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e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
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11institutetext: Departamento de Física, Universidade Federal da Paraíba,
Caixa Postal 5008, 58059-900, João Pessoa, PB, Brazil

# Some remarks on Hayward black hole with a cloud of strings

F. F. Nascimentoe1, addr1  V. B. Bezerrae2,addr1  J. M. Toledoe3,addr1

(Received: date / Accepted: date)

###### Abstract

We obtain the metric corresponding to the Hayward black hole space-time
surrounded by a cloud of strings and investigate the role played by this cloud
on the horizons, geodesics, effective potential and thermodynamics. We compare
the obtained results with the ones of the literature, corresponding to the
Hayward black hole, when the cloud of strings is absent. Also, the question
related to its nature, with respect to regularity, in this scenario, is
examined.

###### Keywords:

Hayward black hole Cloud of strings Thermodynamics

††journal: Eur. Phys. J. C

##  1 Introduction

Black hole solutions of Einstein’s equations were known since the middle
1910s, and then shortly after the formulation of the General Theory of
Relativity. The simplest vacuum solution, which describes the gravitational
field of a static, uncharged, and spherically symmetric body, was obtained by
Schwarzschild schwarzschild1916uber . Its generalization, to include the
presence of electric charge as a source was obtained by Reissner and Nordström
reissner1916eigengravitation ; nordstrom1918een . After nearly fifty years,
Kerr kerr1963gravitational  obtained a generalization of the Schwarzschild
solution, by considering rotation and it took only two years before the metric
of a charged and rotating gravitational body to be obtained, the well-known
Kerr-Newman solution newman1965metric . It is worth emphasizing that the
metrics corresponding to these black hole solutions have a curvature
singularity at $r=0$, whose existence creates some difficulties in the General
Theory of Relativity because, at the singularity point, the physical
quantities diverge and, therefore, the physical laws are not valid.

To solve these difficulties related to the curvature singularity and its
consequences, some black hole solutions have been considered, whose metrics
and curvature invariants have no singularity, that is, they are regular
everywhere, particularly, at the origin. These solutions correspond to what
are called Regular Black Holes. The first solution of a static spherically
symmetric Regular Black Hole dated back to the later 1960s and was obtained by
Bardeen bardeen1968non . Nowadays, there are several different static Regular
Black Hole solutions, among them, we can mention dymnikova1992vacuum ;
dymnikova2003spherically ; ayon1998regular ; ayon1999new ; mars1996models ;
sajadi2017nonlinear ; balart2014regular ; frolov2016notes ;
bronnikov2001regular , in special, the one obtained by Hayward
hayward2006formation . The solution of the field equations presented by
Hayward is free of charge term and its physical aspects are quite like
Bardeen’s solution bardeen1968non . The Hayward black hole solution
hayward2006formation  is static, uncharged and spherically symmetric. It is
worth calling attention to the fact that this solution becomes a de Sitter
space-time at the center of the black hole, and therefore, there is no
singularity at $r=0$, and, additionally, it is asymptotically flat as
$r\rightarrow\infty$.

Originally, the Hayward black hole solution was obtained from the modified
Einstein equations, with the parameter appearing in the solution being related
to the energy level in the near-horizon region of the black hole
hayward2006formation , which can be viewed as a constant acting in this space-
time. On the other hand, the Hayward black hole solution can also be obtained
in the context of a gravity theory coupled with nonlinear electrodynamics, in
which case, the parameter mentioned is no more a universal constant, but a
magnetic charge.

The Regular Black Hole solutions and the interesting consequences arising from
these solutions have inspired further investigations related to such black
holes, as, for example, those regarding particle geodesics abbas2014geodesic ;
chiba2017note ; zhou2012geodesic ; wei2015null , structure and lens effect
perez2018region ; eiroa2011gravitational , thermodynamics
saleh2018thermodynamics ; rodrigue2020thermodynamic ; molina2021thermodynamics
; iguchi2023gravitational  and quasi-normal modes fernando2012quasinormal ;
flachi2013quasinormal ; lin2013quasinormal .

In the later 1970s, Letelier letelier1979clouds  obtained general solutions of
the Einstein equations corresponding to spherically, plane-symmetric and
cylindrically symmetric space-times, by considering a cloud of strings as
sources of the gravitational field. In the first case, namely, when a
spherically symmetric cloud of strings, radially directed, surrounds the
gravitating body, the obtained solution is basically the Schwarzschild black
hole solution slightly modified, in such a way that the metric is similar to
the Schwarzschild one, but with a solid deficit angle which depends on the
parameter associated with the presence of the cloud of strings. Therefore, the
gravitational effects are of global origin, with respect to the cloud. As an
example of these effects, we mention the fact that the radius of the event
horizon is enlarged as compared with the Schwarzschild radius.

Given the possible astrophysical consequences, it is important to investigate
the gravitational consequences when a black hole is immersed in a cloud of
strings. With this aim, several studies concerning different aspects
associated with the physics of black holes surrounded by a cloud of strings
were performed during the last decades, in the context of the General Theory
of Relativity batool2017null ; sood2022thermodynamic ; rodrigues2022bardeen ;
gracca2018effects ; rodrigues2022embedding ; ghaffarnejad2019last , as well as
in different modified versions of this Theory morais2018cloud ;
toledo2019black ; ghaffarnejad2018effects ; herscovich2010black ;
ghosh2014cloud ; cai2020quasinormal . Also, in which concerns the Regular
Black Holes, some studies have been done by considering the presence of a
cloud of stringsrodrigues2022bardeen ; atamurotov2023weak , and some aspects
of the thermodynamics were investigated, with emphasis to the role played by
the cloud of strings.

In this paper, we investigate the role played by a cloud of strings
surrounding a Hayward black hole on the horizons, singular behavior,
geodesics, effective potential, and thermodynamics as compared to the case
where the cloud is absent. Also, some discussion about the regularity is
presented.

The paper is organized as follows. In Sect. 2, we review the Hayward black
hole solution and obtain the Hayward black hole solution in the case in which
this gravitational body is surrounded by a cloud of strings. In Sect. 3, we
focus the discussion on the horizons, singularity, geodesics, and effective
potential. Section 4 is devoted to different aspects of thermodynamics, with
emphasis on the role played by the parameter that codifies the presence of the
cloud of strings. In Section 5, we briefly summarize our results.

