# Electronic processes in collisions between nitrogen impurity ions and
hydrogen atoms

C. C. Jia2,3 Y. Y. Qi4 J. J. Niu1 Y. Wu2 J. G. Wang2 A. Dubois3 N. Sisourat3
<EMAIL_ADDRESS>J. W. Gao1<EMAIL_ADDRESS>1School of Physics, Hangzhou Normal University, Hangzhou 311121, China  2Key
Laboratory of Computational Physics, Institute of Applied Physics and
Computational Mathematics, 100088, Beijing, China  3Sorbonne
Universit$\acute{e}$, CNRS, UMR 7614, Laboratoire de Chimie Physique-
Mati$\grave{e}$re et Rayonnement, 75005, Paris, France 4College of Data
Science, Jiaxing University, Jiaxing 314001, China

###### Abstract

In order to interpret and predict the behavior and properties of fusion
plasma, accurate cross sections for electronic processes in collisions between
plasma impurities and atomic hydrogen are required. In this work, we
investigate the electron capture (or charge exchange), target excitation, and
ionization processes occurring in collision of ${\rm N}^{4+}$ with atomic
hydrogen in a broad energy domain ranging from 0.06 to 225 keV/u. We consider
${\rm N}^{4+}$ ground state ${\rm N}^{4+}(2s)$ and also ${\rm N}^{4+}(2p)$
since the impurities in the edge plasma environment may be excited due to
collisions with electrons and ions/atoms. Total and partial cross sections in
both spin-averaged and spin-resolved cases are calculated using a two-active-
electron semiclassical asymptotic-state close-coupling approach. For electron
capture cross sections the present results show the best overall agreement
with available experimental data for both total and partial cross sections,
and the origins of observed discrepancies are discussed. Furthermore, we
provide new data for target excitation and ionization processes, which are
essential to improve our understanding of this relevant collision system. The
International Atomic Energy Agency (IAEA) has recently published a report
highlighting the importance and the scarcity of such data. Our work therefore
will allow a better modeling and thus understanding of magnetically confined
fusion plasma.

##  I Introduction

In magnetic confinement fusion reactors, the injection of a neutral beam,
typically hydrogen or deuterium, is a standard method to either heat or
diagnose the plasmas [1, 2]. The latter employs emissions, so called Charge
Exchange Recombination Spectroscopy [3, 4], from plasma impurities after
charge transfer with a neutral beam particle. It is routinely used for the
measurement of ion temperature, plasma rotation and impurity densities in
tokamak plasmas [4, 5, 6]. On the other hand, impurity ions are deliberately
seeded into fusion plasmas for a variety of purposes, and it is increasingly
utilized, as large-scale experimental devices like ITER emerge, to
redistribute power from the plasma core to the reactor wall, reducing heat
stress and potential damage to plasma-facing components [7, 8, 9, 10, 11]. To
avoid long time fuel storage by co-deposition, the use of nitrogen, argon,
neon seeding are excellent choices [11], because it behaves as a weak-
recycling, non-intrinsic impurity [12] and its spectroscopic emission can be
observed by a wide range of standard diagnostic tools [13]. In order to
interpret and predict the behavior and properties of impurities in fusion
plasma, a significant amount of modelling, involving the evaluation of
accurate cross sections for electron capture, target excitation, and
ionization of impurities in collisions with atomic hydrogen is required [14,
15].

In the present work, we investigate collisions between N4+ ion in its ground
and first excited states and atomic hydrogen leading to electron capture,

$\displaystyle{\rm N}^{4+}(2s/2p)+{\rm H}(1s)\to{\rm N}^{3+}(n\ell
n^{\prime}\ell^{\prime}~{}^{1,3}L)(n,n^{\prime}\geq 2)+{\rm H}^{+},$ (1)

target excitation,

$\displaystyle{\rm N}^{4+}(2s/2p)+{\rm H}(1s)\to{\rm N}^{4+}(n\ell)(n\geq
2)+{\rm H}^{*}(n\ell^{\prime}),$ (2)

and ionization,

$\displaystyle{\rm N}^{4+}(2s/2p)+{\rm H}(1s)\to{\rm N}^{4+}(n\ell)(n\geq
2)+{\rm H}^{+}+e^{-}.$ (3)

Since the impurities in the edge plasma environment may be excited due to
collisions with electrons and ions/atoms [16], the study of excited
N${}^{4+}(2p)$ in collisions with atomic hydrogen is also of relevance.
However, to our knowledge, no detailed experimental or theoretical
investigation related to excited N${}^{4+}(2p)$ projectile has been reported
so far. For ground state N${}^{4+}(2s)$ ion and atomic hydrogen collisions,
numerous experimental and theoretical studies have been conducted over the
past decade. Total SEC cross sections were measured by Seim et al. [17] in the
energy range of 1.1 - 3.6 keV/u, by Crandall et al. [18] from 1 to 7 keV/u,
and by Phaneuf et al. [19] from 40 to 120 keV/u. In these early experiments,
the cross sections were obtained by passing a beam of N4+ ion through a high-
temperature oven of dissociated hydrogen. Furthermore, using the merged-beam
technique, Huq et al. [20] and Folkerts et al. [21] have measured total SEC
cross sections for the low energy range of 1 - 982 eV/u. Nonetheless, there is
still a lack of experimental data for energies ranging from 10 to 40 keV/u.
For the state-selective SEC cross sections, investigations are more scarce.
McCullough et al. [22] conducted the first measurement using translational
energy spectroscopy (TES) within the energy range of 0.25 - 1.6 keV/u. Bliek
et al. [23] provided the initial insights on spin-resolved state-selective
cross sections in the energy range of 1 - 4 keV/u by means of photon emission
spectroscopy. For ionization of atomic hydrogen by N4+ collisions, the only
available set of experimental data was reported by Shah and Gilbody [24] using
a crossed-beam technique for energies from 30 to 165 keV/u.

On the theoretical side, using molecular orbital (MO) approaches, the SEC
cross sections for N${}^{4+}(2s)$ and H collisions were reported by a number
of authors [25, 26, 21, 27, 28] at various levels of sophistication, for low
impact energies ranging from 10-6 to 10 keV/u. Sidky et al. [29] and Cabrera-
Trujillo et al. [30] extended the cross section calculations to intermediate
energies up to 25 keV/u using the atomic orbital close-coupling method and
electron-nuclear dynamics (END) formalism, respectively. For high impact
energies, calculations using the classical trajectory Monte Carlo (CTMC)
method [31, 32] and the boundary corrected continuum intermediate state
approximation [33, 34] were reported in the energy range of 40 - 200 keV/u.
Despite the intensive investigation, considerable discrepancies still exist
between available experimental and theoretical results. Moreover, no
experimental data and theoretical predictions for target excitation exist.
This lack of data impedes the modeling of fusion plasma.

In the present work, the electron capture, target excitation, and ionization
processes (Eqs.(1-3)) are investigated in a wide energy domain ranging from
0.06 to 225 keV/u. This energy range covers the typical energies of many
astrophysical and laboratory plasmas, including the radiation losses and
neutral beam heating efficiencies in tokamak plasmas. We use a two-active-
electron semiclassical asymptotic-state close-coupling (SCASCC) approach with
a configuration interaction treatment to deal with electronic correlation. The
total and $n\ell$-resolved cross sections in both spin-averaged and spin-
resolved cases are presented and compared with previous experimental
measurements and theoretical calculations where available. Possible reasons
for the disagreements with existing data are also discussed. We therefore
provide a full set of consistent data, which can be employed in various plasma
modeling.

The present paper is organized as follows. In the next section we briefly
outline the SCASCC method used in the present calculations. In Sec. III, we
present a detailed analysis of electron capture, target excitation, and
ionization cross sections, including direct comparisons with available
experimental and theoretical results. A brief conclusion follows in Sec. IV.
Atomic units are used throughout, unless explicitly indicated otherwise.

##  II Theory

In the present work, we have calculated electron capture, target excitation
and ionization cross sections in N4+ and H collisions using a two-active-
electron SCASCC approach, which was previously described, see e.g. in [35, 36,
37, 38]. Here we outline only briefly the main features of the approach. The
two-electron time-dependent Schr$\rm\ddot{o}$dinger equation (TDSE) is written
as

$\small\left[{H-i\frac{\partial}{\partial
t}}\right]\Psi(\vec{r}_{1},\vec{{r}}_{2},t)=0,$ (4)

with the electronic Hamiltonian

$\small\begin{split}H&=\sum_{i=1,2}\left[-\frac{1}{2}\nabla^{2}_{i}+V_{T}(r_{i})+V_{P}({r}_{i}^{p})\right]+\frac{1}{\left|\vec{{r}_{1}}-\vec{{r}_{2}}\right|},\end{split}$
(5)

where $\vec{{r}_{i}}$ and $\vec{r_{i}}^{p}=\vec{{r}_{i}}-\vec{R}(t)$ are the
position vectors of the electrons with respect to the target and the
projectile, respectively. The relative projectile-target position $\vec{R}(t)$
defines the trajectory, with $\vec{R}(t)=\vec{b}+\vec{v}t$ in the usual
straight-line, constant velocity approximation where $\vec{b}$ and $\vec{v}$
are the impact parameter and relative velocity vector, respectively. The term
$V_{T}$ ($V_{P}$) is the electron-target (-projectile) nucleus (and inner
electrons, in the frozen core approximation) potential,

$\begin{split}V_{T}(r_{i})=-\frac{1}{r_{i}},\hskip
5.0ptV_{P}({r}_{i}^{p})=-\frac{5}{r_{i}^{p}}-\frac{2}{r_{i}^{p}}(1+\alpha
r_{i}^{p})e^{-\beta r_{i}^{p}},\end{split}$ (6)

with the variational parameters $\alpha=4.3982$ and $\beta=8.7964$ are taken
from [39, 40].

