# Quasiparticle spectroscopy, transport, and magnetic properties
of Nb films used in superconducting transmon qubits

Kamal R. Joshi  Ames National Laboratory, Ames, IA 50011, U.S.A.  Sunil
Ghimire  Ames National Laboratory, Ames, IA 50011, U.S.A.  Department of
Physics & Astronomy, Iowa State University, Ames, IA 50011, U.S.A.  Makariy A.
Tanatar  Ames National Laboratory, Ames, IA 50011, U.S.A.  Department of
Physics & Astronomy, Iowa State University, Ames, IA 50011, U.S.A.  Amlan
Datta  Ames National Laboratory, Ames, IA 50011, U.S.A.  Department of Physics
& Astronomy, Iowa State University, Ames, IA 50011, U.S.A.  Jin-Su Oh  Ames
National Laboratory, Ames, IA 50011, U.S.A.  Lin Zhou  Ames National
Laboratory, Ames, IA 50011, U.S.A.  Cameron J. Kopas  Rigetti Computing, 775
Heinz Ave., Berkeley, CA 94710, U.S.A.  Jayss Marshall  Rigetti Computing, 775
Heinz Ave., Berkeley, CA 94710, U.S.A.  Josh Y. Mutus  Rigetti Computing, 775
Heinz Ave., Berkeley, CA 94710, U.S.A.  Julie Slaughter  Ames National
Laboratory, Ames, IA 50011, U.S.A.  Matthew J. Kramer  Ames National
Laboratory, Ames, IA 50011, U.S.A.  James A. Sauls  Center for Applied Physics
and Superconducting Technologies, Department of Physics and Astronomy
Northwestern University, Evanston, IL 60208, U.S.A.  Ruslan Prozorov
<EMAIL_ADDRESS>Ames National Laboratory, Ames, IA 50011, U.S.A.
Department of Physics & Astronomy, Iowa State University, Ames, IA 50011,
U.S.A.

(23 July 2022)

###### Abstract

Niobium thin films on silicon substrate used in the fabrication of
superconducting qubits have been characterized using scanning and transmission
electron microscopy, electrical transport, magnetization, quasiparticle
spectroscopy, and real-space real-time magneto-optical imaging. We study
niobium films to provide an example of a comprehensive analytical set that may
benefit superconducting circuits such as those used in quantum computers. The
films show outstanding superconducting transition temperature of $T_{c}=9.35$
K and a fairly clean superconducting gap, along with superfluid density
enhanced at intermediate temperatures. These observations are consistent with
the recent theory of anisotropic strong-coupling superconductivity in Nb.
However, the response to the magnetic field is complicated, exhibiting
significantly irreversible behavior and insufficient heat conductance leading
to thermo-magnetic instabilities. These may present an issue for further
improvement of transmon quantum coherence. Possible mitigation strategies are
discussed.

##  I Introduction

Superconducting qubits are promising candidates for implementing large-scale
quantum computers [1, 2, 3]. The advancement in fabrication and design of
superconducting qubits have demonstrated impressive gate fidelity of up to
99.5 % for two-qubit gates, which is the measure of the ability of a device to
faithfully execute quantum algorithms [4]. However, large-scale devices will
require fidelity well beyond 99.9 % [5]. It was shown that certain impurities
and defects in qubit material tend to shorten the coherence time and lead to
overall lower fidelity operations [6, 7, 8]. For example, microwave loss at
cryogenic temperatures has been attributed to the resonant coupling to two-
level systems (TLS) physically composed of the atomic defects [9, 7].
Therefore, achieving better fidelities requires improved quantum coherence and
that necessiates a better understanding and control over the materials used to
fabricate qubits.

Niobium is the material of choice for various superconducting applications due
to its relatively high transition temperature, $T_{c}=9.35$ K. Niobium thin
films (100-200 nm thick) are used in the superconducting transmon qubits [1].
The “heart” of such qubits is an aluminum Josephson junction used to provide
non-linearity to the circuit and niobium is used to fabricate the vast
majority of the rest of the circuit: the readout resonators, capacitor pads,
and coupling lines in the transmon. The structure must have a high degree of
quantum coherence in all parts to work. Surfaces and interfaces are known to
introduce noise and loss that play a major role in decreasing the coherence
time in transmon qubits [10, 11, 12, 13, 7]. For instance, the presence of
native amorphous oxides with different stoichiometry (NbO, NbO2 and Nb2O5) are
believed to be the host for TLSs, hence decoherence [14, 15, 16, 17]. Another
recently discovered source of potential decoherence in transmons is nano-sized
niobium hydrides [7] (large hydrides in superconducting RF cavities are
believed to be the cause of the so-called Q-disease (a dramatic reduction of
the quality factor above some amplitude of electromagnetic field inside) and
have been known for a long time [18, 19, 20, 21, 22, 23]). While bulk niobium
used in SRF cavities was extensively characterized both in normal and the
superconducting states, the niobium films used in QIS technologies have mostly
been studied in the normal state and, in the superconducting state,
specifically to examine the quality factor behavior and losses at GHz
frequencies [24]. Bulk and film states of niobium are very different in terms
of morphology, purity, and relative length scales involved, and more
conventional studies in the superconducting state are needed.

In this work, we employ various techniques to study thermally excited
quasiparticles, electrical transport, upper and third critical fields,
magnetization and spatial distribution of the magnetic induction in niobium
thin films and suggest the strategies for further optimizations and
improvements.

##  II EXPERIMENTAL

Niobium films, 160 nm thick, were deposited onto [001] high-resistivity
silicon wafers ($>10000$ $\Omega\cdot cm$ using high-power impulse magnetron
sputtering (HiPIMS) in a ultra-high vacuum system with a base pressure
$<1\times 10^{-8}$ Torr. [8].

The film morphology was studied using scanning and transmission electron
microscopy, high-resolution SEM, and TEM. A more detailed investigation of the
transmons from the same batch as in this work, and imaging details, are
available elsewhere [25]. Cross-sectional and plan-view TEM samples were
prepared with a Helios focused ion beam system. The TEM images were acquired
at an acceleration voltage of 200 kV using a Titan Themis. The SEM was used to
map out the larger view of sample surface morphology. The TEM images were used
for quantitative analysis.

To study the low-energy quasiparticles, we used a tunnel-diode resonator for
the precision measurements of the London penetration depth, $\Delta\lambda(T)$
[26, 27], which is then used to calculate the superfluid density, which is
compared to the expectations from the theory.

Four-probe electrical resistivity measurements were performed in a _Quantum
Design_ PPMS. Contacts were made by gluing 25 $\mathrm{\mu m}$ silver wires
using DuPont 4929N conducting silver paste. This technique yields contacts
with contact resistance in the 10 to 100 $\Omega$ range. Transport
measurements of the upper critical field, $H_{c2}$, were performed with a
magnetic field oriented perpendicular to the film plane to avoid the third
critical field, $H_{c3}$, which does not exist in this orientation but is
maximum when the magnetic field is parallel to the film surface [28, 29]. For
measurements in a parallel configuration, the sample was glued on the side of
the plastic cube, similar to those described in Ref. 30. This procedure
provided alignment with the accuracy of about 2 degrees.

Total magnetization was measured using a vibrating sample magnetometer (VSM)
module in _Quantum Design_ PPMS. The advantage of this device is the ability
to sweep the magnetic field continuously at a constant rate.

The two-dimensional distribution of the magnetic induction was mapped in real-
time employing magneto-optical imaging using Faraday Effect (MOFA) in
transparent ferrimagnetic indicators (bismuth-doped iron garnets) placed on
top of the samples. The closed-cycle flow-type optical 4He cryostat exposed
the cooled sample to an Olympus polarized-light microscope. The magnetic
induction on the sample surface polarizes in-plane magnetic moments in the
indicator, and the distribution of this polarization component along the light
propagation is visualized through double Faraday rotation. In the images, only
the magnetic field is visible due to a mirror sputtered at the bottom of the
indicator [31, 32, 33, 34].

##  III Results

Figure 1: (a) Secondary electrons high-resolution SEM wide area view of sample
surface morphology. (b) TEM plan view and (c) TEM cross-sectional view of the
studied Nb thin film. The bulk consists of columnar Nb grains with sizes
ranging between 25 and 75 nm, with an average grain size of about 50 nm. The
surface roughness is of the order of 3 to 5 nm.

Figure 1 shows the morphology of the Nb film revealed by high-resolution
scanning (HR-SEM) and transmission (TEM) electron microscopy. Figure 1(a)
shows the SEM image of a wide view of the top surface morphology, revealing
substantial spatial modulations at two characteristic length scales. Further
details of internal and surface structural texture are revealed by bright-
field plan-view TEM image, Fig.1(b), and cross-sectional view, Fig.1(c). The
contrast variation reveals granular polycrystalline structure on top and
inside the film with grain sizes ranging from 25 nm to 75 nm, with an average
size of about 50 nm. Those grains have a columnar structure along film
thickness, measured at 160 nm as intended by the fabrication. The smaller
droplets observed in secondary electrons contrast, Fig.1(a), represent surface
roughness also seen in the cross-sectional view, Fig.1(c).

