# Spinorial Wheeler–DeWitt wave functions inside a horizon

Nahomi Kan<EMAIL_ADDRESS>National Institute of Technology, Gifu College,
Motosu-shi, Gifu 501-0495, Japan  Takuma Aoyama<EMAIL_ADDRESS>Graduate School of Sciences and Technology for Innovation, Yamaguchi
University, Yamaguchi-shi, Yamaguchi 753–8512, Japan  Kiyoshi Shiraishi
<EMAIL_ADDRESS>Graduate School of Sciences and Technology for
Innovation, Yamaguchi University, Yamaguchi-shi, Yamaguchi 753–8512, Japan

###### Abstract

We revisit the solution of the Wheeler–DeWitt (WDW) equation inside the
horizon of spherical black holes and planar topological black holes in
arbitrary dimensions. For these systems, the solutions of the equations are
found to have the same form. Therefore, Yeom’s Annihilation-to-nothing
interpretation can be applied to each case. We have introduced the Dirac-type
WDW equations into quantum cosmology in a recent paper, so we also apply our
formulation to the quantum theory of the interior of the black hole in order
to obtain the solution of the spinorial wave function. The shape of the wave
packet of the spinorial WDW wave function indicates that the variation of
Yeom’s interpretation holds in this scheme.

###### pacs:

04.20.Gz, 04.20.Jb, 04.50.-h, 04.60.-m, 04.60.Ds, 04.60.Kz, 04.70.Dy,
98.80.Qc.

##  I Introduction

Theoretical study on the black hole singularity problem continues with various
approaches. Although the quantum theory of gravity Kiefer0 ; Kiefer1  itself
has not been completed, many authors apply the quantum cosmological approach
(see, for example, Refs. CAF ; AB ; CS  and references therein), which is a
method of conducting quantum nature of gravity. This is because inside a
symmetric horizon, the roles of the time ($t$) and the radial coordinates
($r$) are exchanged, so that the spacetime of the interior of a black hole is
regarded as a time-dependent universe and can be described by the
Kantowski–Sachs metric KS , for instance.

Recently, many works using the minisuperspace Wheeler–DeWitt (WDW) equation
inside the horizon have appeared BBCCY ; Yeom1 ; Yeom2 ; BCY ; Perry1 ; Perry2
; Hartnoll . In particular, Yeom and his collaborators BBCCY ; Yeom1 ; Yeom2 ;
BCY  have constructed a concrete wave packet and proposed the Annihilation-to-
nothing interpretation based on its typical behavior. One of the motivations
for the present study is to consider further various types of wave packets and
to reveal the properties of the quantum spacetime inside the horizon.

In our previous paper, the authors of the present paper introduced the Dirac
wave equation into quantum cosmology on the basis of the extended
minisuperspace KAHS1 ; KAHS2 . We now apply our formulation to the quantum
theory inside the horizon. Since this spinorial formulation leads to a
positive definite probability density, it can be applied to further
consideration of the singularity problem.

The rest of the paper is organized as follows. In Sec. II, we review the WDW
equations inside the horizon of spherically symmetric black holes and planar
topological black holes in general dimensions. We obtain their fundamental
solutions and construct wave packets using them. In Sec. III, we introduce the
Dirac-type wave equation as an alternative to the WDW equation. We construct
the wave packets and compare them with the wave packets of scalar wave
functions obtained in the previous section. We conclude the present paper with
a summary of results and a discussion in the last section.

##  II A review of the WDW equation inside a black hole

In this section, we review the derivation of the minisuperspace WDW equation
inside a black hole. First, we consider the Einstein–Hilbert action in
$D$-dimensional $(D\geq 4)$ spacetime,111Strictly speaking, the Gibbons–York
terms York ; GH  should be noted here, but is omitted as usual.

$S=\frac{1}{D-2}\int d^{D}x\sqrt{-g}\,R\,,$ (1)

where $g$ is the determinant of the metric tensor $g_{\mu\nu}$, and $R$
denotes the scalar curvature constructed from $g_{\mu\nu}$
$(\mu,\nu=0,1,\ldots,D-1)$.

As the metric of the black-hole interior, we assume the Kantowski–Sachs-type
metric KS , which represents a homogeneous space with spatial section of
topology $\approx S^{D-2}\times\mathbf{R}$,

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-e^{2n(t)}dt^{2}+e^{2\alpha(t)}dr^{2}+r_{s}^{2}e^{2\beta(t)}d\Omega_{D-2}^{2}\,,$
(2)

where $d\Omega_{D-2}^{2}$ is the line element of the $(D-2)$-sphere with unit
radius, and $r_{s}$ is a constant with dimension of length. The metric (2) is
related to the Schwarzschild–Tangherlini interior metric Tangherlini  by, for
example,

$e^{2\beta}=\frac{t^{2}}{r_{s}^{2}}\,,\quad
e^{2\alpha}=\frac{r_{s}^{D-3}}{t^{D-3}}-1\,,\quad
e^{2n}=\left(\frac{r_{s}^{D-3}}{t^{D-3}}-1\right)^{-1}\,,$ (3)

where we should notice that the time and radial coordinates are mutually
exchanged from the definition on the exterior solution.

Now, we derive the WDW equation for the interior of a spherical black hole. We
should note that the lapse function $n(t)$ can be arbitrarily chosen because
of the reparametrization invariance on $t$. Thus, we here choose

$n(t)=\alpha(t)+(D-2)\beta(t)\,.$ (4)

Substituting the metric (2) with the gauge choice (4), we find222The overall
constant factor can be absorbed in the redefinition of the time.

$S\propto\int
dt\left[-2\dot{\alpha}\dot{\beta}-(D-3)\dot{\beta}^{2}+(D-3)r_{s}^{2(D-3)}e^{2\alpha+2(D-3)\beta}\right]\,.$
(5)

Therefore, we obtain the reduced action

$S\propto\int
dt\left[\frac{1}{D-3}(\dot{X}^{2}-\dot{Y}^{2})+(D-3)r_{s}^{2(D-3)}e^{2Y}\right]\,,$
(6)

where $X(t)=\alpha(t)$ and $Y(t)=\alpha(t)+(D-3)\beta(t)$, and consequently we
obtain the following Hamiltonian of the system:

$H(X,Y,\Pi_{X},\Pi_{Y})=(D-3)\left[\frac{1}{4}(\Pi_{X}^{2}-\Pi_{Y}^{2})-r_{s}^{2(D-3)}e^{2Y}\right]\,,$
(7)

where $\Pi_{X}$ and $\Pi_{Y}$ are the canonical momenta conjugate to $X$ and
$Y$, respectively.

Now, the Hamiltonian constraint leads to the (minisuperspace) WDW equation for
the scalar wave function $\Phi(X,Y)$,

$\left(\frac{\partial^{2}}{\partial X^{2}}-\frac{\partial^{2}}{\partial
Y^{2}}+4r_{s}^{2(D-3)}e^{2Y}\right)\Phi(X,Y)=0\,,$ (8)

is obtained after the replacement of
$\Pi_{X}\rightarrow\hat{\Pi}_{X}=-i\frac{\partial}{\partial X}$
$\Pi_{Y}\rightarrow\hat{\Pi}_{Y}=-i\frac{\partial}{\partial Y}$ in the quantum
Hamiltonian constraint $\hat{H}\Phi=0$ Kiefer0 ; Kiefer1 ; Halliwell ; Kiefer3
. Then, we just reproduced the equation presented by Yeom et al. BBCCY ; Yeom1
; Yeom2 ; BCY . Note that this equation is independent of the spacetime
dimensions up to the power of $r_{s}$.

Before discussing the solutions of the WDW equation, we consider another
system, the interior of a planar topological black hole Lemos ; HL ; Vanzo ;
Mann ; Birmingham . The action taken here has a negative cosmological term
added,

$S_{\lambda}=\frac{1}{D-2}\int d^{D}x\sqrt{-g}\,\Bigl{[}R+2\lambda\Bigr{]}\,.$
(9)

The metric of the interior of a planar topological black hole reads

$ds^{2}=-\left(\frac{r_{s}^{D-3}}{t^{D-3}}-\frac{t^{2}}{L^{2}}\right)^{-1}dt^{2}+\left(\frac{r_{s}^{D-3}}{t^{D-3}}-\frac{t^{2}}{L^{2}}\right)dr^{2}+t^{2}d\bar{\Omega}^{2}_{D-2}\,,$
(10)

which is the classical solution of the Einstein equation obtained from varying
the action (9). Here, the length scale $L$ is given by
$L\equiv\left(\frac{2\lambda}{(D-1)(D-2)}\right)^{-1/2}$, and
$d\bar{\Omega}^{2}_{D-2}$ denotes the line element of a flat $(D-2)$
dimensional torus.

Substituting the previous metric (2), but with $d{\Omega}^{2}_{D-2}$ replaced
by $d\bar{\Omega}^{2}_{D-2}$, into the action (9), we obtain

$S\propto\int
dt\left[-2\dot{\alpha}\dot{\beta}-(D-3)\dot{\beta}^{2}+\frac{D-1}{L^{2}}r_{s}^{2(D-2)}e^{2\alpha+2(D-2)\beta}\right]\,.$
(11)

Further setting $X(t)=\alpha(t)-\beta(t)$ and $Y(t)=\alpha(t)+(D-2)\beta(t)$
gives

$S\propto\int
dt\left[\frac{1}{D-1}(\dot{X}^{2}-\dot{Y}^{2})+\frac{D-1}{L^{2}}r_{s}^{2(D-2)}e^{2Y}\right]\,.$
(12)

Therefore, the Hamiltonian for this system is found to be

$H=(D-1)\left[\frac{1}{4}(\Pi_{X}^{2}-\Pi_{Y}^{2})-\frac{1}{L^{2}}r_{s}^{2(D-2)}e^{2Y}\right]\,,$
(13)

and the corresponding WDW equation reads Hartnoll

$\left(\frac{\partial^{2}}{\partial X^{2}}-\frac{\partial^{2}}{\partial
Y^{2}}+\frac{4}{L^{2}}r_{s}^{2(D-2)}e^{2Y}\right)\Phi(X,Y)=0\,.$ (14)

Note that the translation $Y-\ln(L/r_{s})\rightarrow Y$ just turns the
equation (14) into the equation (8).333At the classical level, the
redefinition of time can also absorb the constant factor in front of the
potential term. Thus, we have only to study the WDW equation (8) for both
systems, the interior of a spherical black hole and that of a planar
topological black hole.

Note that the event horizon is located at $X,Y\rightarrow-\infty$, while the
singularity is located at $X\rightarrow\infty$ and $Y\rightarrow-\infty$, in
both cases.

