11institutetext: INFN-LNF, Istituto Nazionale di Fisica Nucleare-Laboratori
Nazionali di Frascati, Frascati, 00044 Roma, Italy 22institutetext: Stefan-
Meyer-Institut für Subatomare Physik, Vienna, 1030, Austria 33institutetext:
University of Vienna, Vienna Doctoral School in Physics, Vienna, 1090, Austria
44institutetext: Department of Physics, Faculty of Science, University of
Zagreb, 10000 Zagreb, Croatia 55institutetext: Horia Hulubei National
Institute of Physics and Nuclear Engineering IFIN-HH Măgurele, Romania
66institutetext: Politecnico di Milano, Dipartimento di Elettronica,
Informazione e Bioingegneria, Milano, 20133, Italy 77institutetext: INFN
Sezione di Milano, 20133, Italy 88institutetext: Physik Department E62,
Technische Universität München, Garching, 85748, Germany 99institutetext:
RIKEN, Tokyo 351-0198, Japan 1010institutetext: Faculty of Physics, Astronomy,
and Applied Computer Science, Jagiellonian University, Krakow, 30-348, Poland
1111institutetext: Center for Theranostics, Jagiellonian University, Krakow,
Poland 1212institutetext: Research Center for Electron Photon Science (ELPH),
Tohoku University, Sendai, 982-0826, Japan 1313institutetext: Centro Ricerche
Enrico Fermi – Museo Storico della Fisica e Centro Studi e Ricerche “Enrico
Fermi”, Roma, 00184, Italy

# Measurements of high-n transitions in intermediate mass kaonic atoms by
SIDDHARTA-2 at DA$\mathrm{\Phi}$NE

F. Sgaramella 11<EMAIL_ADDRESS>M. Tüchler 2233
<EMAIL_ADDRESS>C. Amsler 22 M. Bazzi 11 D. Bosnar 44 M.
Bragadireanu 55 M. Cargnelli 22 M. Carminati 6677 A. Clozza 11 G. Deda 6677 R.
Del Grande 8811 L. De Paolis 11 L. Fabbietti 88 C. Fiorini 6677 I. Friščić 44
C. Guaraldo 11 M. Iliescu 11 M. Iwasaki 99 A. Khreptak 111010 S. Manti 11 J.
Marton 22 M. Miliucci 11 P. Moskal 10101111 F. Napolitano 11 S. Niedźwiecki
10101111 H. Ohnishi 1212 K. Piscicchia 131311 Y. Sada 1212 A. Scordo 11
<EMAIL_ADDRESS>H. Shi 22 M. Silarski 1010 D. Sirghi 11551313 F.
Sirghi 1155 M. Skurzok 10101111 A. Spallone 11 K. Toho 1212 O. Vazquez Doce 11
E. Widmann 22 C. Yoshida 1212 J. Zmeskal 22 C. Curceanu 11

###### Abstract

The SIDDHARTA-2 experiment installed at the DA$\mathrm{\Phi}$NE collider of
INFN-LNF performed, for the first time, measurements of high-n transitions in
intermediate mass kaonic atoms during the data taking campaigns of 2021 and
2022. Kaonic carbon, oxygen, nitrogen and aluminium transitions, which occur
in the setup materials, were measured by using the kaons stopped in the
gaseous helium target cell with aluminium frames and Kapton walls, and are
reported in this paper.
These new kaonic atoms measurements add valuable input to the kaonic atoms
transitions data base, which is used as a reference for theories and models of
the low-energy strong interaction between antikaon and nuclei. Moreover, these
results pave the way for future dedicated kaonic atoms measurements through
the whole periodic table and to a new era for the antikaon-nuclei studies at
low energy.

###### Keywords:

Hadron Spectroscopy – Nuclear Physics

###### pacs:

13.75.Jz and 36.10.-k and 14.40.-n and 29.30.-h and 87.64.Gb and 29.40.Wk

##  1 Introduction

Kaonic atoms represent an ideal tool to study the low-energy regime of Quantum
Chromodynamics (QCD) in the strangeness sector, which cannot be described with
a perturbative approach. They enable to directly access the K-N interaction at
threshold, without the need of an extrapolation as in the case of scattering
experiments, since the relative energy between the kaon and the nucleus is
already at the level of few keV. This access is crucial to constrain and unify
the theoretical descriptions of this interaction.
Kaonic atoms have been studied intensely in the 1970s and 1980s, with
measurements spanning over a wide range of elements in the periodic table,
from lithium to uranium Wiegand:1971zz ; Batty:1979a ; Batty:1977 ;
Backenstoss:1973 ; Backenstoss:1974 ; Batty:1979b ; Barnes:1974 ; Wiegand:1974
; Kunselman:1971 ; Kunselman:1974 ; Batty:1981 ; Cheng:1975 . These
measurements serve as database for low-energy antikaon-nuclei studies,
constraining the theoretical descriptions of the K-N interaction potential
Batty:1997 ; Friedman:1994hx . However, the available data are affected by
large experimental uncertainties and several measurements were proven to be at
variance with more recent measurements employing modern detector technology,
while many more kaonic atoms transitions are not yet measured Friedman:1994hx
. For these reasons, starting in the late 1990s, a new era of kaonic atoms
studies was initiated by the KpX experiment at KEK in Japan, followed by the
DEAR and SIDDHARTA experiments at DA$\mathrm{\Phi}$NE, initially motivated by
the so-called “kaonic hydrogen puzzle” Okada:2007ky ; Iwasaki:1997 ;
Bazzi:2011 , which was eventually solved by these experiments. The puzzle
consisted of the fact that measurements performed on kaonic hydrogen in the
1970s and 1980s had found an attractive-type interaction Davies:1979 ;
Izycki:1980 ; Bird:1983 , in contradiction with the analyses of low-energy
scattering data Martin:1981 ; Kim:1965 ; Sakitt:1965 . Similarly, the early
data for the kaonic helium-4 $2p$ level shift (see Wiegand:1971zz ;
Baird:1983ub ) showed a $>5\sigma$ discrepancy with theoretical expectations
Batty:1997 ; Hirenzaki:2000 , the so-called “kaonic helium puzzle”. The puzzle
was resolved in 2007 by the measurement of the E570 experiment at KEK
Okada:2007ky , later confirmed by the SIDDHARTA collaboration
SIDDHARTA:2012rsv . These examples suggest that the current collection of data
from the old experiments is not reliable, and new measurements with higher
accuracy and state-of-the-art detectors are required to provide well grounded
experimental input to theory.

The SIDDHARTA-2 experiment at the DA$\mathrm{\Phi}$NE collider of Laboratori
Nazionali di Frascati (INFN-LNF) in Italy is performing high precision kaonic
atoms measurements. In 2019, the apparatus was installed on the collider with
the main aim to measure the $2p\rightarrow 1s$ transition in kaonic deuterium.
During the optimization phase of the collider and of the setup performances in
2021 and 2022, data were collected both with a reduced setup called
SIDDHARTINO, which differs in the number of used Silicon Drift Detector (SDD)
arrays, and with the full SIDDHARTA-2 setup. To optimize the setup parameters,
various measurements were performed with different 4He target gas densities
SIDDHARTA2:2022 .
The target cell consists of an aluminium frame with Kapton foil walls and
contains gaseous helium, where the kaons produced by $\phi$-decays are
stopped. As a result, multiple kaonic atoms lines are present in the measured
X-ray spectra, besides those of kaonic helium. Measurements of these high-n
transition energies in the same kaonic atom, combined with the already
performed measurements of low-n levels Friedman:1994hx , can contribute to
separate the one-nucleon interaction from multi-nucleon processes
FRIEDMAN2013170 ; Wycech:2020vpl , and lead to a better and more accurate
understanding of the kaon-nuclei interaction at low energy.
In this work, measurements of several intermediate mass kaonic atoms, such as
kaonic carbon, oxygen, nitrogen, and aluminium high-$n$ transition energies
are reported, which also represent the first measurements ever for the
reported transitions.

##  2 The experimental apparatus

DA$\mathrm{\Phi}$NE Milardi:2018sih , a double ring e+e- collider at INFN-LNF,
provides an ideal environment for precision measurements of kaonic atoms,
since it operates at the centre-of-mass energy of 1.02 GeV/$c^{2}$ and hence
produces $\phi$-mesons almost at rest. The charged kaon pairs from
$\phi$-decays, produced with a branching ratio of $49.1\%$, are therefore
emitted in an almost back-to-back topology with momenta of $\sim$ 127 MeV/c
and low momentum spread ($\delta p/p\sim 0.1\%$). The SIDDHARTINO experimental
apparatus, installed at the interaction point (IP) of the DA$\mathrm{\Phi}$NE
collider, was used for the commissioning of the setup from January 2021 to
July 2021, which was followed by the installation of the full SIDDHARTA-2
apparatus and subsequent data taking in April - July 2022.

Figure 1: Top: Schematic of the SIDDHARTINO experimental apparatus installed
at the DA$\mathrm{\Phi}$NE interaction point Bottom: Detailed view of the kaon
trigger system (cyan), the luminosity monitor (red) and the target cell
(yellow) surrounded by SDDs SIDDHARTA2:2022 .