The Schr$\rm\ddot{o}$dinger equation is solved by expanding the wavefunction
onto a basis set composed of states of the isolated collision partners (i.e.,
asymptotic states),

$\begin{split}\Psi(\vec{r_{1}},\vec{r_{2}},t)=&\sum_{k=1}^{N_{T}}\sum_{l=1}^{N_{P}}a_{kl}^{TP}(t)[\phi_{k}^{T}(\vec{r_{1}})\phi_{l}^{P}(\vec{r_{2}},t)\pm\phi_{k}^{T}(\vec{r_{2}})\phi_{l}^{P}(\vec{r_{1}},t)]e^{-i(E_{k}^{T}+E_{l}^{P})t}\\\
&+\sum_{j=1}^{N_{PP}}a_{j}^{PP}(t)\Phi_{j}^{PP}(\vec{r_{1}},\vec{r_{2}},t)e^{-iE_{j}^{PP}t},\end{split}$
(7)

where $T$ ($P$) and $PP$ superscripts denote states and corresponding energies
for which one or two electrons are on the target (projectile), respectively.
Note that, in the present work, the two-electron states of target (H-) are not
included in the calculations, since the formation of H- is expected to be
negligible compared to the processes in Eqs. (1-3). The $\pm$ sign in the last
part of Eq. (7) stands for the singlet and triplet spin states, respectively,
and the wave functions $\Phi_{j}^{PP}$ are related to the corresponding spin
symmetry. For both electrons, the projectile states contain plane-wave
electron translation factors (ETFs),
$e^{i\vec{v}\cdotp\vec{r_{i}}-i\frac{1}{2}v^{2}t}$, ensuring Galilean
invariance of the results. The insertion of Eq. (7) into (4) results in a
system of first-order coupled differential equations, which can be written in
matrix form as

$i\frac{d}{dt}\textbf{a}(t)=\textbf{S}^{-1}(\vec{b},\vec{v},t)\textbf{M}(\vec{b},\vec{v},t)\textbf{a}(t),$
(8)

where $\textbf{a}(t)$ is the column vector of the time-dependent expansion
coefficients, i.e. $a^{PP}$ and $a^{TP}$ in Eq. (7) and S, M are the overlap
and coupling matrices, respectively. We should emphasize here that we include
explicitly in M all bielectronic couplings, and notably the complex two-center
ones which include the ETF for both electrons. We use the same strategy, i.e.
no neglect of any couplings, to obtain the two-electron (PP) states in the
diagonalisation stage, going well beyond the Hartree-Fock approach. Our
approach represents therefore a configuration interaction (CI) approach.

The coupled equations Eq. (8) are solved using the predictor-corrector
variable-time-step Adams-Bashford-Moulton method for a set of initial
conditions: initial state $i$, and given values of $b$ and $v$. The state-to-
state integral cross sections for a transition are calculated as

$\sigma_{fi}(v)=2\pi\int_{0}^{+\infty}b\left|a_{f}(+\infty)\right|^{2}db.$ (9)

In our approach in which total spin is conserved, the coupled equations Eq.
(8) are solved independently for singlet and triplet symmetries corresponding
to the two possible total spin states of the collision system. The spin-
averaged cross sections are obtained as the sum of the singlet and triplet
results according to their statistical weights, i.e. $\frac{1}{4}$ and
$\frac{3}{4}$, respectively.

##  III Results and discussions

To carry out the computations of cross sections for N4+-H collisions, we first
construct the atomic states of H, N4+ and N3+, to be included in the expansion
Eq. (7). We express these states in terms of Gaussian-type orbitals (GTOs),

$G_{\alpha,\ell,m}(\vec{r})=r^{\ell}e^{-\alpha r^{2}}Y_{\ell,m}(\theta,\phi),$
(10)

for the one-electron ($\phi^{T}$, $\phi^{P}$ in Eq. (7) ) states and products
of the same GTOs for the two-electron ($\Phi^{PP}$) ones. The number of GTOs
and their exponents $\alpha$ are optimized to get correct binding energies of
the relevant states populated during the collision.

In the present calculations, a set of 56 GTOs (12 for $\ell$ = 0, 8 $\times$ 3
for $\ell$ = 1, and 4 $\times$ 5 for $\ell$ = 2) is used on the N5+ center,
and 27 GTOs (9 for $\ell$ = 0 and 6 $\times$ 3 for $\ell$ = 1) are located on
the H center. The basis sets allow the inclusion of 1927 states and pseudo-
states (855 (H, N4+) and 1072 (N3+) states) in the calculations for spin
singlet symmetry, and 1919 (855 (H, N4+) and 1064 (N3+) states) for triplet
symmetry. These states can describe elastic, single electron capture,
excitation, as well as ionization through the inclusion of pseudostates with
energy lying above ionization thresholds.

We have checked our results presented in the following by repeating the
calculations with a larger basis set on the N5+ center built from 64 GTOs (14
for $l$ = 0, 10 $\times$ 3 for $l$ = 1, and 4 $\times$ 5 for $l$ = 2) and on
the H center built from 32 GTOs (11 for $l$ = 0 and 7 $\times$ 3 for $l$ = 1).
For both N${}^{4+}(2s)$ and N${}^{4+}(2p)$ projectiles, similar convergences
of cross sections were obtained. Total and dominant $n\ell$-selective
(N${}^{3+}(2s3\ell$ ${}^{1}\rm{L})$ and N${}^{3+}(2p3\ell$ ${}^{1,3}\rm{L})$)
SEC cross sections from these two sets agree with each other within 2% at low
and intermediate energies, and about 10% for energies above 100 keV/u. For the
cross sections reported in Figs. 6 and 7, the convergence is better than 10%
for the target excitation and about 5% for the ionization at intermediate
energies. In the high energy region, the convergence is evaluated to be about
3% for the dominant channel H ($2p$), and about 20% for the other relative
weak excitation channels and ionization processes. For energies below 1 keV/u,
the convergence for both target excitation and ionization is found to be about
80% at worst where, however, the cross sections are negligibly small compared
to their counterparts at higher energies, see Figs. 6 and 7.

###  III.1 Spin-averaged total and $n\ell$-selective electron capture cross
sections

Figure 1: Spin-averaged total electron capture cross sections of ${\rm
N}^{4+}(2s)$ and H collisions as a function of impact energy. The theoretical
results are from the present calculations (red solid line), MOCC calculations
by Zygelman et al. [28] (green dotted line) and Shimakura et al. [26] (orange
short-dotted line), AOCC calculations of Sidky et al. [29] (blue dash-dot-
dotted line), CTMC calculations of of Olson et al. [31] (yellow solid
pentagons), calculations of the boundary corrected continuum intermediate
state approximation from Purkait et al. [33] (purple dashed line), and END
calculations of Cabrera-Trujillo et al. [30] (black short-dashed line). The
experimental results are from Huq et al. [20] (blue open leftward triangles),
Folkerts et al. [21] (black solid squares), Crandall et al. [18] (purple solid
rightward triangles), Seim et al. [17] (dark-yellow solid circles) and Phaneuf
et al. [19] (black open stars). The present spin-averaged total electron
capture cross sections of ${\rm N}^{4+}(2p)$ and H collisions are also
presented by red dash-dotted lines. Figure 2: Spin-averaged $3\ell$-selective
electron capture cross sections. For N${}^{4+}(2s)$ \+ H($1s$) collisions, the
present calculations are plotted as lines with symbols; the experimental
results of McCullough et al. [22] are shown as symbols and error bars. The
present results for ${\rm N}^{4+}(2p)$ and H collisions are presented by
lines.