Figure 2: Top panels: electrical resistivity of Nb film in magnetic field
parallel to the film plane. (Left) $R\left(H\right)$ scans measured at
different temperatures from 2 to 8 K with 1 K interval. (Right)
$R\left(T\right)$ scans at fixed magnetic fields of, from right to left:
$H=$0, 0.1, 0.3, 0.4, 0.5, 0.7, 1.0, 1.2, 1.4, 1.6, 1.8 and 2 T. Bottom panel:
summary phase diagram showing the third critical field, $H_{c3}$, obtained
from the data shown in the top panels, and the upper critical field, $H_{c2}$,
obtained from the similar measurements with magnetic field oriented
perpendicular to the plane. The inset shows temperature-dependent resistivity
normalized by its value at $T_{c}$. The
$RRR=R(300\>\mathrm{K})/R\left(T_{c}\right)=5$.

Figure 2 presents electrical resistivity measurements as a function of
magnetic field and temperature. The data were collected in parallel and
perpendicular orientations of the film with respect to the applied magnetic
field. In a parallel orientation, the material remains superconducting up to a
so-called third critical magnetic field, $H_{c3}$, surviving in a thin surface
later of the order of the superconducting coherence length, $\xi$ [28, 29,
35]. In niobium, $\xi\approx 10-40$ nm, depending on its purity and degree of
crystallinity [28, 36, 37, 38]. In the perpendicular orientation of the
magnetic field to the film plane, the upper critical field, $H_{c2}$, is the
highest magnetic field possible [28, 29, 39]. Therefore, we obtain these two
critical magnetic fields from the measurements of the same sample, simply
rotating it by 90 degrees without changing the contacts. In the PPMS, the
cooling is achieved via the helium exchange gas when the sample is mounted on
a puck, and so the orientation is not important for thermal equilibrium as
long as the measurements are performed slowly. Indeed, we see the same
$T_{c}=9.35$ K in zero magnetic field in both orientations.

The top left panel of Fig.2 shows the magnetic field - dependent resistance,
$R\left(H\right)$, measured at different temperatures from 2 to 8 K in 1 K
steps. The top right panel shows the temperature-dependent resistance,
$R\left(T\right)$, measured at different magnetic fields, from right to left:
$H=$0, 0.1, 0.3, 0.4, 0.5, 0.7, 1.0, 1.2, 1.4, 1.6, 1.8, and 2 T. In both top
panels, the symbols show the locations of the third critical field,
$H_{c3}\left(T\right)$ (left panel), or critical temperatures,
$T_{c}\left(H_{c3}\right)$ (right panel), an easy pick, considering how sharp
the transitions remain for all curves. Similar measurements were performed in
the perpendicular orientation, yielding the upper critical field,
$H_{c2}\left(T\right)$. The bottom panel of Fig.2 summarizes the results of
the critical fields measurements. The upper curve, obtained from the data in
the top panels, shows the third critical field, $H_{c3}\left(T\right)$. The
agreement between the two types of scans is outstanding. Similarly, the lower
curve shows the upper critical field, $H_{c2}\left(T\right)$.

The inset shows temperature-dependent resistivity normalized by its value at
$T_{c}$ in a broad temperature range up to 300 K. At 300 K, this ratio, called
the residual resistivity ratio, $RRR(300\>\mathrm{K})\approx 5$, is the
commonly used measure of scattering on defects and impurities. For comparison,
the cleanest niobium samples reach $RRR=90000$ [40], and so the value of 5 for
our films is rather low.

Figure 3: (color online) Main panel: normalized superfluid density evaluated
from the measured London penetration depth shown in the insets using the
formula shown in the Y-axis title. The solid red curve is the expectation from
the isotropic weak-coupling BCS theory [41]. The upper inset shows a full-
temperature range variation, whereas the lower left inset shows the data up to
$0.8T_{c}$. The red line is the fit to the isotropic low-temperature
asymptotic BCS formula described in the text. Note very small total variation
of $\Delta\lambda\left(T\right)$ due to significantly reduced effective sample
dimension due to thin-film geometry [42].

We now examine the superconducting state of the studied films. Figure 3 shows
the temperature-dependent superfluid density,
$\rho\left(T\right)=\lambda^{2}\left(0\right)/\lambda^{2}\left(T\right)$,
obtained from the London penetration depth, $\lambda\left(T\right)$, measured
using tunnel-diode resonator [26, 27]. The solid red curve is the expectation
from the isotropic weak-coupling Bardeen-Cooper-Schrieffer (BCS) theory [41].
This measurement is quite difficult due to the extreme thin-film geometry. The
upper inset in Fig.3 shows the full transition curve, which looks flawless,
but cuts off at about $\Delta\lambda\left(T_{cm}\right)=3.3\>\mathrm{\mu m}$,
at which point the film becomes transparent to our 10 MHz RF AC field. This
formally follows from the analysis of the magnetic susceptibility in finite
samples [42]. For our films, the formal calibration gives for the effective
sample size, $R=3\>\mathrm{\mu m}$, which matches perfectly the above
experimental value for the penetration depth at $T_{c}$ when the response
becomes “sample size limited” [42]. This also explains why the measured
$T_{cm}=9.03$ K is a little lower than $T_{c}=9.35$ K detected by the
transport measurements, Fig.2. The lower inset zooms into the lower
temperature behavior. Remarkably, the penetration depth changes only by 10 nm
from the base temperature up to $0.8T_{c}$. And yet we were able to resolve
the whole superfluid density curve shown in the main panel of Fig.3. A clear
exponential attenuation at low temperatures signals a full superconducting
gap. If we fit the data to a low-temperature BCS formula, valid below
$0.3T_{c}$,
$\Delta\lambda/\lambda\left(0\right)=\sqrt{\pi\delta/2t}\exp\left(-\delta/t\right)$
[27], where $\delta=\Delta\left(0\right)/k_{B}T_{c}$ and $t=T/T_{c}$ and use
$\delta$ as a free parameter, we obtain $\delta\approx 1.3$ instead of
isotropic BCS value of $\delta\approx 1.76$. This is most likely associated
with the electronic and superconducting states anisotropy of niobium, and its
multiband nature [38]. In this situation, the range of exponential attenuation
is determined by the smaller gap, specifically on band 2, which also dominates
the thermodynamic response [38]. The theory is further confirmed by the
observed deviation of the experimental superfluid density at elevated
temperatures from the weak coupling expectation, shown by the solid red line
in the main panel of Fig.3. Stronger coupling makes $\rho\left(T\right)$ lie
above the weak coupling values [38].

Figure 4: The VSM measurements of the hysteresis loops at $T=2$ K while
continuously sweeping an applied magnetic field up and down four times, each
time at a different rate, to examine the dynamic effects. The field time
variation and the rate values are shown in the inset. The measurements were
performed in a magnetic field perpendicular to the film plane. Curiously, an
additional hysteretic response appears above the so-called “matching field” of
0.65 T corresponding roughly to 56 nm length scale matching the structural
features found by EM imaging, Fig.1, see text for details.

Figure 4 shows profound features on the $M(H)$ magnetic hysteresis loops
obtained using _Quantum Design_ PPMS-VSM. This instrument allows for the
measurements while sweeping the magnetic field continuously without pausing.
The field time variation is shown in the inset with 200, 100, 50, and 15 Oe/s
rates, more than ten times the span of $dH/dt$. The measurements shown in 4
were performed perpendicular to the film plane. Figure 4 shows additional
dynamic rate-dependent hysteretic response appearing above roughly 0.65 T. In
the simplest approximation, if the average distance between Abrikosov vortices
is $a$, it corresponds to the magnetic induction of $B=\phi_{0}/a^{2}$, where
$\phi=2.07\times 10^{-15}$ Wb is the magnetic flux quantum. If there are some
pinning centers of a characteristic length-scale, $\ell_{p}$, the vortex
pinning is most efficient when the inter-vortex distance matches this scale,
$a\simeq\ell_{p}$. This defines the “matching field”, $H_{m}$. Therefore, we
can estimate $\ell_{p}=\sqrt{\phi_{0}/H_{m}}\approx 56$ nm, where, in our
case, $H_{m}=0.65$ T, Fig.4. This immediately reminds of Fig.1 where this
length scale was determined from the direct EM imaging, providing further
evidence of the importance of the film morphology in determining the magnetic
response. Overall, our films show a much larger magnetic hysteresis compared
to the bulk samples and single crystals [36, 43]. However, a very large
demagnetizing factor and thin-film geometry play a significant role in
diminishing the interval and the magnitude of the reversible magnetic moment,
which is proportional to a very small film volume, whereas irreversible Bean
currents flow in the entire sample [44, 45, 46, 47, 48].

Figure 5: Magneto-optical imaging of the magnetic induction in a thin film
sample at $T=5$ K. Magnetic field was applied perpendicular to the film plane
after cooling the sample in zero field. The first five images show flux
penetration, $H=85,135,210,335,$ and $425$ Oe. After that, the magnetic field
was reduced imaging flux exit and trapping in the sample, $H=230,160,50,$ and
$35$ Oe. The video of this process is available as supplementary material. The
scale bar is shown in the top left image.

