# Nuclear Spin-Depleted, Isotopically Enriched
${}^{\text{70}}$Ge/${}^{\text{28}}$Si${}^{\text{70}}$Ge Quantum Wells

O. Moutanabbir<EMAIL_ADDRESS>Department of Engineering
Physics, École Polytechnique de Montréal, Montréal, C.P. 6079, Succ. Centre-
Ville, Montréal, Québec, Canada H3C 3A7  S. Assali  Department of Engineering
Physics, École Polytechnique de Montréal, Montréal, C.P. 6079, Succ. Centre-
Ville, Montréal, Québec, Canada H3C 3A7  A. Attiaoui  Department of
Engineering Physics, École Polytechnique de Montréal, Montréal, C.P. 6079,
Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7  G. Daligou  Department
of Engineering Physics, École Polytechnique de Montréal, Montréal, C.P. 6079,
Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7  P. Daoust  Department of
Engineering Physics, École Polytechnique de Montréal, Montréal, C.P. 6079,
Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7  P. Del Vecchio
Department of Engineering Physics, École Polytechnique de Montréal, Montréal,
C.P. 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7  S. Koelling
Department of Engineering Physics, École Polytechnique de Montréal, Montréal,
C.P. 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7  L. Luo
Department of Engineering Physics, École Polytechnique de Montréal, Montréal,
C.P. 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7  N. Rotaru
Department of Engineering Physics, École Polytechnique de Montréal, Montréal,
C.P. 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7

###### Abstract

The p-symmetry of the hole wavefunction is associated with a weaker hyperfine
interaction as compared to electrons, thus making hole spin qubits attractive
candidates to implement long coherence quantum processors. However, recent
studies demonstrated that hole qubits in planar germanium (Ge)
heterostructures are still very sensitive to nuclear spin bath. These
observations highlight the need to develop nuclear spin-free Ge qubits to
suppress this decoherence channel and evaluate its impact. With this
perspective, this work demonstrates the epitaxial growth of
${}^{\text{73}}$Ge-depleted isotopically enriched ${}^{\text{70}}$Ge/SiGe
quantum wells. The growth was achieved by reduced pressure chemical vapor
deposition using isotopically purified monogermane
${}^{\text{70}}$GeH${}_{\text{4}}$ and monosilane
${}^{\text{28}}$SiH${}_{\text{4}}$ with an isotopic purity higher than 99.9
$\%$ and 99.99 $\%$, respectively. The quantum wells consist of a series of
${}^{\text{70}}$Ge/SiGe heterostructures grown on Si wafers using a Ge virtual
substrate and a graded SiGe buffer layer. The isotopic purity is investigated
using atom probe tomography following an analytical procedure addressing the
discrepancies in the isotopic content caused by the overlap of isotope peaks
in mass spectra. The nuclear spin background in the quantum wells was found to
be sensitive to the growth conditions. The lowest concentration of nuclear
spin-full isotopes ${}^{\text{73}}$Ge and ${}^{\text{29}}$Si in the
heterostructure was established at 0.01 $\%$ in the Ge quantum well and SiGe
barriers. The measured average distance between nuclear spins reaches 3-4 nm
in ${}^{\text{70}}$Ge/${}^{\text{28}}$Si${}^{\text{70}}$Ge, which is an order
of magnitude larger than in natural Ge/SiGe heterostructures.

††preprint: APS/123-QED

##  I Introduction

Although it was quickly relegated behind silicon (Si) because of its
relatively low bandgap energy, its lack of a stable oxide, and its large
surface state densities, germanium (Ge) is inarguably the material that
catalyzed the transition from what W. Pauli and I. Rabi called the ‘’Physics
of Dirt” [1, 2] to modern-day semiconductor physics and technology [3].
Indeed, the ease by which Ge can then be purified and processed led to the
demonstration of point contact diode mixers for radar reception [4] and of the
point contact and junction transistors [5]. These inventions contributed to
laying the groundwork for what was later coined as the first quantum
revolution. In recent years, there has been a revived interest in Ge-based
materials for integrated photonic circuits [6, 7], sensing [6], high-mobility
electronic s [8], and solid-state quantum computing [9]. The latter, for
instance, aims at capitalizing on the advantageous quantum environment of
holes in Ge, their inherently large and tunable spin-orbit interaction (SOI),
and their reduced hyperfine coupling with nuclear spins to implement
increasingly robust and reliable spin qubits [10, 11, 12, 13, 14, 15, 16, 17,
18, 19]. Indeed, these quantum devices are now considered forefront candidates
for scalable quantum processors [9]. This recent surge in developing Ge qubits
makes one think that Ge may also be a key material in shaping the anticipated
second quantum revolution.

From a fundamental standpoint, it is expected that the hyperfine interaction
to be weaker for holes than electrons due to the p-symmetry of the hole
wavefunction. However, theoretical investigations suggested a hyperfine
coupling that is only one order of magnitude smaller than that of electrons
[20, 21] or of equal strength as in Si [21]. Moreover, the p-symmetry and
d-orbital hybridization of the hole wavefunction leads to an anisotropic
hyperfine coupling that is non-existent for electron spins [20].
Interestingly, recent experimental studies hint at the sensitivity of hole
spin qubits in planar Ge/SiGe heterostructure to nuclear spin bath reporting
an amplitude of the fluctuating Overhauser field of 34.4 kHz, which is
suggested to limit spin dephasing times [22]. Although charge noise is
believed to be the dominant decohering process, these observations call for
the development of nuclear spin-free Ge qubits to elucidate their sensitivity
to hyperfine coupling. Undertaking this research direction requires Ge-based
quantum devices that are depleted of ${}^{\text{73}}$Ge, which is the only Ge
nuclear spin-full stable isotope. This work addresses this very issue and
provides a demonstration of the epitaxial growth of isotopically purified
${}^{\text{70}}$Ge quantum wells (QWs). Note that enriched ${}^{\text{70}}$Ge,
${}^{\text{74}}$Ge, and ${}^{\text{76}}$Ge isotopes were employed in the past
to grow superlattices and self-assembled quantum dots by solid-source
molecular beam epitaxy [23, 24, 25]. Herein, the growth of ${}^{\text{73}}$Ge-
depleted QWs is achieved by hydride precursors using the chemical vapor
deposition (CVD) method, which is broadly adopted in Ge device research
besides being compatible with the processing standards in the semiconductor
industry [9].

Figure 1: Epitaxial growth of isotopically engineered Ge/SiGe
heterostructures. (a) Schematic illustration of four sets of the grown Ge/SiGe
heterostructures. (b) A cross-sectional STEM image of an as-grown Ge/SiGe QW
showing the entire stack consisting of Ge-Vs and SiGe buffer layers grown on a
Si substrate. (c) A close-up cross-sectional STEM image of an isotopically
enriched ${}^{\text{70}}$Ge/SiGe QW. (d) A cross-sectional TEM image showing
that the QW region is defect-free and that the extended defects are confined
at Ge-VS/SiGe interface. (e) The (224) XRD-RSM of a representative CVD-grown
isotopically purified ${}^{\text{70}}$Ge QW heterostructure.

##  II Experimental

The epitaxial growth of isotopically engineered Ge/SiGe QW heterostructures
was carried out on hydrogen-passivated 4-inch (001)-oriented Si wafers in a
reduced-pressure CVD reactor using isotopically purified monogermane
${}^{\text{70}}$GeH${}_{\text{4}}$ (isotopic purity $>$99.9 $\%$) and
monosilane ${}^{\text{28}}$SiH${}_{\text{4}}$ (isotopic purity $>$99.99 $\%$).
The precursors were enriched in a centrifugal setup using natural monogermane
(${}^{\text{nat}}$GeH${}_{\text{4}}$) and SiF${}_{\text{4}}$ as starting gases
[26]. After purification, ${}^{\text{70}}$GeH${}_{\text{4}}$ contains traces
($<$0.006 at.$\%$) of other Ge isotopes: ${}^{\text{72}}$Ge,
${}^{\text{73}}$Ge, ${}^{\text{74}}$Ge, and ${}^{\text{76}}$Ge. Moreover,
chemical contaminants including other hydrides are also negligible, with an
average content being $<$0.06 µmol/mol. Reference Ge/SiGe QW heterostructures
were also prepared following the same growth protocol using conventional
precursors with natural isotopic abundance
(${}^{\text{nat}}$GeH${}_{\text{4}}$ and disilane
${}^{\text{nat}}$Si${}_{\text{2}}$H${}_{\text{6}}$). After annealing in
hydrogen, a 3 µm-thick Ge interlayer, commonly known as Ge virtual substrate
(Ge-VS), was grown on Si using ${}^{\text{nat}}$GeH${}_{\text{4}}$ and a two-
step growth process in the 450-600 °C temperature range. Then follows a
thermal cyclic annealing step (725-875 °C) to improve the Ge-VS quality. A
reverse-graded 1µm-thick Si${}_{\text{1-x}}$Ge${}_{\text{x}}$ layer was then
grown at 600 °C using ${}^{\text{nat}}$GeH${}_{\text{4}}$ and
${}^{\text{nat}}$Si${}_{\text{2}}$H${}_{\text{6}}$ until a uniform Si content
of 18 at.$\%$ is reached. Without interrupting the growth, the
${}^{\text{nat}}$GeH${}_{\text{4}}$ supply was switched to the purified
${}^{\text{70}}$GeH${}_{\text{4}}$ to grow the first
Si${}_{\text{1-x}}$Ge${}_{\text{x}}$ barrier layer (BR1), while keeping all
the other growth parameters unchanged. Thickness and composition of BR1 were
varied in the 0.3-1µm range and x = 0.15-0.18 range to investigate the effect
of the growth time on the isotopic purity of the epilayers. After that the
growth of BR1 was completed, the reactor was then purged in hydrogen for 90 s
before growing the ${}^{\text{70}}$Ge QW layer using
${}^{\text{70}}$GeH${}_{\text{4}}$ supply for a variable growth time of up to
40 s. Next, the reactor was purged in hydrogen for 90 s prior to the growth of
the Si${}_{\text{1-x}}$Ge${}_{\text{x}}$ BR2 layer under identical growth
conditions as BR1. Lastly, a Si capping layer with a few nm thickness was
grown. Fig. 1a illustrates the grown stacks.
Ge/Si${}_{\text{0.18}}$Ge${}_{\text{0.82}}$ (A),
${}^{\text{70}}$Ge/Si${}_{\text{0.18}}$Ge${}_{\text{0.82}}$ (B), and
${}^{\text{70}}$Ge/Si${}_{\text{0.15}}$Ge${}_{\text{0.85}}$ (C) QWs were grown
using this protocol. The
${}^{\text{70}}$Ge/${}^{\text{28}}$Si${}_{\text{0.15}}$${}^{\text{70}}$Ge${}_{\text{0.85}}$
(D) QW was grown following a similar protocol except for the growth of BR1-2
that was performed by changing from
${}^{\text{nat}}$Si${}_{\text{2}}$H${}_{\text{6}}$ to
${}^{\text{28}}$SiH${}_{\text{4}}$ and adjusting the growth conditions to
accommodate the change in the precursor decomposition.