##  2 The metric

In this section, we obtain the metric corresponding to the Hayward black hole
with a cloud of strings and analyze its properties.

###  2.1 The Hayward solution

The metric of the non-singular(regular) black hole obtained by Hayward
hayward2006formation  is given by

$ds^{2}=f(r)dt^{2}-f(r)^{-1}dr^{2}-r^{2}d\Omega^{2},$ (1)

where $l$ and $m$ are positive constants, and
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. The function $f(r)$ is
given by

$f(r)=1-\frac{2mr^{2}}{r^{3}+2l^{2}m}$ (2)

Note that

$\lim_{r\rightarrow 0}\left(1-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)=1,$ (3)

which means that the Hayward metric is a regular (non-singular) solution of
Einstein equations. Furthermore, we can observe this property by calculating
the Kretschmann scalar, which is given by

$\displaystyle K$ $\displaystyle=R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}$
(4)
$\displaystyle=\frac{48m^{2}(r^{12}-4r^{9}g^{3}+18r^{6}g^{6}-2r^{3}g^{9}+2g^{12})}{(r^{3}+g^{3})^{6}},$

In the limit $r\rightarrow 0$, we get

$\displaystyle\lim_{r\rightarrow 0}K$ $\displaystyle=\lim_{r\rightarrow
0}\frac{48m^{2}(r^{12}-4r^{9}g^{3}+18r^{6}g^{6}-2r^{3}g^{9}+2g^{12})}{(r^{3}+g^{3})^{6}}$
(5) $\displaystyle=\frac{96m^{2}}{g^{6}}.$

Using the metric given by Eq. (1), we can obtain the following components of
the Einstein tensor hayward2006formation :

$G_{t}^{\;t}=G_{r}^{\;r}=\frac{12l^{2}m^{2}}{(r^{3}+g^{3})^{2}},$ (6)
$G_{\theta}^{\;\theta}=G_{\phi}^{\;\phi}=-\frac{24(r^{3}-l^{2}m)l^{2}m^{2}}{(r^{3}+g^{3})^{3}},$
(7)

where

$g^{3}\equiv 2l^{2}m.$ (8)

which, through Einstein equations, are proportional to the stress-energy
tensor of the source.

In Hayward’s solution, the function $f(r)$ can also be written as

$f(r)=1-\frac{2m(r)}{r},$ (9)

where

$m(r)=\frac{mr^{3}}{(r^{3}+g^{3})},$ (10)

is the black hole mass, which depends on the radial coordinate and $g^{3}$ is
given by Eq.(8). We can observe that, if $r\rightarrow\infty$,
$m(r)\rightarrow m$. Thus, very far from the Black Hole, the Hayward solution
is similar to the Schwarzschild one.

For $r\rightarrow 0$ (small values of $r$), we can write

$m(r)\approx\frac{mr^{3}}{g^{3}},$ (11)

and then

$f(r)\approx 1-Cr^{2},$ (12)

with $C=\frac{2m}{g^{3}}$ being a positive constant. Observe that the space-
time metric with $f(r)$ given by Eq. (12) is similar to De Sitter space-time.
Thus, the Hayward black hole has an internal core with behavior similar to de
Sitter metric hayward2006formation .

###  2.2 Hayward black hole with a cloud of strings

Now, let us consider the Hayward black hole with a cloud of strings.

A spherically symmetric cloud of strings is described by the stress-energy
tensor letelier1979clouds

$T^{\mu\nu}=\rho\frac{\Sigma^{\mu\beta}\Sigma_{\beta}^{\;\nu}}{(-\gamma)^{1/2}},$
(13)

where $\Sigma^{\mu\nu}$ is a bivector that represents the world sheet of the
strings and is given by

$\Sigma^{\mu\nu}=\epsilon^{ab}\frac{\partial{x^{\mu}}}{\partial{\lambda^{a}}}\frac{\partial{x^{\nu}}}{\partial{\lambda^{b}}},$
(14)

where $\epsilon^{ab}$ is the Levi-Civita bidimensional symbol and
$\epsilon^{01}=-\epsilon^{10}=1$.

The non-null components of the stress-energy tensor of the cloud of strings
are given by letelier1979clouds :

$T_{0}^{\;0}=T_{1}^{\;1}=\frac{a}{r^{2}},$ (15)

$T_{2}^{\;2}=T_{3}^{\;3}=0.$ (16)

Now, let us consider the line element corresponding to a Hayward black hole
with a cloud of strings given by d2022introducing :

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\phi^{2},$ (17)

where, $\nu$ and $\lambda$ are functions of the radial coordinate, $r$, only,
if we assume that the metric is static.

The non-null components of the Einstein tensor obtained by the metric of Eq.
(17) are given by

$G_{t}^{\;t}=e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}.$
(18)

$G_{r}^{\;r}=-e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)+\frac{1}{r^{2}},$
(19)

$G_{\theta}^{\;\theta}=G_{\phi}^{\;\phi}=\frac{1}{2}e^{-\lambda}\left(\frac{\nu^{\prime}\lambda^{\prime}}{2}+\frac{\lambda^{\prime}}{r}-\frac{\nu^{\prime}}{r}-\frac{\nu^{\prime
2}}{2}-\nu^{\prime\prime}\right).$ (20)

Thus, considering that there is no interaction between the cloud of strings
and the black hole, the total stress-energy tensor can be obtained by the
linear superposition of the individual ones. Thus, using Eqs.(18)-(20),
(6)-(7) and (15)-(16) the Einstein equations are given by:

$e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}=\frac{12l^{2}m^{2}}{(r^{3}+2l^{2}m)^{2}}+\frac{a}{r^{2}},$
(21)

$-e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}=\frac{12l^{2}m^{2}}{(r^{3}+2l^{2}m)^{2}}+\frac{a}{r^{2}},$
(22)

$\displaystyle\frac{1}{2}e^{-\lambda}\left(\frac{\nu^{\prime}\lambda^{\prime}}{2}+\frac{\lambda^{\prime}}{r}-\frac{\nu^{\prime}}{r}-\frac{\nu^{\prime
2}}{2}-\nu^{\prime\prime}\right)=$ (23)
$\displaystyle-\frac{24(r^{3}-l^{2}m)l^{2}m^{2}}{(r^{3}+2l^{2}m)^{3}}.$