We now examine the details of the magnetic flux penetration and exit using
direct magneto-optical imaging of the superconducting state using specially
fabricated transparent ferrimagnetic indicators placed on top of samples [31,
49, 50]. Figure 5 shows the sequence of images obtained after cooling in zero
magnetic field to 5 K. The first five frames (left to right, top to bottom)
show increasing magnetic field up to the maximum of $H_{max}=425$ Oe, after
which the magnetic field was reduced, and flux was exiting the sample. There
are two distinct regimes of flux penetration and exit. First, large-scale,
very fast (see the video in supplemental information), dendritic avalanches
similar in their appearance to lightning strikes. Similar behavior was
observed long ago in various films, particularly niobium, and is well-
understood [51, 52]. The physics behind it is straightforward. When first
vortices are pushed inside by the screening currents, these moving vortices
induce an electric field parallel to the currents, resulting in a resistive
state, and energy is dissipated. If there is an insufficient thermal link to
the surrounding and, in the case of films, to the substrate, the released heat
decreases the local critical current in the direction of the vortex motion
creating a self-heating violent unidirectional propagation of bunches of
vortices that follow the heated up trail which needs time to dissipate.
Sometimes they branch out when meeting various obstacles and, in this case,
most likely the grain boundaries. In thicker samples (a few tens of $\mu$m
foils), the avalanches may become “global” leading to a catastrophic collapse
of the entire Bean critical state [50]. The actual local temperature
instabilities have been measured during these processes. In the case of QIS
applications, this insufficient heat channeling may represent a difficult
problem. The second stage of flux penetration is a regular Bean critical state
pattern [44, 45, 47], also somewhat non-uniform, following the granular film
morphology. When the external magnetic field is decreased, the reverse process
happens, and similar dendritic avalanches start to channel out, now shown by a
darker color corresponding to the local suppression of the magnetic induction.
In the end, the mix of uniform and non-uniform trapped flux remains in the
sample as a fingerprint of the intense perturbations occurring when the field
was ramped up and down. The supplemental information shows a real-time movie
of the processes pictured in Fig.5. This highly inhomogeneous magnetic flux
evolution reveals the issues of thermal heat-sinking and morphological
inhomogeneity issues.

##  IV Discussion

Let us review the experimental findings and try to understand all different
results from a common point of view. All measurements indicate that as far as
superconducting properties are concerned, the films are rather far from being
ideal and exhibit various irreversible properties that generally lead to
dissipation, hence quantum decoherence.

One of the most puzzling facts is that while the $RRR=5$ is rather very low,
the superconducting transition temperature, $T_{c}=9.35$ K, is as high as in
good single crystals. According to the theory, $T_{c}$ should be suppressed by
non-magnetic disorder due to anisotropy of the order parameter and a multi-
band nature of the material [38]. However, the same theory predicts saturation
of the $T_{c}$ when the gap averages out at higher scattering rates. Still,
$T_{c}$ should be lower than 9 K, and it isn’t. One natural way to understand
it, supported by the direct EM imaging, Fig.1, is to assume that grains
themselves are quite clean in terms of the scattering rate, but the random
network of grain boundaries creates significant resistance. The boundaries do
not affect high $T_{c}$ inside the grains, though. In the normal state, just
above $T_{c}$, the resistivity is mostly determined by those boundaries.
However, when the grains become superconducting, the boundaries become a
network of Josephson junctions, and the whole sample transitions to zero
resistance. This explains the low $RRR$ and, simultaneously, high $T_{c}$.

The heterogeneous granular structure also explains the measurements of the
third and the upper critical fields, shown in Fig.2. While it is important to
measure these two fields in one experiment and on the same sample, the problem
is that their ratio is about 1.3, whereas a ratio of at least 1.7 is expected
theoretically even in the dirty limit [39]. However, real sample geometry and
surface conditions may significantly affect the $H_{c3}$ [29] beyond the
point-like scattering and smooth semi-infinite surface with perfectly parallel
field considered in Ref.[39]. Therefore, our granular film does not show the
expected maximum value of this ratio. This is further re-enforced by the clear
matching effect at the length-scale of around 50 nm, Fig.4, which correlates
with the granular structure revealed by the film electron microscopy, Fig.1.

Finally, the thermo-magnetic instabilities found in our study indicate an
impeded heat flow and removal in these films. While QIS applications of thin
films do not use an external magnetic field, the transmons are driven at high
frequencies and show elevated dissipation due to morphological
inhomogeneities. In particular, the granular structure may contribute to their
additional distortion of the current flow, and extra resistance [53, 54].
Another possible source is Kapitza resistance due to phonon flow mismatch at
the boundaries and the interfaces [54]. Finally, occasional vortices may enter
the structure due to extreme demagnetization and sharp edges. With or without
vortices, the heat dissipation and cooling may become a serious issue if the
device is located in a high vacuum and can only sink heat through a substrate.

On the bright side, the films show clean exponential attenuation of the weak
magnetic field, signaling a fairly clean superconducting gap. Therefore, they
still have the potential for improvement, and here we suggest some strategies.

###  IV.1 Suggested mitigation strategies

A few suggestions can be made regarding applying such films for
superconducting transmons. (1) To investigate the impeded heat transfer and
dissipation, one may try to measure the coherence time by comparing the same
transmon in vacuum and immersed in liquid helium directly inside a mixing
chamber of a dilution refrigerator. (2) Reducing the substrate thickness may
help to improve heat sinking when transmons are in a vacuum. (3) The most
problematic aspect of these films is that their granular structure may be
fixed by switching to a different deposition method. The alternative methods
producing better Nb films have been known since the 1980s [55, 56]. For
example, molecular beam epitaxy (MBE) was shown to produce Nb films of
excellent crystallinity with $RRR\approx 200$ and, remarkably, $T_{c}=9.45$ K
[55]. In this case, different substrates, better matching Nb thermal
contraction and expansion, such as $\alpha-$Al2O3 and MgO were used. (4) One
may also try to perform post-manufacturing treatment of the prepared films,
for example, using recently introduced electro-annealing that was used for the
optimization of quantum interference devices [57]. (5) Finally, we recently
showed [58] that the introduction of true point-like defects using proton
irradiation expectedly reduces the $T_{c}$ somewhat, to around 9.16 K,
indicating that it affects the grains’ interior. However, it seems to also
inhibit thermo-magnetic instabilities, perhaps because these defects increase
critical current density, thus reducing vortex motion. Uniformly distributed,
dilute point-like defects may be beneficial for the transmons for other
reasons. For example, they may reduce (pin or alter otherwise) the activity of
two-level systems (TLS) and also prevent phase slips in the Josephson network
of grain boundaries.

##  V Conclusions

In summary, a comprehensive characterization of 160 nm thick Nb films on [001]
Si substrate used in superconducting transmons is presented. Electron
microscopy, transport, magnetization, quasiparticle spectroscopy, and real-
space magneto-optical images all show that while these films have outstanding
superconducting transition temperature of $T_{c}=9.35$ K and clean
superconducting gap, their behavior in the magnetic field is complicated
exhibiting significantly irreversible behavior, and insufficient heat
conductance leading to thermo-magnetic instabilities. These may present an
issue for further improvement of transmon quantum coherence. Possible
mitigation strategies are suggested, and the techniques presented here provide
guidelines for comprehensive characterization that may be used to improve
materials for superconducting quantum computing.

###### Acknowledgements.

We thank V. Kogan, M. Zarea, A. Romanenko, and A. Grasselino for discussions
and attention. We thank the Rigetti fabrication team for developing and
manufacturing the studied films. This work was supported by the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Superconducting Quantum Materials and Systems Center (SQMS)
under contract number DE-AC02-07CH11359. The research was performed at the
Ames National Laboratory, operated for the U.S. DOE by Iowa State University
under contract # DE-AC02-07CH11358.