In Fig. 1, we present our calculated spin-averaged total electron capture
cross sections as a function of impact energy in the range 0.06 – 225 keV/u,
together with previous experimental [20, 21, 18, 19], and theoretical [28, 26,
29, 31, 33, 30] results for comparison. Our cross sections show a weak maximum
around 0.1 keV/u, and decrease for increasing energies. The presence of a
shoulder at about 20 keV/u is due to the increasing importance of the
population of high-$\ell$ states, i.e. N${}^{3+}(2s3d)$, at high impact
energies, as discussed below (see Fig. 2). It can be observed that our results
are comparable to the measurements of Folkerts et al. [21], Huq et al. [20],
Crandall et al. [18] and Phaneuf et al. [19] in the respective overlapping
energy regions. Overall, there is a general agreement within the uncertainties
of the experiments. However, our cross sections are slightly higher than the
experimental data of Seim et al. [17] and Crandall et al. [18] for energies
ranging from 1 to 4 keV/u. Moreover, other small discrepancies are seen. For
instance, Folkerts et al. [21] highlighted an apparent dip in their measured
total cross section near 0.1 keV/u, a feature that has not been seen in any
theoretical calculations [28, 26, 29, 31, 33, 30], including our own.
Comparing with available theoretical results, our results are in good
agreement with the CTMC calculations of Olson et al. [31] for energies higher
than 30 keV/u, with results from END formalism of Cabrera-Trujillo et al. [30]
for energies ranging from 2 to 7 keV/u. Below 2 keV/u, our total cross
sections are larger than the END results, as well as the AOCC calculations of
Sidky et al. [29] and the MOCC calculations of Shimakura et al. [26]. The
latter deviation is probably due to insufficient channels included in these
previous calculations [26, 29, 30]. Moreover, the MOCC calculations by
Zygelman et al. [28] were intended primarily for lower energies where
rotational coupling was not included, which results in a further
underestimation of the cross sections. Note finally that according to [29] the
trajectory effect in the AOCC calculations makes about 10% difference of the
cross sections at 0.07 keV/u, and this effect becomes negligible as impact
energy increases to 0.3 keV/u and higher energies.

The present spin-averaged total electron capture cross sections for
N${}^{4+}(2p)$ \+ H($1s$) collisions are also displayed in Fig. 1. The results
are very close to those of N${}^{4+}(2s)$ \+ H($1s$) collisions, except for
energies below 0.5 keV/u. In the latter energy region, the cross sections for
N${}^{4+}(2p)$ \+ H($1s$) collisions exceed those of N${}^{4+}(2s)$ \+ H($1s$)
by up to 15%.

In Fig. 2, we present the spin-averaged $3\ell$-selective electron capture
cross sections for N${}^{4+}(2s)$ \+ H($1s$) collisions, compared with the
experimental data of McCullough et al. [22]. Our results show that the cross
sections for electron capture to N${}^{3+}(2s3d)$ dominate for energies below
0.4 keV/u and above 10 keV/u. In between, the results for electron capture to
N${}^{3+}(2s3p)$ are the dominant channel. The increasing importance of the
population of N${}^{3+}(2s3d)$ around 20 keV/u results in a shoulder-like
structure in the total electron capture cross sections (see Fig. 1). Our
results for electron capture to N${}^{3+}(2s3s)$ and N${}^{3+}(2s3d)$ are in
good agreement with the experimental data [22] in the overlapping energies.
However, for electron capture to N${}^{3+}(2s3p)$, we see important
differences between them, with a maximum about 25% at 1 keV/u. We cannot
firmly conclude on that issue, as the experimental data from McCullough et al.
[22] were determined by normalizing their total cross sections to previous
measurements [18, 17, 20], among which differences of up to 15% exist (see
Fig. 1).

We show also in Fig. 2 the spin-averaged $3\ell$-selective electron capture
cross sections for N${}^{4+}(2p)$ \+ H($1s$) collisions. It can be observed
that the results for N${}^{4+}(2p)$ projectile are almost identical with these
of N${}^{4+}(2s)$ projectile for energies above 4 keV/u. Below, they differ
significantly.

###  III.2 Spin-resolved $n\ell$-selective electron capture cross sections

Figure 3: Cross sections for electron capture to N${}^{3+}(2s3\ell$ ${}^{1}L)$
in N${}^{4+}(2s)$ \+ H($1s$) collisions, the present calculations are plotted
as lines with symbols; the MOCC calculations by Zygelman et al. [28] and
Shimakura et al. [26] are marked by black dash-dotted and blue short-dotted
lines, respectively; the AOCC calculations of Sidky et al. [29] are denoted by
green dot-dotted lines. The experimental results of Bliek et al. [23] are
shown as symbols and error bars.

Fig. 3 shows the present cross sections for electron capture to the dominant
states N${}^{3+}(2s3\ell$ ${}^{1}\rm{L})$ in N${}^{4+}(2s)$ \+ H($1s$)
collisions in the case of singlet total spin symmetry, alongside with the
available theoretical calculations [29, 26, 28] and the only set of existing
experimental data [23]. For electron capture to N${}^{3+}(2s3s$
${}^{1}\rm{S})$, our cross sections are in good agreement with the
experimental data [23] and theoretical calculations [29, 26] in the respective
overlapping energies, except for energies above 4 keV/u, where the MOCC
calculations of Shimakura et al. [26] exceed ours and the AOCC calculations of
Sidky et al. [29]. Note that in [26], an approximate electron translation
factor (up to first order in velocity) was employed which can explain the
observed discrepancy. For electron capture to N${}^{3+}(2s3p$ ${}^{1}\rm{P})$
and N${}^{3+}(2s3d$ ${}^{1}\rm{D})$, our results show the best overall
agreement with the experimental data of Bliek et al. [23]. Some disagreements
are seen when comparing the present results to MOCC calculations of Shimakura
et al. [26] and the AOCC calculations of Sidky et al. [29]. The discrepancy is
probably due to the insufficient channels included in these two calculations,
as discussed above for the total electron capture cross sections. The MOCC
calculations by Zygelman et al. [28], where rotational coupling was not
included, deviate most from all the other results.

Figure 4: Same as Fig. 3 but for electron capture to N${}^{3+}(2s3\ell$
${}^{3}L)$ in N${}^{4+}(2s)$ \+ H($1s$). Note that, in (b), the experimental
results of Bliek et al. [23] for electron capture to N${}^{3+}(2s3s$
${}^{3}\rm{S}+2s3p$ ${}^{3}\rm{P})$ are shown as blue solid squares, and the
corresponding present calculations are plotted as red dashed lines with open
squares.

In Fig. 4, we present our cross sections for electron capture to the dominant
states N${}^{3+}(2s3\ell$ ${}^{3}\rm{L})$ in N${}^{4+}(2s)$ \+ H($1s$)
collisions in the case of triplet total spin symmetry, together with the
available theoretical calculations [29, 26, 28] and the only set of existing
experimental data [23]. The comparison between the present results and the
other available data is similar to the case of singlet total spin symmetry,
except for the electron capture to N${}^{3+}(2s3p$ ${}^{3}\rm{P})$. The
measurements by Bliek et al. [23] only provides the summed cross sections for
electron capture to N${}^{3+}(2s3s$ ${}^{3}\rm{S})$ and N${}^{3+}(2s3p$
${}^{3}\rm{P})$, since the measured emission of the $2s3s$ 3S $\rightarrow$
$2s2p$ 3P transition results from the capture into both N${}^{3+}(2s3s$
${}^{3}\rm{S})$ state and N${}^{3+}(2s3p$ ${}^{3}\rm{P})$ state via the
cascade $2s3p$ 3P $\rightarrow$ $2s3s$ 3S $\rightarrow$ $2s2p$ 3P. However,
the experimental data are smaller than our summed cross sections of
N${}^{3+}(2s3s$ ${}^{3}\rm{S})$ and N${}^{3+}(2s3p$ ${}^{3}\rm{P})$, but in
good agreement with our present results and those from [29, 26, 28] for
electron capture to only N${}^{3+}(2s3p$ ${}^{3}\rm{P})$. Without any other
data, it is clear that further experimental and theoretical efforts are
required to draw definite conclusions on this issue.

Figure 5: Cross sections for electron capture to N${}^{3+}(2p3\ell$
${}^{1,3}L)$ in N${}^{4+}(2p)$ \+ H($1s$) collisions.