Subtracting Eqs. (21) and (22), we obtain

$\lambda=-\nu\Rightarrow\lambda^{\prime}=-\nu^{\prime}.$ (24)

Summing Eqs.(21) and (22) and taking into account Eq.(24), we obtain

$e^{-\lambda}\frac{\lambda^{\prime}}{r}-e^{-\lambda}\frac{1}{r^{2}}+\frac{1}{r^{2}}=\frac{12l^{2}m^{2}}{(r^{3}+2l^{2}m)^{2}}+\frac{a}{r^{2}}.$
(25)

Now, let us write the following relations

$\nu=-\lambda=ln(1+f(r)).$ (26)

Taking into account Eqs.(24) and (26), we can write Eqs.(25) and (23),
respectively, as follows:

$-\frac{1}{r^{2}}(rf^{\prime}+f)=\frac{12l^{2}m^{2}}{(r^{3}+2l^{2}m)^{2}}+\frac{a}{r^{2}},$
(27)

$2\frac{f^{\prime}}{r}+f^{\prime\prime}=48\frac{(r^{3}-l^{2}m)l^{2}m^{2}}{(r^{3}+2l^{2}m)^{3}}.$
(28)

Summing Eqs.(27) and (28) and multiplying by $r^{2}$, we get:

$r^{2}f^{\prime\prime}+rf^{\prime}-f-a-\frac{12l^{2}m^{2}r^{2}}{(r^{3}+2l^{2}m)^{2}}-48\frac{(r^{3}-l^{2}m)l^{2}m^{2}r^{2}}{(r^{3}+2l^{2}m)^{3}}=0,$
(29)

whose solution is given by

$f(r)=\frac{4l^{2}m^{2}-2al^{2}mr-
ar^{4}}{r(r^{3}+2l^{2}m)}+\frac{C_{1}}{r}+rC_{2},$ (30)

where we can adopt $C_{1}=-2m$ and $C_{2}=0$. Substituting Eq.(30) into
Eq.(26), we get

$\nu=-\lambda=ln\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right).$ (31)

Thus, the metric of the Hayward black hole with a cloud of strings is written
as

$\displaystyle ds^{2}$
$\displaystyle=\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)dt^{2}$ (32)
$\displaystyle-\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)^{-1}dr^{2}$
$\displaystyle-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta d\phi^{2}.$

Note that, if $a=0$, we obtain the Hayward metrichayward2006formation  given
by Eq. (1). If $l=0$, the metric of Eq.(32) becomes similar to the Letelier
spacetime letelier1979clouds .

Thus, for the metrics given by Eq. (32), the Kretschmann scalar is

$\displaystyle K$
$\displaystyle=R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}=\frac{4a^{2}}{r^{4}}+\frac{16am}{g^{3}r^{2}+r^{5}}$
(33)
$\displaystyle+\frac{48m^{2}\left(r^{12}-4r^{9}g^{3}+18r^{6}g^{6}-2r^{3}g^{9}+2g^{12}\right)}{\left(r^{3}+g^{3}\right)^{6}},$

If we neglect the could of strings, $a=0$, we obtain the Kretschmann scalar
given in Eq. (4). Now, let us determine the value of these results in the
limits $r\rightarrow 0$ and $r\rightarrow\infty$.

$\displaystyle\lim_{r\rightarrow 0}K$ $\displaystyle=\lim_{r\rightarrow
0}\left(\frac{4a^{2}}{r^{4}}+\frac{16am}{g^{3}r^{2}+r^{5}}\right.$ (34)
$\displaystyle\left.+\frac{48m^{2}\left(r^{12}-4r^{9}g^{3}+18r^{6}g^{6}-2r^{3}g^{9}+2g^{12}\right)}{\left(r^{3}+g^{3}\right)^{6}}\right)=\infty,$

$\displaystyle\lim_{r\rightarrow\infty}K$
$\displaystyle=\lim_{r\rightarrow\infty}\left(\frac{4a^{2}}{r^{4}}+\frac{16am}{g^{3}r^{2}+r^{5}}\right.$
(35)
$\displaystyle+\left.\frac{48m^{2}\left(r^{12}-4r^{9}g^{3}+18r^{6}g^{6}-2r^{3}g^{9}+2g^{12}\right)}{\left(r^{3}+g^{3}\right)^{6}}\right)=0.$

Therefore, from the analysis of the Kretschmann scalar, in the limit
$r\rightarrow 0$, we conclude that the inclusion of the cloud of strings
influences the metric, by destroying the regularity, and as a consequence,
introducing a singularity. In other words, the metric of the Hayward black
hole with a cloud of strings is singular in the origin ($r=0$).

##  3 Black hole horizons, geodesics and effective potential

###  3.1  Black hole horizons

In what follows, we study the horizons of the Hayward black hole space-time,
given by the line element shown in Eq.(32). From now on, let us identify the
function $g(r)$ as

$g(r)=1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}$ (36)

Analyzing the roots of the function $g(r)$, we can obtain a critical black
hole mass, which is given by

$m_{*}=\frac{3}{4}\sqrt{3}(1-a)^{3/2}l,$ (37)

as well as a critical value for the radial coordinate, $r$, shown in what
follows

$\displaystyle r_{*}$
$\displaystyle=-\frac{\sqrt{3}}{2(1-a)^{5/2}}\left[a^{3}-3a^{2}+a\sqrt{1-a}\sqrt[3]{-(1-a)^{9/2}}\right.$
(38)
$\displaystyle\left.+\left(-(1-a)^{9/2}\right)^{2/3}+3a-\frac{(a-1)^{5}}{\left(-(1-a)^{9/2}\right)^{2/3}}-1\right]l.$

Considering only positive values of $r$, if the black hole mass is higher than
the critical mass, $m>m_{*}$, $g(r)$ has two real roots. If $m=m_{*}$, $g(r)$
has a unique real root, which is equal to $r_{*}$. Finally, $g(r)$ has no real
roots for $m<m_{*}$. If we neglect the cloud of strings, $a=0$, the critical
mass is reduced to $m_{*}=(3\sqrt{3}/4)l$ and the critical radius is
$r_{*}=\sqrt{3}l$, quantities already obtained by Hayward hayward2006formation
. The described behavior of $g(r)$ can be observed in Fig. 1.