The fundamental solution of (8) is BBCCY ; Yeom1 ; Yeom2 ; BCY

$\phi_{k}(X,Y)=e^{-ikX}K_{ik}(2r_{s}e^{Y})\,,$ (15)

and general solutions are written by

$\Phi(X,Y)=\int_{-\infty}^{\infty}f(k)\phi_{k}(X,Y)\,dk\,,$ (16)

where the function $f(k)$ denotes the amplitude.

Asymptotics of the modified Bessel function of the second kind with complex
order are known to be Bateman

$K_{ik}(z)\sim\sqrt{2\pi}e^{-\frac{k\pi}{2}}(k^{2}-z^{2})^{-\frac{1}{4}}\sin\left(\frac{\pi}{4}-\sqrt{k^{2}-z^{2}}+k\cosh^{-1}\frac{k}{z}\right)\,,$
(17)

for $k\rightarrow\infty$. According to the analysis of Kiefer Kiefer2 , the
peak of the wave packet tracing the classical path BBCCY ; Yeom1 ; Yeom2 ; BCY

$e^{Y}\cosh X=\mbox{const.}\,,$ (18)

can be constructed with the Gaussian amplitude, whose central value in $k$
takes a relatively large value, owing to the asymptotic form (17).

Nevertheless, we can arbitrarily choose the amplitude $f(k)$, so as to reflect
boundary conditions. The Gaussian or rectangular amplitudes are the other
candidates.444Graphical representation of wave packets of similar wave
functions in quantum cosmology are dealt with Refs. CGS ; ALNW ; KKST  et al.
Another type of the wave packet with an analytical form is considered by Refs.
BBCCY ; Yeom1 ; Yeom2 ; BCY , where the simple choice $f(k)=ik$ leads to

$\Phi_{1}(X,Y)=\int_{-\infty}^{\infty}k\sin
kX\,K_{ik}(2r_{s}^{D-3}e^{Y})\,dk=2\pi r_{s}^{D-3}e^{Y}\sinh
Xe^{-2r_{s}^{D-3}e^{Y}\cosh X}\,.$ (19)

The peak of the absolute square of the wave packet $\Phi_{1}(X,Y)$, which can
be regarded as the probability density Kiefer0 ; Kiefer1 ; Halliwell ; Kiefer3
, exhibits the classical behavior (18) at the boundaries
$X\rightarrow\pm\infty$, as shown in Fig. 1.

Figure 1: Absolute square $|\Phi_{1}|^{2}$ of the scalar wave function for
$r_{s}=1$.

It is worth observing that the quantum effect seems significant in the
vicinity of $X\approx 0$, where classically there is neither a singularity nor
a horizon: The probability density vanishes here $|\Phi_{1}(0,0)|^{2}\approx
0$. One can interpret this wave packet as an annihilation process of the
trajectory from the horizon ($X=-\infty$) and that from the singularity
($X=\infty$). This interpretation is the so-called Annihilation-to-nothing
interpretation proposed by Yeom et al. BBCCY ; Yeom1 ; Yeom2 ; BCY .

In this section, we have reviewed the scalar wave packet of the WDW equation
for the interior of the black hole, leading to Yeom’s interpretation. We have
confirmed that the same interpretation is possible for spherically symmetric
and topological black holes in arbitrary dimensions.

##  III Spinor wave function inside a black hole

In the previous work KAHS1 ; KAHS2 555A similar method has been developed by
Robles-Pérez Perez2 ., the present authors introduced the extended
minisuperspace description with an additional degree of freedom by a method of
the Eisenhart–Duval lift Eisenhart ; Duval ; DGH ; CPC ; Pettini ; Cariglia ;
CGGH ; FK ; Finn ; DSS . Using the covariance in the extended minisuperspace,
we also constructed an associated Dirac equation for spinor wave function. The
introduction of the Dirac-type WDW equation (dubbed spinorial WDW equation) is
originally motivated for obtaining a positive-definite probability density RS
; Ryan ; DHO ; KO ; SC ; YH ; HA ; RAH ; Moniz . The Dirac equation in the
extended minisuperspace has a unique form and conformal covariance.

In this section, we apply the formulation to the present model for a black-
hole interior. In short, the realization of the plan is achieved by finding
the metric of the minisuperspace manifold such that the WDW equation is
written by Laplace equation ($\nabla^{2}\Phi=0$), and drawing down the Dirac
equation ($D\\!\\!\\!\\!/\,\Psi=0$) on that manifold.

The essential idea of the Eisenhart–Duval lift is to introduce the metric
$G_{MN}$ of the extended space with an auxiliary dimension $Z$, that is, in
the present case,

$G_{MN}dX^{M}dX^{N}={4r_{s}^{2(D-3)}e^{2Y}}(-dX^{2}+dY^{2})+dZ^{2}\,,$ (20)

where $X^{M}=(X,Y,Z)$, and consider the Laplace equation in the extended
minisuperspace:

$\frac{1}{\sqrt{-G}}\partial_{M}(\sqrt{-G}G^{MN}\partial_{N}\Phi)=\left[\frac{1}{4r_{s}^{2(D-3)}e^{2Y}}\left(-\frac{\partial^{2}}{\partial
X^{2}}+\frac{\partial^{2}}{\partial Y^{2}}\right)+\frac{\partial^{2}}{\partial
Z^{2}}\right]\Phi=0\,.$ (21)

Here, $G^{MN}$ is the inverse of $G_{MN}$, $G=-(4r_{s}^{2(D-3)})^{2}e^{4Y}$ is
the determinant of $G_{MN}$, and the derivatives are expressed as
$\partial_{M}\equiv\frac{\partial}{\partial X^{M}}$. If we further impose an
additional (first-order) constraint

$\frac{\partial^{2}\Phi}{\partial Z^{2}}=-\Phi\,,$ (22)

we can exactly reproduce the WDW equation (8).

The WDW equation of Dirac-type for a spinor wave function $\Psi$ in the
extended minisuperspace is now naturally introduced by KAHS1 ; KAHS2

$D\\!\\!\\!\\!/\,\Psi\equiv\hat{\gamma}^{M}D_{M}\Psi\equiv\gamma^{A}e_{A}^{M}D_{M}\Psi=0\,,$
(23)

where the constant gamma matrices in the flat spacetime $\gamma^{A}$
($A=1,2,3$) are $\gamma^{1}=\sigma^{1}$, $\gamma^{2}=i\sigma^{2}$, and
$\gamma^{3}=i\sigma^{3}$ ($\sigma^{1}$, $\sigma^{2}$, $\sigma^{3}$ denote the
Pauli matrices). Note that $\\{\gamma^{A},\gamma^{B}\\}=-2\eta^{AB}$, where
$\eta^{AB}=\eta_{AB}=\mbox{diag.}(-1,1,1)$. Here, the dreibein
$e^{A}_{M}=\mbox{diag.}(2r_{s}^{D-3}e^{Y},2r_{s}^{D-3}e^{Y},1)$ is defined
through $\eta_{AB}e^{A}_{M}e^{B}_{N}=G_{MN}$, and
$e_{A}^{M}=\mbox{diag.}((2r_{s}^{D-3})^{-1}e^{-Y},(2r_{s}^{D-3})^{-1}e^{-Y},1)$
is its inverse matrix. Subsequently, we find that
$\\{\hat{\gamma}^{M},\hat{\gamma}^{N}\\}=-2G^{MN}$. The covariant derivative
$D_{M}$ for the spin connection $\omega_{MAB}$ is defined as
$D_{M}\equiv\partial_{M}+\frac{1}{4}\omega_{MAB}\Sigma^{AB}$, where
$\Sigma^{AB}\equiv-\frac{1}{2}[\gamma^{A},\gamma^{B}]$. The spin connection
$\omega_{MAB}$ is given by

$\omega_{MAB}=\frac{1}{2}e^{N}_{A}(\partial_{M}e_{NB}-\Gamma^{L}_{MN}e_{LB})-(A\leftrightarrow
B)\,,$ (24)

where the Christoffel symbol $\Gamma^{L}_{MN}$ is given by

$\Gamma^{L}_{MN}=\frac{1}{2}G^{LP}(\partial_{M}G_{PN}+\partial_{N}G_{PM}-\partial_{P}G_{MN})\,.$
(25)

In the present case, one can find $\omega_{X12}=-\omega_{X21}=-1$.

In the extended minisuperspace presently considered, we find that the Dirac
equation (23) is equivalent to

$\left[\sigma^{1}\frac{\partial}{\partial
X}+i\sigma^{2}\left(\frac{\partial}{\partial
Y}+\frac{1}{2}\right)+i\sigma^{3}(2r_{s}^{D-3})e^{Y}\frac{\partial}{\partial
Z}\right]\Psi=0\,.$ (26)

In order to reduce the equation to that of physical variables $X$ and $Y$, we
choose the additional constraint on $\Psi$:

$\frac{1}{i}\frac{\partial}{\partial Z}\Psi=\Psi\,.$ (27)

Now, the Dirac equation reads in the matrix form,

$\left(\begin{array}[]{cc}-2r_{s}^{D-3}e^{Y}&\frac{\partial}{\partial
X}+\frac{1}{2}+\frac{\partial}{\partial Y}\\\ \frac{\partial}{\partial
X}-\frac{1}{2}-\frac{\partial}{\partial
Y}&2r_{s}^{D-3}e^{Y}\end{array}\right)\left(\begin{array}[]{c}\Psi_{+}\\\
\Psi_{-}\end{array}\right)=\left(\begin{array}[]{c}0\\\
0\end{array}\right)\,,$ (28)

where $\Psi={\Psi_{+}\choose\Psi_{-}}$. The fundamental solutions of the
equation are found to be

$\psi_{\pm,k}\propto\pm K_{ik\pm\frac{1}{2}}(2r_{s}^{D-3}e^{Y})e^{-ikX}\,,$
(29)

and general solutions can be written as

$\Psi_{\pm}(X,Y)=\int_{-\infty}^{\infty}f(k)\psi_{\pm,k}(X,Y)\,dk\,,$ (30)

where $f(k)$ denotes the amplitude.

