# Thurston Geometries in Three-Dimensional New Massive Gravity

Daniel Flores-Alfonso<EMAIL_ADDRESS>Departamento de Física,
CINVESTAV-IPN, A.P. 14-740, C.P. 07000, Ciudad de México, Mexico  Cesar S.
Lopez-Monsalvo<EMAIL_ADDRESS>Conacyt-Universidad Autónoma
Metropolitana Azcapotzalco, Avenida San Pablo Xalpa 180, Azcapotzalco, Reynosa
Tamaulipas, C.P. 02200, Ciudad de México, Mexico  Marco Maceda
<EMAIL_ADDRESS>Departamento de Física, Universidad Autónoma Metropolitana -
Iztapalapa, Avenida San Rafael Atlixco 186, A.P. 55534, C.P. 09340, Ciudad de
México, Mexico

###### Abstract

We show that the three-dimensional Thurston geometries are vacuum solutions to
the 3D new massive gravity equations of motion. We analyze their Lorentzian
counterparts as well.

Thurston geometries, 3D New Massive Gravity

###### pacs:

04.50.Kd, 04.60.Kz

_Introduction._ —Thurston’s conjecture is a statement about how any three-
dimensional manifold can be canonically decomposed into parts. These simpler
parts admit one, and only one, of eight model geometries. Shortly after
Thurston’s conjecture Thurston (1982), Ricci flow was created as an attempt to
prove it Hamilton (1982). The Ricci flow equation has the same form as a heat
equation, in this analogy, the role of the temperature is played by the
metric. The first sketch of proof that the conjecture is true came decades
later Perelman (1159).

In gravitational physics, Thurston geometries enter the picture in 4D
spatially homogeneous cosmological models Taub (1951); Kantowski and Sachs
(1966); Fagundes (1985) and have been used to construct 5D black holes Cadeau
and Woolgar (2001); Hassaïne (2015); Bravo-Gaete and Hassaine (2018). In 3D
gravity, Thurston metrics are interpreted as gravitational instantons. Some
Thurston model geometries are Euclidean vacua of massive gravity theories (see
Ref. Chernicoff  _et al._ (2018) for a recent discussion on some aspects of
exotic massive gravity Özkan  _et al._ (2018)). Let us mention that there are
a number of massive gravity theories in three dimensions, many of which are
closely related to topologically massive gravity (TMG) Deser  _et al._ (1982a,
b); some of these relations are discussed in Özkan  _et al._ (2018). However,
no systematic study has been undertaken so far. In this Letter, we address the
question if all Thurston geometries are vacuum solutions of three-dimensional
new massive gravity (NMG) Bergshoeff  _et al._ (2009).

We are concerned with the following equations of motion

$\Lambda g+G(g)-\frac{1}{2m^{2}}K(g)=0.$ (1)

Here $\Lambda$ is the cosmological constant and we parameterize the NMG
coupling constant as $m^{-2}$, however, there is no need for $m^{2}$ to be
positive. The NMG tensor, $K$, has components

$\displaystyle K_{\mu\nu}=$
$\displaystyle~{}2\square{R}_{\mu\nu}-\frac{1}{2}\nabla_{\mu}\nabla_{\nu}{R}-\frac{1}{2}\square{R}g_{\mu\nu}+4R_{\mu\alpha\nu\beta}R^{\alpha\beta}$
$\displaystyle-\frac{3}{2}RR_{\mu\nu}-R_{\alpha\beta}R^{\alpha\beta}g_{\mu\nu}+\frac{3}{8}R^{2}g_{\mu\nu}.$
(2)

It was established, in a sequence of papers Ahmedov and Aliev (2011a, 2010,
b), that Eq. (1) admits the form of a Klein-Gordon-type equation with a
curvature-squared source term and a constraint equation. This ultimately
allows for all algebraic type $D$ and $N$ solutions of TMG to be mapped into
NMG. The reason behind this map is that for these algebraically special
metrics the TMG equations of motion correspond to a “square root” of the NMG
Klein-Gordon-type equation. In contrast, the converse is not true, that is,
not every type $D$ and $N$ solution of NMG is also a solution of TMG.

In addition, the study of the various compatible geometric structures on
three-dimensional manifolds has been a central theme in contact geometry.
Recently, a class of contact structures has emerged, arising in the context of
six-dimensional supergravity Murcia and Shahbazi (2020), three-dimensional
massive gravity and two-dimensional superconductors Flores-Alfonso  _et al._
(2021). They have been dubbed $\varepsilon$-contact manifolds, where the sign
of $\varepsilon$ indicates if a distinguished vector field in the manifold is
timelike, spacelike or null. For example, the contact sphere with a para-
Sasakian structure is one such manifold where the Maxwell-Chern-Simons field
equations are obtained by extremizing its helicity while identifying the
contact form with the electromagnetic gauge potential.

Previous results show that $\varepsilon$-contact three-manifolds that are also
$K$-contact and $\eta$-Einstein are solutions of NMG. In Ref. Flores-Alfonso
_et al._ (2021) we specialized in a Lorentzian Nil geometry Rahmani (1992).
However, our results are independent of the metric’s signature. In particular,
every Sasakian geometry satisfies the NMG equations of motion. This should be
understood in light of the uniformization theorem of Sasakian three-manifolds
Belgun (2000); Boyer  _et al._ (2006). The three representative metrics of the
theorem are all Thurston model geometries.