We also investigate N${}^{4+}(2p)$ \+ H($1s$) collisions to provide the cross
sections for the dominant $3\ell$-selective electron capture processes. These
cross sections are reported in Fig. 5. In the case of triplet spin symmetry,
the results for electron capture to N${}^{3+}(2p3p$ ${}^{3}\rm{S}$,
${}^{3}\rm{P}$, ${}^{3}\rm{D}$) are dominant in the energy range 0.2 - 10
keV/u, beyond this range, the ones for electron capture to N${}^{3+}(2p3d$
${}^{3}\rm{P}$, ${}^{3}\rm{D}$, ${}^{3}\rm{F}$) are the dominant channel. In
the case of singlet spin symmetry, the cross sections for electron capture to
N${}^{3+}(2p3p$ ${}^{1}\rm{S}$, ${}^{1}\rm{P}$, ${}^{1}\rm{D}$) dominate for
energies below 10 keV/u, while the ones for electron capture to
N${}^{3+}(2p3d$ ${}^{1}\rm{P}$, ${}^{1}\rm{D}$, ${}^{1}\rm{F}$) take over for
higher energies. Note that the cross sections for the singlet case are
significantly lower than for the tripelet case, as for N${}^{4+}(2s)$ \+
H($1s$) collisions

###  III.3 Target excitation and ionization cross sections

Figure 6: Spin-resolved and spin-averaged cross sections of N${}^{4+}(2s)$ \+
H($1s$) collisions for target excitation to (a) H($2s$), (b) H($2p$), (c)
H($n=3$), and ionization processes in (d). The ionization cross sections from
CTMC calculations of Olson et al. [31] and Purkait et al. [32], as well as the
experiment of Shah and Gilbody [24] are also shown in (d). Figure 7: Same as
Fig. 6 but for N${}^{4+}(2p)$ \+ H($1s$) collisions.

We now investigate target excitation of H($1s$) colliding with N${}^{4+}(2s)$
and N${}^{4+}(2p)$, where no previous theoretical or experimental data was
reported. Both spin-resolved and spin-averaged cross sections are presented in
Figs. 6(a-c) for N${}^{4+}(2s)$ projectile and Figs. 7(a-c) for N${}^{4+}(2p)$
projectile. As one can see in Fig. 6, the excitation cross sections are
negligibly small for energies below 1 keV/u. Above this energy, H($2p$)
production is the strongest target excitation channel, showing the dominant
dipolar nature of the interaction with the projectile charge, as the
nonrelativistic limit of the so-called Williams-Weizsäcker virtual photon
method [41, 42]. Similar behavior was recently observed in target excitation
of He+ and H collisions [38]. In addition, a minimum can be observed in the
excitation cross sections around 20 keV/u. This can be assigned to a strong
competition between target excitation and electron capture to N${}^{3+}(2s3d$
${}^{1,3}\rm{D})$, where a maximum can be observed around 20 keV/u (see Fig.
2). The excitation cross sections for N${}^{4+}(2p)$ projectile presented in
Fig. 7 show the same trends as those for N${}^{4+}(2s)$ projectile, and the
same conclusions can be drawn for them.

We next investigate the ionization processes, when the target lost its
electron and the projectile remains in the same charge state after the
collisions, see Eq. (3). Fig. 6(d) shows the present spin-resolved and spin-
averaged cross sections for ionization of atomic hydrogen by N${}^{4+}(2s)$
collisions, together with experimental data from Shah and Gilbody [24] and the
available CTMC results from Olson et al. [31] and Purkait et al. [32] for
comparison. It can be seen that our total spin-averaged cross sections are in
good agreement with the CTMC calculations of Purkait et al. [32] for energies
smaller than 60 keV/u and with the ones of Olson et al. [31] for energies
below 100 keV/u. However, our results lie below the experimental data reported
by Shah and Gilbody [24], by about a factor of 2 at most. The latter
discrepancy may be due to the use of GTOs in the present method, which cannot
describe the exact boundary conditions of continuum states. However, the
experimental data are also larger than the CTMC results. CTMC methods
generally describe well ionization processes. Further investigations are
needed to resolve this discrepancy. Finally, we present the ionization cross
sections for N${}^{4+}(2p)$ projectile in Fig. 7(d) for comparison. These
cross sections are similar to those for N${}^{4+}(2s)$ projectile.

We mention here that some two-electron processes are included in the above
results: the simultaneous projectile excitation and target ionization, which
is included in the total ionization cross sections reported above, contributes
to the total ionization cross sections for about 9% at low impact energies,
15% for energies in the 1 - 10 keV/u range and 2% at high impact energies. The
two-electron process of projectile excitation and capture is included in the
single capture. This process contributes to about 12% at impact energies less
than 0.25 keV/u, 5% at energies from 0.25 keV/u to 2.25 keV/u and 12% - 8% at
higher impact energies. Finally, the projectile excitation and target
excitation process is included in the target excitation. It represents about
20($\pm$ 7)% in the energy range 2 - 20 keV/u, 3% at low impact energies and
1% at high impact energies of the total target excitation cross sections.

##  IV Conclusions

Using a two-active-electron semiclassical asymptotic-state close-coupling
approach with a configuration interaction treatment to deal with electronic
correlation, we have investigated electron capture, target excitation, and
ionization processes in ${\rm N}^{4+}(2s)$ and H collisions. In the energy
region 0.06 - 225 keV/u, total and state-selective cross sections in both
spin-averaged and spin-resolved cases have been reported and compared with
available theoretical and experimental data. For electron capture processes,
in particular for state-selective cross sections, our results show the best
overall agreement with the experimental ones in the different energy ranges
available. This demonstrates the importance of a two-electron treatment taking
into account electronic correlation and the use of extended basis sets within
the close-coupling scheme. We also provide new data for target excitation and
ionization processes, which are essential to improve our understanding of this
relevant collision system. For excitation, H($2p$) production is the strongest
target excitation channel, showing the dominant dipolar nature of the
interaction with the projectile charge. For the ionization cross sections, our
results agree with the experimental data within about a factor of 2. The
latter discrepancy may be due to the use of GTOs in the present method,
further experimental and theoretical works are, however, needed. Our work
provides a complete and consistent set of cross sections over a broad range of
collision energies, which can be used for fusion plasma diagnosis and
modeling. These cross sections will be provided in tabulated form on the
CollisionDB database 111The data has been submitted to CollisionDB, see
https://db-amdis.org/collisiondb.

Furthermore, we also present cross sections for electron capture, target
excitation, and ionization processes in excited ${\rm N}^{4+}(2p)$ \- H
collisions. The results exhibit similarities in magnitude and energy
dependence with those observed in collisions of ${\rm N}^{4+}(2s)$
projectiles. Since the impurities in the edge plasma environment may be
excited due to collisions with electrons and ions/atoms, the reported cross
sections for ${\rm N}^{4+}(2p)$ projectile should be also of relevance in the
studies of properties of fusion plasma.

##  V ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
(Grants No.11934004 and No. 12374229), and the startup project from Hangzhou
Normal University. C.C.J. is supported by a China Scholarship Council (CSC)
scholarship within a CSC-SU program.

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# A Comparative Study of Classical and Quantum Machine Learning Models for
Sentimental Analysis

Diksha Sharma1 Parvinder Singh2 Atul Kumar1<EMAIL_ADDRESS><EMAIL_ADDRESS>1Indian Institute of Technology Jodhpur,
Rajasthan-342037, India  2Central university of Punjab Bathinda,
Punjab-151001, India

###### Abstract

We analyse and classify the sentiments of a text data constructed from movie
reviews. For this, we use the kernel based approach from quantum machine
learning algorithms. In order to compose a quantum kernel, we use a circuit
constructed using a combination of different Pauli rotational gates where the
rotational parameter is a classical non-linear function of data points
obtained from the text data. For analysing the performance of the proposed
model, we analyse the quantum model using decision tree, gradient boosting
classifier, and classical and quantum support vector machines. Our results
show that quantum kernel model or quantum support vector machine outperforms
all other algorithms used for analysis in terms of all evaluation metrics. In
comparison to a classical support vector machine, the quantum support vector
machine leads to significantly better results even with increased number of
features or dimensions. The results clearly demonstrate increase in precision
score by $9.4\%$ using a quantum support vector machine as against a classical
support vector machine if the number of features are $15$.

Quantum Entanglement, SVM, Supremacy, QSVM, Quantum Computing;

##  I Introduction

Machine learning is a branch of artificial intelligence which efficiently uses
the underlying data to generate insightful classifications and predictions
with the help of algorithms- while gradually improving the accuracy using
historical data or information [1]. Machine learning algorithms evolve from
mathematical models reinforcing the prognosis associated with the dataset
which in-turn further facilitates the informed decision process. Such
algorithms help a machine to work efficiently even for analysing massive
quantity of data. With the impact of machine learning algorithms on science
and technology, it is imperative to develop and analyse new machine learning
algorithms. In fact, the last two decades have witnessed an enormous
transformation in terms of algorithms and the area has become a leading
research area across academia and industry. Another area that has impacted
academic and industrial progress is quantum information and computation.
Quantum algorithms- based on the fundamental laws of quantum mechanics- have
shown optimal assurance to offer efficient applications to material science
[2, 3], computational chemistry [4], cryptography [5, 6], traffic navigation
[7, 8], security [9, 10], finance [11], biology [12] and machine learning
[13]. Both, machine learning and quantum computing, deal with large-scale
networks to effectively address critical problems, therefore, a natural
question is to analyse the potentials of quantum computing in the realms of
machine learning, i.e., quantum machine learning. Quantum computing, in
comparison to classical computation, is believed to assist machine learning
algorithms in enumerating distance or similarity between data-points with an
adequate speedup, if not exponential [14, 15]. The advantage can be translated
in terms of time and space complexity or can be obtained using adiabatic
quantum computing for attaining optimum results [16]. Moreover, with the role
of entanglement and non-classical correlations being established as the
reasons for several potential applications in quantum information and
computation [17, 18], quantum machine learning algorithms are conceived to be
much more powerful and effective as compared to their classical counterparts.

