | $>86$ | $16.2_{-1.8}^{+4.0}$ | $5.7_{-0.7}^{+0.6}$ | $5.3_{-0.5}^{+0.4}$ | $4.4_{-0.3}^{+0.5}$ | $5.4_{-0.4}^{+0.3}$ | $5.6_{-0.4}^{+0.4}$
| Illum/BB | $1.8_{-0.4}^{+0.5}$ | $1.82_{-0.29}^{+0.1}$ | $1.27_{-0.2}^{+0.06}$ | $0.84_{-0.11}^{+0.09}$ | $0.97_{-0.04}^{+0.3}$ | $0.77_{-0.07}^{+0.14}$ | $0.52_{-0.06}^{+0.18}$
| Norm | $0.39_{-0.03}^{+0.03}$ | $0.70_{-0.05}^{+0.06}$ | $0.89_{-0.04}^{+0.04}$ | $0.86_{-0.09}^{+0.15}$ | $0.99_{-0.09}^{+0.19}$ | $0.92_{-0.21}^{+0.07}$ | $0.73_{-0.11}^{+0.2}$
simplcut | $f_{\rm sc}$ | $0.773_{-0.015}^{+0.011}$ | $0.417_{-0.03}^{+0.015}$ | $0.293_{-0.017}^{+0.009}$ | $0.163_{-0.004}^{+0.005}$ | $0.132_{-0.009}^{+0.006}$ | $0.132_{-0.003}^{+0.003}$ | $0.110_{-0.011}^{+0.013}$
| $\Gamma$ | $1.867_{-0.009}^{+0.013}$ | $2.124_{-0.018}^{+0.015}$ | $2.283_{-0.025}^{+0.013}$ | $2.305_{-0.016}^{+0.012}$ | $2.315_{-0.013}^{+0.026}$ | $2.381_{-0.021}^{+0.014}$ | $2.42_{-0.06}^{+0.05}$
|  $kT_{\rm e}$ (keV) | $26.4_{-2.1}^{+1.6}$ | $50_{-10}^{+10}$ | $>200$ | $>100$ | $>100$ | $>360$ | $70_{-30}^{+500}$
|  $L_{\rm 0.1-100keV}/L_{\rm Edd}$ (%) | 8.2 | 9.3 | 11.0 | 12.2 | 12.5 | 13.3 | 12.8
constant | ME/LE | $0.985_{-0.006}^{+0.008}$
| HE/LE | $1.057_{-0.019}^{+0.016}$
|  $\chi^{2}$/d.o.f | 3788.8/3386

Note. Best-fit parameters for the model tbabs*simplcut*relconv*refhiden with a
power-law emissivity. Parameters with ∗ are fixed during the fit and the
symbol $P$ denotes the upper or lower boundary. Parameters that are only shown
for Epoch 4 are tied across all observations. The absorption corrected X-ray
flux is calculated with the flux command in XSPEC.


11institutetext:  University of Jyväskylä, Jyväskylä, Finland
11email<EMAIL_ADDRESS>
# Quantum Algorithm Cards: Streamlining the development of hybrid classical-
quantum applications

Vlad Stirbu 11 0000-0001-9462-5922 Majid Haghparast 11 0000-0003-3427-5961

###### Abstract

The emergence of quantum computing proposes a revolutionary paradigm that can
radically transform numerous scientific and industrial application domains.
The ability of quantum computers to scale computations implies better
performance and efficiency for certain algorithmic tasks than current
computers provide. However, to gain benefit from such improvement, quantum
computers must be integrated with existing software systems, a process that is
not straightforward. In this paper, we investigate challenges that emerge when
building larger hybrid classical-quantum computers and introduce the Quantum
Algorithm Card (QAC) concept, an approach that could be employed to facilitate
the decision making process around quantum technology.

###### Keywords:

Q

uantum software, software architecture, software development life-cycle,
developer’s experience, quantum algorithm cards (QACs).

##  1 Introduction

Quantum computers have demonstrated the potential to revolutionize various
fields, including cryptography, drug discovery, materials science, and machine
learning, by leveraging the principles of quantum mechanics. However, the
current generation of quantum computers, known as noisy intermediate-scale
quantum (NISQ) computers, suffer from noise and errors, making them
challenging to operate. Additionally, the development of quantum algorithms
requires specialized knowledge not readily available to the majority of
software professionals. These factors pose a significant entry barrier to
leveraging the unique capabilities of quantum systems.