A schematic view of the SIDDHARTINO experimental apparatus is shown in the top
panel of Fig. 1. To suppress the asynchronous background by taking advantage
of the back-to-back K+K- emission, a kaon trigger (KT) system, consisting of
two scintillators with photomultiplier (PMT) read-out, is employed, one
scintillator being placed below the beam pipe and the other one above,
directly in front of the entrance window of the vacuum chamber, as shown in
the bottom panel of Fig. 1. Only triggered events, where two signals are
detected in coincidence by the two scintillators are detected, while the
others are discarded. To monitor the beam quality and background in real-time,
a luminosity monitor built with plastic scintillators and based on J-PET
technology is implemented Skurzok:2020phi ; Moskal:2021medbio ;
Moskal:2021natcom ; Moskal:2021sciad ; niedzwicki:2017actphyspol . The
luminometer takes advantage of the difference in the time of flight from the
interaction point to the detector, between the Minimum Ionizing Particles
(MIPs), originating from e+ and e- lost by the beams, and kaons. Lead
shielding, placed around the vacuum chamber, further suppresses background
originating from beam losses and consequent bremsstrahlung. Inside the vacuum
chamber, the kaonic atoms are produced inside a cylindrical, lightweight
target cell cooled down to $\sim$ 25 K. The target cell has a height of 125 mm
and a diameter of 144 mm, and consists of high-purity aluminium bars together
with $\sim 150$ $\mu$m thick Kapton sidewalls and a 125 $\mu$m thick Kapton
entrance window.
The X-ray detection system surrounds the target cell. It contains large-area
Silicon Drift Detector arrays, developed by Fondazione Bruno Kessler (FKB) in
Trento, Politecnico di Milano, INFN-LNF and the Stefan Meyer Institute (SMI)
in Vienna. Each array features $2\times 4$ SDD cells with an active area of
0.64 cm2 each. Fig. 2 shows one SDD array. For the SIDDHARTINO setup, 8 SDD
arrays and hence 64 read-out channels with a total active area of 41 cm2 are
employed; for SIDDHARTA-2, 48 arrays are used, with 384 read-out channels and
a total active area of 246 cm2. Each SDD signal is amplified by CUBE 5873732 ,
installed on ceramic carrier structures behind the detectors. The signals are
read out and processed by a dedicated ASIC chip called SFERA Quaglia:2016uox ;
Schembari:2016IEE . For in-situ calibration of the SDDs, two X-ray tubes are
implemented and periodic calibration runs are performed by the activation of
high-purity Ti-Cu foils, mounted on the target cell. The fluorescence lines
from these foils are then used for the online calibration of the detectors
Sgaramella:2022 . The SDD system was optimized and characterized in terms of
its spectroscopic and timing response, showing a linearity better than 3 eV in
the energy range 4 - 14 keV, an energy resolution of ($157.8\pm 0.3\pm 0.2$)
eV at 6.4 keV Miliucci:2021wbj  and a long-term stability at the level of 2-3
eV Sgaramella:2022 , which fulfilled the requirements for the challenging
measurements of light kaonic atoms. The final SIDDHARTA-2 setup for the
planned kaonic deuterium measurement will employ additional measures for the
suppression of background in the form of active, multiple-stage veto systems
revmodphys:2019 , which were not yet implemented during the measurements
reported in this work, to enable a direct comparison with the conditions of
past kaonic helium experiments bazzi:2009 .

Figure 2: Silicon drift detector array for the SIDDHARTA-2 experiment with
eight read-out cells indicated by the grey lines. The array is mounted on a
ceramic carrier.

##  3 Data analysis

Data were accumulated with the helium gas target for a total integrated
luminosity of 76 pb-1. The first run took place with SIDDHARTINO during the
beam optimization phase of DA$\mathrm{\Phi}$NE in Spring 2021 and was
dedicated to optimize the collider and setup performances. Following this run,
a new kaonic helium measurement was performed with the complete SIDDHARTA-2
setup, to assess and adjust the experimental apparatus in view of the kaonic
deuterium measurement planned to be performed in 2023.
Fig. 3 shows the inclusive energy spectrum obtained by summing all the
calibrated data collected with the SDDs. In the spectrum, the fluorescence
X-ray transition lines from the activation of the setup materials by the
particles lost from the beams are be clearly seen. The Cu Kα line was produced
in the calibration foils on the target cell, while the Bi lines originate in
the ceramic carriers behind the SDDs.

Figure 3: Inclusive SDD energy spectrum.

Since most of the background produced by DA$\mathrm{\Phi}$NE is due to
particles lost from the beams due to the Touschek effect, and thus is
asynchronous with the K+K- production, it can be drastically reduced by using
the kaon trigger. Only the X-ray SDD signals within a 5 $\mu$s time window
with respect to the trigger signal can pass the selection (Trigger cut),
resulting in a background reduction of a factor 1.3$\cdot 10^{4}$. The 5
$\mu$s time window was optimized for the front-end electronics which acquires
and processes the signals coming from the SDDs.
The MIPs, originating from the beam-beam and beam-gas interactions can produce
accidental trigger signals when passing through the two scintillators of the
kaon trigger in coincidence. The K+K- pairs detected by the kaon trigger are
identified by time of flight (ToF), allowing to distinguish whether a trigger
signal is produced by kaons or MIPs. The time difference between the trigger
signal and the DA$\mathrm{\Phi}$NE radiofrequency (RF), which provides a
reference for each collision, was measured. Fig. 4 shows the correlation of
the mean time for the two scintillators of the kaon trigger. The K+K- pair
events are clearly distinguishable from the MIPs (ToF cut). Since the $\sim$
370 MHz DA$\mathrm{\Phi}$NE RF cannot be handled by the Constant Fraction
Discriminators (CDF) used to process the kaon trigger signals, half the radio
frequency (RF/2) was used as reference. Thus, every kaon trigger signal can be
associated in time with one of the collisions that occurred over a CDF time
period, and consequently the kaons and MIPs appear in two clusters, as shown
in Fig. 4.

Figure 4: Scatter plot of the time differences between the top (KT up) and
bottom (KT down) scintillators of the kaon trigger and half the
DA$\mathrm{\Phi}$NE radio frequency (RF/2). The coincidence events related to
kaons (high intensity) are clearly distinguishable from MIPs (low intensity).
The double kaon and MIP structures are due to the use of DA$\mathrm{\Phi}$NE
RF/2 as time reference (see text).

Finally, the time information provided by the SDDs was used to further reduce
the electromagnetic background. Fig. 5 shows the time difference between the
kaon trigger signal and the X-ray hits on the SDDs for the SIDDHARTINO runs
and the SIDDHARTA-2 runs. The peaks represent the X-ray events in coincidence
with the kaon trigger, while the flat distribution results from uncorrelated
events. The SDDs’ timing resolution, which depends on the temperature
Miliucci:2022lvn , determines the acceptance window (Drift Time cut). For the
SIDDHARTINO run, the SDDs’ temperature was 170 K and the time window was 950
ns (FWHM) (Fig. 5 top). In the SIDDHARTA-2 run, the SDDs’ temperature was
reduced to 130 K, improving the time window to 450 ns (Fig. 5 bottom). Summing
the data of all runs, this resulted in a further reduction of background by a
factor two.

Figure 5: Distribution of the time difference between the SDD X-rays hit and
the trigger signals during the SIDDHARTINO (top), and the SIDDHARTA-2 (bottom)
runs. The dashed lines represent the drift time window cut used to reduce the
background.

In Fig. 6, the X-ray spectrum of the summed data for the SIDDHARTINO and
SIDDHARTA-2 runs after application of the event selections in Table 1 is
shown. The kaonic atoms signals are now clearly visible. The peaks highlighted
in the figure correspond to the X-ray emissions from kaonic atoms formed in
the helium gas and in the components of the target cell.

Table 1: Data selection steps to reduce the background together with the number of events passing each requirement and the associated background rejection factor Selection | Events | Rejection Factor
---|---|---
No cut | $1.6\cdot 10^{9}$ | //
Trigger cut | 118598 | 1.3$\cdot 10^{4}$
ToF cut | 78423 | 1.5
Drift Time cut | 39163 | 2.0

Figure 6: SDD energy spectrum and fit of SIDDHARTA-2 and SIDDHARTINO summed data after background suppression (see text). The kaonic helium signals are seen as well as the kaonic carbon (KC), oxygen (KO), nitrogen (KN) and aluminium (KAl) peaks. Table 2: Kaonic carbon, oxygen, nitrogen and aluminium transition energies from the fit of the data in Fig. 6. Transition | Energy (eV)
---|---
K-C (6$\rightarrow$5) | 5541.7 $\pm$ 3.1 (stat) $\pm$ 2.0 (syst)
K-C (7$\rightarrow$5) | 8890.0 $\pm$ 13.0 (stat) $\pm$ 2.0 (syst)
K-C (5$\rightarrow$4) | 10216.6 $\pm$ 1.8 (stat) $\pm$ 3.0 (syst)
K-C (6$\rightarrow$4) | 15760.3 $\pm$ 4.7 (stat) $\pm$ 12.0 (syst)
K-O (7$\rightarrow$6) | 6016.0 $\pm$ 60.0 (stat) $\pm$ 2.0 (syst)
K-O (6$\rightarrow$5) | 9968.1 $\pm$ 6.9 (stat) $\pm$ 2.0 (syst)
K-N (6$\rightarrow$5) | 7577.0 $\pm$ 17.0 (stat) $\pm$ 2.0 (syst)
K-N (5$\rightarrow$4) | 14010.6 $\pm$ 8.2 (stat) $\pm$ 9.0 (syst)
K-Al (8$\rightarrow$7) | 10441.0 $\pm$ 8.5 (stat) $\pm$ 3.0 (syst)
K-Al (7$\rightarrow$6) | 16083.4 $\pm$ 3.8 (stat) $\pm$ 12.0 (syst)

The energy of each kaonic atoms transition was obtained from a fit of the
spectrum. The SDDs’ energy response for every transition line reported is
described by the convolution of a Gaussian with an exponential function to
reproduce the incomplete charge collection and electron-hole recombination
effect Miliucci:2022lvn , except for the kaonic helium lines, which were
fitted with Voigt functions to account for the intrinsic linewidth of the
transition due to the possible effect of the strong interaction. For the other
high-n transition lines, the widening induced by strong interaction is known
to be negligible Friedman:1994hx .
In addition, two exponential plus a constant function are used to describe the
background shape. The final fit function properly reproduces the data
distribution, with a $\chi^{2}$/ndf = 1.15 in the energy range from 5 keV to
17 keV.
Several high-$n$ transition energies in intermediate mass kaonic atoms, such
as kaonic carbon, oxygen, nitrogen and aluminium are measured here for the
first time. The discussion about kaonic helium transition energies goes beyond
the scope of this article and for the SIDDHARTINO data is reported in
SIDDHARTA2:2022 . The kaonic carbon, oxygen and nitrogen transitions are the
result of kaons stopped in the Kapton walls, whereas the kaonic aluminium
transitions were produced by kaons stopped in the top and bottom frames of the
target cell. The final results for the kaonic C, O, N and Al transitions are
shown in Table 2. The associated systematic uncertainties take into account
the linearity and stability of the SDDs, as well as the effects produced by
the energy calibration procedure, following the method described in
Sgaramella:2022 ; Miliucci:2021wbj . Thus, various transitions are measured
with a statistical precision better than 10 eV, and for the majority of the
measurements the systematic uncertainty is below 3 eV.