## References

  * Reagor  _et al._ [2016] M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, Quantum memory with millisecond coherence in circuit QED, Physical Review B 94, 014506 (2016).
  * Kjaergaard  _et al._ [2020] M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Superconducting qubits: Current state of play, Annual Review of Condensed Matter Physics 11, 369 (2020), https://doi.org/10.1146/annurev-conmatphys-031119-050605 .
  * Alam and et al [2022] M. S. Alam and et al, Quantum computing hardware for hep algorithms and sensing, arXiv:2204.08605 (2022), arXiv:2204.08605 [quant-ph] .
  * Jurcevic  _et al._ [2021] P. Jurcevic, A. Javadi-Abhari, L. S. Bishop, I. Lauer, D. F. Bogorin, M. Brink, L. Capelluto, O. GÃ¼nlÃ¼k, T. Itoko, N. Kanazawa, A. Kandala, G. A. Keefe, K. Krsulich, W. Landers, E. P. Lewandowski, D. T. McClure, G. Nannicini, A. Narasgond, H. M. Nayfeh, E. Pritchett, M. B. Rothwell, S. Srinivasan, N. Sundaresan, C. Wang, K. X. Wei, C. J. Wood, J.-B. Yau, E. J. Zhang, O. E. Dial, J. M. Chow, and J. M. Gambetta, Demonstration of quantum volume 64 on a superconducting quantum computing system, Quantum Science and Technology 6, 025020 (2021).
  * de Leon  _et al._ [2021] N. P. de Leon, K. M. Itoh, D. Kim, K. K. Mehta, T. E. Northup, H. Paik, B. S. Palmer, N. Samarth, S. Sangtawesin, and D. W. Steuerman, Materials challenges and opportunities for quantum computing hardware, Science 372, eabb2823 (2021), https://www.science.org/doi/pdf/10.1126/science.abb2823 .
  * Schlör  _et al._ [2019] S. Schlör, J. Lisenfeld, C. Müller, A. Bilmes, A. Schneider, D. P. Pappas, A. V. Ustinov, and M. Weides, Correlating decoherence in transmon qubits: Low frequency noise by single fluctuators, Phys. Rev. Lett. 123, 190502 (2019).
  * Lee  _et al._ [2021] J. Lee, Z. Sung, A. A. Murthy, M. Reagor, A. Grassellino, and A. Romanenko, Discovery of nb hydride precipitates in superconducting qubits, arXiv:2108.10385 (2021), arXiv:2108.10385 [quant-ph] .
  * Murthy  _et al._ [2022] A. A. Murthy, J. Lee, C. Kopas, M. J. Reagor, A. P. McFadden, D. P. Pappas, M. Checchin, A. Grassellino, and A. Romanenko, Tof-sims analysis of decoherence sources in superconducting qubits, Applied Physics Letters 120, 044002 (2022), https://doi.org/10.1063/5.0079321 .
  * Müller  _et al._ [2019] C. Müller, J. H. Cole, and J. Lisenfeld, Towards understanding two-level-systems in amorphous solids: insights from quantum circuits, Reports on Progress in Physics 82, 124501 (2019).
  * Oliver and Welander [2013] W. D. Oliver and P. B. Welander, Materials in superconducting quantum bits, MRS Bulletin 38, 816 (2013).
  * Wang  _et al._ [2015] C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. H. Devoret, and R. J. Schoelkopf, Surface participation and dielectric loss in superconducting qubits, Applied Physics Letters 107, 162601 (2015).
  * Dial  _et al._ [2016] O. Dial, D. T. McClure, S. Poletto, G. A. Keefe, M. B. Rothwell, J. M. Gambetta, D. W. Abraham, J. M. Chow, and M. Steffen, Bulk and surface loss in superconducting transmon qubits, Superconductor Science and Technology 29, 044001 (2016).
  * Gambetta  _et al._ [2017] J. M. Gambetta, C. E. Murray, Y. K. K. Fung, D. T. McClure, O. Dial, W. Shanks, J. W. Sleight, and M. Steffen, Investigating surface loss effects in superconducting transmon qubits, IEEE Transactions on Applied Superconductivity 27, 1 (2017).
  * Burnett  _et al._ [2016] J. Burnett, L. Faoro, and T. Lindstrom, Analysis of high quality superconducting resonators: consequences for TLS properties in amorphous oxides, Superconductor Science and Technology 29, 044008 (2016).
  * Niepce  _et al._ [2020] D. Niepce, J. J. Burnett, M. G. Latorre, and J. Bylander, Geometric scaling of two-level-system loss in superconducting resonators, Superconductor Science and Technology 33, 025013 (2020).
  * Verjauw  _et al._ [2021] J. Verjauw, A. Potočnik, M. Mongillo, R. Acharya, F. Mohiyaddin, G. Simion, A. Pacco, T. Ivanov, D. Wan, A. Vanleenhove, L. Souriau, J. Jussot, A. Thiam, J. Swerts, X. Piao, S. Couet, M. Heyns, B. Govoreanu, and I. Radu, Investigation of microwave loss induced by oxide regrowth in high-q niobium resonators, Phys. Rev. Applied 16, 014018 (2021).
  * Premkumar  _et al._ [2021] A. Premkumar, C. Weiland, S. Hwang, B. Jaeck, A. Place, I. Waluyo, A. Hunt, V. Bisogni, J. Pelliciari, A. Barbour, M. Miller, P. Russo, F. Camino, K. Kisslinger, X. Tong, M. Hybertsen, A. Houck, and I. Jarrige, Microscopic relaxation channels in materials for superconducting qubits, Communications Materials 2, 10.1038/s43246-021-00174-7 (2021).
  * Anderson [1959] P. W. Anderson, Theory of dirty superconductors, Journal of Physics and Chemistry of Solids 11, 26 (1959).
  * Arai and Kita [2004] M. Arai and T. Kita, Ab initio calculations of hc2 for nb, nbse2, and mgb2, Journal of the Physical Society of Japan 73, 2924 (2004).
  * Bahte  _et al._ [1998] M. Bahte, F. Herrmann, and P. Schmuser, Magnetization and susceptibility measurements on niobium samples for cavity production, Part. Accel. 60, 121 (1998).
  * Barkov  _et al._ [2013] F. Barkov, A. Romanenko, Y. Trenikhina, and A. Grassellino, Precipitation of hydrides in high purity niobium after different treatments, Journal of Applied Physics 114, 164904 (2013).
  * Barkov  _et al._ [2012] F. Barkov, A. Romanenko, and A. Grassellino, Direct observation of hydrides formation in cavity-grade niobium, Physical Review Special Topics - Accelerators and Beams 15, 122001 (2012).
  * Dhakal  _et al._ [2017] P. Dhakal, G. Ciovati, and G. R. Myneni, Role of thermal resistance on the performance of superconducting radio frequency cavities, Physical Review Accelerators and Beams 20, 032003 (2017).
  * Valente-Feliciano  _et al._ [2022] A. M. Valente-Feliciano, C. Antoine, S. Anlage, G. Ciovati, J. Delayen, F. Gerigk, A. Gurevich, T. Junginger, S. Keckert, G. Keppe, J. Knobloch, T. Kubo, O. Kugeler, D. Manos, C. Pira, T. Proslier, U. Pudasaini, C. E. Reece, R. A. Rimmer, G. J. Rosaz, T. Saeki, R. Vaglio, R. Valizadeh, H. Vennekate, W. V. Delsolaro, M. Vogel, P. B. Welander, and M. Wenskat, Next-generation superconducting rf technology based on advanced thin film technologies and innovative materials for accelerator enhanced performance and energy reach, arXiv:2204.02536 (2022), arXiv:2204.02536 [physics.acc-ph] .
  * Oh  _et al._ [2022] J.-S. Oh, X. Fang, T.-H. Kim, M. Lynn, M. Kramer, M. Zarea, J. A. Sauls, A. Romanenko, S. Posen, A. Grassellino, C. J. Kopas, M. Field, J. Marshall, H. Cansizoglu, J. Y. Mutus, M. Reagor, and L. Zhou, Multi-modal electron microscopy study on decoherence sources and their stability in nb based superconducting qubit, arXiv:2204.06041 (2022), arXiv:2204.06041 [cond-mat.mtrl-sci] .
  * Prozorov and Giannetta [2006] R. Prozorov and R. W. Giannetta, Magnetic penetration depth in unconventional superconductors, Superconductor Science and Technology 19, R41 (2006).
  * Prozorov and Kogan [2011] R. Prozorov and V. G. Kogan, London penetration depth in iron-based superconductors, Reports on Progress in Physics 74, 124505 (2011).
  * Abrikosov [2017] A. A. Abrikosov,  _Fundamentals of the Theory of Metals_ (Dover Publications, Newburyport, 2017).
  * Kogan  _et al._ [2002] V. G. Kogan, J. R. Clem, J. M. Deang, and M. D. Gunzburger, Nucleation of superconductivity in finite anisotropic superconductors and the evolution of surface superconductivity toward the bulk mixed state, Physical Review B 65, 094514 (2002).
  * Liu  _et al._ [2013] Y. Liu, M. A. Tanatar, V. G. Kogan, H. Kim, T. A. Lograsso, and R. Prozorov, Upper critical field of high-quality single crystals of kfe2as2, Phys. Rev. B 87, 134513 (2013).
  * Indenbom  _et al._ [1993] M. V. Indenbom, T. Schuster, M. R. Koblischka, A. Forkl, H. Kronmueller, L. A. Dorosinskii, V. K. Vlasko-Vlasov, A. A. Polyanskii, R. L. Prozorov, and V. I. Nikitenko, Study of flux distributions in high-Tc single crystals and thin films using magnetooptic techniques, Physica C 209, 259 (1993).
  * Prozorov  _et al._ [2005] R. Prozorov, R. W. Giannetta, A. A. Polyanskii, and G. K. Perkins, Topological hysteresis in the intermediate state of type-I superconductors, Phys. Rev. B 72, 212508 (2005).
  * Prozorov [2007] R. Prozorov, Equilibrium topology of the intermediate state in type-I superconductors of different shapes, Phys. Rev. Lett. 98, 257001 (2007).
  * Prozorov  _et al._ [2008] R. Prozorov, A. Fidler, J. Hoberg, and P. Canfield, Suprafroth in type-i superconductors, Nature Physics 4, 327 (2008).
  * Liarte  _et al._ [2017] D. B. Liarte, S. Posen, M. K. Transtrum, G. Catelani, M. Liepe, and J. P. Sethna, Theoretical estimates of maximum fields in superconducting resonant radio frequency cavities: stability theory, disorder, and laminates, Superconductor Science and Technology 30, 033002 (2017).
  * Finnemore  _et al._ [1966] D. K. Finnemore, T. F. Stromberg, and C. A. Swenson, Superconducting properties of high-purity niobium, Physical Review 149, 231 (1966).
  * Lechner  _et al._ [2020] E. M. Lechner, B. D. Oli, J. Makita, G. Ciovati, A. Gurevich, and M. Iavarone, Electron tunneling and x-ray photoelectron spectroscopy studies of the superconducting properties of nitrogen-doped niobium resonator cavities, Physical Review Applied 13, 044044 (2020).
  * Zarea  _et al._ [2022] M. Zarea, H. Ueki, and J. A. Sauls, Effects of anisotropy and disorder on the superconducting properties of niobium, arXiv:2201.07403 (2022), arXiv:2201.07403 [cond-mat.supr-con] .
  * Xie  _et al._ [2017] H.-Y. Xie, V. G. Kogan, M. Khodas, and A. Levchenko, Onset of surface superconductivity beyond the saint-james–de gennes limit, Physical Review B 96, 104516 (2017).
  * Koethe and Moench [2000] A. Koethe and J. I. Moench, Preparation of ultra high purity niobium, Materials Transactions, JIM 41, 7 (2000).
  * Bardeen  _et al._ [1957] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic theory of superconductivity, Physical Review 106, 162 (1957).
  * Prozorov [2021] R. Prozorov, Meissner-london susceptibility of superconducting right circular cylinders in an axial magnetic field, Physical Review Applied 16, 024014 (2021).
  * Stromberg [1965] T. F. Stromberg,  _The superconducting properties of high purity niobium_ , Ph.D. thesis, Iowa State University (1965).
  * Bean [1962] C. P. Bean, Magnetization of hard superconductors, Phys. Rev. Lett. 8, 250 (1962).
  * Bean [1964] C. P. Bean, Magnetization of high-field superconductors, Rev. Mod. Phys. 36, 31 (1964).
  * Brandt [1995] E. H. Brandt, The flux-line lattice in superconductors, Reports on Progress in Physics 58, 1465 (1995).
  * Brandt [1998] E. H. Brandt, Superconductor disks and cylinders in an axial magnetic field. I. Flux penetration and magnetization curves, Phys. Rev. B - Condens. Matter Mater. Phys. 58, 6506 (1998).
  * Brandt [1996] E. H. Brandt, Superconductors of finite thickness in a perpendicular magnetic field: Strips and slabs, Phys. Rev. B 54, 4246 (1996).
  * Young  _et al._ [2005] D. P. Young, M. Moldovan, P. W. Adams, and R. Prozorov, Magneto-optical studies of flux penetratim in super-hard Nb wire, Superc. Sci. Technol. 18, 776 (2005).
  * Prozorov  _et al._ [2006] R. Prozorov, D. V. Shantsev, and R. G. Mints, Collapse of the critical state in superconducting niobium, Phys. Rev. B 74, 220511 (2006).
  * Blanco Alvarez  _et al._ [2019] S. Blanco Alvarez, J. Brisbois, S. Melinte, R. B. G. Kramer, and A. V. Silhanek, Statistics of thermomagnetic breakdown in nb superconducting films, Scientific Reports 9, 3659 (2019).
  * Vestgården  _et al._ [2018] J. I. Vestgården, T. H. Johansen, and Y. M. Galperin, Nucleation and propagation of thermomagnetic avalanches in thin-film superconductors (review article), Low Temperature Physics 44, 460 (2018), https://doi.org/10.1063/1.5037549 .
  * Ramiere  _et al._ [2022] A. Ramiere, C. Z. Antoine, and J. Amrit, Model for hot spots and $q$-slope behavior in granular niobium thin film superconducting rf cavities, Phys. Rev. Accel. Beams 25, 022001 (2022).
  * Mizuno and Aomine [1984] K. Mizuno and T. Aomine, Current induced hotspot in superconducting thin films with different thicknesses, Journal of the Physical Society of Japan 53, 1434 (1984), https://doi.org/10.1143/JPSJ.53.1434 .
  * ichiro Oya  _et al._ [1986] G. ichiro Oya, M. Koishi, and Y. Sawada, High-quality single-crystal nb films and influences of substrates on the epitaxial growth, Journal of Applied Physics 60, 1440 (1986).
  * Wagner [1998] T. Wagner, High temperature epitaxial growth and structure of nb films on alpha-al2o3 (0001), Journal of Materials Research 13, 693 (1998).
  * Collienne  _et al._ [2021] S. Collienne, B. Raes, W. Keijers, J. Linek, D. Koelle, R. Kleiner, R. B. Kramer, J. Van de Vondel, and A. V. Silhanek, Nb - based nanoscale superconducting quantum interference devices tuned by electroannealing, Phys. Rev. Applied 15, 034016 (2021).
  * Tanatar  _et al._ [2022] M. A. Tanatar, D. Torsello, K. R. Joshi, S. Ghimire, M. Zarea, C. J. Kopas, G. Ghigo, J. A. Sauls, and R. Prozorov, Anisotropic superconductivity of niobium from the response to artificial disorder, in preparation (2022).