For the existing base of business applications, classical computing has
already proven its capabilities across a diverse range of solutions. However,
some of the computations they must perform can be accelerated with quantum
computing, much like GPUs are used today. Therefore, quantum systems should
not function in isolation, but they must coexist and inter-operate with
classical systems. To this end, software architects play a crucial role in
achieving seamless integration while simultaneously designing systems that
effectively meet the unique requirements of businesses.

To address the challenges associated with this integration, this paper focuses
on designing hybrid systems that integrate quantum and classical computing,
aiming to overcome architectural, design, and operational hurdles. In doing
so, we look at the software development lifestyle, the technology stack of
hybrid classic-quantum systems, and deployment techniques used today. As a
concrete contribution, we propose quantum algorithm cards as a mechanism to
support decision making process related to quantum technology during the
development and deployment of hybrid classic-quantum applications.

The rest of the paper is organized as follows. In Section 2, we provide the
necessary background for the paper. In Section 3, we address architectural
concerns associated with the development and deployment of hybrid classic-
quantum applications. In Section 4, we introduce the concept of quantum
algorithm cards in detail. The discussion and future plans are drawn in
Section 5.

##  2 Background

Figure 1: Software development lifecycle of a hybrid classical-quantum
system[1].

The software development life-cycle (SDLC) of hybrid classic-quantum
applications consists of a multi-faceted approach, as depicted in Fig. 1. At
the top level, the classical software development process starts by
identifying user needs and deriving them into system requirements. These
requirements are transformed into a design and implemented. The result is
verified against the requirements and validated against user needs. Once the
software system enters the operational phase, any detected anomalies are used
to identify potential new system requirements, if necessary. A dedicated track
for quantum components is followed within the SDLC [2], specific to the
implementation of quantum technology. The requirements for these components
are converted into a design, which is subsequently implemented on classic
computers, verified on simulators or real quantum hardware, and integrated
into the larger software system. During the operational phase, the quantum
software components are executed on real hardware. Scheduling ensures
efficient utilization of scarce quantum hardware, while monitoring
capabilities enable the detection of anomalies throughout the process.

A typical hybrid classic-quantum software system is understood as a classical
program that has one or more software components that are implemented using
quantum technology, as depicted in Fig. 2. A quantum component utilizes
quantum algorithms [3], that are transformed into quantum circuits using a
toolkit like Cirq111https://quantumai.google/cirq or
Qiskit222https://qiskit.org. The quantum circuit describes quantum
computations in a machine-independent language using quantum assembly (QASM)
[4]. This circuit is translated by a computer that controls the quantum
computer in a machine specific circuit and a sequence of operations, such as
pulses [5], that control the individual hardware qubits. The translation
process, implemented using quantum compilers, encompasses supplementary
actions like breaking down quantum gates, optimizing quantum circuits, and
providing fault-tolerant iterations of the circuit. Further, the concept of
distributed quantum computers [6], which interlink multiple distinct quantum
machines through quantum communication networks, emerges as a potential
solution to amplify the available quantum volume beyond what is possible using
a single quantum computer. Nevertheless, the intricacies inherent in the
distributed quantum computers remain hidden from users, as compilers aware of
the distributed architecture of the target system shield them from such
complexities. In essence, the quantum compiler plays a vital role in achieving
the effective execution of generic quantum circuits on existing physical
hardware platforms, making the compilers an active research area in quantum
computing [7].

Figure 2: Quantum computing model: components and interfaces

##  3 Architectural and operational concerns

This section highlights the SDLC stages and the key challenges can be observed
while developing hybrid classic-quantum applications.

Quantum advantage awareness. In accordance with the hybrid classic-quantum
SDLC, it becomes evident that during the decomposition of the system into
smaller software components, a team with limited quantum technology knowledge
might overlook that employing quantum algorithms have the capacity to surpass
the performance of conventional alternatives for algorithmically intensive
tasks.

Quantum algorithm design and implementation. Upon determining the possibility
of quantum advantage aligned with their needs, the team often encounters a
knowledge gap hindering the effective attainment of this objective. The design
of quantum algorithms necessitates comprehension of quantum mechanical
phenomena such as superposition, entanglement, and interference – concepts
that can confound the intuition of those untrained in the field. Although
well-resourced teams will likely have a quantum scientist specialty, similar
to the data scientist role in artificial intelligence, this cannot be assumed
to be generally available. This challenge leaves mainstream developers
grappling to identify the optimal algorithms for their tasks. Ultimately, the
team resorts to selecting algorithms bundled with widely-used quantum
libraries.