Since the inner product of two spinors in the curved minisuperspace can be
defined by

$(\vartheta,\varphi)\equiv\int
dY\sqrt{-G}|G^{XX}|^{1/2}\vartheta^{\dagger}\varphi=2r_{s}^{D-3}\int
dY\,e^{Y}\,\vartheta^{\dagger}\varphi\,,$ (31)

the (positive-definite) probability density $\rho(X,Y)$ should takes the form

$\rho(X,Y)\propto e^{Y}(|\Psi_{+}(X,Y)|^{2}+|\Psi_{-}(X,Y)|^{2})\,.$ (32)

Asymptotics of the modified Bessel function of the second kind with complex
order are known to be Tseng

$\displaystyle K_{ik\pm\frac{1}{2}}(z)$ $\displaystyle\sim$
$\displaystyle\sqrt{2\pi}e^{-\frac{k\pi}{2}\pm
i\frac{\pi}{4}}(k^{2}-z^{2})^{-\frac{1}{4}}\Biggl{[}\sqrt{\frac{k+z}{2z}}\sin\left(\frac{\pi}{4}-\frac{1}{2}\sqrt{\nu^{2}-y^{2}}+k\cosh^{-1}\frac{k}{z}\right)$
(33) $\displaystyle\quad\qquad\qquad\qquad\mp
i\sqrt{\frac{k-z}{2z}}\cos\left(\frac{\pi}{4}-\sqrt{k^{2}-z^{2}}+k\cosh^{-1}\frac{k}{z}\right)\Biggr{]}\,,$

as $k\rightarrow\infty$. This function has a similar oscillatory behavior as
$K_{ik}(z)$ (17). The similarity in asymptotics of $K_{ik}(z)$ and
$\sqrt{z}K_{ik\pm\frac{1}{2}}(z)$ also helps to construct wave packet
solutions for the Dirac-type WDW equation as well as the usual WDW equation.
The different behaviors of the functions $K_{ik}(z)$ and
$\sqrt{z}K_{ik\pm\frac{1}{2}}(z)$ seems to be especially revealed in the
vicinity of $X\approx 0$ in the form of the wave packets. Therefore, the
difference between wave-packet solutions of the usual WDW equation and the
Dirac-type WDW equation is expected to be found in the region of turning point
of the classical trajectory.

We find that the simple choice $f(k)=ik$ in (30) leads to

$\displaystyle(\Psi_{1})_{\pm}(X,Y)$ $\displaystyle=$
$\displaystyle\pm\int_{-\infty}^{\infty}ike^{-ikX}\,K_{ik\pm\frac{1}{2}}(2r_{s}^{D-3}e^{Y})\,dk$
(34) $\displaystyle=$ $\displaystyle\pm\pi\Biggl{[}2r_{s}^{D-3}e^{Y}\sinh
X\mp\frac{1}{2}\Biggr{]}e^{\pm\frac{X}{2}}e^{-2r_{s}^{D-3}e^{Y}\cosh X}\,.$

The probability density
$\rho_{1}(X,Y)=e^{Y}(|(\Psi_{1})_{+}(X,Y)|^{2}+|(\Psi_{1})_{-}(X,Y)|^{2})$ is
exhibited in Fig. 2. The peak of the probability density reproduces the
classical path where $|X|$ is large, but a local hump of the density appears
around $X=0$. It would be appropriate to interpret this feature of the
probability density as “Annihilation-to-Something” process.

Figure 2: The probability density $\rho_{1}(X,Y)$ for $r_{s}=1$.

On the other hand, another simple choice $f(k)=1$ in (30) leads to

$\displaystyle(\Psi_{0})_{\pm}(X,Y)$ $\displaystyle=$
$\displaystyle\pm\int_{-\infty}^{\infty}e^{-ikX}\,K_{ik\pm\frac{1}{2}}(2r_{s}^{D-3}e^{Y})\,dk$
(35) $\displaystyle=$ $\displaystyle\pm\pi
e^{\pm\frac{X}{2}}e^{-2r_{s}^{D-3}e^{Y}\cosh X}\,.$

The probability density
$\rho_{0}(X,Y)=e^{Y}(|(\Psi_{0})_{+}(X,Y)|^{2}+|(\Psi_{0})_{-}(X,Y)|^{2})$ is
exhibited in Fig. 3. The peak of the density rather exactly reproduces the
classical path, even where $X\approx 0$. It can be said that there is “no
annihilation” process.

Figure 3: The probability density $\rho_{0}(X,Y)$ for $r_{s}=1$.

Although only simple amplitude functions have been assumed here, it turns out
that the behavior of the density function in the region where the quantum
effect is expected ($X\approx 0$) has more diverse for spinor waves than for
scalar waves.

In this section, we have introduced the spinorial WDW equation for the
interior of the black hole and constructed the typical wave packets. We have
found that, in our spinorial formulation, we can create wave packets with
various interpretations.

##  IV Summary and discussion

We have solved the spinorial wave function (as well as the scalar wave
function as a comparison) of the interior of the Schwarzschild–Tangherlini and
planar topological black holes in $D$-dimensional spacetime, by using the
extended minisuperspace. We have constructed the wave packets of the spinor
wave function, which correspond to the classical trajectories in the
asymptotic region near the horizon and the singularity.

The behavior of the scalar wave packet near $X\approx 0$ suggests the
“Annihilation-to-nothing” interpretation proposed by Yeom et al. BBCCY ; Yeom1
; Yeom2 ; BCY . The probability density of the spinor wave function is
slightly different from that of the scalar wave function, in the vicinity of
$X=0$ for typical examples of the packets. Thus, we speculate that alternative
interpretation of “no annihilation” or “annihilation-to-something” is
possible.

Since our spinorial formulation leads to a positive definite probability
density, it will be applied to further analysis of the expectation values
Hartnoll  near the singularity (even without using wave packets). We shall
leave the issue for future study.

We should also incorporate possible noncommutativity in the variables of the
minisuperspace (see, for example, Ref. GOR  and references therein). The
approach by considering loop quantum cosmology (see Refs. AB ; CS  and Ref.
OZZWW  for references) (or polymer quantization) may be effective for the
analysis at the singularity. We hope that the study in the aforementioned
directions will be advanced in future.

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# Hybrid quantum convolutional neural networks model for COVID-19 prediction
using chest X-Ray images

Essam H. Houssein
Faculty of Computers and Information
Minia University
Egypt
<EMAIL_ADDRESS>
&Zainab Abohashima
Faculty of Computer Science
Nahda University in Beni-Suef
Egypt
<EMAIL_ADDRESS>
&Mohamed Elhoseny
Department of Computer Science
American University in the Emirates
Dubai, UAE.
<EMAIL_ADDRESS>
&Waleed M. Mohamed
Faculty of Computers and Information
Minia University
Egypt
<EMAIL_ADDRESS>

###### Abstract

Despite the great efforts to find an effective way for COVID-19 prediction,
the virus nature and mutation represent a critical challenge to diagnose the
covered cases. However, developing a model to predict COVID-19 via Chest X-Ray
(CXR) images with accurate performance is necessary to help in early
diagnosis. In this paper, a hybrid quantum-classical convolutional Neural
Networks (HQCNN) model used the random quantum circuits (RQCs) as a base to
detect COVID-19 patients with CXR images. A collection of 6952 CXR images,
including 1161 COVID-19, 1575 normal, and 5216 pneumonia images, were used as
a dataset in this work. The proposed HQCNN model achieved higher performance
with an accuracy of 98.4% and a sensitivity of 99.3% on the first dataset
cases. Besides, it obtained an accuracy of 99% and a sensitivity of 99.7% on
the second dataset cases. Also, it achieved accuracy, and sensitivity of
88.6%, and 88.7%, respectively, on the third multi-class dataset cases.
Furthermore, the HQCNN model outperforms various models in balanced accuracy,
precision, F1-measure, and AUC-ROC score. The experimental results are
achieved by the proposed model prove its ability in predicting positive
COVID-19 cases.

_Keywords_ COVID-19, Convolutional Neural Networks (CNN), Chest X-Ray, Medical
Image Classification, pneumonia, Quantum Circuit, Quantum Computing, SARS-
CoV-2.

Abbreviations |  |  |
---|---|---|---
CNNs | Convolutional Neural Networks | RQCs | Random Quantum Circuits
QC | Quantum Computing | COVID-19 | Coronavirus Disease 2019
ReLU | Rectified Linear Unit | DL | Deep Learning
Conv | Convolutional | CXR | Chest X-Ray
CT | Computed Tomography |  |

##  1 Introduction

Recently, COVID-19 has been rapidly spreading in several countries caused by
infection of human beings with severe acute respiratory syndrome coronavirus
$2$ (SARS-CoV-2) [1]. The ongoing COVID-19 pandemic attacks human health
causing respiratory disease and acute kidney injury [2]. COVID-19 stands for
coronavirus disease 2019, which is a new type of coronavirus and falls as a
subtype of RNA viruses [3]. The genetic structure of COVID-19 is identical to
bat-coronavirus RaTG13, MERS-coronavirus, and SARS-coronavirus by 95%, 50%,
and 82% respectively [4, 5]. The first outbreak of the disease was in China at
the end of 2019. The most common clinical symptoms of the disease are fever,
sore throat, cough, dyspnea, and muscle pain [6]. In March 2020, the World
Health Organization (WHO) reported COVID-19 as a pandemic [1]. The most common
diagnostic tool used for COVID-19 prediction is a real-time "reverse
transcription-polymerase chain reaction" (RT-PCR) [7], even though RT-PCR has
low sensitivity with the early phases [8]. Alternative imaging tools (i.e.,
chest X-ray (CXR) and computed tomography (CT) scans) play an important and
critical role in COVID-19 prediction [9]. Radiologists prefer to use the CXR
to diagnose chest diseases. The CXR has been used to detect infected COVID-19
cases [10]. Thus utilizing the artificial neural networks to detect COVID-19
in CXR has become needed.

The use of Artificial Neural Networks (ANNs) in medical data has significantly
advanced. The CNNs are one of the most powerful, and widespread deep learning
models in pattern recognition and image classification. Furthermore, CNNs have
achieved superior performance in medical image detection [11]. Various works
for the prediction of COVID-19 cases using CNNs structure have been proposed,
these works are presented in detail in section "Related Work". Chowdhury et
al. [12] proposed deep CNNs models to pre-trained their model using transfer
learning to discuss the leverage of artificial intelligence (AI) for
classification COVID-19 on CXR images. Especially that the computational power
of classical ANNs can’t learn large training data with low-cost [13]. In a
particular, it is not capable of generating more kernels (kernel estimation)
with high-dimensional features [14, 15]. Effortlessly, quantum computing is
capable of performing complex kernels in n-dimensional space.