[a]Eamonn Weitz

# Classical vs. quantum corrections to jet broadening in a weakly coupled QGP

###### Abstract

We compute double-logarithmically enhanced corrections to $\hat{q}$ at
relative order $\mathcal{O}(g^{2})$ in the setting of a weakly coupled quark-
gluon plasma, observing how the thermal scale affects the region of phase
space, which gives rise to these corrections. We furthermore clarify how the
region of phase from which these corrections are borne is situated with
respect to that from which the classical corrections arise at relative order
$\mathcal{O}(g)$. This represents a significant step towards our eventual goal
of understanding which class of corrections dominate, thereby pushing forward
our quantitative grasp on the phenomenon of jet quenching in heavy-ion
collisions.

##  1 Introduction

In the context of heavy-ion collisions, jets provide an ideal _hard probe_ of
the quark-gluon plasma (QGP). Through interacting with the QGP, they receive
momentum kicks in the directions transverse to their propagation – _transverse
momentum broadening_. This broadening can be captured by the _transverse
momentum broadening coefficient_ , $\hat{q}=\langle k_{\perp}^{2}\rangle/L$,
which specifies the transverse momentum picked up per unit length, $L$ by a
hard parton propagating through the QGP. See the recent reviews on jets in
heavy-ion collisions [1] or extractions of $\hat{q}$ from data [2] for more
information.

For a weakly coupled QGP, $\hat{q}$ can be expressed in terms of the
transverse scattering kernel

$\hat{q}(\mu)=\int^{\mu}\frac{d^{2}k_{\perp}}{(2\pi)^{2}}k_{\perp}^{2}\mathcal{C}(k_{\perp}),\quad\quad\mathcal{C}(k_{\perp})\equiv(2\pi)^{2}\frac{d\Gamma}{dk_{\perp}^{2}},$
(1)

where $\frac{d\Gamma}{dk_{\perp}^{2}}$ is the rate for a hard parton with
energy, $E\gg T$, the temperature of the plasma, propagating along the $z$
direction to pick up $k_{\perp}$. The cutoff, $\mu$ is installed so as not to
include larger momentum scatterings, which include two hard partons in the
final state. See App. A of [3] for our conventions.

At leading order (LO) in $g$, $\hat{q}$ receives contributions from the hard
$(T)$ [4] and soft $(gT)$ [5] scales, which give rise to the parametric form
(up to logarithms) $\hat{q}\sim g^{4}T^{3}$. The soft contribution is cut off
in the IR by dynamical screening, implemented through Hard Thermal Loop
Effective Theory (HTL) resummation [6]. NLO corrections also come from the
soft scale [7]. Ultrasoft $(g^{2}T)$ modes contribute at $\mathcal{O}(g^{2})$,
for which the perturbative expansion breaks down. We refer to these NLO and
NNLO contributions as _classical corrections_ : they are distributed on the
$T/\omega$ IR tail of the Bose-Einstein distribution, $n_{\mathrm{B}}(\omega)$
and are therefore sourced by the Matsubara zero-mode.

Caron-Huot [7] demonstrated that one may compute the zero-mode contribution to
$\mathcal{C}(k_{\perp})$ in Electrostatic QCD (EQCD) [8], meaning that one can
bypass the somewhat cumbersome HTL computation. More importantly, as a theory
of static modes, EQCD is amenable to study using three-dimensional lattice
simulations, which can thus provide a _non-perturbative_ evaluation of
$\mathcal{C}(k_{\perp})$, summing contributions from the soft and ultrasoft
scales to all orders [9, 10]. Recently, the impact of these classical
corrections on the in-medium splitting rate was assessed [11, 12] and found to
be very relevant. A similar program is well underway for the non-perturbative
determination of classical corrections to the _asymptotic mass_ [13, 14, 15].

These classical corrections are at odds with doubly-logarithmically enhanced
radiative, _quantum corrections_ , appearing at $\mathcal{O}(g^{2})$, first
identified in [16, 17]. There, the leading enhancement is
$\sim\ln^{2}L_{\text{med}}/\tau_{\text{min}}$, with $L_{\text{med}}$ the
length of the medium and $\tau_{\text{min}}\sim 1/T$ the minimum formation
time of the associated radiation. This potentially large double-logarithm can
be resummed [18], with the evolution equations solved numerically in [19, 20].
Interestingly, they also arise in the context of double gluon emission [18,
21], implying that these logarithms are subject to a certain universality.

These corrections come from the _single scattering regime_ where
bremsstrahlung is sourced by a single scattering with the medium. This is in
contrast to the _multiple scattering regime_ , where the bremsstrahlung’s
formation time, $\tau$ is long enough so that it is coherently triggered by
multiple collisions, accounted for through LPM resummation. In [3], we compute
these doubly-logarithmically enhanced corrections in the context of a weakly
coupled QGP, carefully analysing how the thermal scale deforms the region of
phase space from which the double-logarithms emerge.

Figure 1: Depiction of bounds from the integration in Eq. (3). The $(b)$
boundary is defined by $\tau=\sqrt{k^{+}/\hat{q}_{0}}$ and the $(a)$ boundary
by $\tau=k^{+}/\mu^{2}$. Figure taken from [3].