Several characterization techniques were employed to elucidate the basic
properties of the as-grown heterostructures and investigate their isotopic
content. Lattice strain and average content in Ge/SiGe heterostructures were
evaluated from X-ray diffraction (XRD) measurements including reciprocal space
map (RSM) analysis. The microstructure of the grown materials was investigated
by transmission electron microscopy (TEM) and scanning TEM (STEM). The quality
of interfaces, the atomic-level composition, and the isotopic purity were
investigated using atom probe tomography (APT). Additional insights into the
chemical and isotopic compositions are also obtained using secondary ion mass
spectrometry (SIMS). Raman scattering spectroscopy was employed to evaluate
the effects of the isotopic content on phonon scattering in Ge QWs.
Additionally, the uniformity of the growth thickness as well as the optical
signature of quantum confinement were investigated using spectroscopic
ellipsometry (SE).

##  III Results and Discussion

A cross-sectional STEM image of a representative isotopically-engineered Ge QW
heterostructures is shown in Fig. 1b, while the enlarged view of the
${}^{\text{70}}$Ge/Si${}_{\text{0.15}}$${}^{\text{70}}$Ge${}_{\text{0.85}}$ QW
region is displayed in Fig. 1c. The figure shows an 18 nm-thick
${}^{\text{70}}$Ge QW together with BR1 and BR2 layers with thicknesses of 290
nm and 28 nm, respectively. The transition between SiGe barrier layers and
${}^{\text{70}}$Ge QW is of the order of 1-2 nm. To evaluate the structural
quality of the heterostructures, cross-sectional TEM images were acquired
(Fig. 1d). The extended defects are confined to the Si/Ge-VS and Ge-
VS/Si${}_{\text{1-x}}$Ge${}_{\text{x}}$ interfaces, with no defects being
detected in the QW region at the TEM imaging scale. XRD-RSM (224) analysis of
the as-grown heterostructures demonstrates sharp peaks for the SiGe/Ge
substrate, barriers as well as the signature of the strained 18 nm-thick
${}^{\text{70}}$Ge layer, thus suggesting an excellent degree of crystallinity
across the structure (Fig. 1(e)). Here, the variation in composition between
natural and purified SiGe layers (Si${}_{\text{0.18}}$Ge${}_{\text{0.82}}$ vs.
Si${}_{\text{0.15}}$Ge${}_{\text{0.85}}$ as determined by APT) is related to
the difference in composition between the germane precursor supplies.

Figure 2: Vibrational and optical properties of ${}^{\text{70}}$Ge QWs. (a)
Ge-Ge LO vibrational mode recorded for a set of ${}^{\text{70}}$Ge QW grown
with a variable growth time between 4 and 40 s corresponding to a 3-30 nm
thickness range. (b) Ge-Ge LO vibrational mode in two identical
${}^{\text{nat}}$Ge and ${}^{\text{70}}$Ge QW samples. (c) The imaginary
dielectric function of an 18 nm-thick ${}^{\text{70}}$Ge QW structure (green)
and for a reference material consisting of the same layer but without BR2
(blue). (d) CP analysis of the corresponding dielectric function. The CP
energy position is extracted after fitting simultaneously the real and
imaginary part of the second derivative of the dielectric function. Figure 3:
Atom probe tomography of ${}^{\text{70}}$Ge QWs. (a) 3D atom-by-atom APT map
of a ${}^{\text{70}}$Ge/SiGe heterostructure. The 30 × 30 × 30
nm${}^{\text{3}}$ map contains approximately 900,000 atoms. (b) Ge-Ge LO
vibrational mode in two identical ${}^{\text{nat}}$Ge and ${}^{\text{70}}$Ge
QW samples. Mass spectra of the investigated QW samples showing doubly charged
Ge ions (b) and Si ions (c) of 10 million atoms.

A first glimpse into the isotopic content of the as-grown QWs was obtained
from Raman spectroscopy studies. Fig. 2(a) shows Raman spectra around the Ge-
Ge LO mode recorded for a set of QWs grown at a variable growth time between 4
and 40 s corresponding to a 3-30 nm thickness range. The spectra indicate the
presence of two distinct modes. The first is centered around 293.7
cm${}^{\text{-1}}$ corresponding to Ge-Ge LO mode in the SiGe barrier, whereas
the second peak at 305.3 cm${}^{\text{-1}}$ is attributed to the same mode but
in the ${}^{\text{70}}$Ge QW. This assessment is consistent with the observed
increase in the second peak intensity as the QW thickness increases. Note that
the Ge-Ge mode in ${}^{\text{nat}}$Ge QW is detected at 300.1
cm${}^{\text{-1}}$, as demonstrated in Fig. 2(b) comparing two identical
${}^{\text{nat}}$Ge and ${}^{\text{70}}$Ge QW samples. The observed shift
between the two samples is analyzed based on the quasi-harmonic approximation,
which is a valid approximation for semiconductors at room temperature [27,
28]. According to the virtual crystal approximation, a simple harmonic
analysis predicts that the energy of a phonon mode is inversely proportional
to the square root of the average isotopic mass. The average isotopic mass is
given by ⟨ _m_ ⟩ = $\Sigma$${}_{\text{i}}$c${}_{\text{i}}$_m_ ${}_{\text{i}}$,
with c${}_{\text{i}}$ being the fractional composition of an isotope of mass
_m_ ${}_{\text{i}}$. Knowing that the atomic mass of ${}^{\text{nat}}$Ge is
72.63 amu, the measured wavenumbers of Ge-Ge LO mode in the sets of QWs yield
an average atomic mass in ${}^{\text{70}}$Ge QW lattice of 70.17 amu
corresponding to at least 99.6$\%$ enrichment in ${}^{\text{70}}$Ge isotopes.
As discussed below, the growth protocol has a strong effect on the isotopic
purity of the QW.

It is important to mention that the limited spectral resolution ( 1
cm${}^{\text{-1}}$) of the used Raman setup does not allow addressing the
effect of isotopic purification on lattice disorder [28]. Nevertheless, it is
reasonable to conclude that the similarity observed in the full width at the
half maximum of the Ge-Ge peaks in ${}^{\text{nat}}$Ge and ${}^{\text{70}}$Ge
is indicative of a similar crystalline quality, which is consistent with XRD
and TEM studies. To further assess the quality of the grown QWs, SE studies
were carried out on ${}^{\text{70}}$Ge QW samples. For these studies,
reference samples consisting of the same grown layers but without BR2 were
also prepared and investigated. Fig. 2(c) displays the measured spectra for 18
nm ${}^{\text{70}}$Ge QW and the associated reference material. The figure
shows the imaginary dielectric function (left) and the critical point (CP)
analysis of the measured dielectric function. The nature of the lineshape of
the dielectric function of both heterostructures conceals insights into the
quantum confinement in the ${}^{\text{70}}$Ge QW. Note that the penetration
depth of the incident excitation near the E${}_{\text{1}}$ CP is around 20-35
nm for Ge bulk. If one considers a limited spectral range between 1.5-3 eV,
the effect of the underlying materials (SiGe buffer, Ge-VS, and Si substrate)
can be negated as the incident light will not reach and excite them.
Consequently, only the top three layers (${}^{\text{70}}$Ge QW, BR2, and Si
cap) should in principle contribute to the measured dielectric function (Fig.
2(c)). Moreover, the contribution of the 3-5 nm-thick Si cap should be
excluded in the analysis as the E${}_{\text{1}}$ CP of Si is located around
3.4 eV [29], which is outside the measured spectral range.

The second derivative of the dielectric function of the two samples (with and
without the top barrier BR2) is displayed in Fig. 2(d). To unravel the
electronic structure of the analyzed heterostructure, the measured data were
fitted using a generic critical point parabolic band model [30, 29, 31, 32].
The CP energy of the ${}^{\text{70}}$Ge layer without a barrier is evaluated
at 2.156 eV, which is close to the Ge bulk CP of 2.134 eV [31], whereas for
${}^{\text{70}}$Ge QW sample a blueshift is noted yielding a CP energy of
2.233 eV. More importantly, the qualitative difference between both dielectric
functions at 2.17 eV is clear. Indeed, the CP lineshape changes drastically
from 2D Van Hove singularities in the reference structure (green dots) to a
discrete excitonic lineshape in ${}^{\text{70}}$Ge QW (blue dots). This
observed change in CP lineshape and energy is indicative of quantum
confinement and its associated narrowing of the optical transition in Ge [32].