Figure 1: The function $g(r)$ for $l=1$ and different values of $m$: $m=0$,
$m<m_{*}$, $m=m_{*}$ and $m>m_{*}$.

###  3.2 Black hole geodesics

Given the space-time metric, the trajectory of particles and light can be
described by the geodesic motion. The geodesic equations can also be obtained
from the Lagrangian given in the equation

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$

which, in the space-time of the Hayward black hole with a cloud of strings,
can be written as

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}\left[\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)\dot{t}^{2}-\frac{1}{\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)}\dot{r}^{2}\right.$
(39)
$\displaystyle-r^{2}\dot{\theta}^{2}-\left.r^{2}\sin^{2}{\theta}\dot{\phi}^{2}\right],$

where the dot represents the derivative in respect to the proper time $\tau$.
Rescaling the parameter $\tau$, we can define $L=2\mathcal{L}$, which, for
time-like geodesics, is equal to $+1$, for space-like geodesics is equal to
$-1$ and is equal to $0$ for null geodesics chandrasekhar1983mathematical .

The Euler-Lagrange equations are given by

$\frac{d}{d\tau}\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)-\frac{\partial\mathcal{L}}{\partial
x^{\mu}}=0.$ (40)

For $\mu=0$ and $\mu=3$ in Eq.(40), with $\mathcal{L}$ given by Eq.(39), we
get, respectively:

$\dot{t}=\frac{E}{\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)},$ (41)

$\dot{\phi}=-\frac{J}{r^{2}\sin^{2}\theta},$ (42)

where $E$ and $J$ are movement constants which corresponds to the Killing
vectors $\partial_{t}$ and $\partial_{\phi}$, respectively. We can interpret
these constants as the energy $E$ and the angular momentum $J$ of the particle
which is moving nearby the black hole.

Let us restrict the analysis of the geodesics to the equatorial plane of the
black hole, $\theta=\frac{\pi}{2}$. Doing that, the Eqs. (41)-(42) are reduced
to

$\dot{t}=\frac{E}{\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)},$ (43)

$\dot{\phi}=-\frac{J}{r^{2}}.$ (44)

where $\dot{t}$ and $\dot{\phi}$ are the derivatives of $t$ and $\phi$ with
respect to the proper time $\tau$. Substituting Eqs. (43) and (44) into
Eq.(39), we get

$E^{2}=\dot{r}^{2}+V_{eff},$ (45)

where

$V_{eff}=\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)\left(\frac{J^{2}}{r^{2}}+L\right)$
(46)

represents the effective potential for the geodesic motion in the space-time
of a Hayward black hole with a cloud of strings.

Using the relation

$\frac{dr}{dt}\frac{dt}{d\tau}=\frac{dr}{d\tau}\Rightarrow\left(\frac{dr}{dt}\right)^{2}\left(\frac{dt}{d\tau}\right)^{2}=\left(\frac{dr}{d\tau}\right)^{2}\Rightarrow\left(\frac{dr}{dt}\right)^{2}\dot{t}^{2}=\dot{r}^{2}$
(47)

into Eq.(45) and using Eqs.(46) and eq.(43), we get

$\left(\frac{dr}{dt}\right)^{2}=g(r)^{2}\left[1-\frac{g(r)}{E^{2}}\left(\frac{J^{2}}{r^{2}}+L\right)\right].$
(48)

####  3.2.1 Radial movement of a photon

For the radial movement $(J=0)$ of a photon $(L=0)$, Eq.(48) can be written as

$\left(\frac{dr}{dt}\right)^{2}=g(r)^{2}.$ (49)

Substituting Eq.(36) into Eq.(49), we get the relation between the coordinates
$t$ and $r$, which is given by

$\pm t=\int\frac{1}{1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}}dr.$ (50)

Using the Eq.(45), we can obtain the relation between the coordinate $r$ the
proper time $\tau$ for the radial movement of a photon, which is given by

$\left(\frac{dr}{d\tau}\right)^{2}=E^{2},$

$\pm\tau=\frac{r}{E}.$ (51)

####  3.2.2 Radial movement of a massive particle

Now, let us consider the movement of massive particles $(L=1)$ in radial
trajectories $(J=0)$ nearby the black hole. From Eq.(48), we obtain

$\left(\frac{dr}{dt}\right)^{2}=g(r)^{2}-\frac{g(r)^{3}}{E^{2}}.$ (52)

Substituting Eq.(36) into Eq.(52), we can find the relationship between the
coordinates $t$ and $r$ for the radial movement of the particle:

$\pm
t=\int\frac{dr}{\sqrt{\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)^{2}-\frac{\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)^{3}}{E^{2}}}}.$
(53)

From Eq.(45), we get the relationship between the proper time $\tau$ and the
radial coordinate $r$:

$\left(\frac{dr}{d\tau}\right)^{2}=E^{2}-g(r),$

$\pm\tau=\int\frac{dr}{\sqrt{E^{2}-\left(1-a-\frac{2mr^{2}}{r^{3}+2l^{2}m}\right)}}.$
(54)

###  3.3 Effective potential

The behavior of the effective potential ($V_{eff}$) of the geodesic motion,
given by Eq. (46), can tell us about the behavior of a massive particle or a
photon near the black hole. So, in Figs. 2 to 5, we plot the effective
potentials for different values of $a$ and $l$ for time-like and null-like
geodesics. In some figures, we represent, in detail, $V_{eff}$ near the black
hole ($r$ near zero).

In Fig. 2, we represent the effective potential for geodesics ($L=1$ and
$J^{2}=20$). We can observe that, for $l=0$, there is no stable circular
geodesics, since the graphics do not show a local minimum for any value of
$a$. On the other hand, for $l>0$, we can observe the possibility of the
existence of stable circular geodesics, depending on the value of the cloud of
strings parameter $a$.

Figure 2: Effective potential for non-radial time-like geodesics ($L=1$ and
$J^{2}=20$), for different values of $a$ and $l$.

For non-radial time-like geodesics (Fig. 3), we can observe that, in all
cases, $V_{eff}\rightarrow 0$ for regions far from the black hole,
$r\rightarrow\infty$. For $l=0$, there is no stable circular geodesics, since
the graphics do not show local minima. The existence of stable circular orbits
of photons around the black hole depends also on the cloud of strings
parameter, as can be seen in Fig. 3.