On the other hand, the quantum computing (QC) field has proven its significant
role in intractable problems with classical counterparts via quantum supremacy
[16, 17] and with the upgrowth of the concept of quantum computing (QC) and
its improvements in the machine learning field (i.e., learning capacity, run-
time, and learning efficiency) [18]. Quantum computing has also demonstrated a
remarkable influence in machine learning (ML) on near-term quantum computers
[19]. Quantum neural network approaches have been proposed such as [20, 21].
According to our best of knowledge, the start point of QNNs was in 1995. Kak
[22] introduced NNs concepts in the quantum computing world. This study
focuses on the hybrid “quantum-classical” approach. Henderson et al. [23]
presented a new quantum convolutional layer within CNNs based on quantum
circuits to estimate kernel in high-dimensionality. In a similar work [14],
Havlicek et al. proposed a quantum kernel algorithm of support vector machines
to estimate kernel with a quantum circuit and to deal with huge features by
estimating kernel in n-dimensionality space. Mitarai et al. [24] proposed a
new hybrid method called QC learning (QCL) based on a quantum circuit. QCL
works with large datasets for clustering, regression, and classification
tasks.

Therefore, the motivation behind this work is to combine the advantage of the
quantum convolutional layer with CNNs and CXR images as an important tool and
cheapest diagnostic technique of COVID-19 to produce a new proposed HQCNN
model. This model aims to improve the performance of CNNs to detect
coronavirus cases in early phases and to speed up the diagnosis and remedy.
The HQCNN model is measured with two angle encoding gates and a different
number of shots. The better results have been achieved by the Ry angle gate
and 1000 shots. The proposed HQCNN model achieved higher performance with an
accuracy of 98.4%, 99%, and 88.6% on the first dataset( normal and COVID-19
cases), the second dataset (COVID-19 and pneumonia cases), and the third
dataset (normal, COVID-19, and pneumonia cases), respectively. Also, the
higher sensitivity and AUC-ROC scores have been achieved with 99.3%, and 100%,
respectively on the COVID-19 and pneumonia dataset. Besides, the HQCNN model
outperforms various models in specificity, balanced accuracy, precision,
F1-measure, and AUC-ROC score. The experimental results show the ability of
the proposed HQCNN model to classify positive COVID-19 cases. The restrictions
of the HQCNN model are highlighted in Section 5.

The main contributions of this work are:

  * •

A new hybrid CNNs model combined with a quantum circuit called HQCNN for
COVID-19 prediction based chest X-ray images is proposed.

  * •

The comparative performance investigates of HQCNN model using encoding angle
methods and a various number of shots on the quantum device for COVID-19
classification cases.

  * •

HQCNN model is evaluated on binary and multi-class datasets with confirmed
COVID-19 cases.

  * •

An exhaustive and a comparative experimental discussion is presented in terms
of accuracy, balanced accuracy, sensitivity, specificity, precision,
F1-measure, FBeta-measure, AUC-ROC curve, and confusion matrix to evaluate the
performance of the proposed HQCNN model.

The remainder of this paper is organized as follows: Section 2 gives an
overview of recent COVID-19 classification based the CNNs with CXR and CT
images and shows a comparative performance of previous works. Background about
the concept of quantum computing (Qubits, and quantum gates) and CNNs are
presented in Section 3. The hybrid proposed HQCNN model is introduced in
detail in Section 4. Section 5 presents the used datasets, performance
measures, and the experimental results of the hybrid HQCNN model. Finally, the
conclusion of this work and future research are delineated in Section 6.

##  2 Related Work

This section provides various works of COVID-19 classification based on CNNs
that had recently been presented by researchers. Related works are divided
into studies based on CXR images and studies based on CT scans.

In [25], a CNNs model (i.e., VGG19, Xception, MobileNetv2) based on transfer
learning is proposed to extract meaningful biomarkers of coronavirus patients.
Furthermore, the used dataset is collected with 224 COVID-19, 714 pneumonia
(bacterial and viral), and 504 normal CXR images to evaluate the performance
of CNNs models on the classification of medical images. The results are
reported with 96.78% accuracy, 98.66% sensitivity, and 96.46% specificity. A
patch-based technique utilizing statistical analysis with ResNet18 and fully
convolutional-DenseNet103 architectures is presented in [26]. Due to the lack
of CXR images for COVID-19, this technique is used with a small dataset to
classify COVID-19 patients. Moreover, the proposed technique has achieved an
accuracy of 88.9% and a precision of 83.4%. In another study [27], a deep CNNs
model is introduced to detect positive patients of COVID-19, called Truncated
InceptionNet. The inceptionNetV3 is modified with the Truncated model to deal
with limited datasets of COVID-19 to reduce over-fitting. Besides, 162
COVID-19 confirmed cases are added to six several datasets. The proposed model
is evaluated with different datasets to test the ability of the Truncated
InceptionNet model for predicting positive cases of COVID-19 of each dataset.
This model achieved 99.9% accuracy for the fifth dataset with 162
COVID-19,1583 normal, and 4280 pneumonia. The DarkNet technique is presented
for COVID-19 classification on CXR images in [28]. This model is predicted
COVID-19 cases without using a feature extractor algorithms. The DarkNet model
is applied as a classifier for YOLO object detection. This model is used with
binary and multi-class COVID-19 classification. The DarkNet model is achieved
98.08%, 95.13%,95.30%, 96.51% for accuracy, sensitivity, specificity, and
F1-measure, respectively. Besides, the model is achieved 87.02% of accuracy
for a multi-class dataset. In another work [29], a hybrid CNN-LSTM model is
presented using CNNs and long short-term memory to detect COVID-19 cases. The
CNNs are utilized for feature extraction and LSTM for the classification of
images. The CNN-LSTM model is used with 4,575 CXR images, including 1525
COVID-19 images. The CNN-LSTM model has achieved higher results with 99.4%,
99.3%, 99.2%,99.9%, and 99.9% for accuracy, sensitivity, specificity, AUC
score, and F1-measure, respectively. Also, The AUC-ROC curve and confusion
matrix are provided to evaluate the CNN-LSTM model.

In [30], a deep CNN based on Xception model is introduced, called CoroNet. The
proposed model utilized an ImageNet dataset for the pre-trained process. This
model is evaluated on binary and multi-class datasets with 284 COVID-19
images. The proposed CoroNet model accomplished an accuracy of 99%, 95%,89.6%
for binary, three-class, and four-classes datasets. The overall accuracy is
achieved with 89.6%. In this study [31], The VGG16 architecture is used with
transfer learning for COVID-19 classification from CXR images. The two models
are proposed to classify between (COVID-19 vs. healthy) and (COVID-19 vs.
pulmonary) for the first and second models, respectively. The first model
achieved 96% of accuracy and sensitivity, 98% of specificity, and 94% of
F1-measure. The results 98%, 87%, 94%, and 89% for accuracy, sensitivity,
specificity, and F1-measure, respectively, are achieved by the second model.
On the other side, a hybrid GSA-DenseNet121 DL model is presented using a
gravitational search optimizer [32]. The GSA is used to the hyper$-$parameters
tuning of the DenseNet121 model. The random copying is performed to a balance
the used dataset. The GSA-DesneNet121 model is applied with 306 CXR images,
including 99 positive COVID-19 and 207 Non-COVID-19. The proposed model
achieved 98.3% of accuracy. In a similar work [33], a hybrid CNN’s technique
is proposed using a swarm-based optimizer for COVID-19 classification. The
hybrid technique is combined among inception model, Marine Predators, and
fractional-order calculus algorithms. The Inception CNNs model is used for
feature extraction. The Marine Predators algorithm is applied to reduce
extractor features by the selection of meaningful features. The proposed
technique (IFM) is improved using the fractional-order calculus algorithm.
This technique is evaluated on two COVID-19 datasets: the first dataset
includes 200 and the second dataset contains 219 COVID-19 CXR images. The
proposed IFM technique achieved 98.7% of accuracy and 98.2% of f1-score for
the first dataset. The results 99.6%of accuracy, and 99% of f1-score are
obtained for the second dataset.

A new approach [34] is proposed utilizing capsule network and CXR images for
COVID-19 cases detection called the CapsNet model. This model is evaluated on
binary and multi-class datasets. This model has used 1050 CXR images for
COVID-19, normal, and pneumonia classes. For binary dataset, The CapsNet model
is accomplished 97.24%, 97.42%, 97.04%, 97.08%, and 97.24% for accuracy,
recall, specificity, precision, and F1-measure, respectively. Also, the
results 84.22% of accuracy, 84.22% of recall, 91.79% of specificity, 84.61% of
precision, and 84.21% of F1-measure are achieved for the multi-class dataset.
In another separate study [35], a novel system using the EfficientNet model to
diagnose COVID-19 patients. The EfficientNet has used a 10-fold cross-
validation technique for binary and multi-class classification with 1,508 a
total number of CXR images. This model has reported binary class results
99.62% of accuracy, 99.63% recall, 99.64% of precision, and 99.62% of
f1-score. The model achieved 96.70%, 96.69%, 97.59%, and 97.11% for accuracy,
recall, precision, and F1-score, respectively with multi-class classification.

In [36], a 3D CNNs is introduced based on the location-attention model to
distinguish COVID-19 patients from CT images. The dataset images are composed
of 219 COVID-19, 224 pneumonia, and 175 normal images. This model is achieved
86.7% of classification accuracy. In another study with CT images [37], the
CNNs model is proposed with multi-objective differential evolution for tuning
hyper-parameters of CNNs. The results are reported with 0.92% accuracy, 0.90%
for sensitivity, specificity, and F1-measure.

The ResNet50 model is used with multi-view images for the classification of
COVID-19 patients with chest CT scans. The proposed model is achieved an AUC
of 81.9%, the sensitivity of 81.8%, accuracy of 76%, and specificity of 61.5%
in [38]. In [39], a deep learning technique is proposed using DenseNet$-$201
model and transfer learning. The proposed technique is used on 2492 CT images,
including 1262 COVID-19, and 1230 Non-COVID-19 images. The proposed technique
is applied with learned weights of the ImageNet dataset to extract features.
It is achieved accuracy with 97% compared to Resnet152V2, VGG16 models.
Besides, it obtained AUC- ROC score with 0.97%.

The previous studies faced inconsistency issues due to a variety of methods,
images, a variety of data. A brief comparison of the previous research has
been presented in terms of the applied methods, images, data, and
classification performance (as shown in Table 1). Since chest CXR and CT scans
are the second tools to diagnose and detect the COVID-19 pandemic, so the
classification of COVID-19 has become a hot area of research. The researchers
collected datasets from several resources to train and test the models as
shown in Table 1. There is no a standard dataset due to the shortage of
COVID-19 images. Most of the state-of-the-art COVID-19 classification is
designed to fit the currently limited images as [26, 27] and to help in early
diagnosis for COVID-19 cases. There studies focus on prediction confirmed
COVID-19 from COVID-19 and normal datasets such as [28] and another focus on
differentiating confirmed COVID-19 from other pneumonia diseases [27, 38, 32].
Therefore, the prediction of COVID-19 still needs to further accurate and
faster models to assist the radiologists in coming early diagnosis and
treatment.