In addition to the quantum machine learning algorithms, data is also an
important factor to achieve the desired accuracy. In general, data can be
classified as quantitative or qualitative. Among various types of data, text
data plays a key role in daily life, e.g., placing an online order, booking
tickets or buying food online, reviews of movies or stores etc. With the
advent of e-commerce, one of the most desirable requirements is the
availability of reviews or compact surveys for every product before placing an
order. Of course, there is a large amount of data available for different
products but at the same time analysing the available data as positive or
negative for a common man, without any error of biasedness, is also extremely
challenging. The classification or analysis of textual data is therefore known
as sentiment analysis or opinion mining. Sentiment analysis not only helps
customers to fulfill their need and expectation, but also the industry to
improve their products as per the market requirements. Machine learning
algorithms are historically considered to be effective in analysing the text
data as positive or negative statement in the analysis of different cases.
Recently, quantum machine learning algorithms have also been developed for
classification of data but a very few algorithms are comprehensively analyzed
in comparison to classical algorithms [19]. In this paper, we study the use of
quantum as well as classical machine learning algorithms for sentimental
analysis and further compare the classical and quantum machine learning
algorithms in term of different evaluation matrices. In this work, we apply
different quantum machine learning algorithms with multiple feature maps to
increase the overall accuracy. Further, we find that quantum machine learning
algorithms predict the sentiments more precisely with higher accuracy than the
classical machine learning algorithms.

###  I.1 Sentimental Analysis

Sentiments are opinions, thoughts or feelings of an author towards an object,
an entity, or an event. Natural language processing (NLP) provides techniques
to identify the emotion from the text, named as sentimental analysis.
Therefore, extracting the subjective information contained in an opinion is
known as sentiment analysis or opinion mining [20]. Sentiment analysis or
document-level sentiment classification, classifies a sentence by considering
that sentence as a single information unit which includes positive, negative
or neutral statements [21]-implicates the use of data mining and machine
learning to mine the text. Sentiment analysis is used in social media
monitoring, recommendation system, customer service, brand monitoring, stock
market prediction, product analysis to name a few [22, 23, 24, 25].

###  I.2 Machine Learning

In artificial intelligence, machine learning is typically referred as the
process of training a model to make predictions from inter-linkages of the
previously seen data, i.e., machine learning algorithms improve their
performance with the experience. Depending on the application, machine
learning algorithms are classified into three major categories,namely
supervised, unsupervised and reinforcement machine learning algorithms.
Supervised learning algorithms predict the label or class of unknown data
objects using the prior information related to the label of similar objects.
Therefore, supervised learning algorithms take a labeled data as an input in
the training phase, where labels are corresponding outcomes of features.
Whereas in the testing phase, the trained model acts on the unlabeled data and
predict the label. Moreover, based upon the data, supervised machine learning
algorithms are further divided into classification and regression techniques.
For example, if supervised machine learning algorithms are used to predict the
label or class of an unknown data object then it represent classification
techniques; and if supervised machine learning algorithms are used to predict
the future value of real-valued variable then it represent regression
techniques [26]. As opposed to the supervised learning, in unsupervised
learning, the models deal with an unlabeled training data. These models or
algorithms get insights from the data without explicit interventions.
Therefore, unsupervised learning algorithms analyse data to get a pattern
directly from the input data objects. This process is known as
pattern/knowledge discovery or clustering. Depending on the data, another
variant of unsupervised learning is anomaly detection or association analysis
[27]. Moreover, reinforcement learning algorithms are widely used in a game-
like environment where a model or an agent performs a set of action in order
to maximize rewards without the training to solve a given problem. The agent
gets a reward or punishment for each performed action, and the goal is a
cumulative maximum reward with an optimal solution to the problem [28].
Reinforcement learning is used in robotics and automobiles where a machine
performs some actions to complete a task and receives a degree of achievement
as positive reward and risk of an action as negative reward.

The machine learning models have the potential to learn and to discover
patterns to exceptionally predict the class for unknown data objects. For
example, the text data contains reviews or opinions that can be seen as inter-
related words. Therefore, machine learning algorithms are attractive tools to
classify the textual data into positive or negative sentiments. It can extract
the essence of statements and improve the accuracy by getting trained from a
large set of sample data. Further, quantum machine learning algorithms can
also be used to improve the performance measures for sentimental analysis, as
quantum computing offers an exponential speed-up over the classical computing,
due to the existence of long-range of nonlocal correlations between qubits.

##  II Preliminary and Fundamentals

###  II.1 Classical Machine Learning Algorithms

There are number of algorithms which can be classified as supervised learning
algorithms. Sentiment analysis, also known as text classification, uses
supervised learning models to classify a dataset as positive or negative. In
this paper, we readdress supervised learning using different classical machine
learning algorithms such as support vector machine, decision tree, and
gradient boosting. For demonstrating the sentiment classification, Ye et al.
used support vector machine and naïve bayes, and demonstrated the support
vector machine to be more efficient for accurate classification [29].
Similarly, Pang et al. also classified a datset using naïve bayes, maximum
entropy classification and support vector machine algorithms and deduced that
support vector machine performs as the best algorithm in comparison to other
algorithms [30]. There are other instances to support the efficiency of
support vector machine algorithms in outperforming other classical algorithms
in terms of accuracy [31, 32]. Gradient boosting classifier is also used to
perform sentiment analysis for achieving significant better results [33].
Moreover, decision tree and naïve bayes algorithm used by Zuo for sentiment
classification of the review dataset [34] leading to decision tree algoritms
outperforming the naïve bayes algorithms. The classical machine learning
algorithms used in this article for sentiment analysis are the well
established algorithms known for their best performance in the literature.
In this section, we briefly describe different classical supervised machine
learning algorithms. For our purpose, we will mainly explore the support
vector machine, decision tree, and gradient boosting algorithms.

####  II.1.1 Decision Tree

Decision tree-influenced by the human classificational method- is based on
divide and conquer approach. It classifies the data by recursively molding the
features into a tree structure. A decision tree is built upside down with the
top node representing the root of tree, inductively splits into several
branches connecting the internal nodes and ends up in the leaf node. Each leaf
node is a decision node representing a particular class.

The emplacement of features at a root or internal node highly impacts the
algorithm’s classification performance. Therefore, attribute selection
measures select the splitting criterion or attribute that optimally divides
the data into smaller chunks until an individual class-label is attained.
Decision tree primarily uses three attribute selection measures- information
gain, gain ratio and gini index. Information gain works in a greedy manner by
selecting a feature that yields highest information. Information gain for a
dataset $Y$ for any feature $X$ is the reduction of Shannon’s entropy of $Y$
given $X$ from the uncertainty of $Y$, i.e.,

$ID(Y;X)=H(Y)-H(Y|X)$

Clearly, information gain also represents the mutual information between $Y$
and $X$. Gain ratio is the successor of information gain as it further removes
the restriction of using only categorical values. It penalizes each attribute
by dividing the information gain with entropy of that attribute; known as
split-information. Further, gini index is a CART algorithm (Classification And
Regression Tree) that splits the dataset based on the feature with minimum
impurity such that

$Gini(Y)=1-\sum_{i=1}^{n}p_{i}^{2}$

where $p_{i}$ is the probability of a feature in dataset belonging to a class.
The main objective of designing a decision tree is to provide a set of
decision rules to ambiguously classify features to get a robust classification
model.

####  II.1.2 Support Vector Machine (SVM)

Support Vector Machine (SVM) is one of the robust classification algorithms
based on the statistical methods introduced by Vapnik [35]. SVM classifies
data points to different classes by finding a hyperplane also known as the
decision surface. The decision surface is chosen in a way that classifies data
points with maximum margin in between the support vectors where support
vectors are nearest points to the hyperplane. Hyperplane further creates a
boundary among data objects in a multi-dimensional feature space. The position
and orientation of the decision surface is clearly influenced by support
vectors. SVM further classifies non-separable input data points by mapping
those to the higher dimensional space to ensure distinct classifications using
the decision surface. One can use the linear SVM for sentiment classification,
because it shows significant performance [36]. The decision boundary equations
are represented as

$\vec{W}\cdot\vec{x_{i}}+b\geq 1\hskip 42.67912pt\mbox{when}\hskip
14.22636pty_{i}=+1$ $\vec{W}\cdot\vec{x_{i}}+b\leq-1\hskip
42.67912pt\mbox{when}\hskip 14.22636pty_{i}=-1$

where $x_{i}$ is the support vector, $W$ is normal to hyperplane, $b$ is
offset and classification of support vector $y_{i}\in\\{-1,1\\}$. The
optimization problem of maximizing the margin therefore turns out to be
minimizing the value $\absolutevalue{\absolutevalue{W}}^{2}/2$ while
satisfying the constraint $y_{i}(W\cdot x_{i}+b)\geq 1$.