In the following, the isotopic content of the grown QWs is discussed based on
APT studies. Fig. 3(a) shows a representative 3D 30 × 30 × 30
nm${}^{\text{3}}$ atom-by-atom APT map of a ${}^{\text{70}}$Ge QW. The map
indicates that the QW region contains mainly the ${}^{\text{70}}$Ge isotope,
but traces of other isotopes can also be seen. Before quantifying and
discussing the level of these contaminants, the recorded mass spectra are
described first, as shown in Fig. 3(b,c). The figures exhibit the mass spectra
recorded for a set of four QW samples labeled A, B, C, and D, as illustrated
in Fig. 1(a). These samples were grown under different conditions. In sample
A, the QW was grown using ${}^{\text{70}}$GeH${}_{\text{4}}$, whereas the SiGe
barriers were grown using ${}^{\text{nat}}$GeH${}_{\text{4}}$. In the other
three samples, the growth of both barriers and QWs was conducted using
${}^{\text{70}}$GeH${}_{\text{4}}$. However, the change from
${}^{\text{nat}}$GeH${}_{\text{4}}$ to ${}^{\text{70}}$GeH${}_{\text{4}}$
occurred during the growth of the underlying SiGe layer at a variable
thickness from the interface with the QW: 290 nm (B), 1000 nm (C), and 1890 nm
(D). This means that the changes from ${}^{\text{nat}}$GeH${}_{\text{4}}$ to
${}^{\text{70}}$GeH${}_{\text{4}}$ took place at different times during the
growth of SiGe buffer layer prior to the QW growth in these samples (B: 8 min,
C: 24 min, and D: 29 min). In the case of sample D, the growth of SiGe
barriers was conducted using the isotopically purified precursor
${}^{\text{28}}$SiH${}_{\text{4}}$ instead of
${}^{\text{nat}}$Si${}_{\text{2}}$H${}_{\text{6}}$. The growth rate was higher
for this sample due to a higher GeH${}_{\text{4}}$ supply required for the
growth optimization using ${}^{\text{28}}$SiH${}_{\text{4}}$ precursor. The
obtained APT mass spectra are compared in Fig. 3(b,c) showing the spectra of
doubly charged Ge ions (Fig. 3(b)) and doubly charged Si ions (Fig. 3(c)).
Each spectrum contains 10 million atoms from the selected region which
includes most of the top barrier, the full QW and its interfaces, and a part
of the bottom barrier. Note that this includes the QW interfaces and the local
fluctuations in the isotopic purity observed near these interfaces, as shown
in Fig. 4.

The mass spectrum of sample A shows peaks associated with all five Ge isotopes
at intensities close to the natural abundance of each isotope as most of the
signal originates from the barriers grown with
${}^{\text{nat}}$GeH${}_{\text{4}}$ (Fig. 3(b)). However, in samples B, C, and
D, the APT spectra clearly show enrichment in ${}^{\text{70}}$Ge isotope as
the peaks related to other isotopes have significantly diminished.
Interestingly, the level of this contamination from other isotopes is
intimately related to the growth protocol. Indeed, the level of Ge isotope
cross-contamination becomes lower the longer the time, relative to the moment
of the QW growth, of the transition from ${}^{\text{nat}}$GeH${}_{\text{4}}$
to ${}^{\text{70}}$GeH${}_{\text{4}}$. This indicates that the detection of
${}^{\text{70}}$Ge${}^{\text{++}}$, ${}^{\text{73}}$Ge${}^{\text{++}}$,
${}^{\text{74}}$Ge${}^{\text{++}}$, and ${}^{\text{76}}$Ge${}^{\text{++}}$
peaks is a manifestation of the reservoir effect, meaning that
${}^{\text{nat}}$GeH${}_{\text{4}}$ used to grow the much thicker Ge-VS and
SiGe-VS still resides in the growth reactor for an extended period of time.
This leads to the undesired incorporation of the nuclear spin-full
${}^{\text{73}}$Ge isotope into the growing QW structure. Herein, it is shown
that an early introduction of ${}^{\text{nat}}$GeH${}_{\text{4}}$ can
eliminate this contamination to a great extent. Ideally, the growth of the
entire stack Ge-VS/SiGe-VS/BR1/Ge/BR2 should be done using
${}^{\text{70}}$GeH${}_{\text{4}}$, but the process can be costly. Similarly,
Fig. 3(c) shows that the use of ${}^{\text{28}}$SiH${}_{\text{4}}$ to grow the
SiGe barriers leads to a significant reduction, more than 30-fold, of the
amount of ${}^{\text{29}}$Si isotope in the heterostructure. Since the hole
wavefunction in Ge QW is expected to leak to the SiGe barriers, it is also
important to suppress the hyperfine interactions that may result from the
presence of ${}^{\text{29}}$Si isotope.

Figure 4: Isotopic purity and nuclear spin background in ${}^{\text{70}}$Ge
QWs. (a) The evolution of ${}^{\text{70}}$Ge content across sample B as
measured by SIMS (dotted lines) and APT with tail-corrected analysis (solid
lines) from ${}^{\text{72}}$Ge, ${}^{\text{74}}$Ge, and ${}^{\text{76}}$Ge
signals. (b) The depth profile of the isotopic purity across the investigated
heterostructures. (c) The concentration profiles of nuclear spin-full species
${}^{\text{73}}$Ge and ${}^{\text{29}}$Si in the investigated
heterostructures. (d) The evolution as a function of the thickness of the
average distance between nuclear spins in the investigated heterostructures.

The local isotopic purity and 3D distribution of isotopes can be obtained from
APT. However, since the peaks of heavier isotopes are embedded in the tails of
the lighter isotopes (Fig. 3(b)), it is important to carefully analyze and
model the mass spectra to separate the tails and the peaks to accurately
quantify the isotopic content in the heterostructures. Herein, SIMS analyses
were carried out to validate the APT isotope mapping method. Since all
non-${}^{\text{70}}$Ge isotopes originate from a natural Ge source, one can
use the content measured for each isotope to estimate the ${}^{\text{70}}$Ge
purity by making a projection of the overall contamination based on the
natural distribution of isotopes. As shown in Fig. 4(a), SIMS data provide
estimates derived from ${}^{\text{72}}$Ge, ${}^{\text{74}}$Ge, and
${}^{\text{76}}$Ge signals coinciding almost perfectly which each other and
the estimate for the ${}^{\text{70}}$Ge purity gained from considering the
signal from all Ge isotopes. For APT, however, a difference was observed (data
not shown) when estimating based on doubly charged ${}^{\text{72}}$Ge,
${}^{\text{74}}$Ge, or all of the isotopes. This discrepancy is caused by the
aforementioned overlap of isotope peaks in the mass spectra (Fig. 3(b)). To
address this issue, a Monte-Carlo approach is implemented where the tails are
fitted locally around the peak region, and the peak and tail are decomposed
multiple 100s or 1000s times to find the average content of the peak and the
error created by the decomposition. The resulting estimate using the tail-
corrected data for doubly-charge ${}^{\text{72}}$Ge, ${}^{\text{74}}$Ge, and
${}^{\text{76}}$Ge match SIMS data, as shown in Fig. 4a (solid line).

Using the same Monte Carlo approach, we can quantify the ${}^{\text{70}}$Ge
purity in all samples. The result is shown in Fig. 4(b) highlighting once more
the differences between the samples in terms of isotopic purity near the QW
caused by the difference in time passed between the onset of
${}^{\text{70}}$GeH${}_{\text{4}}$ growth and the QW growth. Furthermore, both
SIMS and APT data consistently show that the 90 s growth interruption at the
QW interfaces, introduced to promote the growth of sharper interfaces, leads
to an accumulation in ${}^{\text{nat}}$Ge at the interface. For the growth of
sample D, the top barrier was grown without interruption thus suppressing the
isotopic cross-contamination at the interface. Maintaining the
${}^{\text{70}}$Ge purity is important to achieve a nuclear spin-depleted
interface and BR1.

A more accurate evaluation of the nuclear spin background is obtained from APT
analyses displayed in Fig. 4(c). The figure outlines the total concentration
profiles of nuclear spin-full isotopes ${}^{\text{73}}$Ge and
${}^{\text{29}}$Si across the investigated heterostructures. It is noticeable
that in
${}^{\text{nat}}$Si${}^{\text{nat}}$Ge/${}^{\text{70}}$Ge/${}^{\text{nat}}$Si${}^{\text{nat}}$Ge
(sample A) the nuclear spin concentration drops from 6 at.$\%$ in the SiGe
barriers down to 0.1 at.$\%$ in the QW. This background is further reduced to
0.02 at.$\%$ in QWs of samples B and C and even below 0.01 at.$\%$ in sample D
consisting of
${}^{\text{28}}$Si${}^{\text{70}}$Ge/${}^{\text{70}}$Ge/${}^{\text{28}}$Si${}^{\text{70}}$Ge.
Besides providing the isotopic composition profiles, APT also allows
extracting the atomic-level spatial distribution of individual nuclear spin-
full species ${}^{\text{73}}$Ge and ${}^{\text{29}}$Si, as displayed in Fig.
4(d), The figure shows the depth evolution of the average distance between
neighboring nuclear spins across the investigated heterostructures. To obtain
these profiles, a model of SiGe lattice was generated from APT maps [33] on
which the distribution of each isotope was imprinted thus allowing the
calculation of the distance between nuclear spins in a lattice plane-by-
lattice plane fashion. The uncertainty in these calculations was assessed by
sampling 10 different models. The obtained result demonstrates that the
average distance between nuclear spins is the lowest in the QW for all
samples, but it remains sensitive to the growth conditions. For instance, in
${}^{\text{nat}}$Si${}^{\text{nat}}$Ge barriers (A) the obtained average
distance is 0.3-0.4 nm, whereas it increases by one order of magnitude to 3-4
nm in isotopically
pure${}^{\text{28}}$Si${}^{\text{70}}$Ge/${}^{\text{70}}$Ge/${}^{\text{28}}$Si${}^{\text{70}}$Ge
heterostructure (D).