Table 1:  Comparison of classification performance for existing related work.

Work |  Images |  Type |  Method |  Data |  ACC.(%)
---|---|---|---|---|---
[25] |  224 COVID-19 714 pneumonia 504 normal |  CXR |  MobileNetV2 |  imbalanced |  96.78
[26] |  180 COVID-19 191 normal 131 others |  CXR |  ResNet18 |  imbalanced |  88.9
[27] |  162 COVID-19 |  CXR |  CNNs |  imbalanced |  99.9
[28] |  125 COVID-19 500 normal |  CXR |  DarkNet |  imbalanced |  98.08
[29] |  1525 COVID-19 1525 normal 1525 pneumonia |  CXR |  CNN$+$LSTM |  balanced |  99.4
[30] |  284 COVID-19 330 bacterial 310 normal 327 Viral |  CXR |  Xception |  balanced |  89.6
[31] |  250 COVID-19 2753 pneumonia 3520 normal |  CXR |  VGG16 |  imbalanced |  96
[32] |  99 COVID-19 207 Non-COVID-19 |  CXR |  DenseNet121$+$GSA |  balanced |  98.3
[33] |  200 COVID-19 1,675 Non-COVID-19 |  CXR |  Inception+FO-MPA |  imbalanced |  98.7
[33] |  219 COVID-19 1,341 Non-COVID-19 |  CXR |  Inception+FO-MPA |  imbalanced |  99.6
[34] |  1050 COVID-19 1050 normal |  CXR |  Capsule Network |  balanced |  97.24
[34] |  1050 COVID-19 1050 normal 1050 pneumonia |  CXR |  Capsule Network |  balanced |  84.22
[35] |  504 COVID-19 500 normal |  CXR |  EfficientNet |  balanced |  99.62
[35] |  504 COVID-19 500 normal 504 pneumonia |  CXR |  EfficientNet |  balanced |  96.70
[36] |  219 COVID-19 175 normal 224 pneumonia |  CT |  3D-DL |  balanced |  86.7
[37] |  73 COVID-19 |  CT |  CNNs$+$MODE |  - |  92
[38] |  368 COVID-19 127 pneumonia |  CT |  ResNet50 |  imbalanced |  76
[39] |  1262 COVID-19 1230 Non$-$COVID-19 |  CT |  DenseNet201 |  balanced |  97

##  3 Preliminaries

In this section, an introduction about some of the concepts quantum computing
used in this work like the quantum bit, quantum gates, and architecture of
convolutional neural networks (CNNs). If the reader is familiar with these
concepts may skip this subsection.

###  3.1 Quantum computing

Quantum computing relies on postulates and characteristics of quantum
mechanics (i.e., quantum bits, interference, superposition, and entanglement)
to information processing. Quantum computing gives us the ability to solve
complex problems better and faster than classical computing [40, 41]. Qubit is
the small unit to process information in a quantum computer like the bit in
classical computing. A qubit can also be in one-state, zero-state, or both of
states at the same time known as linear superposition. The qubit is a state
vector in Hilbert space.

$|\psi\rangle=\left(\begin{array}[]{l}\theta\\\
\delta\end{array}\right)=\theta|0\rangle+\delta|1\rangle$ (1)

Where $\theta$ and $\delta$ are the probability amplitudes that represented by
complex numbers and $|\theta^{2}|+|\delta^{2}|$ = 1

From the postulates of quantum mechanics, any unitary transformation (unitary
matrix) is a quantum gate. Matrix to be unitary, the following condition must
be verified:

$UU^{\dagger}=U^{\dagger}U=I$ (2)

where $U^{\dagger}$ is the conjugate transpose of a matrix $U$ and $I$ is an
identity matrix. Quantum gates can be classified based on numbers of qubits:
one-qubit gates, two-qubits gates, and multiple-qubits gates [42, 43].
Firstly, from the most popular and widely used gate in one-qubit gates is a
Hadamard gate or square-root of NOT gate. The use of the Hadamard gate to
present qubits into superposition. Pauli gates also are one-qubit gates.
Secondly, two-qubits gates which work on $4x4$ unitary matrices. For example,
Controlled NOT, and swap gate. Lastly, multiple-qubits gates which work on
multiple qubits as $2^{n}$ x $2^{n}$ unitary matrices such as Toffoli and SWAP
gates. Table 2 shows quantum gates with circuit representation, unitary
matrix, the number of qubits, and the operation of the gate.

Table 2: summarizes standard quantum computing gates.

Gate |  Notation |  Matrix |  Qbit |  Use
---|---|---|---|---
Hadamard |  |  12(111-1) |  1 |  Create superposition state between two quantum bits.
Pauli-X |  |  (0110) |  1 |  Flip quantum bit from state to another.
Pauli-Y |  |  (0-ii0) |  1 |  Make $\pi$-rotation for quantum bit around the Y-axis.
Pauli-Z |  |  (100-1) |  1 |  Make $\pi$ -rotation for quantum bit around the Z-axis.
Swap |  |  (1000001001000001) |  2 |  Swap two quantum bits states.
Toffoli |  |  (1000000001000000001000000001000000001000000001000000000100000010) |  3 |  Controlled-Controlled NOT (CC Not) gate. Flip target quantum bit if both control two quantum bits equal one.

###  3.2 Convolutional Neural Networks

Convolutional Neural Networks (CNNs) [44] inspired by convolution operation to
produce a convolutional (Conv) layer that is the heart of the CNNs. As
illustrated in Figure 1, the basic architecture of CNNs is similar to a multi-
layer perceptron starts with the input image and sequence of hidden layers to
predict labels over the output layer. The layers of CNNs are explained in
detail in Section 4. The CNNs play a critical role in computer vision
applications such as image classification [27], image segmentation [45],
object detection [46], and signal processing [47]. Various models had been
developed based on the concepts of CNNs as ResNet50, and VGG-19.

Figure 1: Illustrates the general architecture of the CNNs: The CNNs consists
of feature extraction and classification parts. First of all, the input image
is pushed to the Conv layer as a matrix of pixels to extract features and
patterns from images by apply kernels on the input image. Then, the pooling
layer is applied to reduce the number of features and fast training time. The
next layer is applied convolution operation on convoluted features with the
kernels to extract more invariant and local features and so on. With the last
pooled feature map flatten layer has been applied. Finally, fully connected
layers have been used for train and classification.

##  4 Proposed HQCNN Model

The proposed hybrid quantum-classical CNNs model (HQCNN) aims to improve CNNs
classification for medical images and predict COVID-19 and healthy patients in
early stages. The main idea of the HQCNN model is based on hybrid computation
to enhance the performance of classical learning [48]. The proposed model
consists of two parts: first, the quantum part has utilized the quantum Conv
layer, which is proposed by Henderson et al. [23]. Second, the classical part
with CNNs structure. As shown in Figure 2, the HQCNN model has one quantum
Conv layer, three Conv layers followed by the rectified linear unit activation
function, two max-pooling layers, and followed by two fully connected layers.

Figure 2: Block diagram for proposed HQCNN model.

The layers of the model are presented in detail in the following steps.

###  4.1 Quantum convolutional layer

The proposed HQCNN model has used quantum convolution that is presented by the
Maxwell Henderson et al. [23]. Quantum convolution works as small random
quantum circuits (RQCs) to calculate convolution operation and can implement
on near-term quantum hardware. This circuit has applied with local locations
of input images to extract elementary and informative features. The quantum
convolution layer consists of three phases: encoding, random quantum circuit,
and decoding(as shown in Figure 2).

####  4.1.1 Encoding

Up till now, the encoding data to quantum is a challenge in quantum machine
learning (QML) [49]. Several encoding methods have been discussed in [50]. In
this work, angle encoding has been used to transform input data into rotation
angles of quantum states. The angle rotations gates are the simplest encoding
methods to access data into a quantum circuit. In this study, due to needing
the single quantum gate for each entry to encode data. These gates are
corresponding to encode classical pixel data to a quantum state. Rotation
matrices are rotation operators of Pauli-matrices in the form of exponential
around $X$, $Y$, $Z$ axes (as seen in Table 1). Three rotation gates $Rx$,
$Ry$, $Rz$ are a single quantum bit rotation via angle $\alpha$ around $X$,
$Y$, and $Z$ axes, respectively. The RY and RX rotation gates can be
represented by the following equations.

$R_{x}(\alpha)=\left(\begin{array}[]{cc}\cos\left(\frac{\alpha}{2}\right)&-i\sin\left(\frac{\alpha}{2}\right)\\\
-i\sin\left(\frac{\alpha}{2}\right)&\cos\left(\frac{\alpha}{2}\right)\end{array}\right)$
(3)
$R_{y}(\alpha)=\left(\begin{array}[]{cc}\cos\left(\frac{\alpha}{2}\right)&-\sin\left(\frac{\alpha}{2}\right)\\\
\sin\left(\frac{\alpha}{2}\right)&\cos\left(\frac{\alpha}{2}\right)\end{array}\right)$
(4)

####  4.1.2 Random Quantum Circuit

A quantum circuit is a series of quantum unitary operations (or gates) and
measurements connected via wires (Qubits). Just like the classical Conv layer,
quantum Conv layer composed of quantum kernels apply to the input image. The
main idea of quantum convolution utilizing random quantum circuits to split
input image into small local locations to extract meaningful features. The
advantages of the quantum circuit in quantum convolution works with a few
quantum bits and shallow-depth of quantum circuits.

####  4.1.3 Measurement

The measurement phase is also known as the decoding phase. Decoding is
measuring quantum data to transform into classical form. Pauli matrices can
use as measurement methods [51]. In the HQCNN model, the Pauli-Z gate is used
to the decoding phase (as seen in Table 2).

###  4.2 Convolutional layer

Conv layer is the pivotal and significant layer in the feature extraction part
of CNN. This layer performs convolution operation on input features with
kernels to extract invariant and informative features from images as
convoluted features map to the next layer. Convolution operation computes dot
products between small local locations of the input image and kernels[28, 52].

$(F*K)(i,j)=\sum_{m}\sum_{n}K(m,n)F(i-m,j-n)$ (5)

Where $*$ is the convolution operation to produce a convoluted feature map,
$F$ is the input feature, and $K$ is kernel or filter. Conv layer has been
followed by the ReLU function transformation to add activation values into the
model network. The function returns $0$ for all negative values and returns
the maximum value for all positive values. The ReLU function can be defined by
the following formula:

$ReLU(x)=max(0,x)$ (6)

Where $x$ is an input value.