##  4 Conclusions

The SIDDHARTA-2 experiment performed high precision measurements of a series
of intermediate mass kaonic atoms transitions, which also represent the first
measurements ever for the reported transitions. Kaonic carbon, oxygen,
nitrogen and aluminium X-ray transitions in the 5 - 16 keV energy range were
measured during the 2021 and 2022 data taking campaigns, by using kaons
stopped in the setup materials.
These new data enrich the kaonic atoms transitions database, which is used as
input and as test-bed for theories and models of kaon-nuclei interactions at
low energies, a field which is still far from being fully understood. The new
data added by SIDDHARTA-2 can stimulate a revival of the theoretical activity
in the field, towards a better understanding of the strong interaction with
strangeness and of the role played by multi-nucleon absorption processes
Friedman:1994hx ; Wycech:2020vpl , with implications extending from particle
and nuclear physics to astrophysics revmodphys:2019 ; Merafina:2020ffb ;
Curceanu:2020kkg ; Drago:2019tbs .
The series of measurements reported in this paper also show the potential of
DA$\mathrm{\Phi}$NE and SIDDHARTA-2 technologies to address high precision
kaonic atoms measurements along the whole periodic table, within a future
program which was already put forward by the scientific community
Curceanu:2021oqj ; Curceanu:2021jgq .

## Acknowledgments

We thank C. Capoccia from LNF-INFN and H. Schneider, L. Stohwasser, and D.
Pristauz- Telsnigg from Stefan Meyer-Institut for their fundamental
contribution in designing and building the SIDDHARTA-2 setup. We thank as well
the DA$\mathrm{\Phi}$NE staff for the excellent working conditions and
permanent support. We acknowledge support from the SciMat and qLife Priority
Research Areas budget under the program Excellence Initiative—Research
University at the Jagiellonian University. Part of this work was supported by
the Austrian Science Fund (FWF): Doctoral program No. W1252-N27, as well as
[P24756-N20 and P33037-N]; the EXOTICA project of the Minstero degli Affari
Esteri e della Cooperazione Internazionale, PO21MO03; the Croatian Science
Foundation under the project IP-2018-01-8570; the EU STRONG-2020 project
(Grant Agreement No. 824093; the EU Horizon 2020 project under the MSCA (Grant
Agreement 754496); the Japan Society for the Promotion of Science JSPS KAKENHI
Grant No. JP18H05402; the Polish Ministry of Science and Higher Education
grant No. 7150/E-338/M/2018 and the Polish National Agency for Academic
Exchange (grant no PPN/BIT/2021/1/00037).

## References

  * (1) C. E. Wiegand, R. H. Pehl, Measurement of Kaonic X Rays from He-4, Phys. Rev. Lett. 27 (1971) 1410–1412.  doi:10.1103/PhysRevLett.27.1410.
  * (2) C. Batty, et al., Measurement of kaonic and pionic X-rays from liquid helium, Nucl. Phys. A 326 (1979) 455–462.  doi:10.1016/0375-9474(79)90403-2.
  * (3) C. Batty, et al., Measurement of kaonic X-rays from Li, LiH AND Be, Nucl. Phys. A 282 (1977) 487–492.  doi:10.1016/0375-9474(77)90072-0.
  * (4) G. Backenstoss, et al., K- mass and K- polarizability from kaonic atoms, Phys. Lett. B 43 (1973) 431–436.  doi:10.1016/0370-2693(73)90391-2.
  * (5) G. Backenstoss, et al., Intensities and strong interaction attenuation of kaonic X-rays, Nucl. Phys. B 73 (1974) 189–201.  doi:10.1016/0550-3213(74)90014-5.
  * (6) C. Batty, et al., Measurement of strong interaction effects in kaonic atoms, Nucl. Phys. A 329 (1979) 407–428.  doi:10.1016/0375-9474(79)90384-1.
  * (7) P. Barnes, et al., Measurement of kaonic X-rays from Al, Si, Ni and Cu, Nucl. Phys. A 231 (1974) 477–492.  doi:10.1016/0375-9474(74)90511-9.
  * (8) C. Wiegand, G. Godfrey, Measurements of x rays and $\gamma$ rays from stopped kaons, Phys. Rev. A 9 (1974) 2282.  doi:10.1103/PhysRevA.9.2282.
  * (9) R. Kunselman, Kaon mass measurement from kaonic atom x-ray energies; the 4f-3d kaonic transition in chlorine, Phys. Lett. B 34 (1971) 485–487.  doi:10.1016/0370-2693(71)90660-5.
  * (10) R. Kunselman, Negative - kaon mass, Phys. Rev. C 9 (1974) 2469.  doi:10.1103/PhysRevC.9.2469.
  * (11) C. Batty, et al., Nuclear quadrupole deformation effects on pionic and kaonic X-rays, Nucl. Phys. A 355 (1981) 383–402.  doi:10.1016/0375-9474(81)90534-0.
  * (12) S. Cheng, et al., K- mass from kaonic atoms, Nucl. Phys. A 254 (1975) 381–395.  doi:10.1016/0375-9474(75)90224-9.
  * (13) C. Batty, E. Friedman, A. Gal, Strong interaction physics from hadronic atoms, Physics Reports 287 (1997) 385–445.  doi:10.1016/S0370-1573(97)00011-2.
  * (14) E. Friedman, A. Gal, C. J. Batty, Density dependent K- nuclear optical potentials from kaonic atoms, Nucl. Phys. A 579 (1994) 518–538.  doi:10.1016/0375-9474(94)90921-0.
  * (15) S. Okada, et al., Precision measurement of the 3d —$>$ 2p x-ray energy in kaonic He-4, Phys. Lett. B 653 (2007) 387–391.  doi:10.1016/j.physletb.2007.08.032.
  * (16) M. Iwasaki, et al., Observation of Kaonic Hydrogen ${K}_{\alpha}$ X Rays, Phys. Rev. Lett. 78 (1997) 3067–3069.  doi:10.1103/PhysRevLett.78.3067.
  * (17) M. Bazzi, et al., A new measurement of kaonic hydrogen X-rays, Phys. Lett. B 704 (3) (2011) 113.  doi:10.1016/j.physletb.2011.09.011.
  * (18) J. Davies, et al., Observation of kaonic hydrogen atom X-rays, Phys. Lett. B 83 (1979) 55–58.  doi:10.1016/0370-2693(79)90887-6.
  * (19) M. Izycki, et al., Results of the search forK-series X-rays from kaonic hydrogen, Z Physik A 297 (1980) 11––15.  doi:10.1007/BF01414238.
  * (20) P. Bird, et al., Kaonic Hydrogen atom X-rays, Nucl. Phys. A 404 (1983) 482–494.  doi:10.1016/0375-9474(83)90272-5.
  * (21) A. Martin, Kaon-nucleon parameters, Nucl. Phys. B 179 (1981) 33–48.  doi:10.1016/0550-3213(81)90247-9.
  * (22) J. Kim, Low-Energy ${K}^{-}-p$ Interaction and Interpretation of the 1405-MeV $Y_{0}{}^{*}$ Resonance as a $\overline{K}N$ Bound State, Phys. Rev. Lett. 14 (1965) 29–30.  doi:10.1103/PhysRevLett.14.29.
  * (23) M. Sakitt, et al., Low-Energy ${K}^{-}$-Meson Interactions in Hydrogen, Phys. Rev. 139 (1965) B719–B728.  doi:10.1103/PhysRev.139.B719.
  * (24) S. Baird, et al., Measurements on Exotic Atoms of Helium, Nucl. Phys. A 392 (1983) 297–310.  doi:10.1016/0375-9474(83)90127-6.
  * (25) S. Hirenzaki, et al., Chiral unitary model for the kaonic atom, Phys. Rev. C 61 (2000) 055205.  doi:10.1103/PhysRevC.61.055205.
  * (26) M. Bazzi, et al., Measurements of the strong-interaction widths of the kaonic 3He and 4He 2p levels, Phys. Lett. B 714 (2012) 40–43.  doi:10.1016/j.physletb.2012.06.071.
  * (27) D. Sirghi, et al., A new kaonic helium measurement in gas by SIDDHARTINO at the DA$\Phi$NE collider, J. Phys. G: Nucl. Part. Phys. 49 (2022) 055106.  doi:10.1088/1361-6471/ac5dac.
  * (28) E. Friedman, S. Okada, Feasibility guidelines for kaonic atom experiments with ultra-high-resolution x-ray spectrometry, Nuclear Physics A 915 (2013) 170–178.  doi:https://doi.org/10.1016/j.nuclphysa.2013.07.005.
  * (29) S. Wycech, B. Loiseau, An Advantage of ”Upper Levels”, Acta Phys. Polon. B 51 (2020) 109.  doi:10.5506/APhysPolB.51.109.
  * (30) C. Milardi, et al., Preparation Activity for the Siddharta-2 Run at DA$\Phi$NE, in: 9th International Particle Accelerator Conference, IPAC2018, Vancouver BC Canada, 2018.  doi:10.18429/JACoW-IPAC2018-MOPMF088.
  * (31) M. Skurzok, et al., Characterization of the SIDDHARTA-2 luminosity monitor, JINST 15 (10) (2020) P10010.  doi:10.1088/1748-0221/15/10/P10010.
  * (32) P. Moskal, et al., Simulating NEMA characteristics of the modular total-body J-PET scanner—an economic total-body PET from plastic scintillators, Phys. Med. Biol. 66 (2021) 175015.  doi:10.1088/1361-6560/ac16bd.
  * (33) Moskal, P., Gajos, A., Mohammed, M. and others, Testing CPT symmetry in ortho-positronium decays with positronium annihilation tomography, Nature Communications 12 (2021) 5658.  doi:10.1038/s41467-021-25905-9.
  * (34) P. Moskal, et al., Positronium imaging with the novel multiphoton PET scanner, Science Advances 7 (2021) eabh4394.  doi:10.1126/sciadv.abh4394.
  * (35) S. Niedźwiecki, et al., J-PET: A New Technology for the Whole-body PET Imaging, Acta Phys. Pol. B 48 (2017) 1567.  doi:10.1126/sciadv.abh4394.
  * (36) L. Bombelli, C. Fiorini, T. Frizzi, R. Nava, A. Greppi, A. Longoni, Low-noise cmos charge preamplifier for x-ray spectroscopy detectors, in: IEEE Nuclear Science Symposuim Medical Imaging Conference, 2010, pp. 135–138.  doi:10.1109/NSSMIC.2010.5873732.
  * (37) R. Quaglia, F. Schembari, G. Bellotti, A. D. Butt, C. Fiorini, L. Bombelli, G. Giacomini, F. Ficorella, C. Piemonte, N. Zorzi, Development of arrays of Silicon Drift Detectors and readout ASIC for the SIDDHARTA experiment, Nucl. Instrum. Meth. A 824 (2016) 449–451.  doi:10.1016/j.nima.2015.08.079.
  * (38) F. Schembari, et al., SFERA: An Integrated Circuit for the Readout of X and $\gamma$-Ray Detectors., IEEE Trans. Nucl. Sci. 63 (2016) 1797.  doi:10.1109/TNS.2016.2565200.
  * (39) F. Sgaramella, et al., The SIDDHARTA-2 calibration method for high precision kaonic atoms x-ray spectroscopy measurements, Physica Scripta 97 (11) (2022) 114002\.  doi:10.1088/1402-4896/ac95da.
  * (40) M. Miliucci, et al., Silicon drift detectors system for high-precision light kaonic atoms spectroscopy, Measur. Sci. Tech. 32 (9) (2021) 095501.  doi:10.1088/1361-6501/abeea9.
  * (41) C. Curceanu, C. Guaraldo, M. Iliescu, M. Cargnelli, R. Hayano, J. Marton, J. Zmeskal, T. Ishiwatari, M. Iwasaki, S. Okada, D. L. Sirghi, H. Tatsuno, The modern era of light kaonic atom experiments, Rev. Mod. Phys. 91 (2019) 025006\.  doi:10.1103/RevModPhys.91.025006.
  * (42) Bazzi, M. and others (SIDDHARTA collaboration), Kaonic helium-4 X-ray measurement in SIDDHARTA, Phys. Lett. B 681 (2009) 310–314.  doi:10.1016/j.physletb.2009.10.052.
  * (43) M. Miliucci, et al., Large area silicon drift detectors system for high precision timed x-ray spectroscopy, Measur. Sci. Tech. 33 (9) (2022) 095502.  doi:10.1088/1361-6501/ac777a.
  * (44) M. Merafina, F. G. Saturni, C. Curceanu, R. Del Grande, K. Piscicchia, Self-gravitating strange dark matter halos around galaxies, Phys. Rev. D 102 (8) (2020) 083015.  arXiv:2007.03024, doi:10.1103/PhysRevD.102.083015.
  * (45) C. Curceanu, et al., Kaonic Atoms to Investigate Global Symmetry Breaking, Symmetry 12 (4) (2020) 547.  doi:10.3390/sym12040547.
  * (46) A. Drago, M. Moretti, G. Pagliara, The equation of state of dense matter: Stiff, soft, or both?, Astron. Nachr. 340 (1-3) (2019) 189–193.  doi:10.1002/asna.201913586.
  * (47) C. Curceanu, et al., Kaonic Atoms Measurements at DA$\Phi$NE: SIDDHARTA-2 and Future Perspectives, Few Body Syst. 62 (4) (2021) 83.  doi:10.1007/s00601-021-01666-5.
  * (48) C. Curceanu, et al., Fundamental physics at the strangeness frontier at DA$\Phi$NE. Outline of a proposal for future measurements (4 2021).  arXiv:2104.06076.