Figure 3: Effective potential for non-radial null-like geodesics ($L=0$ and
$J^{2}=20$), for different values of $a$ and $l$.

In which concerns the behavior of radial time-like geodesics, Fig. 4 shows us
that the stability of radial movement does not occur for $l=0$. On the other
hand, for $l>0$, there will always be stable geodesics.

Figure 4: Effective potential for radial time-like geodesics ($J^{2}=0$ and
$L=1$), for different values of $a$ and $l$.

Finally, we can observe in Fig. 5 that, for radial null-like geodesics
($J^{2}=0$ and $L=0$), the effective potential is constant and equal to zero.

Figure 5: Effective potential for radial null-like geodesics ($J^{2}=0$ and
$L=0$), for different values of $a$ and $l$.

##  4 Black Hole thermodynamics

In this section, we study the thermodynamics of the Hayward black hole with a
cloud of strings by examining the behavior of the mass, Hawking temperature,
and heat capacity as a function of entropy.

###  4.1 Black hole mass

Let $r_{h}$ be the radius of the horizon, thus we have that $g(r_{h})=0$,
where $g(r)$ is given by Eq.(36). Thus, we can write the mass of the black
hole in terms of $r_{h}$ through the following equation:

$m=\frac{(1-a)r_{h}^{3}}{2(r_{h}^{2}+(a-1)l^{2})},$ (55)

which is written in terms of the parameter that takes care of the presence of
the cloud of strings, namely, $a$. Note that if $a=0$, we recover the mass of
the regular Hayward black hole, without the cloud of strings, in terms of the
horizon radius. By setting $l=0$ and $a=0$ we recover the mass of the
Schwarzschild black hole.

The area of the horizon can be calculated by

$A=\int\sqrt{-g}d\theta d\phi=4\pi r_{h}^{2}.$ (56)

On the other hand, the entropy of the black hole can be calculated through the
area law bekenstein1973black , using the relation

$S=\frac{A}{4}=\pi r_{h}^{2}.$ (57)

Thus, we can write the mass parameter as a function of the entropy as

$m=\frac{(1-a)S^{3/2}}{2\pi^{3/2}(\frac{S}{\pi}+(a-1)l^{2})}.$ (58)

Figure 6: Black hole mass as a function of the entropy $m(S)$ for different
values of $a$ and $l$.

In Figure 6, we represent the behavior of the mass parameter, $m$, as a
function of the entropy of the black hole, $S$, in different situations. Note
that, for the Schwarzschild black hole, ($a=0$ and $l=0$), the mass parameter
presents only positive values for positive values of the entropy,$S$. Similar
behavior is obtained when ($l=0$), thus we return to the Schwarzschild black
hole scenario, but now, with a cloud of strings, $0<a<1$. If we consider the
Hayward black hole, it is possible to notice that the mass parameter has
positive and negative values depending on the parameters of the black hole.
This is also repeated when we consider the cloud of strings in Hayward’s black
hole space-time. It is important to note that the cloud of string parameter
modifies the phase transition point for Hayward black hole surrounded by this
cloud.

###  4.2 Hawking temperature

The surface gravity $(\kappa)$ for the Hayward black hole with a cloud of
strings can be calculated using the following expression:

$\kappa=\frac{g^{\prime}(r)}{2}\left|\frac{}{}_{r_{h}}\right.,$ (59)

with ′ denoting the derivative with respect to the radial coordinate. Hawking
showed that the black hole emits radiation and its corresponding temperature,
Hawking temperature, for stationary space-time, is given by
hawking1975particle :

$T_{\kappa}=\frac{\kappa}{2\pi}.$ (60)

It is worth remembering that for vacuum space-time with spherical symmetry,
the first law of thermodynamics gives

$dm=T_{f}dS\rightarrow T_{f}=\frac{dm}{dS},$ (61)

where $m$ and $S$ are the total energy and entropy of the system,
respectively, and $T_{f}$ is the temperature of the black hole predicted in
the first law. Taking into account the area law given by Eq.(57), the first
law of thermodynamics does not provide a correct way to calculate the
temperature of regular black holes ma2014corrected ; maluf2018thermodynamics .

Thus, the Hawking temperature, $T_{\kappa}$ and $T_{f}$, for the Hayward black
hole obtained through equations Eqs. (60) and (61), respectively, are not
equivalent, that is, the first law, given by Eq.(61), is not appropriate to
calculate the correct temperature for regular black holes since such black
holes are not vacuum solutions.

Using Eq. (60) and with $\kappa$ given by (59), it is possible to calculate
the Hawking temperature, $T_{\kappa}=T$, for the Hayward black hole with the
cloud of strings:

$T=\frac{mr_{h}(r_{h}^{3}-4l^{2}m)}{2\pi(r_{h}^{3}+2l^{2}m)^{2}}.$ (62)

Putting Eq.(55) into Eq.(62), we obtain the following result

$T=\frac{(1-a)[r_{h}^{2}+3(a-1)l^{2}]}{4\pi r_{h}^{3}}.$ (63)

Substituting $r_{h}=(\frac{S}{\pi})^{1/2}$ in Eq.(63), we finally obtain the
Hawking temperature expression as a function of entropy for the Hayward black
hole with the cloud of strings:

$T=\frac{(1-a)\sqrt{\pi}[3(a-1)l^{2}+\frac{S}{\pi}]}{4S^{3/2}}.$ (64)

Figure 7: Black hole temperature as a function of entropy $T(S)$ for different
values of $a$ and $l$.

In figure 7, we represent the behavior of the temperature parameter, $T$, as a
function of the entropy of the black hole, $S$, in different situations. Note
that for the Schwarzschild space-time ($a=0$ and $l=0$) the temperature
parameter presents only positive values for $S>0$. Analogously, for the
Letelier space-time ($l=0$ and $0<a<1$). When we consider the Hayward space-
time ($l=1$ and $a=0$), it is already possible to notice that the temperature
parameter will present positive and negative values depending on the black
hole parameters. This is also repeated when we consider the cloud of strings
in Hayward spacetime ($l=1$ and $0<a<1$).