###  4.3 Max-pooling layer

Max-pooling layer has used to reduce computational learning by selection the
most important and valuable features. Convoluted feature maps have been
divided into small regions based on the stride number. The max-pooling idea
has taken the maximum value of each small location to produce pooled feature
maps. Before the classification part, flatten is used to link two parts in
CNNs by converting the max-pooled feature map into a 1-dimensional array.

###  4.4 Fully Connected Layer

The Fully Connected (FC) layer is the second part of the CNNs structure. The
fully connected layers perform the classification process after the flattening
layer by applied weights to predict classes. The dropout layer is used after
the FC layer to reduce the overfitting of the model (as shown in Figure 2).

Figure 3 illustrates the flowchart of the HQCNN model: the proposed HQCNN
model has nine layers, one quantum Conv layer, three classical Conv layers
with ReLU function, two max-pooling layers, two FC layers with ReLU function,
and one output layer is applied with a softmax activation function. The
quantum Conv layer is used encoding method with RY rotation gate and decoding
with the Pauli-Z gate. The three classical Conv layers are used with 2D CNNs
and are combined with the max-pooling layer. Each max-pooling layer is
performed with a 2 x 2 kernel size and the stride equal to $2$, followed by a
dropout layer with a 0.2 dropout rate. The two FC layers are applied with 300
and 100 neurons for the first and second layers, respectively. The total
parameters for the HQCNN model are 120,394. Table 3 shows the summary layers
of the proposed HQCNN model.

Figure 3: Flowchart for proposed HQCNN model. Table 3: Shows the summary of
the HQCNN model.

Layer |  Type |  Units |  Kernel Size |  Input size |  No. of parameters
---|---|---|---|---|---
1 |  Quantum Conv |  4 |  2 x 2 |  (28 x 28 x 1) |  -
2 |  Conv2D |  16 |  2 x 2 |  (14 x 14 x 4) |  272
3 |  MaxPooling2D |  - |  2 x 2 |  (14 x 14 x 4) |  -
4 |  Conv2D |  16 |  2 x 2 |  (7 x 7 x4) |  1040
5 |  Conv2D |  32 |  2 x 2 |  (7 x 7 x 32) |  2080
6 |  MaxPooling2D |  - |  2 x 2 |  (7 x 7 x 8) |  -
7 |  FC1 |  300 |  - |  (288) |  86,700
8 |  FC2 |  100 |  - |  (300) |  30,100
9 |  Output |  2 |  - |  (100) |  202

##  5 Experimental Results and Discussion

This section introduces the used images dataset, the performance measures to
evaluate the HQCNN model, and the analysis and discussion of experimental
results.

###  5.1 Dataset

The used CXR COVID-19 images in this study are collected from two collections.
Firstly, COVID-19 image dataset [53] is made by Joseph Cohen. This dataset
includes COVID-19, viral, MERS, SARS, and pneumonia. For this study, only 670
CXR COVID-19 images are adapted. Secondly, the augmented COVID-19 dataset[54]
is made by Walid El-Shafai. This dataset includes CXR and CT COVID-19 images
with augmentation. For this study, only 491 the CXR COVID-19 images without
augmentation are adapted. The pneumonia dataset [55] is used for normal and
pneumonia (viral and bacterial) CXR images. The pneumonia dataset includes
5216 train images (1341 normal and 3875 pneumonia) and 624 test images (234
normal and 390 pneumonia). The images are divided into three datasets to
evaluate the ability of the proposed HQCNN model to detect COVID-19 cases. The
first binary dataset (D1) including 1161 COVID-19 and 1575 normal images. The
second binary dataset (D2) includes 1161 COVID-19 and 4247 pneumonia images.
The third multi-class dataset (D3) includes 1161 COVID-19, 1575 normal, and
4247 pneumonia images. Table 4 summarizes the used CXR images in this work.
Figure 4 shows samples from the used images.

Table 4: Summarizes of three datasets are used in this work.

dataset |  COVID-19 |  normal |  pneumonia |  Train |  Test |  Total
---|---|---|---|---|---|---
D1 |  1161 |  1575 |  - |  1010 COVID-19 1341 normal |  151 COVID-19 234 normal |  2736
D2 |  1161 |  - |  5216 |  1000 COVID-19 3875 normal |  161 COVID-19 390 pneumonia |  5377
D3 |  1161 |  1575 |  5216 |  1341 normal 1000 COVID-19 3875 pneumonia |  161 COVID-19 390 pneumonia 234 normal |  6952

Figure 4: Samples of used images.

###  5.2 Performance measures

The most common measures for evaluating classification models are confusion
matrix with measures metrics like accuracy, specificity (or true-negative
rate), sensitivity (Recall or true-positive rate), precision, and F1-measure.
Measures metrics can compute by terms TP, TN, FP, FN as the following
formulas:

  * •

True Positives (TP): The proposed HQCNN model correctly predicts COVID-19
cases and labelled as COVID-19.

  * •

True Negatives (TN): The proposed HQCNN model correctly predicts normal cases
and labelled as normal.

  * •

False Positive (FP): The proposed HQCNN model incorrectly predicts normal
cases and labelled as normal.

  * •

False Negatives (FN): The proposed HQCNN model incorrectly predicts COVID-19
cases and labelled as COVID-19.

$Accuracy(Acc.)=TP+TN/(TP+TN+FP+FN)$ (7) $Sensitivity(Sns.)=TP/TP+FN$ (8)
$Specificity(Spc.)=TN/TN+FP$ (9) $Precision(Prc.)=TP/TP+FP$ (10)
$F1-measure=2*(Precision*Recall)/(Precision+Recall)$ (11)
$BalancedAccuracy=1/2(TP/TP+FP+TN/TN+FN)$ (12) $\centering\begin{split}FBeta-
measure=(1+\beta^{2})(Precision*Recall)/(\beta^{2}*\\\
(Precision+Recall))\end{split}\@add@centering$ (13)
$FalsePositiveRate=FP/FP+TN$ (14)

Receiver Operating Characteristic (ROC) curve identifies as ROC-Area Under
Curve (ROC-AUC) is the method to measure the ability of the classifier to
predict the right labels. The ROC is a curve between the true-positive rate
(sensitivity) and the false-positive rate (1 $-$ sensitivity). Fbeta-measure
is the weighted harmonic average of precision and recall of the HQCNN model.

###  5.3 Discussion

Experimental results have been implemented on Google Colaboratory by using
PennyLane and TensorFlow with Keras. PennyLane is a cross-platform Python
library for hybrid quantum-classical computation [48]. The main aim of the
HQCNN model is to use the concept of the quantum circuit to enhance the
performance of the classical CNNs model and the prediction of infected cases
of COVID-19 with higher results. RQC works with (2x2) small squares of an
input image. Each (2x2) square of the image is encoded into a quantum state by
the RY gate. The decoding of the quantum state into classical produces a new
four features of a small square. Figure 5 shows samples of normal, pneumonia,
and COVID-19 images, which are examined by the RQC. The first column shows the
input image with 28 x 28 size and divided by 250. The quantum Conv layer
assigns a feature map into four channels. The quantum Conv layer is used only
as a preprocessing layer to CXR images. The train and test phases will be
completed on quantum preprocess data with CNNs layers.

Figure 5: Samples of normal, COVID-19, and pneumonia images are processed by
random quantum circuit.

At the beginning of the evaluation, the proposed model is evaluated with RY
and RX gates to encode data into RQC. RY and RX gates have been used with a
different number of shots (500 and 1000). As to be noted from Table 5, the RY
gate (1000 shots) achieved higher results 97.6%, 99.3%, 96.5%, 94.9, and 97%
for accuracy, sensitivity, specificity, precision, and F1-measure,
respectively.

Table 5: The performance of HQCNN model with rotation gates and number of
shots on D1.

Rotation gate | Shots | ACC.(%) | Sns.(%) | Spc.(%) | Prc.(%) | F1-measure(%)
---|---|---|---|---|---|---
RY | 1000 | 97.6 | 99.3 | 96.5 | 94.9 | 97
RX | 95.8 | 97.3 | 94.8 | 92.4 | 94.8
RY | 500 | 96.8 | 98 | 96.1 | 94.2 | 96.1
RX | 94.5 | 98 | 92.3 | 89.1 | 96.1

As shown in Figure 6(a), the RY gate (1000 shots) achieved 97.9 % of balanced
accuracy. Besides, the RY gate (1000 shots) obtained the largest AUC-ROC with
99.7% closer to $1$. The test loss curve of the HQCNN model with RY (1000
shots) continues to reduce until the end of the test phase. The test loss with
RY (1000 shots) is 0.07%. In addition, the test loss with RY (500 shots) is
0.09%, the RX achieved 0.9% and 0.1% of test loss for $1000$ and $500$ shots,
respectively (see Figure 7). Figure 8 shows the confusion matrix for RY and RX
with a different number of shots. The RY (1000 shots) has predicted the
highest number of positive cases of COVID-19 (150 of 151 cases), as shown in
Figure 8(c). Increasing the number of shots will calculate the better
estimation of the output statistic of observables (i.e., expectation value and
variance).

As mentioned above, increasing the number of shots increases the performance
of the model. Besides, the encoding method (RY) can be improved the overall
results of the model. So, the experimental results will be completed with an
encoding angle (RY gate) and 1000 shots.

(a)

(b)

Figure 6: (a) comparative analysis of balanced accuracy, FBeta-
measure($\beta$=0.5), and FBeta-measure($\beta$=2) for D1. (b) visualization
of the ROC curve.

(a)

(b)

Figure 7: Visualization of the learning curve for both (a) test accuracy and
(b) test loss for D1 with 20 epochs.

(a)

(b)

(c)

(d)

Figure 8: The confusion matrix of the proposed HQCNN with RY and RX gates. (a)
RY(500 shots), (b) RX(500 shots), (c) RY (1000 shots), and (d) RX (1000
shots).

The proposed HQCNN model results are compared with the CNNs, multi-layer
perceptron (MLP), support vector machines(SVMs), k-nearest neighbours (KNN),
AdaBoost, random forest (RF), bagging classifier (BC), decision tree (DT),
Gaussian Naive Bayes (GNB), and XGBoost classifiers. In classical CNN, the
quantum layer has been replaced by the classical Conv layer with four filters.
The HQCNN model is optimized with the Adam algorithm with a learning rate of
0.0001. The performance of HQCNN model is evaluated with measures metrics as
represented in Equations 7, 8, 9, 10, and 11. Besides, balanced accuracy 12 to
calculate the accuracy in an imbalanced dataset, Fbeta-measure 13, and the
AUC-ROC curve. The experimental results of HQCNN are divided into four parts
to measure the ability of the proposed model for the classification of
COVID-19 patients. The first part presents the classification results of the
HQCNN model between normal and COVID-19 cases 5.3.1. The second part discusses
the experimental results of the HQCNN model between COVID-19 and pneumonia
patients 5.3.2. The last part debates the performance of the HQCNN model for a
multi-class dataset among normal, COVID-19, and pneumonia patients 5.3.3. The
last part compares the HQCNN model to CXR existing studies in this work 5.3.4.