Availability and cost of quantum hardware. Upon successful implementation of
the components and confirmation of quantum advantage, the team faces the task
of selecting suitable hardware for executing the quantum tasks efficiently.
This decision necessitates a comprehensive grasp of the most fitting qubit
implementation technology, contingent on the interconnections among the qubits
within the generic quantum circuit. While quantum compilers have the
capability to convert the initial circuit into a version optimized for the
specific machine, the considerations of hardware availability and associated
expenses remain pivotal factors to be addressed for each application’s
context.

##  4 Quantum algorithm card proposal

Addressing the previously highlighted architectural and operational
considerations, our proposal recommends incorporating a Quantum Algorithm Card
(QAC) in conjunction with quantum algorithm implementations. This artifact
purpose is two fold: first, it serves as a repository for insights into the
algorithm’s implementation, and secondly, it conveys critical information for
the users that rely on the implementation to realize the application specific
needs. As a starting point, the QAC contains the following sections: the
overview containing administrative information and high level overview of the
algorithm purpose, the intended purpose describes the tasks for which the
algorithm provide optimal performance, the usage details conveys information
about how the implementation can be used and integrated into a larger system,
the performance metrics includes information that is useful to evaluate the
results of the algorithm and monitoring for deviations, the limitations
conveys the known situations for which the use of the algorithm is not
suitable, the  reference refers to the canonical document that introduced the
algorithm, and, finally, the caveats should include relevant information that
the user should be aware. Table 1 provides an outline of the proposed QAC
elements, the corresponding content and the recipients of the information.
It’s important to note that these elements are not exhaustive and can be
tailored to suit the specifics of each corresponding quantum algorithm.

Table 1:  Quantum algorithm card: elements, content and recipients (e.g. T -
technology management and architects, D - software developers, O - operations)
Quantum Algorithm Card
---
Element | Description | Target
Overview |  \- Provider/designer/maintainer information. \- Brief description of the algorithm’s purpose, key features and functionalities. \- Algorithm’s high-level architecture or approach, complexity |  T D O
Intended use |  \- Clear description of the tasks the algorithm is designed for. \- Specific scenarios for which the quantum algorithm is intended. |  T
Usage details |  \- Information about algorithm usage, e.g. inputs and outputs. \- Quantum volume needed to run the algorithm. |  D O
Performance metrics |  \- Metrics and explanations used to evaluate the algorithm’s performance. \- Decision thresholds, variation approaches, and any relevant quantitative measures. |  T O
Limitations |  \- Clear articulation of the algorithm’s limitations and potential failure modes. \- Known scenarios where the algorithm might not perform well or could provide incorrect results. |  T D
References |  \- Citations to relevant research papers and resources. |  T D
Caveats |  \- Situations that users should avoid. |  D

##  5 Discussion and Future Work

Serving as a facilitator, the QAC artifact aims to enhance communication
across common specializations within the team to support decision making. For
example, technology managers and architects need specific information during
the high-level implementation phase; software engineers need to know how the
integrate the quantum technology into the larger classical system; operations
needs to know how to execute and monitor the quantum components in production
environment. Consequently, it is imperative that the card’s content is
conveyed in a language that is easily comprehensible by the intended audience,
which are users and not developers of quantum technology.

The initial evaluation of the concept was performed, as an internal exercise
based on a paper prototype [8] of Grover’ search algorithm [9], on a target
group that has both classic and quantum software development skills. The
concept was found to be useful, especially for the developers that have
artificial intelligence and machine learning background, as they were already
familiar with similar concepts like Model Cards [10], and Data Cards [11].
However, as the classic and quantum disciplines are rather different, there is
a fine line that needs to the considered carefully when deciding the depth of
content about the quantum algorithm should be included, otherwise the card
becomes a communication impediment rather than facilitator.

Further work is needed to validate the concept with external target groups. We
are also planning to develop a Python toolkit that aims to streamline the
collection of information included, and to automate the generation of QACs.
The Quantum Algorithm Cards Toolkit (QACT) will enable the developers and
implementers of quantum algorithms share their metadata and metrics with
developers, researchers, and other stakeholders.

## Acknowledgement

This work has been supported by the Academy of Finland (project DEQSE 349945)
and Business Finland (project TORQS 8582/31/2022).

## References

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