###  4.3 Heat capacity

Heat capacity provides information about the thermodynamic stability of a
system. We can calculate the heat capacity of the Hayward black hole with the
cloud of strings from the following expression:

$C=T\frac{\partial S}{\partial T}=T\left(\frac{\partial T}{\partial
S}\right)^{-1}.$ (65)

Substituting Eq.(64) into Eq.(65), we find the following expression for the
heat capacity as a function of the black hole entropy:

$C=-\frac{2S[3(a-1)l^{2}\pi+S]}{9(a-1)l^{2}\pi+S}.$ (66)

Figure 8: Heat capacity as a function of the entropy $C(S)$ for different
values of $a$ and $l$.

The behavior of the heat capacity as a function of entropy, for different
values of $l$ and $a$, is given in Figure 8. For $l=0$, Eq.(66) reduces to
$C=-2S$, as expected for the case of Schwarzschild spacetime. In this case,
the heat capacity is negative for $S>0$, which indicates an unstable
thermodynamic system.

When we take into account Hayward ($l=1$,$a=0$) and the Hayward space-time
with the cloud of strings ($l=1$,$0<a<1$), there are values for the entropy
for which the heat capacity acquires positive values. This means that the
Hayward black hole can be unstable or stable depending on the values of the
entropy considered. It is important to point out that the string parameter
plays an important role in the behavior of the heat capacity, modifying the
phase transition point. Thus, unlike the Schwarzschild black hole, which is
thermodynamically unstable, the Hayward black hole and the Hayward black hole
with a cloud of strings present regions of stability, depending on the chosen
parameters.

##  5 Concluding remarks

In conclusion, we can say that the cloud of strings has an interesting role in
different aspects related to the Hayward black hole surrounded by this cloud.

Firstly, let us analyze the Kretschmann scalar in the limit $r\rightarrow 0$,
whose result is $\infty$. This means that the inclusion of the cloud of
strings turns the black hole singular. In other words, the metric of the
Hayward black hole with a cloud of strings is singular at the origin.

With relation to the possible roots of $g(r)$, and considering a critical
mass, $m_{*}$, given by Eq(37), three different scenarios are present, namely:
(i) the black hole mass is higher than the critical mass, $m>m_{*}$, thus
$g(r)$ has two real roots; (ii) the black hole mass is equal to the critical
mass, $m=m_{*}$, then, $g(r)$ has a unique real root, which is equal to
$r_{*}$; and (iii) $m<m_{*}$, in which case, $g(r)$ has no real roots.

The effective potential ($V_{eff}$) of the geodesic motion, given by Eq. (46),
is represented in Figs. 2 to 5, for different values of $a$ and $l$ for time-
like and null-like geodesics. As shown, in some cases, in which $l=0$, there
are no stable circular geodesics, irrespective of the values of $a$. On the
other hand, for $l>0$, we can observe the possibility of the existence of
stable circular geodesics, depending on the value of the cloud of strings
parameter $a$.

For non-radial time-like geodesics (Fig. 3), we can observe that, in all
cases, $V_{eff}\rightarrow 0$ for regions far from the black hole,
$r\rightarrow\infty$. For $l=0$, there is no stable circular geodesics.
Otherwise, there are stable circular orbits of photons around the black hole,
depending on the presence of the cloud of strings, as can be seen in Fig. 3.

In which concerns the behavior of radial time-like geodesics, shown in Fig. 4,
for $l>0$, there exist stable geodesics in all situations.

The behavior of the mass parameter, m, as a function of the entropy of the
black hole, S, in different situations, are very similar, as we can see in
Fig. 6. It is worth calling attention to the fact that preserves the shape,
irrespective of the values of the parameter that codifies the presence of the
cloud of strings. The only difference is a shift to the left, as the values of
this parameter increase.

When the Schwarzschild black hole space-time (a = 0 and l = 0) is taken into
account, the temperature parameter presents only positive values for $S>0$.
The same behavior occurs when a cloud of strings is added to Schwarzschild
black hole space-time, namely, $l=0$ and $0<a<1$. Now, if the Hayward black
hole is considered, $l=1$ and $a=0$, the temperature will assume positive and
negative values depending on the black hole parameters. The same situation
appears when the presence of a cloud of string is considered in Hayward.

The heat capacity assumes positive values or negative ones, depending on the
values of entropy, when the cloud of strings is present. This means that the
Hayward black hole with a cloud of strings can be unstable or stable depending
on depending on the values of the entropy considered. This behavior is
analogous to the one obtained when the cloud is absent. Otherwise, the role of
the cloud of strings is to shift the graphs to the left.

It is important to point out that the cloud of strings parameter plays an
important role in the behavior of the heat capacity, modifying the points of
the phase transitions, as well as on the Hawking temperature, black hole mass,
effective potential, geodesics and horizons, and removing the regular
character of the Hayward regular black hole solution, by removing it and
turning the solution singular.

###### Acknowledgements.

V.B. Bezerra is partially supported by CNPq-Brazil ( Conselho Nacional de
Desenvolvimento Científico e Tecnológico) through Research Project No.
307211/2020-7. F. F. Nascimento and J. M. Toledo acknowledge Departamento de
Fisica, Universidade Federal da Paraiba, for hospitality.

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# The Quantum-Extended Church-Turing Thesis in Quantum Field Theory

Cameron<EMAIL_ADDRESS>Physics Department, University of
Connecticut, Storrs, CT 06269, USA

The quantum-Extended Church-Turing thesis has been explored in many physical
theories including general relativity but lacks exploration in quantum field
theories such as quantum electrodynamics. Through construction of a
computational model whose gate set mimics the interactions of QED, we
demonstrate that one of the defining features of quantum field theory,
particle creation and annihilation, is not likely to violate the quantum-
Extended Church-Turing thesis. Through this computational model, it is shown
that particle creation is likely only another form of quantum parallelism.
However, whether or not the quantum-Extended Church-Turing thesis will hold
for all computational devices in quantum field theories is still not known.
For example, we briefly examine certain interactions in quantum
electrodynamics which may create multi-qubit gates. These gates may have
exponential complexity at the cost of being exponentially weak. This may in
turn allow for computational advantage over traditional gate sets such as
Clifford+T.

##  1 The Church-Turing Thesis

The Church-Turing thesis states that every solvable decision problem can be
calculated on a Turing machine [1, 2, 3]. This thesis was later extended to
complexity theory by proposing that any efficient calculation done by a
physical system can be efficiently computed by a Turing machine, where an
efficient computation is a computation which can be performed in polynomial
time [4].