##  IV Conclusion

In summary, this work demonstrates the epitaxial growth of nuclear spin-
depleted, isotopically enriched ${}^{\text{70}}$Ge QWs. The growth was
achieved on Si wafers using enriched precursors
${}^{\text{70}}$GeH${}_{\text{4}}$ and ${}^{\text{28}}$SiH${}_{\text{4}}$ in a
reduced-pressure CVD system. The crystalline quality of the grown
heterostructures was confirmed by XRD and electron microscopy studies. The
critical point of the grown QWs exhibits a discrete excitonic lineshape at
2.233 eV indicative of quantum confinement. The isotopic purity and the
distribution of the nuclear spin background were investigated using APT. In
this regard, a Monte Carlo approach was introduced to solve the discrepancies
in APT analyses caused by the overlap of isotope peaks in the recorded mass
spectra. These analyses demonstrate that the isotopic content is very
sensitive to the growth conditions including any growth interruption. The
latter was found to induce an accumulation of natural Ge isotopes at the
growth interface leading to lower ${}^{\text{70}}$Ge content. To evaluate the
distribution of the residual nuclear spin background, a lattice model was
constructed to map the average distance between the two nuclear spin-full
isotopes ${}^{\text{73}}$Ge and ${}^{\text{29}}$Si. These studies showed that
the distance between nuclear spins reaches 3-4 nm in
${}^{\text{70}}$Ge/${}^{\text{28}}$Si${}^{\text{70}}$Ge, which is an order of
magnitude higher than in natural Ge/SiGe heterostructure. Additionally, the
lowest concentration of ${}^{\text{73}}$Ge and ${}^{\text{29}}$Si contaminants
in the heterostructure was established at 0.01$\%$ in both QW and barriers of
${}^{\text{70}}$Ge/${}^{\text{28}}$Si${}^{\text{70}}$Ge heterostructure. These
insights constitute a valuable input to improve the design and theoretical
modeling of spin qubits by providing quantitative, atomic-level details on
nuclear spin distribution.

METHODS.

X-ray diffraction (XRD) measurements performed using a Bruker Discover D8. A 3
bounces Ge(220) 2-crystals analyzer was placed in front of the XRD detector
during the XRD (004) and (224) reciprocal space map (RSM) analysis. The
microstructure of the grown materials was investigated by transmission
electron microscopy (TEM). TEM specimens were prepared in a Thermo Fisher
Helios Nanolab 660 dual-beam scanning electron microscope using a gallium-
focused ion beam (FIB) at 30, 16, and 5 kV. Electron beam-induced carbon and
platinum were locally deposited on the sample to protect the imaged region
from being damaged by the ion-beam milling during the thinning of the TEM
lamella. TEM and scanning TEM (STEM) analyses were carried out on a Thermo
Scientific Talos F200X S/TEM system with an acceleration voltage of 200 kV.

Insights into the quality of interfaces, the atomic-level composition, and the
isotopic purity were obtained using atom probe tomography (APT). APT specimens
were prepared in a FEI Helios Nanolab 660 dual-beam scanning electron
microscope using a gallium-focused ion beam (FIB) at 30, 16, and 5 kV. A
120-150 nm-thick chromium capping layer was deposited on the samples before
FIB irradiation to minimize the implantation of gallium ions into the imaged
region. APT studies were performed in a LEAP 5000XS tool. The LEAP 5000XS
utilizes a picosecond laser to generate pulses at a wavelength of 355 nm. For
the analysis, all samples were cooled to a temperature of 25 K. The
experimental data were collected at laser powers of 3-6 pJ. Additional
insights into the chemical and isotopic compositions are also obtained using
secondary ion mass spectrometry (SIMS).

Raman scattering analyses were performed at room temperature using a 633 nm
excitation laser. Additionally, the uniformity of the growth thickness as well
as the optical signature of quantum confinement were investigated using
spectroscopic ellipsometry (SE). SE measurements were carried out at room
temperature, using a variable angle spectroscopic RC2-XI ellipsometer
manufactured by J. A. Woollam Co. The variable angle spectroscopic
ellipsometer system covers the 0.5–6 eV range. All heterostructures were
measured between 70° and 80° angles of incidence with a 1° step. A noticeable
increase in the sensitivity of the SE parameters ($\Psi$ and $\Delta$) was
observed around 76-77°, which is very close to the Brewster angle for Si and
Ge. Thus, during the optical modeling, special care was accorded to the
modeling near this angle.

ACKNOWLEDGEMENTS. The authors thank J. Bouchard for the technical support with
the CVD system. O.M. acknowledges support from NSERC Canada (Discovery Grants,
Alliance International Quantum, and CQS2Q Consortium), Canada Research Chairs,
Canada Foundation for Innovation, Mitacs, PRIMA Québec, and Defense Canada
(Innovation for Defense Excellence and Security, IDEaS), the European Union’s
Horizon Europe research and innovation programme under grant agreement No
101070700 (MIRAQLS), and the US Army Research Office Grant No.
W911NF-22-1-0277.

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2023

[1,2]Florian Rehm

1]CERN, Geneva, Switzerland

2]RWTH Aachen University, Aachen, Germany

3]DESY, Hamburg, Germany

# Precise Image Generation on Current Noisy Quantum Computing Devices

*<EMAIL_ADDRESS>Sofia Vallecorsa<EMAIL_ADDRESS>Kerstin Borras<EMAIL_ADDRESS>Dirk Krücker<EMAIL_ADDRESS>Michele Grossi<EMAIL_ADDRESS>Valle Varo<EMAIL_ADDRESS>[  [  [

###### Abstract

The Quantum Angle Generator (QAG) is a new full Quantum Machine Learning model
designed to generate accurate images on current Noise Intermediate Scale
(NISQ) Quantum devices. Variational quantum circuits form the core of the QAG
model, and various circuit architectures are evaluated. In combination with
the so-called MERA-upsampling architecture, the QAG model achieves excellent
results, which are analyzed and evaluated in detail. To our knowledge, this is
the first time that a quantum model has achieved such accurate results. To
explore the robustness of the model to noise, an extensive quantum noise study
is performed. In this paper, it is demonstrated that the model trained on a
physical quantum device learns the noise characteristics of the hardware and
generates outstanding results. It is verified that even a quantum hardware
machine calibration change during training of up to 8% can be well tolerated.
For demonstration, the model is employed in indispensable simulations in high
energy physics required to measure particle energies and, ultimately, to
discover unknown particles at the Large Hadron Collider at CERN.

###### keywords:

Full Quantum Generative Model, Quantum Image Generation, Detailed Quantum
Inference Evaluation, Quantum Noise Study, Quantum Circuit Entanglement Study,
Quantum Hardware Training

##  1 Introduction

Quantum computing has the potential for a new paradigm in future computing to
accelerate tasks or even handle classically unsolvable problems
unsolvable_problems . In the current Noise Intermediate Scale Quantum (NISQ)
era, quantum devices suffer from non-negligible hardware errors, limited
connectivity and a limited number of qubits NISQ . While practical quantum
advantage is currently extremely difficult to accomplish, finding the best
suited algorithms to effectively combat the problems of NISQ-era devices
remains a widely researched topic. Quantum Machine Learning (QML) is a domain
which achieves acceptable results on NISQ devices due to the observed
robustness against noise qml_data .

High Energy Physics (HEP) experiments, such as those at the Large Hadron
Collider (LHC) at CERN, require enormous amounts of simulated data for
deriving high precision physics results bigdata_2021 . To handle this demand,
gigantic quantities of computing hardware resources are necessary, which has
led to the creation of the world’s largest computing grid operated by CERN
WLCG_webpage . To alleviate this strain on computational resources, Machine
Learning (ML) models have been developed that exhibit remarkable speed-ups
over current Monte Carlo-based simulations while maintaining the required
level of accuracy. In general, QML simulations represent a promising approach
to address the only further increasing simulation demands in the future
QML_speedup ; qml_hep . QML employs quantum circuits which exploit the quantum
properties of superposition and entanglement, which possess the potential to
outperform neural networks, their classical analogue quantum_supremacy . In
addition, QML might have the advantage of learning more complex distributions
with fewer parameters than classical ML due to their wider accessible phase
space.

Encoding the classical data into qubit states on quantum computers is a non-
trivial task state_preparation2 . Currently, many encoding techniques exist,
each exhibiting specific advantages and disadvantages, and in practice,
identifying “the best” encoding technique remains application dependent
state_preparation2 . To achieve a potential quantum advantage over classical
computing, theoretical studies suggest that at least linear scaling from
qubits to features is required data_encoding . On the other side, models which
employ better than linear scaling encoding techniques have drawbacks, making
them unsuitable for generating precise images on NISQ devices. For example,
amplitude encoding can only generate probability distributions and not
absolute pixel entries, i.e. energy values.