# Quantum Computing Education for Computer Science Students: Bridging the Gap
with Layered Learning and Intuitive Analogies
††thanks: This research was conducted as part of the QCloud QuantumEd project
led by Munster Technological University and funded by the EOSC Future project
$INFRAEOSC-03-2020$ \- Grant Agreement Number $101017536$. This publication
was supported in part by the CyberSkills HCI Pillar 3 Project 18364682. Dr
Murray and Dr Mjeda acknowledge support from Science Foundation Ireland co-
funded from the European Regional Development Fund under Grant
$13/RC/2077\\_P2$ and $13/RC/2094_{P}2$ respectively.

1st Anila Mjeda  Cyber Skills and Department of Computer Science
Munster Technological University
Cork, Ireland
<EMAIL_ADDRESS>2nd Hazel Murray  Cyber Skills and Department of Computer
Science
Munster Technological University
Cork, Ireland
<EMAIL_ADDRESS>
###### Abstract

Quantum computing presents a transformative potential for the world of
computing. However, integrating this technology into the curriculum for
computer science students who lack prior exposure to quantum mechanics and
advanced mathematics remains a challenging task. This paper proposes a
scaffolded learning approach aimed at equipping computer science students with
essential quantum principles. By introducing foundational quantum concepts
through relatable analogies and a layered learning approach based on classical
computation, this approach seeks to bridge the gap between classical and
quantum computing. This differs from previous approaches which build quantum
computing fundamentals from the prerequisite of linear algebra and
mathematics. The paper offers a considered set of intuitive analogies for
foundation quantum concepts including entanglement, superposition, quantum
data structures and quantum algorithms. These analogies coupled with a
computing-based layered learning approach, lay the groundwork for a
comprehensive teaching methodology tailored for undergraduate third level
computer science students.

###### Index Terms:

Quantum computing, Computer science, education, curriculum development,
quantum, pedagogy, teaching, layered learning, analogies, scaffolding.

##  I Introduction

Quantum computing is an emerging field with the potential to revolutionize the
world of computing. Its rapid advancement into a mainstream and commercial
technology [1, 2] makes it essential to equip computer science students with
the skills and knowledge to harness its potential. However, the typical
computer science student has no prior knowledge of quantum mechanics and
students often struggle to grasp the quantum computing concepts which are
fundamentally different from classical computing. Hence, it is crucial to
develop teaching and learning approaches tailored to teach quantum computing
concepts to computer science students.

This paper proposes a layered learning approach, emphasizing the grounding of
quantum concepts in classical computation and intuitive analogies. We posit
that to teach quantum computing effectively to computer science students
without a background in quantum mechanics, we need a layered (scaffolded)
approach that builds on the existing knowledge of classical computing and
underpins the quantum computing upskilling with foundational knowledge. In
this paper, a curriculum introducing fundamental quantum computing topics is
outlined which is scaffolded from classical computing concepts. This differs
from existing approaches which either require advanced pre-requisites in
physics or ground their foundations in linear algebra [3].

Analogies are a powerful teaching methodology for conveying details of complex
concepts. They are particularly valuable to quantum mechanics where concepts
are often at odds with our classical interpretations. However, Didics [4]
found that educators often rely on spur-of-the-moment creation of analogies
which, at times, may not accurately represent the chosen concept. Therefore,
in the second part of this paper we outline an initial collection of analogies
which can be used to explain core quantum topics including entanglement,
superposition, quantum data structures and quantum cryptography algorithms.
This approach ensures a cohesive and accessible understanding of quantum
concepts, rather than relying on the creation of ad-hoc analogies during
classes.

Overall, the goal of this paper is to assist educators in their creation and
delivery of accessible and informative quantum education to computer science
students. Bridging the gap between quantum computing and tradition computer
science allows computing students to contribute to the next generation of the
computation which will revolutionise how we interpret and analyse data,
consider cybersecurity and understand the world around us.

The paper is laid out as follows: Section II provides a background on existing
quantum education literature. Section III describes current education
approaches including a review of existing international quantum education
curriculum content. Section IV outlines our suggested layered learning
curriculum for introducing quantum foundations to computer science students.
In each of the layers, we build on classical foundations and describe useful
analogies to convey the increasingly complex topics. Table I provides
intuitive descriptions of two pivotal quantum algorithms. Tables II–V include
a collection of tailored analogies for the explanation of quantum data
structures, superposition, quantum gates and entanglement respectively. The
paper finishes with a description of future work and conclusions in Section V.

##  II Literature review

Higher education for careers in quantum industry has predominantly been the
domain of PhD programs in physics departments, typically with a focus on
proof-of-principle quantum experiments [5]. Engaging a diverse range of degree
subjects and levels can significantly expand the talent pool and enhance
participation and retention in the quantum workforce. This being especially
poignant as we seek to transition into marketable quantum products that
address real-world challenges [6, 5]. Computer science education spaces appear
to be the natural milieu where to invest in the development of the current and
future workforce [7, 3] but the current reality presents significant
challenges. The conceptual and mathematical foundations established in physics
courses tend to serve as the basis of quantum computing [5, 8] while in many
cases software students can enter and finish a computer science degree with no
previous physics education and a limited mathematics background. This gap in
foundational knowledge can create difficulties in grasping the complex
terminologies and approaches commonly used in existing teaching resources and
scientific papers, acting as a substantial barrier [9, 8, 10].

In response to these challenges, introductory lectures on quantum computing
have been developed by several researchers [11, 12, 7, 9, 13, 14, 8],
including the lectures from CERN [9] which were underpinned by the principles
of minimizing the prerequisites and emphasizing the practical implementation
of any quantum protocols and algorithms discussed in the course. When teaching
a quantum computing course without prerequisites in physics or mathematics,
Temporão et al. [3] observed that a significant portion of the curriculum
focused on fundamental Linear Algebra and essential concepts of quantum
physics. In an effort to make quantum computing accessible to a wider audience
by eliminating prerequisites to join the course, [8] employ a visual
representation alongside a spiral curriculum. In their visual representation
[8] they replace bra-ket notation with the analogy of a white ball
representing $\ket{0}$, and a black ball representing $\ket{1}$ and only after
the core concepts are well understood do they introduce students to the
mathematical bra-ket notation. Carrascal et al. [7] propose a teaching roadmap
for quantum computing that begins with an understanding of how information is
represented in classical computers, emphasizing concepts such as probability,
wavefunctions, and measurement. The curriculum then transitions to testing
quantum gates, proceeds to quantum programming, and concludes with an
exploration of established quantum algorithms [7]. In a somewhat outlier
approach to teaching quantum science, [15] advocate an artistic methodology,
incorporating gamification and theatre projects as engaging strategies to
render quantum science more accessible to the general public.