####  5.3.1 The experimental results on D1

As shown in Table 6, the performance HQCNN model outperforms CNN and other
models. The hybrid proposed model has predicted COVID-19, and normal patients
with 98.4% of accuracy, 99.3% of sensitivity, 97.8% of specificity, 96.7% of
precision, and 98% of F1-measure. The MLP model outperforms the HQCNN model in
measure of sensitivity, it achieved a sensitivity of 100%. Besides, as
observed from Figure 9(a) proposed hybrid model has achieved higher balanced
accuracy and Fbeta-measure with 98.6% and 98.4%, respectively. Additionally,
The HQCNN model scored the largest AUC of ROC curve with 0.99%(near to 1)
compared by various models, as shown in Figure 10(a). The confusion matrices
of binary D1 is displayed in Figure 11. The confusion matrix of the HQCNN
model, among of 151 COVID-19 images, one patient is misclassified by the HQCNN
model. From a total of 234 normal cases, five cases are misclassified by the
HQCNN model.

Table 6: Comparison of classification performance between proposed HQCNN and
different models on D1.

Model | ACC.(%) | Sns.(%) | Spc.(%) | Prc.(%) | F1-measure(%)
---|---|---|---|---|---
HQCNN | 98.4 | 99.3 | 97.8 | 96.7 | 98
CNN | 96.8 | 98.6 | 95.7 | 93.7 | 96
MLP | 92.9 | 100 | 88.4 | 84.8 | 91.7
CNN+SVM(RBF) | 94.5 | 98.6 | 91.8 | 88.6 | 93.4
CNN+SVM(Poly) | 93.5 | 99.3 | 89.7 | 86.2 | 92.3
CNN+KNN | 95.5 | 98.6 | 93.5 | 90.8 | 94.6
CNN+AdaBoost | 94.5 | 99.3 | 91.4 | 88.2 | 93.4
CNN+RF | 94.2 | 98.6 | 91.4 | 88.1 | 93.1
CNN+BC | 94.5 | 98.6 | 91.8 | 88.6 | 93.4
CNN+DT | 95 | 99.3 | 92.3 | 89.2 | 94
CNN+GNB | 94.8 | 98.6 | 92.3 | 89.2 | 93.7
CNN+XGBoost | 95 | 99.3 | 92.3 | 89.2 | 94

(a)

(b)

(c)

Figure 9: Comparative analysis of balanced accuracy, FBeta-
measure($\beta$=0.5), and FBeta-measure($\beta$=2) for (a) D1, (b) D2, and (c)
D3.

####  5.3.2 The experimental results on D2

Here, Table 7 reports the results of the HQCNN model compared to various
models on D2. the HQCNN model achieved 99%, 99.3%, 98.9%, 99.7%, and 97.5% for
accuracy, F1-measure, precision, sensitivity, and specificity, respectively.
Besides, better-balanced accuracy has been achieved with 98.6%, 98% for HQCNN,
and CNN+KNN models, respectively as shown in Figure 9(b). From Figure 10(b),
the proposed HQCNN model achieved AUC-ROC score 100% (equal to 1). The HQCNN
model distinguished between COVID-19 and pneumonia with higher efficiency. The
confusion matrices of D2 have been shown in Figure 12. From a total of 161
COVID-19 cases, four cases are misclassified by the HQCNN model. Among 390
pneumonia images, one patient is misclassified by the HQCNN model.

Table 7: Comparison of classification performance between proposed HQCNN and
different models on D2.

Model | ACC.(%) | Sns.(%) | Spc.(%) | Prc.(%) | F1-measure(%)
---|---|---|---|---|---
HQCNN | 99 | 99.7 | 97.5 | 98.9 | 99.3
CNN | 97.4 | 99.7 | 91.9 | 96.7 | 98.2
MLP | 96.3 | 99.7 | 88.1 | 95.3 | 97.4
CNN+SVM(RBF) | 97.8 | 99.7 | 93.1 | 97.2 | 98.4
CNN+SVM(Poly) | 96.9 | 99.7 | 90 | 96 | 97.8
CNN+KNN | 98.5 | 99.2 | 96.8 | 98.7 | 98.9
CNN+AdaBoost | 97 | 99.2 | 91.9 | 96.7 | 97.9
CNN+RF | 96.5 | 97.6 | 93.7 | 97.4 | 97.5
CNN+BC | 97.8 | 99.7 | 93.1 | 97.2 | 98.4
CNN+DT | 97 | 99.2 | 91.9 | 96.7 | 97.9
CNN+GNB | 97.8 | 98.9 | 95 | 97.9 | 98.4
CNN+XGBoost | 98 | 99.2 | 95 | 97.9 | 98.5

(a)

(b)

Figure 10: The AUC-ROC curve of QHCNN and different models on (a) D1, and (b)
D2.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 11: The confusion matrix of the proposed HQCNN and different models for
D1.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 12: The confusion matrix of the proposed HQCNN and different models on
D2.

####  5.3.3 The experimental results on D3

To further evaluate, the imbalanced multi-class dataset is used to evaluate
HQCNN performance. The dataset is combined with normal, COVID-19, and
pneumonia images. Table 8 presents the classification results of the HQCNN
model and other models on an imbalanced multi-class dataset. The best results
have been evaluated with 88.6%, 88.7%, 89.3%, and 88.8% for accuracy,
sensitivity, precision, and F1-measure. Besides, the HQCNN model achieved
90.4% of balanced accuracy compared to various models. In Figure 13, The AUC-
ROC curve has been exhibited with three normal, COVID-19, and pneumonia
classes. It can be noted that the HQCNN distinguished among three classes with
higher AUC values 0.97%, 0.99%, and 0.97% for normal, COVID-19, and pneumonia
classes. The confusion matrices are shown in Figure 14. The confusion matrix
of the HQCNN model, among 161 COVID-19 images, four patients are
misclassified. From a total of 234 normal cases, 25 cases are misclassified.
From 390 pneumonia patients, 60 patients are misclassified by the HQCNN model.

Table 8: Comparison of classification performance between proposed HQCNN and
different models on D3.

Model | ACC.(%) | Sns.(%) | Prc.(%) | F1-measure(%)
---|---|---|---|---
HQCNN | 88.6 | 88.7 | 89.3 | 88.8
CNN | 87.7 | 87.7 | 88.8 | 88.2
MLP | 77.9 | 78 | 81.5 | 74
CNN+SVM(RBF) | 82.3 | 82.2 | 83.1 | 80.9
CNN+SVM(Poly) | 84.2 | 84.2 | 84.7 | 83.2
CNN+KNN | 83.4 | 82 | 83.6 | 82.5
CNN+AdaBoost | 78.8 | 78.8 | 79.8 | 77.3
CNN+RF | 81.4 | 81.2 | 81.2 | 80.3
CNN+BC | 82 | 82.2 | 83.3 | 80.8
CNN+DT | 83.8 | 83 | 83.8 | 82.9
CNN+GNB | 82.9 | 82.9 | 83.4 | 81.8
CNN+XGBoost | 82.2 | 82.1 | 83.3 | 80.7

####  5.3.4 Comparison with CXR existing studies

To boost the effectiveness of the HQCNN model in classifying the COVID-19
cases, the HQCNN model is compared to CXR existing studies, as shown in Table
9. The HQCNN model outperforms previous studies with CXR images in terms of
sensitivity and F1-measure. The Truncated InceptionNet and CNN+LSTM models
outperform the HQCNN model with a term of accuracy. The Truncated InceptionNet
achieved 99.9% accuracy for one dataset but, the overall accuracy was 98.77%.
The Truncated InceptionNet is based on a large InceptionNetV3 model. The
CNN+LSTM has 20 layers, including 12 Conv layers. The HQCNN model achieved
better results with nine layers, including four Conv layers (one quantum layer
and three classical layers).

Table 9: Comparison with CXR existing studies.

Study |  COVID-19 Images |  Class |  ACC.(%) |  Sns.(%) |  F1-measure(%)
---|---|---|---|---|---
MobileNetV2 [25] |  224 |  multi |  96.7 |  98.66 |  -
ResNet18 [26] |  180 |  multi |  88.9 |  85.9 |  84.4
Truncated InceptionNet [27] |  162 |  multi |  99.9 |  98 |  98
DarkNet [28] |  125 |  binary |  98.08 |  95.13 |  96.51
DarkNet [28] |  125 |  multi |  87.02 |  85.35 |  87.37
CNN+LSTM [29] |  1512 |  binary |  99.4 |  99.3 |  99.9
CoroNet [30] |  280 |  binary |  99 |  99.3 |  98.5
CoroNet [30] |  280 |  multi |  95 |  96.9 |  95.6
VGG16 [31] |  250 |  binary |  96 |  96 |  94
GSA-DenseNet121 [32] |  99 |  binary |  98.38 |  98.5 |  98
Inception+FO-MPA[33] |  200 |  binary |  98.7 |  - |  98.2
Inception+FO-MPA[33] |  219 |  binary |  99.6 |  - |  99
CapsNet[34] |  1050 |  binary |  97.24 |  97.42 |  97.24
CapsNet[34] |  1050 |  multi |  84.22 |  84.22 |  84.21
EfficientNet[35] |  504 |  binary |  99.62 |  99.63 |  99.62
EfficientNet[35] |  504 |  multi |  96.70 |  96.69 |  97.11
HQCNN |  1161 |  binary |  99 |  99.7 |  99.3
HQCNN |  1161 |  multi |  88.6 |  88.7 |  88.8

In summary, as mentioned in 5.3.1, 5.3.2, 5.3.3, and 5.3.4, the proposed HQCNN
model achieved a sensitivity of 99.3%, 99.7%, and 88.7% on D1, D2, D3,
respectively. Table 10 shows the results of the HQCNN model with each class on
D1, D2, and D3. In D1, the COVID-19 class is predicted with high accuracy of
99.3%, sensitivity of 97%, and F1-measure of 98%. In normal cases, The HQCNN
model obtained 100% sensitivity. In imbalanced binary D2, the COVID-19 class
is categorized with 97.5%, 99%, and 98% for accuracy, sensitivity, and
F1-measure, respectively. The pneumonia class achieved 100% precision, 99.7%
accuracy, and 99% for sensitivity and F1-measure. Furthermore, in the
imbalanced multi-class D3, the higher results are obtained with COVID-19
class. It classified the COVID-19 cases with an accuracy of 97.5%, sensitivity
of 99%, and F1-measure of 98%. Besides, with pneumonia category achieved a
high 92% sensitivity. The lower sensitivity is achieved with 77% in normal
cases. The higher precision is achieved with 100% in pneumonia patients.