##  2 Double Logarithmic Corrections and the Thermal Scale

The correction from [16] emerges in the standard dipole picture

$\delta\hat{q}_{\text{\cite[cite]{[\@@bibref{Number}{Liou:2013qya,Blaizot:2013vha}{}{}]}}}(\mu)=4\alpha_{s}C_{R}\hat{q}_{0}\int^{\mu}\frac{d^{2}k_{\perp}}{k_{\perp}^{2}}\int\frac{dk^{+}}{k^{+}},$
(2)

where $k^{+}\equiv k_{\perp}^{2}\tau$ is the energy of the bremsstrahlung and
$\hat{q}_{0}$ is the LO transverse momentum broadening coefficient, stripped
of the Coulomb logarithm as is done in the _harmonic oscillator approximation_
(HOA). Here we can explicitly see that one of the logarithms comes from a
soft, $dk^{+}/k^{+}$ divergence, with the other coming from a collinear,
$d^{2}k_{\perp}/k_{\perp}^{2}$ divergence. For what follows, it turns out to
be more convenient to work with $\tau$ and $k^{+}$

$\delta\hat{q}_{\text{\cite[cite]{[\@@bibref{Number}{Liou:2013qya,Blaizot:2013vha}{}{}]}}}(\mu)=\frac{\alpha_{s}C_{R}}{\pi}\hat{q}_{0}\int_{\tau_{\text{min}}}^{\mu^{2}/\hat{q}_{0}}\frac{d\tau}{\tau}\int_{\hat{q}_{0}\tau^{2}}^{\mu^{2}\tau}\frac{dk^{+}}{k^{+}}=\frac{\alpha_{s}C_{R}}{2\pi}\hat{q}_{0}\ln^{2}\frac{\mu^{2}}{\hat{q}_{0}\tau_{\text{min}}}.$
(3)

The limits above come from integrating over the triangle presented in Fig. 1.
Boundary $(a)$ arises from the need to cut off non-diffusive momentum
exchanges above the scale $\mu$. The line $(b)$ is then defined by
$k_{\perp}^{2}\equiv\hat{q}_{0}\tau$, marking the boundary with the _deep LPM
regime_ in which multiple scatterings occur. Above boundary $(b)$ there is no
longer a double-logarithmic enhancement as the $k^{+}$ integrand changes as
$1/k^{+}\rightarrow 1/\sqrt{k^{+}}$. Finally, boundary $(c)$ is an artefact of
the _instantaneous approximation_ : scatterings between the jet and medium are
assumed to take place instantaneously compared to the formation time
associated with the radiation. The result from [16] is recovered upon
identifying $\mu$ with the _saturation scale_ ,
$Q_{s}\equiv\hat{q}L_{\text{med}}$.

12345

Figure 2: Deformation of the double-logarithmic phase space with the inclusion
of thermal effects. Regions 1 and 2 form the “few scattering” regime, over
which we integrate to get Eq. (4) whereas we integrate over regions 3 and 4,
the “strict single scattering” regime to get Eq. (5). Region 5 then gives rise
to the $\mathcal{O}(g)$ corrections to $\hat{q}$, calculated in [7]. Figure
taken from [3].

In a weakly coupled QGP, as soon as the energy overlaps with the temperature
scale, one needs to account for more medium effects than those captured by
these instantaneous, spacelike interactions. Specifically, by taking
$T>\mu\gg\sqrt{g}T$111The $\mu>T$ case is studied in [3]. and replacing
$1\rightarrow(1+2n_{\text{B}}(k^{+}))$ in the $k^{+}$ integrand of Eq. (3), we
find

$\delta\hat{q}(\mu)^{\mathrm{few}}=\frac{\alpha_{s}C_{R}}{2\pi}\hat{q}_{0}\bigg{\\{}\ln^{2}\frac{\mu^{2}}{\hat{q}_{0}\tau_{\mathrm{int}}}-\frac{1}{2}\ln^{2}\frac{\omega_{\mathrm{T}}}{\hat{q}_{0}\tau_{\mathrm{int}}^{2}}\bigg{\\}}\quad\text{with}\;\omega_{\text{T}}=\frac{2\pi
T}{e^{\gamma_{E}}}\quad\text{for}\;\;\frac{\omega_{\mathrm{T}}}{\mu^{2}}\ll\tau_{\mathrm{int}}\ll\sqrt{\frac{\omega_{\mathrm{T}}}{\hat{q}_{0}}}.$
(4)

In doing so, we account for the Bose-Einstein stimulated emission of the
radiated gluon as well as the absorption of a gluon from the medium. We will
comment shortly on the purpose of $\tau_{\text{int}}$. But can these
additional effects be neglected in a way that is consistent with single
scattering, for instance, by demanding that
$\hat{q}_{0}\tau^{2}_{\text{min}}\gg T$ in Eq. (3)222This demand is motivated
by the fact that $n_{\text{B}}(k^{+})$ is exponentially suppressed for
$k^{+}\gg T$.? It turns out that the answer is no [3]: such a choice of
$\tau_{\text{min}}$ would necessarily allow for formation times associated
with the deep LPM regime, where $\tau_{\mathrm{LPM}}\gtrsim 1/g^{2}T$. Note
that the correction in Eq. (4) corresponds to integrating over the 1 and 2
regions in Fig. 2.

The requirement $1/g^{2}T\gg\tau_{\mathrm{int}}\gg 1/gT$ means that processes
where a _few scatterings_ occur are included in Eq. (4). Indeed,
$\tau_{\mathrm{int}}$ defines a border with what we have identified as a
_strict single scattering_ regime, where the formation time is _a priori_
consistent with single scattering, i.e $\tau\ll 1/g^{2}T$. This region is
characterised by so-called _semi-collinear_ processes [22], where timelike as
well as spacelike exchanges are allowed to occur. The leading contribution
from this region is given by integrating over the 3 and 4 regions in Fig. 2
and yields

$\delta\hat{q}_{\text{semi}}(\mu)=\frac{\alpha_{s}C_{R}}{2\pi}\hat{q}_{0}\ln^{2}\frac{\mu^{2}\tau_{\text{int}}}{\omega_{\mathrm{T}}},$
(5)

where we have taken the HOA. Adding Eqs. (4), (5), we then find

$\delta\hat{q}(\mu_{\perp})_{\text{dlog}}=\frac{\alpha_{s}C_{R}}{4\pi}\hat{q}_{0}\ln^{2}\frac{\mu^{4}}{\hat{q}_{0}\omega_{\mathrm{T}}}.$
(6)

As well as the disappearance of $\tau_{\mathrm{int}}$, we note the absence of
an IR cutoff, $\tau_{\mathrm{min}}$; looking to Fig. 2, the double-logarithm
is instead cut off by the scale $\omega_{\mathrm{T}}$. Thus, the thermal scale
plays an extremely important role in this context.

##  3 Relation to Classical Corrections

As well as double-logarithmic corrections at $\mathcal{O}(g^{2})$, we also
find _power law corrections_ when integrating over regions 3 and 4

$\delta{q}_{\mathrm{PL}}=\frac{\alpha_{s}C_{R}}{2\pi}\hat{q}_{0}\frac{4T\ln\left(\frac{\mu^{2}\tau_{\text{int}}}{k^{+}_{\text{IR}}e}\right)}{k^{+}_{\text{IR}}},$
(7)

where $k^{+}_{\mathrm{IR}}$ is an IR cutoff on the energy. Power law
corrections of this kind are usually discarded as they are unphysical – they
always cancel against other power law corrections coming from adjacent regions
of phase space. Here, we use this fact to our advantage; in the calculation of
the $\mathcal{O}(g)$ corrections, power law corrections should appear, with
$k^{+}_{\mathrm{IR}}$ instead acting as a UV cutoff there.

In more detail, one can use causality properties of $\mathcal{C}(k_{\perp})$,
also revealed in [7], to carry out the $k^{+}$ integral by analytically
continuing into the $k^{+}$ complex plane. $k^{+}_{\mathrm{IR}}$ then appears
as the radius of the arc of the deformed contour, with the arc lying between
the zeroth and first Matsubara modes. There is no dependence on
$k^{+}_{\mathrm{IR}}$ in [7] as the $1/k^{+}_{\mathrm{IR}}$ terms go to zero
and can thus be safely neglected. Nevertheless, we have indeed computed these
arc contributions explicitly and shown that they cancel exactly against the
result from Eq. (7), further confirming how the region from which the
classical corrections emerge is connected to that associated with the
logarithmically-enhanced quantum corrections.

##  4 Conclusion and Outlook

We have studied how, in the setting of a weakly coupled QGP, the thermal scale
affects the double-logarithmic phase space, originally identified in [16, 17].
In more detail, we showed how the scale, $\omega_{\mathrm{T}}$ cuts off this
region of phase space and furthermore, how the region, which gives rise to the
classical corrections, computed in [7] fits in comparison.

In obtaining Eq. (6), we have taken the HOA, neglecting a neighbouring region
phase space, which permits both single and multiple scattering processes. To
properly deal with such a region, we would need to solve an LPM resummation
equation, derived in [3] (see also [23]), differential in the transverse
momentum picked up by the parton. We foresee that the use of the _improved
opacity expansion_ [24] could allow us to arrive at an approximate solution of
this equation but we leave such an endeavour to future work.

## Acknowledgements

We thank Jacopo Ghiglieri for collaboration on the original work [3].