####  II.1.3 Gradient Boosting

Ensemble learning uses multiple machine learning models that are strategically
combined to solve a computational problem more efficiently,i.e., ensemble
learning increase the overall performance of classification or prediction
model. In ensemble learning, many weak learners are combined to create a model
with high performance measures. Boosting is an ensemble learning technique
that boosts the weak learner into a stronger one. Gradient Boosting algorithm
is also known as gradient boosting machines where a number of weak learners
are trained over the training data and compositely form a strong learner with
greater accuracy. Weak learners are models that can predict the test data
slightly better than random guessing. For this, gradient boosting optimizes
the loss function using gradient decent by collaborating multiple weak
learners. Here, the gradient is the derivative of a function with input
parameters that needs to be optimized. The loss function can be mathematically
written as

$F=\arg\min_{\alpha}\sum_{i=1}^{n}L(y_{i},\alpha)$

where $\alpha$ is the predicted value, $y$ is the actual value, and $L$ is a
loss function chosen for the classification.

###  II.2 Evaluation Metrics

Evaluating a classification model is an essential step to measure its
performance over the unknown data objects. As machine learning algorithms are
applied to diverse academic domains including the very sensitive ones such as
the medical domain where a misclassification or an incorrect prediction can
have severe implications, performance evaluation is one of the most important
ingredients in machine learning. Hence, modeling evaluation matrices are
critical parameters for the success of machine learning algorithms where
evaluation matrices quantify the model by providing different measures like
accuracy, precision, recall etc. In order to evaluate the performance of a
classification model, instances of correct prediction or classification are
recorded. For instance, confusion matrix is an example of a evaluation matrix
and can be used to record the classification and misclassification records.
The confusion matrix is constructed based on the test data results of a model.
Figure 1 demonstrates a format of a confusion matrix showing that the
confusion matrix contains correct classifications and misclassifications
records in the form of four important values, i.e., TP - True Positive, FP -
False Positive, FN - False Negative and TN - True Negative. A brief
description of these values are enumerated below as

  1. 1.

True Positive (TP) contains number of data points that are actual positive and
also predicted as positive,.i.e, correctly classified by the algorithm as the
class of concern;

  2. 2.

False Positive (FP) represent data points that are actual negative but
predicted as positive, i.e., data points are misclassified as the class of
concern;

  3. 3.

False Negative (FN) contains number of data points that are actual positive
but computed as negative i.e., data points are misclassified as not the class
of concern; and

  4. 4.

True Negative (TN) represents the value for the data points that are actual
negative and also predicted as negative, i.e, correctly classified by the
algorithm as not the class of concern.

Further, the performance of classification model in term of accuracy ,
precision, recall and F1- score is evaluated using all these values, namely

Figure 1: Confusion matrix for performance evaluation for a problem involving
two classes

  * •

Accuracy :  Accuracy is evaluated as the ratio of total number of data
instances which are classified accurately to the total number of
classification done.

$Accuracy=\frac{TP+TN}{TP+FP+FN+TN}$ (1)

  * •

Precision :  Precision evaluates the total number of truly predicted positives
for the positive class or concerned class. Precision and false positive are
inversely proportional, therefore, as the value for precision increases,
number of instances with false positive decreases. In the similar way,
precision can be defined for the negative class as well. In the nutshell,
precision measures the exactness or reliability of a model.

$\displaystyle Precision(P)=\frac{TP}{TP+FP}$ $\displaystyle
Precision(N)=\frac{TN}{TN+FN}$ (2)

  * •

Recall :  Recall determines the predicted positive class which was actually
positive. In other words, recall measures the sensitivity of the model. Higher
the recall, lesser will be the number of predicted false negative or false
positive instances.

$\displaystyle Recall(P)=\frac{TP}{TP+FN}$ $\displaystyle
Recall(N)=\frac{TN}{TN+FP}$ (3)

  * •

F1- Score :  F1- Score combines precision and recall values to yield a single
measure for the performance of the model. F1-score measures the testing
accuracy of the model, defined as the harmonic mean of the precision and the
recall.

$F1-Score=2*\left(\frac{Precision*Recall}{Precision+Recall}\right)$ (4)

  * •

Receiver Operating Characteristics (ROC) Curve :  In order to visualize the
performance of classification model, receiver operating characteristics (ROC)
curve is a useful tool. The ROC curve primarily measures the efficacy of the
model in the prediction of true positive while neglecting the incidence of
false positive. In ROC curve, the false positive rate is plotted with respect
to true positive rate.

$\displaystyle TruePositiveRate(TPR)=\frac{TP}{TP+FN}$ $\displaystyle
FalsePositiveRate(FPR)=\frac{FP}{TN+FP}$ (5)

###  II.3 Quantum Computing and Quantum Machine Learning Algorithms

The statistical nature of quantum theory puzzled many physicist including
Einstein which further prompted him and couple of his colleagues to raise
questions on foundations of quantum mechanics, now known as Einstein-Podolsky-
Rosen (EPR) paradox [37]. However, J. S. Bell proposed another thought
experiment by designing an inequality whose violation - the Bell inequality-
confirms the presence of correlations between subsystems of a composite system
contradicting the very assumptions of locality and realism as advocated by EPR
[38]. The fundamental debate has now translated to the practical
implementation of fault tolerant quantum computer which can utilize the
nonlocal aspects of quantum theory to perform efficient tasks which are either
not possible using classical resources or may require exponential time and
memory even with the use of best of classical resources. Of course, there may
be several instances of NP-hard problems which would also be quantumly hard
but then one do not expect quantum computers to provide solutions to all
problems. Nevertheless, a quantum computer can efficiently solve many
instances of classically hard problems. The technological developments have
ensured a rapid advancements towards realization of a quantum computer, at
least with more than 100 qubits. Even with the intricate challenges of
controlling, measuring, and accessing the quantum information, the future of
quantum computation to address large-scale computational issues is highly
promising. Quantum computing has the capability to embrace machine learning by
strengthening the analysis through quantum machine learning algorithms. In
this section, we briefly discuss foundational aspects of quantum computing and
quantum machine learning algorithms that we use for sentiment analysis.

####  II.3.1 Fundamentals of Quantum Computing

Quantum computation uses the fundamental aspects of quantum theory for the
paradigm shift in computing from classical to quantum regime [39, 40]. For a
two-level system, the fundamental unit is termed as a qubit with can be
represented as an arbitrary linear superposition of the two orthogonal basis
vectors. To be precise, a qubit- using computational basis states $\ket{0}$
and $\ket{1}$\- can be described as $\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$
where $\alpha$, $\beta$ $\in$ $\mathbb{C}$ and
$\absolutevalue{\alpha}^{2}+\absolutevalue{\beta}^{2}=1$. For multiple qubits,
the inherent nature of superposition leads to another interesting quantum
mechanical phenomena known as entanglement [41]. For two qubit pure entangled
systems, the four maximally entangled Bell states are extensively used for
optimal communication and computational tasks [42, 43]. For more than two
qubit systems, there exist multiple classes of entanglement due to the complex
nature of multiqubit entanglement [44]. The entangled systems violating the
Bell or Bell-type inequalities exhibit nonlocal correlations with no classical
analogues. The nonclassical correlations are not limited to entangled systems
only, but can be extended to separable ones as well [45, 46]. It has been well
established now that entanglement and nonclassical correlations offer
efficient advantages to quantum computation over classical resources.

Analogous to the classical circuits, quantum circuits also comprise of wires
and quantum gates. However, the only constraint quantum gates need to satisfy
is that the operator representing the gate must be a unitary operator- unlike
classical gates quantum gates are reversible [47]. For example, the single
qubit quantum gates are X-gate, Y-gate and Z-gate; commonly known as Pauli
gates. One of the most valuable single qubit quantum gate is the Hadamard gate
or H-gate which evolves a single qubit state into a superposition state.