When exploring effective teaching strategies for quantum computing to software
students, the intuition is to first examine the methods employed in teaching
complex quantum concepts to physics students [16]. To clarify the often
counter-intuitive phenomena of quantum physics, educators frequently rely on
simplifying and idealizing complex processes, incorporating thought
experiments, analogies, and various representations. Particularly in the
context of quantum physics, where the quantum phenomena do not align with our
macroscopic experiences and understanding, the use of analogies becomes
crucial [16].

Didics [4] identifies five primary aims for employing analogies in teaching
quantum theory in physics: introducing a new topic, clarifying taught
concepts, capturing students’ attention, increasing class participation, and
comparing classical and quantum physics. Using analogies to clarify concepts
accounts for half of all analogies used. Interestingly, there was no
systematic use of analogies, with 90% of them being developed spontaneously
[4]. Furthermore, the study reports that the analogies used often rely heavily
on specific shared cultural backgrounds, such as national sayings and proverbs
[4].

Drawing parallels with the teaching of complex quantum concepts in physics,
analogies are also employed as a valuable tool in quantum education [17]. For
instance, the concept of superposition, is often taught using the coin toss
analogy, where a coin in mid-air represents a superposition of heads and
tails. Depending on the students’ backgrounds, be it in physics, mathematics,
or engineering, the analogy is then complemented by connecting it to concepts
like photon or electron spins, by demonstrations such as the Stern-Gerlach
experiment, use of mathematical-symbolic representations, such as vectors, and
graphical representations such as the Bloch sphere and unit circles [17].

Informed by the existing literature, we posit in this paper the importance of
developing effective pedagogical strategies that cater to students with
diverse backgrounds, particularly those lacking prerequisites in physics and
mathematics. This perspective has guided our work in adopting strategies that
minimize prerequisites and emphasize practical implementations to facilitate
understanding and engagement. Additionally, this has informed our focus on the
development of analogies to teach complex quantum concepts rather than relying
on analogies developed on the spur of the moment during classes which tend to
not offer a consistent and accessible approach to the understanding of quantum
phenomena.

##  III Teaching Approaches

###  III-A Current Approaches

In existing quantum computing education the status quo is to teach students
with a physics background some computing topics. However, as quantum computing
evolves into a main stream technology with practical and commercial
applications, the importance grows for traditional computer science students
to gain exposure to this technology.

The authors reviewed existing modules and programmes which aim to provide an
introduction to quantum computing for computer science students from eight
different universities [18, 19, 20, 21, 22, 23, 24, 25].

The typical layout for these modules begins with a motivation for the study of
quantum computing followed by a review of linear algebra and complex vector
spaces in the context of quantum information. This introduction is then
followed by the core quantum concepts including quantum bits, quantum gates
and quantum properties (entanglement, superposition, measurement). It then
typically continues with an analysis of quantum algorithms (such as Deutsch-
Jozsa algorithm, Grover’s algorithm, Shor’s algorithm). Finally most modules
finish with quantum communication and quantum cryptography applications.

In each of these modules, the goal was to teach quantum computing without a
requirement for a background in quantum mechanics. However, instead of a basis
in physics, all of these module except [25] based the fundamentals on linear
algebra. In fact in 2022 Temporāo et al. [3] proposed and tested an
introductory quantum computing course where every concept is based on applied
linear algebra. This layered learning approach demonstrated that quantum
computing can be taught without a prerequisite of physics or advanced quantum
mechanism.

While this is an effective layered approach, many computer science students
struggle with mathematics and mathematics anxiety [26]. Women in particular
are affected by the impact maths anxiety has on their vocational interests and
an effort should be made to avoid further excluding women from this emerging
domain [27]. We postulate that quantum computing can also be taught using
computer-science domain specific fundamentals and real world analogies. While
it is beneficial, similar to classical computing, students can have strong
computing skills irrespective of mathematics skills. The development of
quantum computation will require expertise from multiple fields including
computer science, engineering, mathematics and physics. Computer scientists
can bring expertise to the quantum computing field that augment and complement
the skill sets mathematicians and physicists bring.

From the courses we observed, many courses include strict prerequisites: “A
strong undergraduate background in linear algebra, discrete probability, and
theory of computation. No background in physics is required.” [24]. The
current positioning of quantum computing education for computer scientists as
fundamentally of mathematical basis has the potential to exclude a large
cohort of computer scientists without this mathematical inclination [28].

In this paper we discuss the Layered Learning aspects of our proposed approach
with a particular focus on classical computing and analogies as the basis for
explaining fundamental concepts rather than mathematics or physics.

###  III-B “Layered Learning” and Use of Analogies in Education

Traditionally, scaffolding in learning is recognized as a supportive method
aimed at fostering student learning. It is a layered learning approach that
involves providing structured support to learners as they progress toward
mastering a new concept or skill. Similar to the support structure used in
construction, scaffolding helps students by breaking down complex tasks into
smaller, more manageable steps.

Initially introduced by Wood et al. [29] in their exploration of tutoring’s
influence on children’s problem-solving skills, the concept of scaffolding has
been extensively examined and extended by numerous researchers [30, 31, 32,
33, 34, 35, 36]. Even though the majority of empirical studies investigating
scaffolding tend to be limited in scale and are often characterized by
descriptive approaches, research in this area has contributed valuable
insights and findings. Foremost, the findings indicate that scaffolding is an
effective approach for fostering the students’ metacognitive and cognitive
activities [36]. For a theoretical treatment of the scaffolding as a
technique, the reader is directed to [30, 35].

In this paper, we present a scaffolded approach to teaching quantum computing
concepts. Henceforth, we will refer to scaffolding as the ’Layered Learning
Approach.’ This adjustment aims to enhance cross-disciplinary understanding
and simplify explanations.

The use of analogies in STEM education has a long history and a strong
theoretical and empirical basis. Analogies have been recognized as an
essential feature of scientific reasoning and discovery, as scientists often
use analogies to generate hypotheses, test predictions, and communicate
findings [37, 38, 39, 40, 41]. To give one example, Rutherford’s analogy of
imagining the atom as a miniature solar system[42] was so effective that it
remains the dominant imagery that comes to mind when thinking of or
illustrating an atom. The power of analogies stands in relating complex
concepts to familiar situations or phenomena and their use can foster the
development of higher order thinking skills, such as analysis, synthesis,
evaluation, and creativity [41]. For a systematic mapping study of use of
analogies in science education, the reader is directed to [43].

The advantages are contingent upon effective analogies, while, spur-of-the-
moment, unplanned analogies, even if well-intentioned, can be misconstrued and
prove misleading [41]. Therefore, we propose using a series of domain-specific
analogies to facilitate the learning of key concepts in quantum computing
among computer science students.

We propose using these analogies within a Layered Learning Approach that
begins with revisiting classical computing foundations, emphasizing strong
knowledge of algorithms, data structures, and complexity theory, followed by
introducing fundamental quantum principles like qubits, quantum gates,
superposition, and entanglement through simplified explanations and analogies
from everyday life.

At a glance:

  * •

Layer 1: Classical Foundations. Here we start by reinforcing classical
computing concepts to ensure students have a strong understanding of
algorithms, data structures, and complexity theory. This also where we
emphasize the classical-quantum hybrid nature of quantum computing.

  * •

Layer 2: Quantum Foundations. The students are introduced to fundamental
quantum principles such as qubits, quantum gates, superposition, and
entanglement. Simple analogies and where possible visual aids are used to make
these abstract concepts more accessible.

##  IV Layered Learning

###  IV-A Layer 1: Classical Foundations

In this foundational layer, we lay the groundwork for understanding quantum
computing by reinforcing classical computing concepts. This approach serves as
a bridge between the students’ existing knowledge and the world of quantum
computation. It focuses on ensuring that students have a robust understanding
especially of those aspects of classical computing that are vital for
comprehending the nuances and potential of quantum computing.

####  IV-A1 Algorithms

We begin by revisiting and reinforcing core classical algorithmic concepts
which are the backbone of computing. Students delve into algorithms for
searching, sorting, and problem-solving. For instance, we explore classical
sorting algorithms like quicksort and mergesort. Students also engage in
interactive discussions on fundamental algorithms that underpin different
applications. Case studies relying on challenges such as route planning and
network optimization (e.g. the classic problem of finding the shortest path in
a graph), are used. Classical algorithms such as Dijkstra’s or Bellman-Ford
are explored, laying the groundwork for understanding how quantum algorithms
can offer enhancements in areas like optimization.

####  IV-A2 Complexity Theory

Complexity theory examines the efficiency and computational limits of
algorithms. Students delve into concepts like time and space complexity. This
knowledge is essential for evaluating the performance of classical algorithms
and understanding the potential improvements quantum algorithms can offer. For
example, the students are guided through problems that are hard to solve
efficiently by exploring the concept of NP-completeness in classical
complexity theory. This is used to contextualise the benefits of quantum
algorithms, such as say Shor’s algorithm [44] for factorization.

Figure 1:  Grover’s algorithm amplifies the wave function at the point we are
searching for.

Overall, the introduction of quantum algorithms and quantum programming in
this layer, serves to contextualise how quantum computing can significantly
enhance functionality in various application domains, from searching databases
more efficiently to breaking classical cryptographic systems. In this layer,
we transition from the fundamental principles of quantum computing to the
practical application of quantum algorithms. We introduce students to quantum
algorithms such as Grover’s [45] and Shor’s [44] algorithms (see Table I), and
we emphasize the importance of hands-on coding. We recommend students are
initially introduced to quantum programming via python for example using
Qiskit or Ket, a language they would already be familiar with, before other
languages such as Silq and Q# are introduced.