The limitations of the HQCNN model in this work are:

  * •

The size of the COVID-19 image is used with 28x28 pixels which are divided
into 4x4 pixel.

  * •

The architecture of the HQCNN model is a small to compatible with 4 qubits in
the quantum simulator.

  * •

The used random quantum circuit works as preprocessing layer for images.

  * •

The size of the used images in the training phase is small.

Table 10: The HQCNN model performance with each class on D1, D2, and D3.

Class | ACC.(%) | Sns.(%) | Prc.(%) | F1-measure(%)
---|---|---|---|---
|  D1 |  |  |
normal | 97.8 | 100 | 98 | 99
COVID-19 | 99.3 | 97 | 99 | 98
|  D2 |  |  |
COVID-19 | 97.5 | 99 | 98 | 98
pneumonia | 99.7 | 99 | 100 | 99
|  D3 |  |  |
normal | 89.3 | 77 | 89 | 83
COVID-19 | 97.5 | 99 | 98 | 98
pneumonia | 94.6 | 92 | 85 | 88

(a)

(b)

(c)

Figure 13: The AUC-ROC curve of QHCNN and different models for D3.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 14: The confusion matrix of the proposed HQCNN and different models on
D3.

##  6 Conclusion and Future Work

In this work, a new hybrid quantum-classical CNNs (HQCNN) model has been
proposed for COVID-19 classification with chest radiography images. The aim of
the HQCNN model combines the random quantum circuit with CNNs and to enhance
the performance. The HQCNN model has been used the random quantum circuit as a
quantum convolution layer to compute convolution operation on a quantum
device. The HQCNN model has been evaluated on binary and multi-class COVID-19
datasets, it outperformed classical CNNs and various models. It achieved an
accuracy of 99% and 88.6%, a sensitivity of 99.3%, and 88.7% for binary, and
multi-class datasets, respectively. The future work will be highly focused on
using different encoding methods (i.e., amplitude encoding) with further
quantum convolutional layers. Also, we will enhance the HQCNN architecture to
overcome on limitations of the proposed model.

## Author Contribution statement

Essam H. Houssein: Supervision, Methodology, Conceptualization, Formal
analysis, Writing - review & editing. Zainab Abohashima: Software, Resources,
Data Curation, Writing - original draft. Mohamed Elhoseny: Conceptualization,
Formal analysis, Writing - review & editing. Waleed M. Mohamed:
Conceptualization, Formal analysis, Writing - review & editing. All authors
read and approved the final paper.

## Conflict of interest

The authors declare that there is no conflict of interest. Non-financial
competing interests.

## Compliance with ethical standards

This article does not contain any studies with human participants or animals
performed by any of the authors.

## Acknowledgments

The authors would like to thank Pennylane’s team members for helpful
discussions, especially, Dr Andrea Mari for valuable discussions.

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Cloudy model | Equilibrium chemistry
---|---
Parameters | Priors | Posteriors | Parameters | Priors | Posteriors
$\log_{10}P_{0}$ (bar) | $\mathcal{U}(-6,3)$ | $-1.78^{+2.72}_{-2.30}$ |  $\log_{10}P_{0}$ (bar) | $\mathcal{U}(-6,3)$ | $-2.33^{+2.49}_{-2.07}$
$R_{p}\leavevmode\nobreak\ (R_{\rm J})$ | $\mathcal{N}(1.49,0.08)$ | $1.500^{+0.049}_{-0.050}$ | $R_{p}\leavevmode\nobreak\ (R_{\rm J})$ | $\mathcal{N}(1.49,0.08)$ | $1.500^{+0.038}_{-0.051}$
$M_{\rm p}\leavevmode\nobreak\ (M_{\rm J})$ | $\mathcal{N}(1.03,0.12)$ | $0.99^{+0.11}_{-0.11}$ | $M_{\rm p}\leavevmode\nobreak\ (M_{\rm J})$ | $\mathcal{N}(1.03,0.12)$ | $1.01^{+0.10}_{-0.10}$
$T\leavevmode\nobreak\ (\rm K)$ | $\mathcal{U}(500,2500)$ | $2132^{+252}_{-398}$ | $T\leavevmode\nobreak\ (\rm K)$ | $\mathcal{U}(500,2500)$ | $2099^{+271}_{-410}$
$\log_{10}P_{\rm c}$ (bar) | $\mathcal{U}(-6,3)$ | $0.25^{+1.80}_{-2.00}$ |  $\log_{10}P_{\rm c}$ (bar) | $\mathcal{U}(-6,3)$ | $-0.36^{2.13}_{-2.25}$
$f_{\rm haze}$ (dex) | $\mathcal{U}(-3,6)$ | $3.68^{+1.04}_{-1.22}$ |  $f_{\rm haze}$ (dex) | $\mathcal{U}(-3,6)$ | $4.63^{+0.74}_{-0.58}$
$\delta$ (ppm) | $\mathcal{U}(-2000,2000)$ | $92^{+185}_{-172}$ |  $\delta$ (ppm) | $\mathcal{U}(-2000,2000)$ | $10^{+207}_{-164}$
- | - | - | C/O | $\mathcal{U}(0.1,1.6)$ | $1.00^{+0.38}_{-0.47}$
- | - | - | [M/H] | $\mathcal{U}(-1,3)$ | $-0.21^{+0.83}_{-0.53}$
Free chemistry w/o TiO&VO | Free chemistry with TiO&VO
Parameters | Priors | Posteriors | Parameters | Priors | Posteriors
$\log_{10}P_{0}$ (bar) | $\mathcal{U}(-6,3)$ | $-1.41^{+2.34}_{-2.30}$ |  $\log_{10}P_{0}$ (bar) | $\mathcal{U}(-6,3)$ | $-1.81^{+2.50}_{-2.29}$
$R_{p}\leavevmode\nobreak\ (R_{\rm J})$ | $\mathcal{N}(1.49,0.08)$ | $1.507^{+0.045}_{-0.049}$ | $R_{p}\leavevmode\nobreak\ (R_{\rm J})$ | $\mathcal{N}(1.49,0.08)$ | $1.510^{+0.031}_{-0.044}$
$M_{\rm p}\leavevmode\nobreak\ (M_{\rm J})$ | $\mathcal{N}(1.03,0.12)$ | $1.00^{+0.10}_{-0.10}$ | $M_{\rm p}\leavevmode\nobreak\ (M_{\rm J})$ | $\mathcal{N}(1.03,0.12)$ | $1.01^{+0.10}_{-0.10}$
$T\leavevmode\nobreak\ (\rm K)$ | $\mathcal{U}(500,2500)$ | $1997^{+319}_{-631}$ | $T\leavevmode\nobreak\ (\rm K)$ | $\mathcal{U}(500,2500)$ | $1639^{+571}_{-726}$
$\log_{10}P_{\rm c}$ (bar) | $\mathcal{U}(-6,3)$ | $0.24^{+1.71}_{-1.76}$ |  $\log_{10}P_{\rm c}$ (bar) | $\mathcal{U}(-6,3)$ | $0.17^{+1.74}_{-3.29}$
$f_{\rm haze}$ (dex) | $\mathcal{U}(-3,6)$ | $3.10^{+1.30}_{-1.66}$ |  $f_{\rm haze}$ (dex) | $\mathcal{U}(-3,6)$ | $1.88^{+2.66}_{-3.04}$
$\delta$ (ppm) | $\mathcal{U}(-2000,2000)$ | $-54^{+227}_{-165}$ |  $\delta$ (ppm) | $\mathcal{U}(-2000,2000)$ | $146^{+263}_{-465}$
$\rm MMR_{Na}$ (dex) | $\mathcal{U}(-10,0)$ | $-4.02^{+2.41}_{-3.50}$ |  $\rm MMR_{Na}$ (dex) | $\mathcal{U}(-10,0)$ | $-5.08^{+2.94}_{-2.92}$
$\rm MMR_{K}$ (dex) | $\mathcal{U}(-10,0)$ | $-4.49^{+2.12}_{-3.08}$ |  $\rm MMR_{K}$ (dex) | $\mathcal{U}(-10,0)$ | $-5.03^{+2.63}_{-2.93}$
$\rm MMR_{H_{2}O}$ (dex) | $\mathcal{U}(-10,0)$ | $-6.64^{+2.13}_{-2.01}$ |  $\rm MMR_{H_{2}O}$ (dex) | $\mathcal{U}(-10,0)$ | $-6.23^{+2.29}_{-2.24}$
$\rm MMR_{CH_{4}}$ (dex) | $\mathcal{U}(-10,0)$ | $-6.34^{+2.37}_{-2.33}$ |  $\rm MMR_{CH_{4}}$ (dex) | $\mathcal{U}(-10,0)$ | $-5.77^{+2.50}_{-2.56}$
$\rm MMR_{CO}$ (dex) | $\mathcal{U}(-10,0)$ | $-5.44^{+2.64}_{-2.68}$ |  $\rm MMR_{CO}$ (dex) | $\mathcal{U}(-10,0)$ | $-5.30^{+2.68}_{-2.79}$
$\rm MMR_{CO_{2}}$ (dex) | $\mathcal{U}(-10,0)$ | $-5.90^{+2.46}_{-2.50}$ |  $\rm MMR_{CO_{2}}$ (dex) | $\mathcal{U}(-10,0)$ | $-5.89^{+2.64}_{-2.44}$
- | - | - |  $\rm MMR_{TiO}$ (dex) | $\mathcal{U}(-10,0)$ | $-4.84^{+1.98}_{-3.36}$
- | - | - |  $\rm MMR_{VO}$ (dex) | $\mathcal{U}(-10,0)$ | $-6.78^{+1.98}_{-1.95}$
131313$P_{0}$: reference pressure; $R_{\rm p}$: planet radius at the reference
pressure; $M_{\rm p}$: planet mass; $T$: temperature of the isothermal
atmosphere; $P_{\rm c}$: cloud-top pressure; $f_{\rm haze}$: Rayleigh-like
scattering factor; C/O: atmospheric carbon-to-oxygen ratio; [M/H]: atmospheric
metallicity; MMR: mass mixing ratio.