####  II.3.2 Quantum Support Vector Machine (QSVM)

Supervised quantum models are summarised as a kernel method where analysis or
classification of the data is performed in a high dimensional Hilbert space
accessible through inner products [48]. Quantum support vector machine
represents a quantum-classical hybrid approach where kernel value is computed
using a quantum computer and the classification process is performed on a
classical machine. For computing the kernel values, classical data must be
converted into quantum states. Therefore, first the classical data points are
transformed to quantum states using feature mapping techniques, and then the
kernel values are estimated. In fact, the kernel is analogous to calculating
inner products between quantum states [49] such that

$K(\vec{x_{i}},\vec{x_{j}})=\absolutevalue{\bra{\psi_{\vec{x_{i}}}}\ket{\psi_{\vec{x_{j}}}}}^{2}$
(6)

where $\vec{x_{i}},\vec{x_{j}}$ $\in$ dataset and
$\absolutevalue{\bra{\psi_{\vec{x_{i}}}}\ket{\psi_{\vec{x_{j}}}}}^{2}$ can be
further defined as

$\absolutevalue{\bra{\psi_{\vec{x_{i}}}}\ket{\psi_{\vec{x_{j}}}}}^{2}=\absolutevalue{\bra{0^{n}}\mathcal{U}^{\dagger}_{\psi(\vec{x_{i}})}\mathcal{U}_{\psi({\vec{x_{j}}})}\ket{0^{n}}}^{2}$
(7)

From Eq. (7), one clearly see that kernel values can be practically computed
using quantum circuits by evolving the initial state $\ket{0}$ under the
influence of a unitary operator $\mathcal{U}_{\psi(x)}$ and then finally
measuring the state in computational basis. Eventually, the computed kernel
value is passed to the classical SVM for classification, i.e.,

$class=sign\left(\sum_{i=1}^{d}y_{i}\alpha_{i}K(\vec{x_{i}},\vec{z})+b\right)$
(8)

where $b$ represents offset. Here, if the sign of classical SVM is positive
then the state belongs to a positive class and if the sign of classical SVM is
negative then the state belongs to a negative class [35].

Figure 2: The proposed pipeline for analysis of the text data

##  III Related Work

The fusion of machine learning with quantum computing projects a promise to
develop several new and exciting algorithms- including the variational
algorithms- for optimal speed-up to large-scale data driven tasks [50, 13,
51]. In this section, we briefly discuss some of the algorithms developed in
the field of quantum machine learning.

Quantum computing benefits classical supervised and unsupervised machine
learning algorithms. Rebentrost et al. have shown the implementation of
quantum version of the support vector machine using phase estimation and
quantum matrix inversion algorithm for maximizing the speed [52]. Another
significant quantum support vector machine was proposed by Havelick et al.
which was also implemented on IBMQ open-source software [49]. The proposal was
based on evaluating the kernel of classical support vector machine quantum
mechanically analogous to computing the inner product of the quantum states.
Lu et al. [53] demonstrated the construction of a quantum decision tree where
the von-Neumann entropy determines the split over the node and fidelity
measures the distance between two quantum states. On the experimental front,
the efficient use of quantum computing in classifying high-dimensional vectors
and mathematical routines was further established using an entanglement based
classification [54]. Schuld et al. introduced a quantum pattern classification
algorithm based on k-nearest neighbours where the distance between states is
measured using the Hamming distance [55]. Moreover, they also utilized the
proposed algorithm for classifying Modified National Institute of Standards
and Technology (MNIST) dataset [55]. The results facilitated Ruan et al. to
design a quantum k-nearest neighbour algorithm for calculating the Hamming
distance between training inputs and testing state [56]. On similar lines,
Wiebe et al. also presented a quantum algorithm for nearest neighbour and
k-means clustering algorithm, evolving quantum computing into the realms of
unsupervised machine learning [57]. For this, they have proposed two methods
for calculating the distance between vectors using a quantum computer, namely
inner product method and euclidean method. Another hybrid k-means clustering
algorithm was proposed by Sarma et al. for finding k-subsets having less
dissimilarity among its members [58]. A standard k-means clustering algorithm
is based on finding the distance usually taken as euclidean distance.
Therefore, in this case, euclidean distance is evaluated using quantum
circuits and classification of data points is executed classically. They have
also implemented the algorithm on the IBMQ open-source software and
demonstrated its accuracy to be better then the classical k-means algorithm.
There are several other instances of significant contributions to quantum
machine learning algorithms for supervised as well as unsupervised learning in
comparison to classical algorithms [59, 60, 14, 61, 62, 63, 64].

The first instance of understanding the neural networking of human brain using
quantum computing was demonstrated by Kak [65]. Menneer and Narayanan further
developed the idea to discuss a practical approach of using quantum computing
in neural networks [66]. In order to obtain the quantum inspired neural
network, they first trained a number of neural networks with respect to the
training inputs and then evaluated the superposition of these networks.
Ventura and Martinez further proposed a method for quantum associative memory
based upon Grover’s algorithm [67] and shown the results leading to
exponential storage capacity of associative memory [68]. Based on the results
of Ventura and Martinez, Zhou et al. developed a model for the quantum
associative neural network to generate a quantum binary decision for quantum
array to store patterns [69].

Evidently, in all quantum and classical algorithms, dataset plays a crucial
role. The performance of algorithms always depend on the type of data points,
dimension of the dataset, outliers etc. In view of this, Huang et al. [70]
further established the power of the data by considering different quantum
kernels and their comparisons with classical algorithms and the advantages of
quantum algorithms when the geometric difference is very high.

Clearly, the performance of an algorithm can be analysed by testing it over
different types of real-valued datasets. The quantum support vector machine
was originally tested over the ad-hoc numerical dataset. The sentiment
analysis of an unseen data using quantum algorithms is still at a nascent
stage. Therefore, in this study, we perform a comparative sentiment analysis
using quantum and classical algorithms based on different performance metrics.
Depending on the number of features, our analysis quantifies the advantages of
quantum machine learning algorithms for classifying the IMDB review data. We
demonstrate that the quantum support vector machine is a much better and
efficient algorithm in comparison to the best of classical algorithms.

##  IV BASIC STRUCTURE OF PROPOSED APPROACH

Figure 3: Pre-processing steps for useful feature extraction from the raw
dataset

In this work, we analyse sentiments of textual data considering its
applications in spam filtering, intention mining, product analysis and market
research, to name a few. Fig. 2 clearly depicts our approach through for the
sentimental analysis. For this, we use the IMDb movie review dataset
downloaded from Kaggle [71] in which the reviews are stored in text format and
a single review contains 200 or more words. Moreover, the dataset contains
40,000 reviews, out of which 20,019 belong to the negative class and 19,981
belong to the positive class of sentiments. For our purpose, corresponding to
each review, a negative (0) or positive (1) label is stored under the label
column. As the raw text data generally includes redundant information with no
adequate contribution to the classification process, the pre-processing of
data such as removing special characters and punctuation (@,#,$\%$,:, and many
more), lower-casing, removing stop-wards, and limmatizing the words becomes
very significant. For example, Fig:3 shows the pre-processing steps and their
effects on the text data.

Figure 4: The word cloud of frequently appeared words in the whole documents

In order to convert reviews into numerical form, the next important step is to
perform vectorization of the pre-processed text data as accepted by machine
learning algorithms. In the proposed strategy, we use CountVectorizer, also
known as Bag of Words (BoW). The BoW evaluates the frequency associated with
each token that occurs in a document. For this, BoW creates a set of words
present in the whole dataset and computes the frequency for each token by
analyzing the occurrence of words in a document. BoW may also contain
frequently appearing words in the created set, therefore, we do not consider
words appearing in more than $80\%$ of documents and less than $20\%$ of
documents. BoW further produces a sparse matrix where some of the tokens
acquire a large value and others do not. Hence, we normalize the resulted
dataset by dividing feature values with the square root of the sum square of
all features, given as

$f^{{}^{\prime}}_{i}=\frac{f_{i}}{\sqrt{\sum_{i=1}^{d}f_{i}}}$ (9)

Here, $f_{i}$ is the feature value, and $d$ represents dimensions of the data.
The Fig. 4 represents a cloud of words where the size of words shows their
occurrence in the dataset.

###  IV.1 Feature Selection

The dataset originally contained a total of $57,658$ unique words with most of
them appearing less than 10 times in whole documents, which may lead to over-
fitting. For a better analysis of the text data, we specifically require those
features that show their significance during classification. For feature
selection, we first use the recursive feature elimination (RFE) method which
recursively eliminates features not contributing to the classification. RFE
assigns weights to features using an estimator or model. In this paper, we use
the random forest model as an estimator for RFE. The random forest classifier
is based on the concept of bootstrap aggregating and uses bagging for features
that make it immune to overfitting. We further optimize parameters for the
number of trees to be used and opt for the Gini index as a splitting criterion
and constraints to use the square root of total number of features in a random
forest. Random forest thus selects the best decision tree through which we
obtain best features. RFE recursively performs this process on all features
randomly and assigns weights to features on their appearances. Corresponding
to all features obtained from RFE, we further characterize these features for
their significance in classification. Therefore, we also use Lasso (Lease
Absolute Shrinkage Selector Operator), which regularizes features by
penalizing. For this, we use the normalized data $(x_{i},y_{i})$, where
$i=1.....n$, to approximately compute $y_{i}$ while minimizing the cost
function. As a key point, Lasso adds the sum of absolute value of a
coefficient parameter to the cost function given as

$w^{{}^{\prime}}=argmin[C(w)+\sum_{i=0}^{d}\lambda\absolutevalue{w_{i}}]$ (10)

Here $C(w)$ is the cost function and $w$ is the coefficient parameter which is
determined by $\lambda$. Specifically, $\lambda$ is a hyper-parameter that
controls the complexity and needs to be determined. The high value of
$\lambda$ does not allow the coefficient parameter to attain a significant
value; on the other hand, if $\lambda$ is $0$, the model overfits. Therefore,
we optimize the value for $\lambda$ to get the best essential features.
Moreover, we allow the coefficient parameter to attain a negative value in
order to get features from negative text reviews. Lasso gives us the
coefficient with respect to features.