TABLE I: Examples Illustrating the Benefits of Quantum Algorithms Name of Algorithm |  Example Illustrating Benefits
---|---
Grover’s Algorithm |  In the classical world, searching for a specific item in an unsorted database of $N$ items would require $O(N)$ time. Grover’s algorithm [45] offers a quadratic speedup, performing the task in approximately $\sqrt{N}$ steps. To do this, it represents the list we are searching through as a wave-function where the different points on the function are the different items we are searching through. Grover’s algorithm operates on the whole list at the same time and forces the amplitude at the point we are looking for to double each time it operates on the wave-function (see Figure 1). This is repeated over and over again until the amplitude of the point we want is significantly higher than all the other points’ amplitudes. This means that when we measure, the output is more likely to be the point we are searching for than any other point. An analogy for this process is to imagine busy librarians who know they need to find a certain book in an unsorted library. One of the librarians is running around helping another customer, when they pass a book that looks like the book they were searching for. Not having time to stop, they quickly place a post-it note in front of the book to remind themselves to come back later to check if its right. If lots of different librarians all do this as they run around the library floor, then eventually a librarian who comes in and just picks the book with all the post-it notes in front of it has a high probability of having chosen the right book. If classically it would have taken 1000 searches for the librarians to find the right book, Grover’s algorithm allows them to find it with just 32 post-it notes.
Shor’s Algorithm |  Factoring large numbers into their prime components can be slow with classical algorithms, especially for numbers with many digits. Shor’s algorithm [44], however, factors large numbers exponentially faster, with significant implications for cryptography. Consider a large number like 143; classically factoring it involves trying various divisors until we guess the correct prime factors, 11 and 13. Instead of this trial and error approach, Shor’s algorithm can efficiently find the prime factors without needing to try each option one by one. Shor’s algorithm takes advantage of superposition to simultaneously complete a computation for all our guesses at the same time. This superposition can be again represented by a wave-function where each point on the wave-function is a different guess. Shor’s algorithm allows us to simultaneously apply operations across the whole wave-function to complete computations that will reveal whether our guess is a factor or not. However, only one of the computations will result in the correct answer. Shor’s algorithm ensures that all the wrong answers will destructively interfere with each other so that only the correct answer is left which we can then measure and read out. It would take a classical computer 300 trillion years to break a RSA-2048 bit encryption key, whereas with Shor’s algorithm it could take a quantum computer 10 seconds.

####  IV-A3 Data Structures

We revisit classical data structures such as arrays, linked lists, and trees
and analyse their respective trade-offs when used in classical computing.
Subsequently the students are introduced to the ‘equivalent’ quantum data
structures and their computational advantages are discussed (see Table II).

TABLE II: Comparison of Data Structures Classical Data Structure |  Quantum Data Structure |  Computational Advantages in Quantum Computing
---|---|---
Array |  Quantum Array |  1\. Superposition: Quantum arrays can store multiple values simultaneously in superposition, allowing for parallel processing and speeding up certain search and manipulation tasks. Superposition in quantum arrays is like having a number of different answers to a question all at once. It means the array can hold and work with multiple possibilities together, which helps quantum computers quickly explore many options at the same time. 2. Quantum Fourier Transform: Quantum arrays can leverage the Quantum Fourier Transform for efficient operations such as pattern matching and signal processing. The Quantum Fourier Transform (QFT) is a mathematical operation in quantum computing that helps analyze the frequencies or patterns within a set of data. It’s like a ”magic” tool that lets quantum computers quickly figure out the hidden patterns in a list of numbers. Imagine you have a list of numbers, and some of these numbers follow a particular pattern. For example, you might have a list of numbers that represent sound waves, and you want to find the main frequency (pitch) of the sound. In classical computing, you would need to perform a series of mathematical operations to figure out this frequency. It might take a lot of time and calculations. Now, with the Quantum Fourier Transform, a quantum computer can do this much faster. It takes your list of numbers and ”magically” extracts the important information about the pattern, like the main frequency of the sound, in a single step. It’s like having a super-efficient way to find patterns in data.
Linked List |  Quantum Linked List |  1\. Entanglement: Quantum linked lists can exploit entanglement to establish non-local connections between elements, potentially enabling faster traversal and search operations. 2. Coherent Superposition: Quantum linked lists can be in a coherent superposition state, offering opportunities for solving problems like database searching with greater efficiency.
Tree |  Quantum Tree |  1\. Quantum Parallelism: Quantum trees can perform simultaneous operations on multiple branches through superposition, making tree-based algorithms more efficient, such as searching and decision trees. 2. Grover’s Algorithm: Quantum trees can take advantage of algorithms like Grover’s to accelerate search operations and improve performance in various applications.

####  IV-A4 Classical-Quantum Hybrid Nature

We emphasize how quantum computing complements classical computing rather than
replacing it entirely. Exploring the hybrid nature of quantum computing,
students learn how classical and quantum components can collaborate to solve
complex problems more effectively. They’re encouraged to draw parallels from
daily life, like hybrid vehicles using both gasoline and electricity for
efficiency in areas lacking complete electric infrastructure. Similarly, in
cooking, chefs optimize their process by combining conventional stove-tops
with modern tools like sous vide cookers, improving both experience and
results. This fusion mirrors how quantum computing integrates classical and
quantum elements to enhance computational capabilities, particularly for
solving intricate problems more efficiently.

###  IV-B Layer 2: Quantum Foundations

In our approach to teaching quantum computing to computer science students new
to quantum mechanics, the second layer focuses on establishing quantum
foundations. This layer is vital, as it highlights core principles without
assuming prior quantum knowledge. Using simple analogies bridges the gap
between quantum mechanics and the practical world of quantum computing,
enhancing accessibility and engagement for computer science students.

####  IV-B1 Qubits

A fundamental quantum concept to begin with is the Qubit [46], which serves as
the quantum analog of classical bits. Qubits have the unique property of
existing in a state of superposition, allowing them to represent both 0 and 1
simultaneously. This concept often proves challenging for students, so clear
and relatable explanations are essential. To make qubits more understandable,
we draw analogies from everyday experiences. For instance, we compare a qubit
in superposition to a spinning coin showing both heads and tails at once. In
Table III, we provide a collection of analogies for teaching purposes. Visual
aids, such as diagrams representing qubit states as vectors, can also aid
comprehension. Figure 2 illustrates the Bloch Sphere, where a qubit’s state is
represented. For example, a qubit which has a half a chance of measuring as a
0 and half a chance of measuring as a 1 can be visualised as sitting on the
equator of the globe, half way between 0 and 1.

All the analogies illustrate the concept of superposition by emphasizing the
idea that qubits can represent multiple states at once and that measurement
results in the selection of one of those states.

TABLE III: Superposition Analogies Analogy |  Description
---|---
Coin Toss Analogy |  Qubits are often compared to coin tosses. In classical computing, a coin has two states heads (H) or tails (T), similar to a classical bit (0 or 1). In contrast, a qubit can be thought of as a spinning coin. When it is in superposition, it is both heads and tails simultaneously, and when measured, it randomly collapses to one of these states.
Radio Tuner Analogy |  Compare a qubit in superposition to a radio tuner. In this analogy, when the radio is in superposition, it is simultaneously tuned to multiple stations. When you turn the dial (measure), you instantly lock onto one station, even though all stations are playing together.
Quantum Dice Analogy |  Imagine a quantum die with multiple faces. When you roll the die it is in superposition (any face is up at any one time), when it stops it will randomly land on one face, collapsing the superposition and providing our “measurement”.
Polarized Glasses |  Imagine you are at a 3D movie theater, wearing polarized glasses. These glasses allow light waves moving in one direction to pass through while blocking those in other directions. Initially, without the glasses, the picture may appear blurry or indistinct, much like being in a superposition. However, when you put on the 3D glasses, it is similar to a quantum measurement; it collapses the state of the picture, revealing only one outcome: the clear image you went there to see.
Music Player Analogy |  Think of a quantum bit like a music player that simultaneously plays multiple songs. When you press play (measure), it randomly selects and plays one song.
Football Analogy |  Imagine you’re watching a football match, and a team scores a goal. In classical computing, it’s like knowing that the goal was either scored (1) or not scored (0), just like a regular coin toss. However, with qubits, it’s as if the ball is still spinning in the air even after it’s scored. It’s in a superposition state, being both ”goal” and ”no goal” at the same time. When you measure it, like checking the replay, it randomly settles on either ”goal” or ”no goal,” just as a qubit randomly collapses to one of its possible states when measured.
$|i\rangle$$|-i\rangle$$|0\rangle$$|1\rangle$$|+\rangle$$|-\rangle$ Figure 2:
A representation of the Bloch sphere

Similarly, we propose analogies to explain quantum logic gates (Table IV). We
also utilise the visual aid of the Bloch sphere (Figure 2) and the analogy of
an ice skater on the surface of the sphere to represent the effect each of the
gates has on the spin (phase) and position (state) of qubits (Figure 3).