## References

  * [1] L. Apolinário, Y.-J. Lee and M. Winn,  _Prog. Part. Nucl. Phys._ 127 (2022) 103990 [2203.16352].
  * [2] Q.-F. Han, M. Xie and H.-Z. Zhang,  _Eur. Phys. J. Plus_ 137 (2022) 1056 [2201.02796].
  * [3] J. Ghiglieri and E. Weitz,  _JHEP_ 11 (2022) 068 [2207.08842].
  * [4] P.B. Arnold and W. Xiao,  _Phys.Rev._ D78 (2008) 125008 [0810.1026].
  * [5] P. Aurenche, F. Gelis and H. Zaraket,  _JHEP_ 0205 (2002) 043 [hep-ph/0204146].
  * [6] E. Braaten and R.D. Pisarski,  _"Phys. Rev. D"_ 45 (1992) 1827.
  * [7] S. Caron-Huot,  _Phys.Rev._ D79 (2009) 065039 [0811.1603].
  * [8] E. Braaten and A. Nieto,  _Phys.Rev._ D51 (1995) 6990 [hep-ph/9501375].
  * [9] M. Panero, K. Rummukainen and A. Schäfer,  _Phys.Rev.Lett._ 112 (2014) 162001 [1307.5850].
  * [10] G.D. Moore and N. Schlusser,  _Phys. Rev. D_ 101 (2020) 014505 [1911.13127].
  * [11] G.D. Moore, S. Schlichting, N. Schlusser and I. Soudi,  _JHEP_ 10 (2021) 059 [2105.01679].
  * [12] S. Schlichting and I. Soudi,  _Phys. Rev. D_ 105 (2022) 076002 [2111.13731].
  * [13] G.D. Moore and N. Schlusser,  _Phys. Rev. D_ 102 (2020) 094512 [2009.06614].
  * [14] J. Ghiglieri, G.D. Moore, P. Schicho and N. Schlusser,  _JHEP_ 02 (2022) 058 [2112.01407].
  * [15] J. Ghiglieri, G.D. Moore, P. Schicho, N. Schlusser and E. Weitz, 7, 2023 [2307.09297].
  * [16] T. Liou, A. Mueller and B. Wu,  _Nucl.Phys._ A916 (2013) 102 [1304.7677].
  * [17] J.-P. Blaizot, F. Dominguez, E. Iancu and Y. Mehtar-Tani,  _JHEP_ 1406 (2014) 075 [1311.5823].
  * [18] J.-P. Blaizot and Y. Mehtar-Tani,  _Nucl. Phys. A_ 929 (2014) 202 [1403.2323].
  * [19] P. Caucal and Y. Mehtar-Tani,  _Phys. Rev. D_ 106 (2022) L051501 [2109.12041].
  * [20] P. Caucal and Y. Mehtar-Tani,  _JHEP_ 09 (2022) 023 [2203.09407].
  * [21] P. Arnold and S. Iqbal,  _JHEP_ 1504 (2015) 070 [1501.04964].
  * [22] J. Ghiglieri, G.D. Moore and D. Teaney,  _JHEP_ 03 (2016) 095 [1509.07773].
  * [23] E. Iancu,  _JHEP_ 10 (2014) 095 [1403.1996].
  * [24] J.a. Barata, Y. Mehtar-Tani, A. Soto-Ontoso and K. Tywoniuk,  _JHEP_ 09 (2021) 153 [2106.07402].


# GGLasso - a Python package for General Graphical Lasso computation

Fabian Schaipp  Oleg Vlasovets  Christian L. Müller

††margin:  DOI: https://doi.org/https://doi.org Software •
https://doi.orgReview • https://github.com/fabian-
sp/GGLassohttps://doi.orgRepository • https://doi.orgArchive Submitted: 18
October 2021
Published: License
Authors of papers retain copyright and release the work under a Creative
Commons Attribution 4.0 International License
(https://creativecommons.org/licenses/by/4.0/https://doi.orgCC BY 4.0).

## Summary

We introduce GGLasso, a Python package for solving General Graphical Lasso
problems. The Graphical Lasso scheme, introduced by (Friedman, Hastie, and
Tibshirani 2007) (see also (Yuan and Lin 2007; Banerjee, El Ghaoui, and
D’Aspremont 2008)), estimates a sparse inverse covariance matrix $\Theta$ from
multivariate Gaussian data
$\mathcal{X}\sim\mathcal{N}(\mu,\Sigma)\in\mathbb{R}^{p}$. Originally proposed
by (Dempster 1972) under the name Covariance Selection, this estimation
framework has been extended to include latent variables in (Chandrasekaran,
Parrilo, and Willsky 2012). Recent extensions also include the joint
estimation of multiple inverse covariance matrices, see, e.g., in (Danaher,
Wang, and Witten 2013; Tomasi et al. 2018). The GGLasso package contains
methods for solving the general problem formulation:

$\displaystyle\min_{\Theta,L\in\mathbb{S}_{++}^{K}}\quad\sum_{k=1}^{K}\left(-\log\det(\Theta^{(k)}-L^{(k)})+\langle
S^{(k)},\Theta^{(k)}-L^{(k)}\rangle\right)+\mathcal{P}(\Theta)+\sum_{k=1}^{K}\mu_{1,k}\|L^{(k)}\|_{\star}.$
(1)

Here, we denote with $\mathbb{S}_{++}^{K}$ the $K$-product of the space of
symmetric, positive definite matrices. Moreover, we write
$\Theta=(\Theta^{(1)},\dots,\Theta^{(K)})$ for the sparse component of the
inverse covariances and $L=(L^{(1)},\dots,L^{(K)})$ for the low rank
components, formed by potential latent variables. Here, $\mathcal{P}$ is a
regularization function that induces a desired sparsity structure. The above
problem formulation subsumes important special cases, including the single
(latent variable) Graphical Lasso, the Group, and the Fused Graphical Lasso.

## Statement of need

Currently, there is no Python package available for solving general Graphical
Lasso instances. The standard single Graphical Lasso problem (SGL) can be
solved in scikit-learn (Pedregosa et al. 2011). The skggm package provides
several algorithmic and model selection extensions for the single Graphical
Lasso problem (Laska and Narayan 2017). The package regain (Tomasi et al.
2018) comprises solvers for single and Fused Graphical Lasso problems, with
and without latent variables. With GGLasso, we make the following
contributions:

  * •

Proposing a uniform framework for solving Graphical Lasso problems.

  * •

Providing solvers for Group Graphical Lasso problems (with and without latent
variables).

  * •

Providing a solver for – what we call – _nonconforming GGL_ problems where not
all variables need to be present in every instance. We detail a use case of
this novel extension on synthetic data.

  * •

Implementing a block-wise ADMM solver for SGL problems following (Witten,
Friedman, and Simon 2011) as well as proximal point solvers for FGL and GGL
problems (Zhang et al. 2019, 2020).

In the table below we give an overview of existing functionalities and the
GGLasso package.

| scikit-learn | regain | GGLasso | comment
---|---|---|---|---
SGL | yes | yes | yes | new: block-wise solver
SGL + latent | no | yes | yes |
GGL | no | no | yes |
GGL + latent | no | no | yes |
FGL | no | yes | yes | new: proximal point solver
FGL + latent | no | yes | yes |
GGL nonconforming (+latent) | no | no | yes |

## Functionalities

### Installation and problem instantiation

GGLasso can be installed via pip.

    pip install gglasso

The central object of GGLasso is the class glasso_problem which streamlines
the solving or model selection procedure for SGL, GGL, and FGL problems with
or without latent variables.

As an example, we instantiate a single Graphical Lasso problem (see the
problem formulation below). We input the empirical covariance matrix S and the
number of samples N. We can choose to model latent variables and set the
regularization parameters via the other input arguments.

# Import the main class of the packagefrom gglasso.problem import
glasso_problem# Define a SGL problem instance with given data Sproblem =
glasso_problem(S, N, reg = None, reg_params = {’lambda1’: 0.01}, latent =
False)

As a second example, we instantiate a Group Graphical Lasso problem with
latent variables. Typically, the optimal choice of the regularization
parameters are not known and are determined via model selection.

# Define a GGL problem instance with given data Sproblem = glasso_problem(S,
N, reg = "GGL", reg_params = None, latent = True)

Depending on the input arguments, glasso_problem comprises two main modes:

  * •

if regularization parameters are specified, the problem-dependent default
solver is called.

  * •

if regularization parameters are _not_ specified, GGLasso performs model
selection via grid search and the extended BIC criterion (Foygel and Drton
2010)).

problem.solve()problem.model_selection()

For further information on the input arguments and methods, we refer to the
https://gglasso.readthedocs.io/en/latest/problem-
object.htmlhttps://doi.orgdetailled documentation.

Figure 1: Illustration of the latent SGL: The estimated inverse covariance
matrix $\hat{\Omega}$ decomposes into a sparse component $\hat{\Theta}$
(central) and a low-rank component $\hat{L}$ (right).

### Problem formulation

We list important special cases of the problem formulation given in Equation
1. For a mathematical formulation of each special case, we refer to the
https://gglasso.readthedocs.io/en/latest/math-
description.htmlhttps://doi.orgdocumentation.

#### Single Graphical Lasso (_SGL_):

For $K=1$, the problem reduces to the single (latent variable) Graphical Lasso
where

$\mathcal{P}(\Theta)=\lambda_{1}\sum_{i\neq j}|\Theta_{ij}|.$

An illustration of the single latent variable Graphical Lasso model output is
shown in Figure 1.