Fig. 5 shows the comparison curve for Lasso and RFE. The figure clearly
indicates that RFE shows features to be either significant or not by analyzing
0 or 1 value corresponding to features; whereas Lasso shows the coefficient
value for features. The figure further demonstrates that features attaining a
0 value for RFE also get a coefficient value nearly equal to 0 and therefore
get the last preference while selecting features.

Figure 5: Feature Selection: Using RFE and Lasso, where RFE gives 0 and 1
values for features and Lasso gives the coefficient values

Finally, we construct the database using selected features and proceed with
the analysis using classical machine learning algorithms. In contrast to
classical machine learning algorithms, for the quantum machine learning
algorithms, we first construct a model for mapping classical features into
quantum states. Therefore, we now proceed to discuss the model for mapping
classical data to quantum states in the following subsection.

###  IV.2 Feature Mapping

Once the features are selected as per the requirements of the near-term
quantum computer, we proceed with the development of a feature mapping model.
For feature mapping, the classical data $\vec{x_{i}}$ $\in$ $\mathbb{R}^{n}$
is embedded into quantum states $\ket{\phi(\vec{x_{i}})}$ through the unitary
transformation $\mathcal{U}_{\phi(\vec{x})}$ such that

$\ket{\phi(\vec{x_{i}})}=\bigotimes_{i=1}^{n}\mathcal{U}_{\phi(\vec{x})}(x_{i})$
(11)

Fig. 6 represents the circuit implementation for the unitary transformation.
We initiate the circuit with $\ket{0}$ states and create a superposition state
by applying Hadamard gates on all qubits. The number of features determines
the number of qubits used in the circuit. The superposition state is then
evolved as shown in Fig. 6. Here ZZ gate is a combination of CNOTs and
rotation gate along Z axis such that the angle is determined by a function
given as

$\phi(\vec{x_{i}},\vec{x_{j}})=(\pi-x_{i})(\pi-x_{j})$ (12)

where $\phi(\vec{x})$ is a classical non-linear function which depends on
feature values $x_{i}$ and $x_{j}$ of a document associated with the dataset.
The CNOT gates entangle a qubit with its successor in a linear form. Moreover,
UC gate is a combination of $R_{x}$ and $R_{y}$ gates, where $R_{x}$ is
rotation gate with a fixed angle; whereas $R_{y}$ is a feature value $(x_{i})$
dependent gate. Eventually, the designed feature map transforms classical data
points into higher dimensional space ($\mathbb{C}^{2n}$-dimensions) and
facilitates computing the kernel value for the quantum support vector machine.

Figure 6: The feature map circuit used to embed classical data into the
quantum state

###  IV.3 Classification

We now proceed to classify sentiment text data using quantum and classical
machine learning algorithms. For implementing the quantum machine learning
algorithms, we use IBM quantum systems. Considering that the current IBM
quantum systems permit 5-queries at a time therefore, we start with a subset
of the data with 150 samples for efficiently implementing it over the
available near-term quantum computer. As the dataset is small, we train and
test the data multiple times using K-fold stratified cross-validation to
develop a generalized model and preserve the percentages of samples from both
positive and negative classes during the sampling of the dataset. In addition,
we also use 10-folds which samples the dataset 10-times into train and test
data points. Further, for a classical support vector machine, we tune the
kernel function and analyze the linear kernel ($k(x,y)=x^{T}y+C$) which
performs the best for text classification dataset where $C$ is the
regularization parameter. The value for $C$ is also fine-tuned, but it does
not contribute significantly in classification, and hence is considered as
1.0.

To compare the classification performances of models, we use the quantum
support vector machine where kernel values are evaluated using the feature map
circuit. Further, measurements on each qubit are performed in the
computational basis and measurement results are translated to the classical
SVM. We further analyze parameters $C$ and $\gamma$ for classification where
$\gamma$ is curvature of hyperplane and find that for quantum SVM also these
parameters do not contribute efficiently towards classification. In fact, the
use of $C$ and $\gamma$ leads to a decrease in accuracy. Another factor that
plays an important role in classification is repetition of feature mapping
circuits. For our analysis, we required to repeat the feature map circuit
twice for the five-dimensional data in order to achieve a better accuracy. The
number of repetitions may keep changing with the dimensions of data. A similar
observation can be made for the number of shots. As stated above, the
performance of models is measured using a confusion matrix, precision, recall
and F1-score.

##  V Results

For our analysis, we train the classical and quantum support vectors using
parameters specified in the previous subsection- the classical models were
implemented in Anaconda platform using scikit-learn and the quantum model was
implemented on IBM-quantum experience using qasm-simulator [72]. Moreover,
classical supervised algorithms such as decision tree and gradient boosting
classifier are also used for comparing the performance evolution of the
models. Fig. 7 clearly demonstrates the advantage of using quantum support
vector machine in terms of accuracy, precision, recall and F1-score. As
discussed above, the performance of a classification model can not ascertained
by analysing the accuracy only. The factors such as precision and recall
further play a significant role in determining specificity and sensitivity of
the model, respectively.

Figure 7: Performance measures of classical and quantum support vector
machines

Table:2 further summarizes minimum accuracy, maximum accuracy and standard
deviation obtained using both classical and quantum SVMs. Moreover, Table:2
shows that the use of quantum support vector machine results in significantly
better results in comparison to all other classical supervised models used in
this paper. Among the classical models, classical support vector machine is
found to be the highest performing model which can be used as a benchmark.

| CSVM | QSVM
---|---|---
Minimum Accuracy | 40 | 47
Maximum Accuracy | 80 | 93
Standard Deviation Accuracy | 12.98 | 15.24
Table 1: A comparison between classical and quantum support vector machines
based on Accuracy

Models | Accuracy | Precision | Recall | F1-Score
---|---|---|---|---
Classical Support Vector Machine | 64.1 | 67.96 | 63.99 | 59.84
Decision Tree | 51.99 | 52.77 | 51.99 | 51.98
Gradient Boosting Classifier | 60.66 | 61.94 | 60.66 | 60.70
Quantum Support Vector Machine | 68.6 | 69.49 | 68.66 | 67.25

Table 2: Average values of K-Fold cross validation Accuracy, Precision,
Recall, and F1-Score

Finally, we also evaluate the performance of models using ROC-AUC values as
depicted in Fig. 8 where AUC stands for area under ROC curve. The ROC curve
represents a plot- for best ROC-AUC value from k-fold runs- for true positive
rate (TPR) versus false positive rate (FPR) where the TPR and FPR are
predicted probabilities for test data points by respective models. Therefore,
our results suggest that the quantum support vector machine will predict any
unseen instance as positive with the highest probability if that instance is
actually positive.

Figure 8: The ROC-curve between quantum and classical model at their best ROC-
AUC values

Furthermore, Fig. 9 clearly demonstrates the advantages of using quantum
support vector machine with increased number of features as compared to
classical support vector machine. In fact, for the use of 13 features, we
obtain a highly significant difference of $12.6\%$ in the precision value
using a QSVM as opposed to a CSVM. For the classical vector machine, we
observe an increase in precision score near 8 feature region and for the
quantum support vector machine, we witness the highest precision score around
12-14 feature region. The quantum kernel technique used in this paper, can
potentially be used for a correlated dataset.

Figure 9: Precision score of classical and quantum SVMs with increasing number
of features

##  VI Conclusion

In this work, we performed a comparative analysis to classify the text data of
IMDb movie reviews using quantum support vector machine in comparison to
classical machine learning models such as decision tree, gradient boosting
classifier and classical support vector machine. Our results have shown
quantum support vector machine to outperform all classical models in terms of
different evaluation metrics. The analysis presented here clearly demonstrates
the efficiency of quantum quantum support vector machine in successfully
classifying an unseen data with maximum prediction probability. Among all
classical models, the classical support vector machine is found to be the most
efficient in sentiment analysis of the used data. Therefore, we
comprehensively analyse the classical and quantum support vector machines with
increased number of features or dimensions to ascertain the advantages of
quantum support vector machine for achieving significantly better precision
score and other metrics. For our analysis with thirteen features, the
precision score using the quantum support vector machine is $12.6\%$ higher
than the case where classical support vector is used. The results indicate
that the quantum support vector machine performs optimally with small training
and hence can be further utilized in health care industries where higher
prediction probability and sensitivity of the model are of utmost importance
with small datasets. A future problem of particular interest would be to
optimize the kernel to further reduce the computational time with an improved
efficiency of algorithms.

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