Figure 3: Ice skater spinning on the Bloch sphere ice rink. TABLE IV: Quantum Gate Analogies Quantum Gate |  Analogy & Description
---|---
X-Gate (Pauli-X Gate) |  Quantum “Bit Flipper”: Think of the Pauli-X gate as a quantum ”bit flipper”. Just as in classical computing, where a NOT gate flips the value of a bit (from 0 to 1 or vice versa), the Pauli-X gate flips the state of a qubit from $\ket{0}$ to $\ket{1}$ or from $\ket{1}$ to $\ket{0}$. If an ice skater is spinning on the North pole (state $\ket{0}$ on the Bloch sphere), applying the Pauli-X gate will make them appear on the south pole (state $\ket{1}$ in the Bloch sphere) spinning in the same direction she was before (i.e. the state changes but not the phase).
Z-Gate (Pauli-Z Gate) |  “Phase Flipper”: The Pauli-Z gate is often compared to a “phase flipper“. It changes the phase of the qubit without altering its state. Now let us imagine that the ice skater is spinning very quickly clockwise on the south pole (state $\ket{1}$). Applying the Pauli-Z gate, will make the ice skater abruptly transition to an anticlockwise spin while still located at the same spot on the ice (i.e. the phase changes but not the state). Specifically, the Pauli-Z gate maps $\ket{1}\rightarrow-\ket{1}$ and leaves $\ket{0}$ unchanged.
Y-Gate (Pauli-Y Gate) |  “Bit and Phase Flipper”: The Pauli-Y gate can be likened to a “bit and phase flipper”. It changes both the state and the phase of a qubit. It is a combination of a X gate (state change) and Z gate (phase change). It specifically results in ($\ket{0}\rightarrow i\ket{1}$) and ($\ket{1}\rightarrow-i\ket{0}$).
Hadamard Gate |  Quantum “Coin Tosser”: The Hadamard gate is like a quantum “coin tosser”. It can create a superposition by equally weighting $\ket{0}$ and $\ket{1}$ states. Imagine it as flipping a fair coin, where it’s equally likely to land heads or tails. When you measure, it is as if you observe the result of the coin toss. If the ice skater is spinning clockwise on the North Pole ($\ket{0}$), the Hadamard gate would cause them to appear on the $\ket{+}$ point on the equator while still spinning clockwise. If the Hadamard gate is applied again, the ice skater will reappear on the north pole still spinning clockwise. If they were instead spinning clockwise at the south pole ($\ket{1}$), the Hadamard gate would make them appear on the far side of the equator ($\ket{-}$) spinning anti-clockwise, and as you might expect, reapplying the Hadamard gate, would make the skater reappear at the south pole spinning clockwise.
CNOT Gate (Controlled-X) |  “Remote Control”: The CNOT gate is often compared to a “remote control” that flips one qubit based on the state of another. Think of two light bulbs where one is the control. If the control bulb is on, the second bulb changes its state (so if its on it turns off or vice versa), and if the control bulb is off, the second bulb stays the same. We can also think of two ice skaters Alice and Bob, where Alice is the control skater. Think of Alice as someone that Bob looks up to and gets cues from. If Alice is in the $\ket{1}$ state and Bob goes through the CNOT gate, Bob will change his state (for example if he is on the North pole he’ll appear at the South pole and vice-versa). If Alice is on the North pole (state $\ket{0}$) and Bob goes through a CNOT gate, nothing happens because Alice has not given the cue to move.
Toffoli Gate |  “Double Remote Control”: The Toffoli gate is like a “double remote control”. It requires two control qubits to flip a target qubit. Think of it as having two light switches, and both must be in the “on” position to turn on a light bulb. In this case, we would need two ice skaters to serve as controls: Alice and Alex, while Bob needs “approval” from both of them before he can change his state. For example, Both Alice and Alex need to be in the $\ket{1}$ state (South Pole) when Bob goes through the CNOT gate for Bob to change his state.

####  IV-B2 Superposition and Entanglement

Superposition is a defining characteristic of quantum systems and we explain
it through the analogies we provided when explaining the qubit (Table III).

Entanglement111In the Everettian (quantum physicist Hugh Everett III
(1930–1982)) view of quantum physics the concept of entanglement is described
via the universal wave function, in other words, positing that the quantum
state of the whole universe is ’interlinked’ and can be ’captured’ in one wave
function., another challenging concept, can be illustrated by discussing the
behaviour of entangled particles. Analogies to twin particles sharing a
connection, such that when you measure one, you instantly know the state of
the other, can be used to simplify this idea. To accommodate for different
educational backgrounds, other analogies closer to every-day life are provided
in Table V.

TABLE V: Entanglement Analogies Analogy |  Description
---|---
Twin Telepathy |  Imagine two telepathic twins separated at birth. When one twin thinks of something, the other twin instantly knows what they’re thinking, no matter how far apart they are. This mysterious connection between the twins mirrors the instantaneous communication between entangled particles. Measuring one particle reveals the state of the other, no matter how far apart they are in space.
Dance Practice |  Picture a dancer practicing a routine in front of a mirror. When the dancer changes their move, the mirror does so instantly, maintaining perfect synchronization. This illustrates the instantaneous correlation between entangled particles. When you observe one particle and change its state, you will know that the other particle has changed its state accordingly.
Interlocking Puzzles |  Imagine two puzzle pieces that are designed to fit together perfectly. No matter how you separate them and shuffle them with other pieces, they always seem to find each other and connect. In a similar way, entangled particles are intrinsically connected, and when you measure one, the other instantly aligns itself, even when they are separated.
Blue and Red Pool Balls |  Imagine two pool balls, one blue and one red, moving on the pool table. At a certain point in time they bump into each other. Even though before their encounter they were moving in directions and speeds completely unrelated to each other, the moment they make contact (bump) they become “entangled” and their trajectories will mirror each others (for example they can move in precisely opposite directions). When you know which way the red ball is going, you would instantly know also the trajectory of the blue ball. It is as though their collision created a mysterious connection, and the state of one ball instantly determines the state of the other. Needless to say, that in physics this type of “mysterious connection” of the two pool balls is not explained by entanglement, but metaphorically this analogy captures the idea of quantum entanglement.
Left and Right Shoes |  I take off my left and right shoes and put them in two separate boxes. I shuffle the boxes and give one to you. If I open my box and I have the left shoe then you don’t need to open your box to know that you have the right shoe. This is an example of classical correlation. When two quantum particles are entangled, we have a quantum correlation that means we can learn even more information about the second particle than would ever be possible for a classical system.

####  IV-B3 Measurement

The final foundational concept is measurement. In classical mechanics, looking
at something does not change its state. However, in quantum mechanics, a qubit
can be in a superposition of both 0 and 1 at the same time but when measured
it must collapse to either 0 or 1. The quantum measurement collapse can be
likened to trying to observe the natural behaviour of wild animals at night
via flash photography. As you attempt to capture the positions of the animal
herd, the sudden burst of light alters their natural behaviour and they freeze
in a given position. This property, has applications for network security. If
we send classical bits from one place to another, we have no way to know
whether they were observed/eavesdropped by a malicious user. However, if we
communicate using qubits, a malicious user who observes the qubits will
collapse the wave function and we will know that the message was intercepted.

One interesting thing to note, is that we are collapsing the wave function for
the property we are measuring, this is called the basis that we are measuring
with respect to. Looking at the Bloch Sphere in Figure 2, two bases we can
measure with respect to are the $\ket{+}$ or $\ket{-}$ basis and the $\ket{0}$
or $\ket{1}$ basis. If we are measuring with respect to the $\ket{+}$,
$\ket{-}$ basis, we are asking which will it collapse to $\ket{+}$ or
$\ket{-}$? If we are measuring with respect to the $\ket{0}$, $\ket{1}$ basis,
we are asking which will it collapse to $\ket{0}$ or $\ket{1}$?

When we measure with respect to one basis all other bases (even if previously
measured move back into a superposition state). Think of it like two clowns
looking over each of your shoulders. When you turn to look at one clown, the
other one starts changing its face and costume. You spin to look at this clown
and now they freeze in position but the other one starts moving again. No
matter how quickly you turn you can’t see both clowns at the same time. This
phenomenon is at the core of the famous Heisenberg Uncertainty Principle.

To facilitate quantum computing access, quantum simulators can be used to
demonstrate quantum computing concepts in a controlled environment and to
allow students to experiment and visualize quantum processes.

When possible, access to cloud-based quantum computing platforms (e.g., Amazon
Braket, IBM Quantum Experience, Microsoft Azure Quantum) facilitates hands-on
experience for students to run quantum programs on actual quantum processors.

The students are encouraged to actively participate in discussions, solve
problems, and collaborate on quantum projects. Interactive tutorials, quizzes,
and assignments are used to reinforce learning.

One example or interactive learning is encouraging students to develop their
own analogies for quantum concepts. This utilization of analogies not only
enhances the comprehension of intricate scientific concepts but also plays a
pivotal role in fostering creative thinking among students. Encouraging
learners to explore complex ideas through analogies serves as a catalyst for
their creative cognition. Analogies act as bridges, connecting unfamiliar or
abstract concepts to relatable and tangible experiences [43]. When students
are prompted to decipher scientific theories or abstract notions by drawing
parallels to everyday phenomena, it sparks their imagination and ingenuity.
This approach prompts them to think “out of the box”, fostering the ability to
envision connections and solutions beyond conventional boundaries.

Moreover, involving students in the process of crafting and dissecting
analogies cultivates their critical thinking and problem-solving skills [41].
By encouraging them to construct domain-specific analogies, educators empower
students to exercise their creativity and analytical reasoning. Engaging in
the creation of analogies requires students to discern fundamental
characteristics and relationships between dissimilar concepts, honing their
abilities to identify patterns and similarities. This practice not only aids
in comprehending complex topics but also nurtures a mindset that values
imaginative thinking and innovative problem-solving—an indispensable skill set
for their academic and professional endeavors. Ultimately, encouraging the use
and analysis of analogies stimulates students’ intellectual curiosity and
encourages them to approach challenges with resourcefulness and adaptability
[37, 38, 39, 40, 41].

##  V Conclusions and Future Work

The layered learning approach proposed in this paper offers a structured and
practical framework for teaching quantum computing concepts to computer
science students. By emphasizing foundational knowledge and practical
applications for quantum computing, we aim to address the challenge of
teaching quantum computing to students unfamiliar with quantum mechanics. The
use of good analogies has shown promise in teaching scientific subjects and we
present an collection of analogies relevant to quantum computing to help
bridge the gap between quantum computing and the “typical” computer science
student.

Through the presented teaching approaches, students are introduced to quantum
principles by first cementing their knowledge of classical foundations such as
algorithms, data structures, and complexity theory before moving onto quantum
foundations including qubits, quantum gates, superposition, and entanglement.
The goal is to teach these quantum topics and provide computer science
students an insight into the realm of quantum computing without the
prerequisite of extensive quantum mechanics knowledge.

The paper underscores the importance of shifting educational focus to the
benefits of quantum computing for real-world applications, demonstrating the
potential of quantum computing in domains like cryptography and optimization.

###  V-A Future Work

Refinement of Analogies. We plan to further explore and refine domain-specific
analogies that effectively illustrate complex quantum concepts to diverse
student cohorts. This includes the development of new analogies based on
students’ feedback and understanding; and, the use of our interdisciplinary
network of quantum researchers to increase the diversity of domain-specific
analogies.

Evaluation and Assessment. It is also necessary to conduct comprehensive
assessments and evaluations to measure the effectiveness of the proposed
teaching methodologies. This involves collecting data on student learning
outcomes, engagement levels, and understanding through pre-and post-
assessments.

Expanded Curriculum Development. We are currently pursuing the development of
practical teaching materials for quantum computing and the enhancement of the
curriculum to encompass quantum computing topics such as quantum error
correction, quantum machine learning algorithms, and quantum cryptography.