#### Group Graphical Lasso (_GGL_):

For

$\mathcal{P}(\Theta)=\lambda_{1}\sum_{k=1}^{K}\sum_{i\neq
j}|\Theta_{ij}^{(k)}|+\lambda_{2}\sum_{i\neq
j}\left(\sum_{k=1}^{K}|\Theta_{ij}^{(k)}|^{2}\right)^{\frac{1}{2}}$

we obtain the Group Graphical Lasso as formulated in (Danaher, Wang, and
Witten 2013).

#### Fused Graphical Lasso (_FGL_):

For

$\mathcal{P}(\Theta)=\lambda_{1}\sum_{k=1}^{K}\sum_{i\neq
j}|\Theta_{ij}^{(k)}|+\lambda_{2}\sum_{k=2}^{K}\sum_{i\neq
j}|\Theta_{ij}^{(k)}-\Theta_{ij}^{(k-1)}|$

we obtain Fused (also called Time-Varying) Graphical Lasso (Danaher, Wang, and
Witten 2013; Tomasi et al. 2018; Hallac et al. 2017).

#### Nonconforming GGL:

Consider the GGL case in a situation where not all variables are observed in
every instance $k=1,\dots,K$. GGLasso is able to solve these problems and
include latent variables. We provide the mathematical details in the
https://gglasso.readthedocs.io/en/latest/math-description.html#ggl-the-
nonconforming-casehttps://doi.orgdocumentation and give an
https://gglasso.readthedocs.io/en/latest/auto_examples/plot_nonconforming_ggl.html#sphx-
glr-auto-examples-plot-nonconforming-ggl-pyhttps://doi.orgexample.

### Optimization algorithms

The GGLasso package implements several methods with provable convergence
guarantees for solving the optimization problems formulated above.

  * •

_ADMM_ : for all problem formulations we implemented the ADMM algorithm (Boyd
et al. 2011). ADMM is a flexible and efficient optimization scheme which is
specifically suited for Graphical Lasso problems as it only relies on
efficient computation of the proximal operators of the involved functions
(Danaher, Wang, and Witten 2013; Tomasi et al. 2018; Ma, Xue, and Zou 2013).

  * •

_PPDNA_ : for GGL and FGL problems without latent variables, we included the
proximal point solver proposed in (Zhang et al. 2019, 2020). According to the
numerical experiments in (Zhang et al. 2020), PPDNA can be an efficient
alternative to ADMM especially for fast local convergence.

  * •

_block-ADMM_ : for SGL problems without latent variables, we implemented a
method which solves the problem block-wise, following the proposal in (Witten,
Friedman, and Simon 2011). This wrapper simply applies the ADMM solver to all
connected components of the empirical covariance matrix after thresholding.

Figure 2: Runtime comparison for SGL problems of varying dimension and sample
size at three different $\lambda_{1}$ values. The left column shows the
runtime at low accuracy, the right column at high accuracy.

### Benchmarks and applications

In our example gallery, we included benchmarks comparing the solvers in
GGLasso to state-of-the-art software as well as illustrative examples
explaining the usage and functionalities of the package. We want to emphasize
the following examples:

  * •

https://gglasso.readthedocs.io/en/latest/auto_examples/plot_benchmarks.html#sphx-
glr-auto-examples-plot-benchmarks-pyhttps://doi.orgBenchmarks for SGL
problems: our solver is competitive with scikit-learn and regain. The newly
implemented block-wise solver is highly efficient for large sparse networks
(see Figure 2 for runtime comparison at
https://gglasso.readthedocs.io/en/latest/auto_examples/plot_benchmarks.html#calculating-
the-accuracyhttps://doi.orglow and high accuracy, respectively).

  * •

https://gglasso.readthedocs.io/en/latest/auto_examples/plot_soil_example.html#sphx-
glr-auto-examples-plot-soil-example-pyhttps://doi.orgSoil microbiome
application: following (Kurtz, Bonneau, and Müller 2019), we demonstrate how
latent variables can be used to identify hidden confounders in microbial
network inference.

  * •

https://gglasso.readthedocs.io/en/latest/auto_examples/plot_nonconforming_ggl.html#sphx-
glr-auto-examples-plot-nonconforming-ggl-pyhttps://doi.orgNonconforming GGL:
we illustrate how to use GGLasso for GGL problems with missing variables.

## Acknowledgements

We thank Prof. Dr. Michael Ulbrich, TU Munich, for supervising the Master’s
thesis of FS that led to the development of the software. We also thank Dr.
Zachary D. Kurtz for helping with testing of the latent graphical model
implementation.

## References

Banerjee, Onureena, Laurent El Ghaoui, and Alexandre D’Aspremont. 2008. “Model
selection through sparse maximum likelihood estimation for multivariate
gaussian or binary data.” _J. Mach. Learn. Res._ 9: 485–516.
https://doi.org/10.5555/1390681.1390696.

Boyd, Stephen, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein.
2011. “Distributed Optimization and Statistical Learning via the Alternating
Direction Method of Multipliers.” _Found. Trends Mach. Learn._ 3 (1). Hanover,
MA, USA: Now Publishers Inc.: 1–122. https://doi.org/10.1561/2200000016.

Chandrasekaran, Venkat, Pablo A. Parrilo, and Alan S. Willsky. 2012. “Latent
Variable Graphical Model Selection via Convex Optimization.” _Ann. Statist._
40 (4). Institute of Mathematical Statistics: 1935–67.
https://doi.org/10.1214/11-aos949.

Danaher, Patrick, Pei Wang, and Daniela M. Witten. 2013. “The Joint Graphical
Lasso for Inverse Covariance Estimation Across Multiple Classes.” _J. R. Stat.
Soc. B_ 76 (2). Wiley: 373–97. https://doi.org/10.1111/rssb.12033.

Dempster, Arthur P. 1972. “Covariance selection.” _Biometrics_ 28 (1). JSTOR:
157–75. https://doi.org/10.2307/2528966.

Foygel, Rina, and Mathias Drton. 2010. “Extended Bayesian Information Criteria
for Gaussian Graphical Models.” In _Advances in Neural Information Processing
Systems_ , edited by J. Lafferty, C. Williams, J. Shawe-Taylor, R. Zemel, and
A. Culotta. Vol. 23. Curran Associates, Inc.
https://proceedings.neurips.cc/paper/2010/file/072b030ba126b2f4b2374f342be9ed44-Paper.pdf.

Friedman, J., T. Hastie, and R. Tibshirani. 2007. “Sparse Inverse Covariance
Estimation with the Graphical Lasso.” _Biostatistics_ 9 (3). Oxford University
Press (OUP): 432–41. https://doi.org/10.1093/biostatistics/kxm045.

Hallac, David, Youngsuk Park, Stephen Boyd, and Jure Leskovec. 2017. “Network
Inference via the Time-Varying Graphical Lasso.” In _Proceedings of the 23rd
ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_.
ACM. https://doi.org/10.1145/3097983.3098037.

Kurtz, Zachary D, Richard Bonneau, and Christian L Müller. 2019.
“Disentangling microbial associations from hidden environmental and technical
factors via latent graphical models.” _bioRxiv_ , 2019.12.21.885889.
https://doi.org/10.1101/2019.12.21.885889.

Laska, Jason, and Manjari Narayan. 2017. “skggm 0.2.7: A scikit-learn
compatible package for Gaussian and related Graphical Models,” July. Zenodo.
https://doi.org/10.5281/zenodo.830033.

Ma, Shiqian, Lingzhou Xue, and Hui Zou. 2013. “Alternating Direction Methods
for Latent Variable Gaussian Graphical Model Selection.” _Neural Comput._ 25
(8). MIT Press - Journals: 2172–98. https://doi.org/10.1162/neco_a_00379.

Pedregosa, F., G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M.
Blondel, et al. 2011. “Scikit-Learn: Machine Learning in Python.” _J Mach
Learn Res_ 12: 2825–30.

Tomasi, Federico, Veronica Tozzo, Saverio Salzo, and Alessandro Verri. 2018\.
“Latent Variable Time-Varying Network Inference.” In _Proceedings of the 24th
ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_.
ACM. https://doi.org/10.1145/3219819.3220121.

Witten, Daniela M., Jerome H. Friedman, and Noah Simon. 2011. “New Insights
and Faster Computations for the Graphical Lasso.” _J. Comput. Graph. Statist._
20 (4). Taylor & Francis: 892–900. https://doi.org/10.1198/jcgs.2011.11051a.

Yuan, M., and Y. Lin. 2007. “Model Selection and Estimation in the Gaussian
Graphical Model.” _Biometrika_ 94 (1). Oxford University Press (OUP): 19–35.
https://doi.org/10.1093/biomet/asm018.

Zhang, Ning, Yangjing Zhang, Defeng Sun, and Kim-Chuan Toh. 2019. “An
Efficient Linearly Convergent Regularized Proximal Point Algorithm for Fused
Multiple Graphical Lasso Problems,” February.
http://arxiv.org/abs/1902.06952v1.

Zhang, Yangjing, Ning Zhang, Defeng Sun, and Kim-Chuan Toh. 2020. “A Proximal
Point Dual Newton Algorithm for Solving Group Graphical Lasso Problems.” _SIAM
J. Optim._ 30 (3): 2197–2220. https://doi.org/10.1137/19M1267830.