The extended Church-Turing thesis held until the appearance of Shor’s
algorithm, which allows quantum computers to factor large numbers in
exponentially less time than the best classical algorithm [5, 6]. This
algorithm implies that certain computational problems may be efficiently
solved on a quantum computer but not on a classical Turing machine. In
response to this development, the quantum-Extended Church-Turing thesis was
formulated, which proposes that any efficient calculation can be efficiently
done by a quantum Turing machine [7].

##  2 qECT in General Relativity

Armed with a violation of the extended Church-Turing thesis, physicists sought
out new ways to use physics to violate the quantum-Extended Church-Turing
thesis. Many of the most promising approaches to date have been found through
utilizing time dilation in general relativity.

In general relativity, time dilation allows for an observer to initialize a
computing machine outside of a black hole, and then cross the event horizon in
order to view the result of an infinitely long computation in constant time
[8, 9]. This allows the observer to perform calculations which would otherwise
take infinite time on a Turing machine. As the results of these infinitely
long computational problems cannot be computed by a Turing machine, this
violates the Church-Turing thesis. However, Susskind proposed solving this
through altering the Church-Turing thesis to require the physical system to be
able to communicate with the holographic boundary of space. This modification
prevents the event horizon of a black hole from being used to violate the
Church-Turing thesis [10].

Similarly to how the properties of general relativity have been leveraged to
test the quantum-Extended Church-Turing thesis, we will investigate if the
properties of quantum field theory may allow for computational advantage over
traditional quantum computers.

##  3 Computing Machines in Quantum Electrodynamics

Quantum field theory is the most accurate description of the physical world
which has been verified through experiment [11, 12, 13]. The widely successful
Standard Model of particle physics is a quantum field theory which
encapsulates the strong, weak, and electromagnetic forces, leaving out only
gravity [14, 15, 16]. Although the study of quantum computing brings computer
science a step closer to the fundamental laws of physics, there are still
properties of quantum field theory which are not shared by quantum mechanics.

Foremost of these properties is particle creation and annihilation. This
property allows quantum field theory to create new particles, or new computing
subsystems, during a computation. At a first glance, this property is
reminiscent of a nondeterministic Turing machine, which can branch to create
new Turing machines and calculate each possible solution to a problem in
parallel [17]. However, if particle creation in quantum field theory indeed
behaved this way, it would violate the quantum-Extended Church-Turing thesis,
assuming $P\neq NP$. We will instead demonstrate that this property is likely
only another form of quantum parallelism, a property already present in
traditional quantum computers.

One of the simplest quantum field theories is Quantum Electrodynamics (QED).
QED only describes electromagnetic interactions, disregarding the Strong and
Weak force, as well as gravity. This theory simply couples a fermionic field
to a U(1) gauge boson. The Lagrangian of quantum electrodynamics is [14],

$\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}+m)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-ie\bar{\psi}\gamma^{\mu}A_{\mu}\psi$
(1)

We will use this physical theory to inspire a computational model,
constructing a gate set which mimics the interactions of QED. In this
computational model, we will make use of the spin degree of freedom of the
photon as a qubit. Therefore, computational gates which create new photons
will also create new qubits.

In quantum field theory, not only can we have a superposition of states, but
we can also have a superposition of particle number. This can be illustrated
through the following two states,

$\frac{1}{\sqrt{2}}(A_{1}(x)-A_{2}(x))\ket{\Omega}\approx\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$
(2)

$\frac{1}{2}(A_{1}(x)-A_{2}(x))+\frac{1}{2}(A_{1}(x)A_{1}(y)+A_{2}(x)A_{2}(y))\ket{\Omega}\approx\frac{1}{2}(\ket{0}-\ket{1})+\frac{1}{2}(\ket{00}+\ket{11})$
(3)

The operator $A_{1}(x)$ creates a horizontally polarized photon at position
$x$, and the operator $A_{2}(y)$ creates a vertically polarized photon at
position $y$. In this example, horizontally polarized photons are defined as
the $\ket{0}$ computational basis state, and vertically polarized photons are
defined as the $\ket{1}$ computational basis state.

Equation 2 shows a photon at position x in a $\ket{-}$ state, a computational
state which can be easily obtained in a normal quantum computer. However,
equation 3 demonstrates a superposition of particle number, which includes
both a single photon in the state $\ket{-}$ in superposition with a bell state
$\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$. Wavefunctions such as this second
example allow quantum field theory to realize states in a way which cannot be
realized using quantum mechanics, as quantum mechanics preserves particle
number [18].

##  4 Example Computer in QED

Since calculating the exact interactions and time evolution of an arbitrary
state in QED can be incredibly complex, we will instead use a simpler gate
based model as a guide. The gate set of this computational model will mimic
the interactions of QED.

The time evolution operator in quantum field theory is found by exponentiating
the Hamiltionian to obtain the time evolution operator $U(t)$ [14].

$U(t)=e^{-i\mathcal{H}t}$ (4)

Perturbation theory allows us to expand the time evolution operator in orders
of the coupling constant $e$. In this way we can investigate different
interactions which take place in QED. For example, we can see the structure of
the following interaction which is present at fourth order in perturbation
theory.

$\mathcal{O}(e^{4})a^{\dagger}_{p}a^{\dagger}_{q}a_{r}a_{s}\in\frac{(-i\mathcal{H}t)^{4}}{4!}$
(5)

This interaction has the form of a two-qubit gate and is somewhat similar to
those used to generate two-qubit gates in superconducting architectures [19,
20]. Due to this, we will include a two-qubit gate in the gate set of the QED
computational model. This interaction can be visually represented through the
following Feynman diagram, where time is along the x-axis and space is along
the y-axis.

Figure 1. A Feynman diagram whose form suggests a two qubit gate.

In addition to two-qubit gates, the following interaction is present at sixth
order.

$\mathcal{O}(e^{6})a^{\dagger}_{p}a^{\dagger}_{q}a^{\dagger}_{r}a^{\dagger}_{s}a_{t}a_{u}\in\frac{(-i\mathcal{H}t)^{6}}{6!}$
(6)

Due to this interaction, we will similarly adopt a novel gate into the gate
set which can output four qubits from two input qubits, allowing for the
computational model to utilize particle creation. The interaction can be
expressed diagramatically using the following Feynman diagram,

Figure 2. A Feynman diagram in Quantum Electrodynamics which allows for the
creation of new qubits.