## References

  * [1] Y. Lu, A. Sigov, L. Ratkin, L. A. Ivanov, and M. Zuo, “Quantum computing and industrial information integration: A review,” _Journal of Industrial Information Integration_ , p. 100511, 2023.
  * [2] M. Hasanovic, C. Panayiotou, D. Silberman, P. Stimers, and C. Merzbacher, “Quantum technician skills and competencies for the emerging quantum 2.0 industry,” _Optical Engineering_ , vol. 61, no. 8, pp. 081 803–081 803, 2022.
  * [3] G. P. Temporão, T. B. Guerreiro, P. S. Ripper, and A. M. Pavani, “Teaching quantum computing without prerequisites: a case study,” in _2022 IEEE International Conference on Quantum Computing and Engineering (QCE)_. IEEE, 2022, pp. 673–676.
  * [4] N. Didiş, “The analysis of analogy use in the teaching of introductory quantum theory,” _Chemistry Education Research and Practice_ , vol. 16, no. 2, pp. 355–376, 2015.
  * [5] M. F. Fox, B. M. Zwickl, and H. Lewandowski, “Preparing for the quantum revolution: What is the role of higher education?” _Physical Review Physics Education Research_ , vol. 16, no. 2, p. 020131, 2020.
  * [6] M. Kaur and A. Venegas-Gomez, “Defining the quantum workforce landscape: a review of global quantum education initiatives,” _Optical Engineering_ , vol. 61, no. 8, pp. 081 806–081 806, 2022.
  * [7] G. Carrascal, A. A. Del Barrio, and G. Botella, “First experiences of teaching quantum computing,” _The Journal of Supercomputing_ , vol. 77, no. 3, pp. 2770–2799, 2021.
  * [8] J. Liu and D. Franklin, “Introduction to quantum computing for everyone: experience report,” in _Proceedings of the 54th ACM Technical Symposium on Computer Science Education V. 1_ , 2023, pp. 1157–1163.
  * [9] E. F. Combarro, S. Vallecorsa, L. J. Rodríguez-Muñiz, Á. Aguilar-González, J. Ranilla, and A. Di Meglio, “A report on teaching a series of online lectures on quantum computing from cern,” _The Journal of Supercomputing_ , vol. 77, no. 12, pp. 14 405–14 435, 2021.
  * [10] T. Kushimo and B. Thacker, “Investigating students’ strengths and difficulties in quantum computing,” in _2023 IEEE International Conference on Quantum Computing and Engineering (QCE)_ , vol. 3. IEEE, 2023, pp. 33–39.
  * [11] Ö. Salehi, Z. Seskir, and İ. Tepe, “Teaching quantum computing to an audience beyond physicists: A case study over 22 workshops in 10 countries,” _arXiv preprint arXiv:2010.13552_ , 2020.
  * [12] M. Mykhailova and K. M. Svore, “Teaching quantum computing through a practical software-driven approach: Experience report,” in _Proceedings of the 51st ACM technical symposium on computer science education_ , 2020, pp. 1019–1025.
  * [13] S. Khodaeifaal, “Updated and adapted curriculum and pedagogy of physics with the fourth industrial revolution and quantum revolution: From waves principles to quantum mechanics fundamentals,” in _2022 IEEE International Conference on Quantum Computing and Engineering (QCE)_. IEEE, 2022, pp. 653–668.
  * [14] N. A. Davis and B. R. La Cour, “Quantum computing for the faint of heart,” in _2022 IEEE International Conference on Quantum Computing and Engineering (QCE)_. IEEE, 2022, pp. 669–672.
  * [15] S. Chitransh, D. Fischer, N. Kawalek, N. LaRacuente, J. Markman, S. P. Gaunkar, and U. Zvi, “A multidisciplinary, artistic approach to broadening the accessibility of quantum science,” in _2022 IEEE International Conference on Quantum Computing and Engineering (QCE)_. IEEE, 2022, pp. 701–708.
  * [16] S. Aehle, P. Scheiger, and H. Cartarius, “An approach to quantum physics teaching through analog experiments,” _Physics_ , vol. 4, no. 4, pp. 1241–1252, 2022.
  * [17] S. Seegerer, T. Michaeli, and R. Romeike, “Quantum computing as a topic in computer science education,” in _Proceedings of the 16th Workshop in Primary and Secondary Computing Education_ , 2021, pp. 1–6.
  * [18] The University of Warwick, “Quantum Computing - CS419-15,” accessed 5 May 2024. [Online]. Available: https://warwick.ac.uk/fac/sci/dcs/teaching/modules/cs419/#description
  * [19] University of Kent, “Introduction to Quantum Computing & Quantum Cryptography - COMP8220,” accessed 5 May 2024. [Online]. Available: https://www.kent.ac.uk/courses/modules/module/COMP8220
  * [20] Imperial College London, “Quantum Computing - 70021,” accessed 5 May 2024. [Online]. Available: https://www.imperial.ac.uk/computing/current-students/courses/70021/
  * [21] Durham University, “Quantum Computing - COMP4117,” accessed 5 May 2024. [Online]. Available: https://apps.dur.ac.uk/faculty.handbook/2022/UG/module/COMP4117
  * [22] University of Oxford, “Quantum Information: 2023-2024,” accessed 5 May 2024. [Online]. Available: https://www.cs.ox.ac.uk/teaching/courses/2023-2024/qi/#:~:text=The%20aim%20of%20this%20course,is%20divided%20into%20four%20parts
  * [23] Ludwig Maximilian University of Munich, “Introduction to Quantum Computing,” accessed 5 May 2024. [Online]. Available: https://www.nm.%ifi.lmu.de/teaching/Vorlesungen/2024ss/quantum-computing/
  * [24] Carnegie Mellon University , “Quantum Computation and Quantum Information 2018 - 15-859BB,” accessed 5 May 2024. [Online]. Available: https://www.cs.cmu.edu/~odonnell/quantum18/
  * [25] The university of Sydney, “Foundations of Quantum Computing - OLET2610,” accessed 5 May 2024. [Online]. Available: https://www.sydney.edu.au/units/OLET2610
  * [26] M. I. Núñez-Peña, M. Suárez-Pellicioni, and R. Bono, “Effects of math anxiety on student success in higher education,” _International Journal of Educational Research_ , vol. 58, pp. 36–43, 2013.
  * [27] H. E. Levy, L. Fares, and O. Rubinsten, “Math anxiety affects females’ vocational interests,” _Journal of Experimental Child Psychology_ , vol. 210, p. 105214, 2021.
  * [28] F. Neri, “Teaching mathematics to computer scientists: Reflections and a case study,” _SN Computer Science_ , vol. 2, no. 2, p. 75, 2021.
  * [29] D. Wood, J. S. Bruner, and G. Ross, “The role of tutoring in problem solving,” _Journal of child psychology and psychiatry_ , vol. 17, no. 2, pp. 89–100, 1976.
  * [30] R. D. Pea, “The social and technological dimensions of scaffolding and related theoretical concepts for learning, education, and human activity,” in _Scaffolding_. Psychology Press, 2018, pp. 423–451.
  * [31] J. C. Richardson, S. Caskurlu, D. Castellanos-Reyes, S. Duan, M. S. U. Duha, H. Fiock, and Y. Long, “Instructors’ conceptualization and implementation of scaffolding in online higher education courses,” _Journal of Computing in Higher Education_ , pp. 1–38, 2022.
  * [32] B. J. Reiser and I. Tabak, “Scaffolding,” in _The Cambridge Handbook of the Learning Sciences, Second Edition_. Cambridge University Press, 2014, pp. 44–62.
  * [33] S. Shin, T. A. Brush, and K. D. Glazewski, “Designing and implementing web-based scaffolding tools for technology-enhanced socioscientific inquiry,” _Journal of Educational Technology & Society_, vol. 20, no. 1, pp. 1–12, 2017.
  * [34] T. A. Brush and J. W. Saye, “A summary of research exploring hard and soft scaffolding for teachers and students using a multimedia supported learning environment,” _The Journal of Interactive Online Learning_ , vol. 1, no. 2, pp. 1–12, 2002.
  * [35] B. R. Belland, “Scaffolding: Definition, current debates, and future directions,” _Handbook of research on educational communications and technology_ , pp. 505–518, 2014.
  * [36] J. Van de Pol, M. Volman, and J. Beishuizen, “Scaffolding in teacher–student interaction: A decade of research,” _Educational psychology review_ , vol. 22, pp. 271–296, 2010.
  * [37] D. Gentner and M. Jeziorski, “The shift from metaphor to analogy in western science,” in _Metaphor and Thought_ , A. Ortony, Ed. Cambridge: Cambridge University Press, 1993, pp. 447–480.
  * [38] L. Jonāne, “Analogies in science education,” _Pedagogika_ , vol. 119, no. 3, pp. 116–125, 2015.
  * [39] S. Glynn, “Analogies, role in science learning. encyclopedia of science education,” 2015.
  * [40] A. García-Carmona, “The use of analogies in science communication: Effectiveness of an activity in initial primary science teacher education,” _International Journal of Science and Mathematics Education_ , vol. 19, no. 8, pp. 1543–1561, 2021.
  * [41] Z. Keri and H. S. Elbatarny, “The power of analogy-based learning in science.” _HAPS Educator_ , vol. 25, no. 1, pp. 13–20, 2021.
  * [42] T. E. o. E. Britannica, “Rutherford model,” Encyclopedia Britannica, Jun. 2023, accessed 15 November 2023. [Online]. Available: https://www.britannica.com/science/Rutherford-model
  * [43] H. Pedro and E. Edinson, “Use of analogies in science education, asystematic mapping study,” in _CS & IT Conference Proceedings_, vol. 11, no. 10. CS & IT Conference Proceedings, 2021.
  * [44] P. W. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in _Proceedings 35th annual symposium on foundations of computer science_. Ieee, 1994, pp. 124–134.
  * [45] L. K. Grover, “A fast quantum mechanical algorithm for database search,” in _Proceedings of the twenty-eighth annual ACM symposium on Theory of computing_ , 1996, pp. 212–219.
  * [46] A. O. Pittenger, _An introduction to quantum computing algorithms_. Springer Science & Business Media, 2012, vol. 19.