At present, there exist several quantum generative models, for example, the
Quantum Circuit Born Machine (QCBM) born_machine , Quantum Variational
Autoencoders Qvariational_autoencoder  or variations of quantum Generative
Adversarial Networks qgan_survey ; 1QGAN_hep ; QuGAN . They all face
limitations. Some models either do not scale well in terms of qubits required
relative to the number of encoded features, or they do not achieve a
satisfying level of fidelity. The Quantum Angle Generator (QAG) presented in
this paper and first introduced in reference ACM_IEEE_QAG , aims to overcome
these problems. Employing angle encoding with linear scaling of qubits to
features, it achieves extremely accurate results for a real-world problem on
current physical noisy quantum devices.

The paper content is structured as follows. First, the HEP use case is
motivated and the training data is defined. Next, the QAG model and the
employed angle encoding technique are presented. Then, multiple circuit
architectures are compared and the advantages of the best ones are
highlighted. An in-depth accuracy analysis of the model follows to highlight
its excellent precision. The quantum hardware noise behavior is evaluated,
including training and inference executed on real quantum devices. Lastly,
conclusions are drawn and summarized.

##  2 High Energy Physics Simulations

Simulations remain a crucial component of HEP analysis to evaluate the results
obtained by the processed data of the experiments. Currently, simulations are
performed predominantly with Monte Carlo methods such as the Geant4 toolkit
Geant4 . However, Monte Carlo simulations are very hardware resource demanding
and occupy half of the worldwide LHC Computing Grid RoadmapHEP . Future LHC
experiments will require more simulations due to more energetic particles,
simultaneous collisions and detectors constructed with higher granularity.
However, the projected budget for hardware development and computing resources
cannot keep pace with these increasing demands detector_simulations ;
LHC_computing_demand . As a result, ML alternatives to Monte Carlo methods are
being actively researched. Initial prototypes predict significant reductions
in simulation time and hardware resources while retaining acceptable levels of
accuracy ju2021performance ; biscarat2021towards ; sela2022deep . This
research goes one step beyond classical ML. Since HEP data sets are generally
created by underlying quantum mechanical effects, performing the computations
on quantum devices which likewise make use of quantum effects has the
potential to substantially enhance the simulations in accuracy and in terms of
sustainable computing.

Figure 1: An example 3D calorimeter shower image. A primary particle enters
the calorimeter at $(x,y,z)$ = $(13,13,0)$ and generates a secondary particle
shower along the z-axis.

Electromagnetic calorimeters are constructed as high granularity sensor grids
to measure the energy of photons, electrons, and positrons through complex
particle shower generation processes in space and time calorimeter_cern . They
constitute a key component of HEP detectors to measure the energy of the
particles produced in the interaction process and occupy most of the
simulation time RoadmapHEP . Calorimeter outputs can be interpreted at lowest
order as static and spatial 3D images, which we call “shower images”: the
value of each pixel corresponds to the energy measured in a specific
calorimeter cell. The initial data from reference data_set  consists of
$25\times 25\times 25$ pixel images. An example of a 3D shower image is
visualized in figure 1. To reduce the dimensionality, the images are averaged
along two spatial axes (x- and y-direction), resulting in a one-dimensional
representation that is further downsampled to eight pixels by averaging three
contiguous pixels along the z-direction. Although the initial data set
provides many different energies, for simplicity this study focuses on images
recorded by particles in the energy range of [225, 275] GeV. The data set is
split into a training and test set, each consisting of approximately 1 000
samples. The downsampled data is available in reference quantum_data_set , and
an example image is illustrated later in this paper in figure 6(a).

##  3 The Quantum Angle Generator

The QAG represents a QML model which employs the well established technique of
angle encoding encoding_classifiers ; angle_encoding_application  to generate
extremely precise images. It scales linearly with the number of encoded
features. Thus, the generation of $n$ features requires $n$ qubits. In this
study, the number of features corresponds to the number of pixels. A
comprehensive description of the QAG model, its objective training function,
and an evaluation of several quantum circuits are provided below.

Figure 2: The structure of the Quantum Angle Generator.

###  3.1 Model Description

The QAG model consists of variational quantum circuits trained by an objective
function. The model structure is visualized in figure 2. All qubits are
initialized in the basis state $|0\rangle$. The state preparation function
implements a Hadamard (H) gate to constitute superposition, followed by a
y-rotational (Ry) gate to introduce randomness so that the model can draw new
samples at each execution. For this, the Ry gate angles $\Omega$ are randomly
drawn from a [-1, 1] uniform distribution and pixel-wise multiplied by the
pixel standard deviations present in the training data to obtain correct pixel
energy variations. To account for all various primary particle energies, all
angles $\Omega$ are multiplied by a random value between [-0.25, 0.25].

The unitary transformation consists of quantum circuits and constitutes the
trainable part of the QAG model. Various circuit architectures were tested as
documented in section 3.3.

To convert the quantum states back into classical energy values via angle
encoding, the model must be executed multiple times and the quantum states
measured. The number of executions is commonly denoted as the number of shots
$nb_{\textrm{shots}}$. It is counted how often state $|0\rangle$ is measured.
The scalar intersection $I$ of the vertical axis on the Bloch sphere (z-axis)
and the angle $\theta$ is calculated with:

$\displaystyle I$
$\displaystyle=2\cdot\frac{\textrm{counts}(|0\rangle)}{nb_{\textrm{shots}}}-1\,,$
(1) $\displaystyle\theta$ $\displaystyle=\arcsin(I)\,.$

(a)

(b)

Figure 3: (a) Decoding of an example state $\Psi$ into an angle $\theta$. (b)
Some representative angles $\theta$ and its corresponding energy values.

The angle $\theta$ operates in the x-z-plane of the Bloch sphere and is
defined as zero in the $|+\rangle$ state. Rotating $\theta$ clockwise leads to
positive angles. The decoding process is visually illustrated in figure 3(a)
with an example state $|\Psi\rangle$, its intersection $I$ and the
corresponding angle $\theta$ for a single qubit example. The angle $\theta$
can then be transformed into a pixel energy $E$ by the linear change of ranges
equation:

$E=\frac{(E_{\textrm{max}}-E_{\textrm{min}})(\theta-\theta_{\textrm{min}})}{\theta_{\textrm{max}}-\theta_{\textrm{min}}}\,.$
(2)

$E_{\textrm{max}}$, $E_{\textrm{min}}$, $\theta_{\textrm{max}}$ and
$\theta_{\textrm{min}}$ are defined in figure 3(b): the minimum energy
$E_{\textrm{min}}=0\,$MeV is set to $\theta=-\pi/2$ and the maximum energy
$E_{\textrm{max}}=0.6\,$MeV is set to $\theta=+\pi/2$. With
$E_{\textrm{min}}=0$ and $\theta_{\textrm{max}}=-\theta_{\textrm{min}}$
equation 2 can be simplified into:

$E=\left({\frac{E_{\textrm{max}}}{2\cdot\theta_{\textrm{max}}}}\right)\cdot(\theta+\theta_{\textrm{max}})\,.$
(3)

For multi-qubit quantum circuits: the final quantum states of all qubits are
measured, and the outcomes are decoded independently. Although the qubit
results are individually decoded, the state of each qubit is entangled with
others due to the gates applied within the variational quantum circuit, as
lined out later.

It is worth to be noted that the angle $\theta$ and, therefore, the
corresponding decoded energy $E$ remain in discrete values. Since, $\theta$
depends in value and accuracy on the number of shots $nb_{\textrm{shots}}$:
the larger $nb_{\textrm{shots}}$, the better the achievable energy precision
and resolution. Fortunately, with present quantum devices, the
$nb_{\textrm{shots}}$ can be easily chosen to be large. Currently, on IBMQ
devices the maximum possible number of shots is $nb_{\textrm{shots}}=$ 100
000. For the simplified calorimeter use case, this is more than sufficient.
For comparison, in a previous classical reduced precision ML research in
reference reduced_precision , it is demonstrated that already $256$ discrete
energy levels are sufficient for correctly reproducing the full-size
calorimeter shower image. In this reduced precision research, the parameters
of the neural network are quantized from a larger format (floating point 32)
down to a smaller number format (integer 8). This study will show that 512
shots provide a sufficient resolution. The detailed image generation process
for the QAG model, as described in this subsection, is provided as an
algorithm in the appendix A.

###  3.2 Training Objective Function

The QAG model is trained with two losses employed as objective functions. The
first one is the Mean Maximum Discrepancy (MMD) loss MMD1 ; MMD2 , already
successfully applied by other quantum models, e.g. the QCBM born_machine .
Training exclusively with the MMD loss resulted in good average shower
distributions. However, when exploring the generated images in more detail,
for example in the pixel correlation, the model did not perform
satisfactorily. Therefore, a second correlation (Corr) loss is added to help
learn the patterns present in the training data (e.g., image pixel
correlations). The Corr loss is calculated by the pixel-wise mean squared
error (MSE) between the pixel correlation values present in the training data
and the ones inside the generated data. The pixel-wise correlations are
illustrated in figure 6(b).

To train the QAG model, the Simultaneous Perturbation Stochastic Approximation
(SPSA) optimizer SPSA  is employed, which only requires two optimization steps
per epoch. The hyperparameters for training were found by extensive
hyperparameter searches employing the Optuna optuna  library. All tests in
this study are executed in Qiskit version 0.26.2. The models are trained for
500 epochs, containing one batch. The dynamic MMD loss weight starts at a
value of one and decays linearly with $-0.001\cdot$epoch, starting from epoch
100. Opposite, the Corr loss weight increases by the same value starting at
zero. The dynamical training batch size is set to generate one image in the
first hundred epochs and afterward to 20 images to calculate the Corr loss
between multiple images. Each quantum job contains 512 shots for training and
inference. The generator SPSA optimizer learning rate is set to $c_{0}=1$ with
an exponential learning rate decay of $0.006$ starting from epoch 50. All
these settings showed the best performance in the tests.