As is conventionally done in quantum computing, we can view these gates in a
circuit diagram. For example,

Figure 3. Use of the particle creation gate depicted in a circuit diagram,
which may be generated from the interaction in Figure 2.

This ability to create new qubits differentiates this computational model from
a traditional quantum computer. This brings about the question, does the
addition of this particle creation gate bring computational advantage to
devices in quantum field theory?

##  5 Equivalence to a Qutrit Quantum Computer

To answer the previous question, we will now demonstrate that the computation
model constructed can be simulated by a quantum computer with an exponential
number of ancilla qutrits. Therefore, these particle creation gates would not
allow for a violation of the quantum-Extended Church-Turing thesis.

We can demonstrate the equivalence of this QED gate set to a traditional
quantum computer as follows. Each spin degree of freedom in quantum
electrodynamics can be translated to 3 states. The first two states are common
to quantum computers, $\ket{0}$, $\ket{1}$, however we also include a third
state $\ket{\Omega}$, which indicates that the particle does not exist.

Figure 4. The same circuit as in Figure 3, simulated on a quantum computer
with access to ancilla qubits initialized in the $\ket{\Omega}$ state.

Re-examining the state from Section 3,
$\ket{\psi}\approx\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})+\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$,
we can re-express this state as the qutrit state,

$\frac{1}{2}(A_{1}(x)-A_{2}(x))+\frac{1}{2}(A_{1}(x)A_{1}(y)+A_{2}(x)A_{2}(y))\ket{\Omega}\approx\frac{1}{\sqrt{2}}(\ket{0\Omega}+\ket{1\Omega})+\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$
(7)

In addition to being able to express a superposition of particle number using
qutrits, and since the computing machine is limited to a polynomial amount of
time, only an exponential number of particles can be created with the gates
included. In this way, all states which can be reached in exponential time can
be expressed with a finite number of qutrits.

The gate set outlined in the previous sections translate to two-qutrit and
four-qutrit gates, which can be efficiently simulated using Clifford+T gates.
In this way, any state and any operator which can be efficiently computed in
the computational model put forth in Section 4, can be efficiently simulated
by a qutrit quantum computer with an exponential number of ancilla qutrits.
Therefore, these particle creation gates would not allow for a violation of
the quantum-Extended Church-Turing thesis.

Through understanding that qutrit states in superposition corresponds to a
superposition of particle number in the computational model, this
computational model suggests that particle creation acts only as another form
of quantum parallelism. This finding demonstrates that particle creation, one
of the defining aspects of quantum field theory, will likely not allow for a
violation of the quantum-Extended Church-Turing thesis.

##  6 Exponentially Small Multi-Qubit Operations

Although particle creation may not violate the quantum-Extended Church-Turing
thesis, this does not imply that other aspects of quantum field theory will
not. There are still other possible mechanisms which may still allow for
devices in quantum field theory to gain computational advantage over quantum
computers.

We can discover such an example by investigating the limitations of the
constructed gate set. The four-qutrit particle creation gate shown previously
can be simulated efficiently by Clifford+T gates since it has a fixed size.
However, quantum electrodynamics allows for interactions which scale with the
system size, and therefore are likely more difficult to simulate with a
quantum computer. These interactions are exponentially suppressed by the
coupling constant $e$, suggesting that these interactions may be best modeled
by a new class of gates best described as _exponentially suppressed multi-
qubit_ gates.

A few examples of these high order Feynman diagrams are shown below for a
7-qubit case.

Figure 5. Example Feynman diagrams which may allow for multi-qubit gates whose
magnitude is suppressed exponentially by the coupling constant
$\mathcal{O}(e^{N}$).

Each vertex suppresses the amplitude of the diagram by a factor of $e$, the
coupling constant of electromagnetism. Therefore, the amplitude of an
$N$-qubit operation which leveraged this type of interaction must scale as
$e^{N}$.

As a first approach to modeling these interactions, it may be useful to
examine multi-qubit operators $U$ which satisfy the following constraint,

$Tr(\mathcal{I}-U)\leq 2^{-n}$ (8)

Where $n$ is the number of qubits to which the gate is applied. These
operators may have up to $\mathcal{O}(2^{2n})$ parameters, which makes them
exponentially difficult to exactly realize with the traditional Clifford+T
gate set. However, it may be possible to approximate these gates using
traditional gate sets.

Examining the successes of quantum computing indicate that access to these
gates may provide computational advantage. The Gottesman-Knill theorem
demonstrates that the computational speedup of quantum computers does not
arise solely from the properties of entanglement or superposition [21].
Furthermore, Shor’s algorithm instead suggests that quantum speedups arise
from exponentially small finely tuned gates [5]. In this way, access to these
gates may allow for greater computational power than a traditional gate set
such as Clifford+T.

If computational advantage is identified with these gates, then it is
important to remember that the quantum-Extended Church-Turing thesis is only
violated if these gates can be realized in QED or the Standard Model.
Althought these gates are inspired by exponentially weak mutiparticle
interactions in QED, symmetries and other properties of quantum field theories
often constrain physically realizable interactions.

##  7 Conclusion and Further Directions

In this paper, we construct a computational model to investigate the
computational power of quantum field theories. Using this model, we
demonstrate that particle creation, one of the defining properties of quantum
field theory, is not likely to violate the quantum-Extended Church-Turing
thesis, as the states and gates of this computational model can be efficiently
simulated by a qutrit quantum computer with exponentially many ancilla.

Although particle creation may not allow for computational advantage over a
quantum computer, other properties of quantum field theories such as
exponentially weak multi-qubit interactions may still allow for computational
advantage.

To begin investigating this, it may be useful to examine if the class of
operators which satisfy $Tr(\mathcal{I}-U)\approx 2^{-n}$ allow for
computational advantage in any known problems, or if these gates can be
approximated in polynomial time by traditional gate sets. Since these
operators have exponentially many tunable parameters, it is possible that they
may be able to access parts of the state space which would be impossible for
normal gate sets to efficiently access.

Even if computational advantage is identified through this multi-qubit gate
set, these gates must still be constructed in a quantum field theory such as
QED or the Standard Model in order to claim a violation of the quantum-
Extended Church-Turing thesis. However, seeing as Shor’s algorithm finds a
computational speedup through exponentially small, finely tuned gates, the
computational power of this gate set may be of interest.

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