###  3.3 Quantum Circuit Study

The ideal circuit should contain a certain, optimized to be minimal, number of
parameters to achieve a sufficient level of accuracy. Different circuit
architectures were compared to each other based on characteristic numbers
expressing the power of the circuit. The circuits employ trainable rotational
gates and two-qubit entanglement gates. As the angle encoding primarily uses
the qubits y-axis component, we predominantly employ Ry gates. For some
circuit architectures, we test if additional z-rotational gates (Rz gates) or
deeper circuits with depth 2 (denoted as d2) can further improve the results.
We use two-qubit controlled-not gates (cx gates) native on IBM Quantum (IBMQ)
ibmq  devices; while other entanglement gates are compositions of multiple
native gates of the hardware. Keeping an eye on the goal of executing the
training on a real quantum device, the absolute number of decomposed gates
should be kept as small as possible.

The characteristic circuit numbers used in this study are the number of
trainable parameters $N_{p}$, the expressibility $X$ and the entanglement
capability $E$. Larger circuits with more trainable parameters are potentially
capable of achieving more accurate results. However, it might be that a
plateau is reached at some point where the same task can be solved with
similar accuracy by a smaller circuit, which is being investigated here. The
definitions for $X$ and $E$ are from reference expressibility : $X$ describes
how well the circuit can represent the pure states of the representative
Hilbert space. For a single qubit the expressibility exhibits how many states
of the Bloch sphere can be represented. In this paper, we measure $1-X$: the
closer to $1$ the better the expressibility of the model, while the closer to
$0$ the worse. The entanglement capability $E$ is a measure that expresses the
ability of a circuit to generate entangled states between the qubits.
Likewise, $E$ ranges from $0$ to $1$, where $1$ represents the best achievable
value. The circuit architectures under study are introduced in the appendix B.
Their corresponding characteristic numbers and theoretical potential are
provided in the appendix C. In the following, the circuits are evaluated for
the calorimeter use case.

Figure 4: The MSE uncertainty (the smaller, the better) of the various trained
circuit architectures as a function of $N_{p}$ (left), $X$ (middle) and $E$
(right), see text for explanations. Green rectangles mark optimal areas.

We start by interpreting the results displayed in figure 4. The MSE accuracy
metric is calculated by taking the pixel-wise MSE between the average Geant4
and QAG images. The training is repeated $25$ times for each circuit and the
mean and standard deviations are plotted. To prevent the influence of
outliers, the best and worst two trials are discarded in this analysis. The
MSE is given as a function of: $N_{p}$ on the left, $X$ in the middle, and $E$
on the right. By inspecting the plots, it can be recognized that the MSE does
not correlate with any of the characteristic circuit values in the plots,
neither do the characteristic values correlate among themselves, as shown in
the appendix C.

The MERA-up, MERA-up_d2, and MERA-up_Rz architecture perform best with the
lowest MSE. This is consistent with the observation that they maintain a high
$X$ and $E$, as provided in the appendix B. The error bars provide a hint
about the training stability. It can be observed that the better the average
MSE of a model, the smaller its standard deviation.

All in all, the MERA-up_Rz circuit clearly performs best considering the
characteristic circuit values and the achieved accuracy in training. However,
with the emphasis on a low number of $N_{p}$, the plain MERA-up circuit
performs almost as well, while needing only half the number of parameters.
Therefore, the following studies employ the MERA-up circuit for training the
QAG model.

##  4 In-depth Accuracy Analysis

In this section, we analyze the results of the QAG model operating the MERA-up
circuit architecture. We showcase typical accuracy metrics for the calorimeter
simulation in HEP.

(a) Figure 5: The unweighted loss values the training epochs. The blue lines
correspond to the MMD losses and the green lines to the Corr losses. The thick
solid line represents the mean loss across trials, the colored band around
indicates the standard deviation (STD), and the thin line denotes the loss of
the best trial’s loss.

###  4.1 Training Evaluation

In a first step, the statistical trends of the objective functions during
training are investigated. In figure 5, the unweighted loss functions
(excluding loss weights) are plotted as a function of the training epochs. The
mean of twenty training repetitions is visualized as a thick solid line and
the standard deviation (STD) as a colored band. The Corr loss starts
influencing the training only at epoch 100 because its weight is set to zero
before.

The MMD loss of all training repetitions converges stable, and the STD band
narrows towards the end of the training. Overall, the MMD and Corr loss
converge smoothly without strong oscillations, which is a desirable
characteristic for stable (Q)ML training. The MMD loss contributes far more
than the Corr loss throughout the training. However, the Corr loss plays a
significant role in achieving good physics accuracy in the generated shower
images.

###  4.2 Inference Evaluation

In the following, the accuracy in inference is evaluated. The generated images
of the best trained model are compared to the Geant4 test data, which consists
of 980 images. Likewise, there are 980 images generated by the QAG model to
create the following accuracy metrics. The details about how the accuracy
metrics are calculated are provided in the appendix D.

1\. Average calorimeter shower shape: The first metric represents the
calorimeter shower shape displayed in figure 6(a). The shower shape is
perfectly reproduced by the QAG model. The MSE corresponds to $0.00059\pm
0.00037$, which is extremely close to zero, indicating a very good accuracy.

(a)

(b)

(c)

Figure 6: (a) Visualization of the average calorimeter shower shapes. The
energy is given in an arbitrary unit (a.u.) due to image downsampling. The
pixel-wise correlation plot for (b) Geant4 and (c) the QAG model. The
correlation ranges between -1 and 1; a value of 1 indicates a perfect positive
correlation.

2\. Pixel-wise correlation: The second metric corresponds to the pixel-wise
image correlation. The positive or negative correlation patterns between all
the pixels are determined. The baseline represents the correlation from Geant4
in figure 6(b). The correlation for the generated data of the QAG model is
presented in figure 6(c). It can be derived that the overall correlation
pattern is accurately reproduced by the QAG model. Like the Geant4 data, it
consists of a larger and more compact positively correlated group of pixels.
The other pixels are negatively correlated. Inspecting the particular details,
different color shades indicate some minor deviations. However, the achieved
correlation precision by QAG is astonishing. Therefore, it can be concluded,
that the quantum circuits are capable to reproduce complex correlation
patterns through substantial entanglement strategies, as present in the MERA-
up architecture.

3\. Energy sum: The energies contained in all pixels calculated for the
individual images represent the third accuracy metric. In figure 7(a), the
energy sum histogram of the Geant4 images reveals a Gaussian shape and is
correctly reproduced visually by the QAG model, which is also confirmed by the
mean $\mu$ and the standard deviation $\sigma$.

(a)

(b)

(c)

Figure 7: (a) The energy sum histogram, the four k-means clusters present in
the images of (b) the Geant4, and (c) the QAG images.

4\. k-means Clusters: This metric evaluates if the QAG model can correctly
represent specific image modes. The Geant4 data is clustered with the k-means
algorithm k_means  to find four clusters or image modes as illustrated in
figure 7(b). Cluster 0 deposits substantially larger fractions of energy in
earlier calorimeter cells than in higher cluster numbers. Further, the
particles from clusters 0 and 1 contain a larger energy fraction, estimated by
the integral area below the curves. Here we are interested in whether the QAG
model can reproduce this behavior. The four clusters of the QAG images are
provided in figure 7(c). A similar structure can be observed, which indicates
a good accuracy in reproducing the energy contents and image modes on average.

5\. Pixel-wise energy distribution: The last metric is used to examine the
distributions of the energy content of each pixel, as illustrated in figure 8.
Overall, the histograms of the QAG model match those of the Geant4 model. Even
pixels with non-Gaussian energy distributions in the Geant4 model are
correctly reproduced by the QAG model. For example, the longer tail towards
smaller energies on the left side of pixel 4 is equally present in the
histogram for the QAG model. The large histogram overlaps indicate that not
only the averages are reproduced with high accuracy, but also the energy
distributions returned for each individual pixel are correct.

Figure 8: The energy deposition histograms for Geant4 and the QAG model. Each
histogram represents the energy distribution for an individual pixel, as
labeled above each plot. The x-axes represent the energies on a consistent
scale across all histograms. The y-axes count the hits per energy bin, with
each hit corresponding to the energy contained in a pixel for a single image.
Note the varying y-axis scales.

##  5 Quantum Noise Study

In the current NISQ era, relatively high hardware error levels are one of the
primary limitations to effectively employing algorithms on real quantum
devices. Similar as in the classical case, QML models appear to be noise
resilient to some degree of hardware errors noise_robust_circuits ;
QGAN_noise_study ; QGAN_ACAT_noise_study ; noise_robust_training . In the
following, the robustness of the QAG model to simulated noise is tested in
inference and training. Furthermore, training and inference are executed on
real quantum devices with measured noise levels and compared to the results
with simulated noise.

###  5.1 Inference

In a first step, quantum noise is applied only to the inference of a model
trained without noise. Inference is performed using three different noise
configurations: simulated noise at varying levels, simulated noise derived
from the later used hardware, and finally, with the real quantum hardware. In
the simulated noise configurations, each qubit noise is modeled with the same
readout measurement error, and each inter-qubit connection with the same two-
gate (CNOT) error. In the combined noise model both are on the same level. In
contrast, the hardware noise levels can vary widely for individual qubits as
well as for gates.

Multiple noise configurations and error levels from zero up to 15% are tested.
The MSE is utilized as the accuracy measure. The results are illustrated in
figure 9. For each configuration, the average value of 20 generated images is
plotted in dependence of the noise level as a solid line and the standard
deviation as a colored band around the mean. The gray horizontal line serves
as accuracy reference for the noise-free configuration.

Figure 9: Inference noise study. On the x-axis the noise levels are provided
and on the y-axis the MSE as accuracy metrics (the lower, the better).
Inference is executed with various noise configurations and on real quantum
hardware. At each point, twenty images are generated.

All configurations maintain sufficient accuracy up to approximately 1.5% of
noise. The configuration with readout noise only (green) is most robust and
maintains a stable accuracy of up to 8% of noise. CNOT noise only (blue) and
both readout noise and CNOT noise combined (orange) experience a stronger
impact. As expected, the combined configuration (orange) performs worst. As a
side note, current quantum devices have average noise levels below 5%, which
is expected to gradually decrease further. But as discussed in the following,
the noise levels are unstable and can sometimes spike up. Therefore, wider
noise ranges were investigated.

Next, the inference is run by loading the real hardware noise model from
ibmq_montreal into the simulator. The ibmq_montreal device consists of a
Falcon r4 processor with 27 qubits. The average readout noise over the qubits
employed at the time of the test is 2.51% and the average gate noise level is
0.97%. The explicit noise model, containing the entries for each qubit, is
provided in the appendix in figure 11(b). The result is included in figure 9
(blue triangle). The noise level position (x-axis) is determined as the
average of readout and CNOT noise. Although the noise levels of the qubits
vary strongly, the measured accuracy of the hardware noise simulation agrees
well with the simulated noise in mean and standard deviation within the
uncertainties. This suggests that a model trained without any noise would
theoretically be able to run inference on noisy hardware without a significant
drop in accuracy.

Finally, the inference is executed on the real ibmq_montreal device. The
result is plotted as a red triangle in figure 9. The accuracy on the real
hardware is worse than predicted by the simulation, as indicated by a larger
MSE value. The decomposition of the circuit to the real hardware includes swap
operations, which imply additional two-qubit entanglement gates for the
quantum circuit. It is possible that these are not included in the hardware
noise simulation and lead to higher noise influence on real hardware and thus
to worse results.

###  5.2 Training

The noise study is repeated to include noise also during training. We study if
the QAG model can learn to compensate for noise in training, especially when
running on the real quantum device. Besides, we investigate with which noise
values the model can still maintain a reasonable accuracy.

Figure 10: Training noise study. On the x-axis the noise levels are provided
and on the y-axis the MSE as accuracy metrics (the lower, the better).
Training is executed with various noise configurations and on real quantum
hardware. The training with fake noise levels are repeated ten times.

The results are provided in figure 10. The configurations with readout noise
(green) and CNOT noise (blue) maintain a similar level of accuracy until
approximately 3%. The accuracy of the combination of readout and CNOT noise
(orange) decreases marginally from 1% of noise and further with larger noise
levels. However, at 3% of noise, its accuracy is still close to the noiseless
case, staying within one standard deviation. This indicates that training the
model with noise makes the QAG model more robust than only applying noise to a
trained model in inference only.

Next, two quantum devices are simulated. ibmq_montreal and ibm_cairo, which
are both 27 qubit devices, but ibm_cairo has the more advanced Falcon r5.11
processor. Likewise, the training is repeated ten times and the results are
added as blue and orange triangles in figure 10. It can be noted that the
average accuracy of the training with hardware noise performs slightly worse
than the simulated combined noise model (orange line). The strong overlap
between the error from the noise simulation (orange band) and the hardware
noise simulation (orange and blue error bar) indicates that the accuracy
difference is statistically not significant.

Finally, the entire training was executed on the real quantum device. First,
the training was carried out on ibmq_montreal. During training, around epoch
280, an unpredicted significant noise change occurred and the readout noise of
one qubit increased to 8%, as shown in figure 11. As a result, the MMD loss
spikes up, as shown in figure 12(a). This negatively influenced the training.
However, after the noise change during training, the model recovered and
adapted to the new noisy environment. In the still remaining number of epochs
the loss decreased to a modest level. The training was repeated on the best
performing ibm_cairo device without having a hardware calibration change
during training, and the training losses are shown in figure 12(b).

(a)

(b)

Figure 11: The noise levels of the ibmq_montreal device are shown (a) before
and (b) after the calibration change during training. The lower x-axes
displays qubit indices as provided at the IBMQ hardware, while the upper
x-axes represents their usage in the quantum circuit (e.g., in figure 13,
counting from top to bottom). Inference is performed with the noise model (b),
where it is notable that the readout noise of qubit number five on the
hardware increased from two to over eight percent. Note the different y-axis
scales.

The results of both hardware training are included as red and green triangles
in figure 10. The error bars correspond to the accuracy deviations within 50
generated validation images. It can be observed that the average accuracy of
the ibmq_montreal hardware training (red) is visibly worse than the noise
simulation (blue). This is most likely due to the large noise increase and the
fact that the model did not have enough remaining epochs to fully recover.
This is also indicated by the still decreasing losses towards the end of the
training. As mentioned above, the training was repeated on the more stable
machine ibm_cairo. This time, the training was completed without calibration
changes and with about $1$% lower hardware noise level (readout noise $0.86$%
and CNOT noise $0.89$%). The accuracy of the hardware training on ibm_cairo
(green) is only marginally worse than the one from the simulator (orange).
This indicates that the simulated and real hardware results behave similarly
for low hardware noise levels. This fulfills the expectations derived from the
pure simulation that exhibit only statistically insignificant variations in
the very good accuracy at these low noise levels.

(a)

(b)

Figure 12: The losses during training on the real quantum device for (a)
ibmq_montreal and (b) ibm_cairo. Shown are the unweighted MMD and Correlation
loss as a function of the training epochs.

Comparing the absolute MSE magnitude of the real hardware training ($\approx
0.002$ and $\approx 0.003$) with the one from inference from the previous
section ($\approx 0.005$), the accuracy improved, suggesting that the QAG
model is able to adapt its parameters to the noisy hardware to improve its
precision. This is also confirmed by the ibmq_montreal training, where the
accuracy worsened entirely after the calibration change, but then recovered.
In the appendix in figure 15(a), the average shower image created by the
ibmq_montreal device is visualized, and in figure 15(b) by ibm_cairo. The
shower image of ibm_cairo agrees well with Geant4, whereas ibmq_montreal
exhibits some deficiencies because of the missing remaining training epochs
after the calibration.

##  6 Conclusion

The results of this study clearly demonstrate that the newly developed QAG
model is capable of generating images with good precision, as measured with a
variety of validation metrics. This includes correctly reproducing average
values, but most importantly, also complex pixel-wise correlations with the
chosen optimal MERA-up quantum architecture. These results reveal that the QAG
model with a good entangled circuit is capable of learning intrinsic
correlation patterns from the training data.

Our study exhibit the significant impact that quantum hardware noise can have
on the accuracy of quantum machine learning models. The results evidence that
training the models with noise leads to better performance (stable until 3%
noise) because the QAG model adapts to the underlying noise behavior and
converges faster in contrast to the situation of applying noise in inference
only (stable until 1.5% noise). This was also verified on the real hardware by
the ibm_cairo training. Furthermore, our study shows that the QAG model is
robust and can produce accurate results even with significant hardware
calibration changes with up to 8% noise, as demonstrated by the training on
ibmq_montreal. Overall, the newly developed QAG model demonstrates that
training quantum machine learning models with realistic quantum hardware noise
can lead to robust models and accurate results, which is of great importance
for the future development of real world quantum machine learning
applications.

#### Acknowledgment

The authors would like to express their sincere gratitude to Simon Schnake and
Alexis-Harilaos Verney-Provatas for their help, advice, and proofreading,
which improved the quality and outcome of this paper.
This work has been sponsored by the Wolfgang Gentner Programme of the German
Federal Ministry of Education and Research and by the CERN Quantum Technology
Initiative. Additionally, we would like to acknowledge partial funding from
the BMBF Project ”NiQ: Noise in Quantum Algorithms” within the BMBF program
”Anwendungsnetzwerke für das Quantencomputing“ and from the Deutsches
Elektronen-Synchrotron DESY, a member of the Helmholtz Association (HGF).
Access to the IBM Quantum Services was obtained through the IBM Quantum Hub at
CERN. The views expressed are those of the authors and do not reflect the
official policy or position of IBM or the IBM Q team.

##  Appendix A QAG Inference Algorithm

Algorithm 1 shows the pseudo code for generating one image using the QAG model
from section 3.1. Here the MERA-up architecture (appendix B) is used as an
example. The standard deviation $\sigma_{\text{pixel}}$ for the energy
distribution of each pixel is obtained from training data.

Algorithm 1 Quantum Angle Generator Inference

Input: $nb_{\textrm{shots}}$ $\ignorespaces\triangleright$ Number of
measurements

Input: $\sigma_{\text{pixel}}$ $\ignorespaces\triangleright$ Array of the
pixel standard deviations

Output: $E$ $\ignorespaces\triangleright$ Array of pixel energies forming the
image

Begin

Initialize all 8 qubits to state $|0\rangle$

Randomly draw value $v$ from uniform distribution $[-0.25,0.25]$

for all index, qubits do

Apply Hadamard gate H to qubit

Randomly draw $\Omega$ from uniform distribution $[-1,1]$

$\Omega_{\text{pixel}}$ = $\Omega\cdot
v\cdot\sigma_{\textrm{pixel}}[\textrm{index}]$

Apply rotation Ry($\Omega_{\textrm{pixel}}$) to qubit

end for

Apply the MERA-up unitary transformation to all qubits

Measure the qubits $nb_{\textrm{shots}}$ times

Count number of times state $|0\rangle$ is measured
$\rightarrow\textrm{counts}(|0\rangle)$

Compute $\theta$ according to equation 1

Compute pixel energies $E$ according to transformation 3

Return $E$

End

##  Appendix B Quantum Circuit Architectures under Study

The quantum circuit architectures investigated in this paper are summarized in
figure 13. In general, it was observed that hierarchical architectures perform
best while maintaining a reasonable number of quantum gates and parameters.
Specifically, we examine the Tree Tensor Network (TTN) architecture and Multi-
scale Entanglement Renormalization Ansatz (MERA) introduced in reference
TTN_MERA . Multiple variations of these circuits are tested: 1) circuits with
a depth two (naming scheme ”d2”). These circuits contain two layers, with all
circuit gates placed twice, to evaluate if deeper circuits perform better. 2)
Circuits additional with Rz-gates are employed after each Ry-gate (naming
scheme ”Rz”) to assess if rotations around an additional axis can improve the
accuracy. Both architecture variants double the number of parameters of the
initial circuit.

Figure 13: The basic quantum circuit architectures under study. Top left: TTN
architecture, top right: Linear entanglement architecture, bottom: the full
circuit corresponds to the MERA architecture, with the red highlighted section
corresponding to the MERA-up architecture.

For the MERA architecture, further variations are analyzed where only the
right circuit half from the original architecture is implemented, denoted as
the MERA upsampling (MERA-up) circuit. In the MERA-up circuit, the information
is upsampled from the central qubit and spread to all the other qubits,
similar to what happens in classical ML generative models, for example,
transpose convolutional neural networks. The left half of the MERA circuit,
the MERA downsampling (MERA-down) architecture, is not tested because it would
rather compress the information as required in classification tasks that are
not used in this paper. The last architecture contains a simple linear
entanglement strategy.

In reference expressibility , multiple complex four-qubit architectures are
compared. However, these architectures are not exploited in our study. The
reason is most of them contain many more gates and parameters, and they do not
scale well for more than four qubit circuits. Also, many architectures employ
parameterized two-qubit rotational gates which we found, taking the
decomposition into account, not as effective as the combination of separate
rotation and entanglement gates.

##  Appendix C Characteristic Circuit Numbers

(a)

(b)

(c)

Figure 14: Study of potential correlations between the three characteristic
circuit numbers. On the left: $X$ versus $E$. In the middle: $N_{p}$ versus
$X$. On the right: $N_{p}$ versus $E$.

The results of evaluating the characteristic circuit numbers (number of
parameters $N_{p}$, expressibility $X$ and entanglement capability $E$) are
shown in figure 14. In figure 14(a) $E$ is plotted as a function of $X$. The
MERA_Rz and MERA-up_d2_Rz architecture perform best and lie almost on top of
each other in the top right corner, directly followed by MERA-up_Rz and
TTN_Rz. The Linear architecture and the basic TTN perform worst. It can be
clearly noted that circuits with more gates (Rz, d2) perform better. $N_{p}$
is studied in figure 14(b) and figure 14(c). It can be seen that the best two
circuits – MERA_Rz and MERA-up_d2_Rz – contain by far the largest number of
parameters. However, a more limited number of parameters is desirable to
address NISQ hardware limitations. The MERA-up_Rz and TTN_Rz architectures
contain approximately half the number of parameters compared to the MERA_Rz
and MERA-up_d2_Rz circuits but perform almost similarly accurately. Therefore,
they may be preferred in practice. Further reducing the number of parameters,
the MERA-up architecture with only $N_{p}=23$ parameters maintains adequate
values for $E$ and $X$ and is the baseline circuit for the more detailed
studies. All characteristic circuit numbers measured in the circuit study are
summarized in table 1. In the table, it is evident that the MERA-up circuit
maintains a considerably lower training time on the classical hardware
compared to models with a higher number of parameters. The training was
executed on a cluster equipped with 32 Intel Xeon Gold 6130 CPU cores running
at 2.10GHz.

Table 1: The characteristic circuit values $N_{p}$, $X$ and $E$ are listed. Additionally, the statistics of the MSE accuracy metrics and the training times $t$ in minutes are provided for the individual circuits for repeating the training 25 times on the calorimeter data set. Circuit Name | $N_{p}$ | $X$ | $E$ | MSE |  $t$ in min
---|---|---|---|---|---
Linear | 16 | 0.8191 | 0.261 | 0.00113 $\pm$ 0.0008 | 35.0 $\pm$ 0.2
TTN | 29 | 0.8068 | 0.462 | 0.00245 $\pm$ 0.0012 | 41.5 $\pm$ 0.6
TTN_Rz | 58 | 0.9912 | 0.918 | 0.00440 $\pm$ 0.0022 | 85.9 $\pm$ 1.5
MERA | 45 | 0.9564 | 0.855 | 0.00220 $\pm$ 0.0011 | 51.3 $\pm$ 0.5
MERA_Rz | 90 | 0.9997 | 0.959 | 0.00634 $\pm$ 0.0036 | 121.5 $\pm$ 9.6
MERA-up | 23 | 0.9377 | 0.894 |  0.00059 $\pm$ 0.0004 | 39.4 $\pm$ 2.9
MERA-up_d2 | 46 | 0.9715 | 0.906 | 0.00038 $\pm$ 0.0002 | 50.5 $\pm$ 0.2
MERA-up_Rz | 46 | 0.9961 | 0.926 | 0.00047 $\pm$ 0.0005 | 75.2 $\pm$ 7.5
MERA-up_d2_Rz | 92 | 0.9999 | 0.954 |  0.00094 $\pm$ 0.0008 | 116.2 $\pm$ 4.3

##  Appendix D Definitions of Accuracy Metrics

In this section, we provide detailed formulas for the inference accuracy
metrics discussed in section 4.2.

###  D.1 MSE of the Average Shower Shape

The Mean Squared Error (MSE) for the average calorimeter shower shape is
derived from the difference between the average images of Geant4 and the QAG
model, calculated over 980 generated inference images and the corresponding
980 Geant4 images. In the subsequent equations, $k$ iterates over the $N=$ 980
images and $i$ denotes the ith pixel of each image. Hence,
$E_{k,\,i}^{\textrm{G4}}$ (Geant4) and $E_{k,\,i}^{\textrm{QAG}}$ (QAG model)
correspond to the energy of pixel $i$ from image $k$. The average pixel
energies $\overline{E}_{i}^{\textrm{G4}}$ and
$\overline{E}_{i}^{\textrm{QAG}}$ are computed as follows:

$\overline{E}_{i}^{\textrm{G4}}=\frac{1}{N}\sum_{k=1}^{N}E_{k,\,i}^{\textrm{G4}}\hskip
14.22636pt\text{and}\hskip
14.22636pt\overline{E}_{i}^{\textrm{QAG}}=\frac{1}{N}\sum_{k=1}^{N}E_{k,\,i}^{\textrm{QAG}}.$
(4)

The MSE is then calculated by iterating over the $M=$ 8 individual pixels by
the index $i$:

$\text{MSE}=\frac{1}{M}\sum_{i=1}^{M}\Big{(}\overline{E}_{i}^{\textrm{G4}}-\overline{E}_{i}^{\textrm{QAG}}\Big{)}^{2}.$
(5)

As the MSE corresponds to an error, the smaller the value, the better the
accuracy of the QAG model.

###  D.2 Energy Sum Histogram

Each histogram entry (or hit) is the total energy $\hat{E}$ of a shower image:

$\hat{E}^{\textrm{G4}}=\sum_{i=1}^{M}E_{i}^{\textrm{G4}}\hskip
14.22636pt\text{and}\hskip
14.22636pt\hat{E}^{\textrm{QAG}}=\sum_{i=1}^{M}E_{i}^{\textrm{QAG}},$ (6)

where $i$ is iterating over the $M=8$ pixels.

The means of the distributions for our data sets, Geant4 and QAG, are denoted
as $\mu^{\textrm{G4}}$ and $\mu^{\textrm{QAG}}$, respectively. Similarly, the
standard deviations are represented by $\sigma^{\textrm{G4}}$ and
$\sigma^{\textrm{QAG}}$.

The means $\mu$ of the total energy for the $N=980$ images are then calculated
using the well-known equation:

$\mu^{\textrm{G4}}=\frac{1}{N}\sum_{i=1}^{N}\hat{E}_{\textrm{i}}^{\textrm{G4}}\hskip
14.22636pt\text{and}\hskip
14.22636pt\mu^{\textrm{QAG}}=\frac{1}{N}\sum_{i=1}^{N}\hat{E}_{\textrm{i}}^{\textrm{QAG}}.$
(7)

And the standard deviations, denoted by $\sigma$, are computed as:

$\begin{split}\sigma^{\textrm{G4}}&=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(\hat{E}_{\textrm{i}}^{\textrm{G4}}-\mu^{\textrm{G4}})^{2}}\\\
\\\ &\text{and}\\\ \\\
\sigma^{\textrm{QAG}}&=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(\hat{E}_{\textrm{i}}^{\textrm{QAG}}-\mu^{\textrm{QAG}})^{2}}\end{split}$
(8)

A closer agreement between the means and standard deviations of the two
distributions indicates a higher accuracy of the QAG model.

##  Appendix E Full Training on Quantum Device

(a)

(b)

(c)

Figure 15: On the left, the average shower images of the QAG model are trained
on ibmq_montreal and in the middle on ibm_cairo. The correlation plot on the
right is created for the ibm_cairo training.

The average shower images of the quantum hardware training are correctly
reproduced, as provided in figure 15. The correlation plot for the ibm_cairo
training is provided in figure 15(c). The overall correlation pattern is
correctly reproduced. All in all, the model trained on ibm_cairo shows a good
performance.

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