_Thurston Geometries._ —In this section, we summarize and discuss the results
of evaluating the NMG equations of motion in each of the Thurston geometries:
$E^{3}$, $S^{3}$, $H^{3}$, $E^{1}\times S^{2}$, $E^{1}\times H^{2}$,
$\widetilde{{\rm{SL}}(2,\mathbb{R})}$, Nil, and Sol Thurston (gt3m, 1997);
Scott (1983).

Let us consider first, the three Thurston geometries formed by a direct
product $E^{1}\times E^{2}=E^{3}$, $E^{1}\times S^{2}$ and $E^{1}\times
H^{2}$. Euclidean space $E^{3}$ is flat and thus to be a solution of NMG it
only requires

$\Lambda=0,$ (3)

in other words $m^{2}$ is unspecified. For the remaining product spaces we
write the scalar curvature as

$R=\pm\frac{2}{\ell^{2}},$ (4)

where the plus sign is for $E^{1}\times S^{2}$. As NMG vacua the length scale
$\ell$ is fixed by the cosmological constant. Moreover, the quadratic-
curvature coupling constant is also fixed by $\Lambda$

$\ell^{2}=\pm\frac{1}{2\Lambda},\quad m^{2}=\Lambda,$ (5)

where, once more, the plus sign is for $E^{1}\times S^{2}$. Lorentzian
counterparts of these spaces are (A)dS${}_{2}\times S^{1}$ which were found to
be a solution of Eq. (1) in Refs. Clément (2009); Ahmedov and Aliev (2011b),
as long as Eqs. (5) hold.

Although Euclidean space $E^{3}$ is a product space $E^{1}\times E^{2}$ it is
also a space of constant (sectional) curvature $\kappa$. Evidently, $\kappa$
vanishes for $E^{3}$. The other two spaces with constant $\kappa$ are $S^{3}$
($\kappa>0$) and $H^{3}$ ($\kappa<0$). We write their scalar curvature as

$R=\pm\frac{6}{\ell^{2}},$ (6)

where the minus sign is for $H^{3}$. As it is for $E^{3}$, $m^{2}$ is unfixed
by the NMG equations of motion, however there is a constraint in the parameter
space given by

$\Lambda=\pm\frac{1}{\ell^{2}}-\frac{1}{4\ell^{4}m^{2}}.$ (7)

This equation determines the value of $\ell$ in terms of the NMG couplings and
corresponds to Eq. (3.9) of Ref. Bakas and Sourdis (2011) for the plus sign.
Moreover, these are the same equations (A)dS spacetimes must satisfy as NMG
vacua, cf. Bergshoeff  _et al._ (2009); Liu and Sun (2009); Ayón-Beato  _et
al._ (2009). However, since $\ell^{2}$ must be positive then the range of
values of the coupling constants are restricted. Notice that when $m^{-2}=0$
we recover Einstein gravity and the familiar conditions
$\ell^{2}=\pm\Lambda^{-1}$.

Thus far, we have discussed five of the eight Thurston model geometries. These
five geometries are closely related to the three representative geometries of
the uniformization theorem for Riemann surfaces, i.e., $H^{2}$, $E^{2}$, and
$S^{2}$. We considered the trio of product spaces and the trio with constant
sectional curvature, $\kappa$, where $E^{3}$ appears in both triplets. We now
advance by looking at yet another trio $\widetilde{{\rm{SL}}(2,\mathbb{R})}$,
Nil, and $S^{3}$ which are all line bundles over the representative Riemann
surfaces mentioned above.

This new triplet appears in a version of the Uniformization Theorem that
exists in three dimensions for Sasakian manifolds Belgun (2000). In short,
compact Sasakian three-manifolds admit only three different types of
geometries. The representatives of these three types are precisely Nil,
$S^{3}$, and $\widetilde{{\rm{SL}}(2,\mathbb{R})}$. Furthermore, these
correspond to the only three Thurston models on compact Seifert bundles with
nonzero Euler number Scott (1983). They are all contact manifolds satisfying
the $\eta$-Einstein condition Okumura (1962)

${\rm Ric}(g)=\lambda g+\nu\eta\otimes\eta,$ (8)

where $\eta$ is the contact form, $g$ is the metric, and $\lambda,\nu$ are
some constants. When $\nu=0$ the manifold is also an Einstein manifold, this
is the case for $S^{3}$. Just as three-manifolds are Einstein manifolds if and
only if they have constant $\kappa$, Sasakian three-manifolds are
$\eta$-Einstein if and only if they have constant $\phi$-holomorphic sectional
curvature $k$ Okumura (1962); Tanno (1969); Boyer  _et al._ (2006).

Now, let us recall that a contact manifold possesses a special hyperplane
distribution in its tangent bundle. In three-manifolds this contact
distribution is a plane distribution. We mention this now because sectional
curvature is a function of plane distributions. In three-manifolds any tangent
space is three dimensional. Thus, there are at each point of the manifold
$\binom{3}{2}=3$ independent planes. We say that a manifold has constant
sectional curvature if for any plane at any point $\kappa$ takes the same
value. In Sasakian manifolds, we say that a manifold has constant
$\phi$-holomorphic sectional curvature if the sectional curvature restricted
to the contact distribution is constant Tanno (1969). Here $\phi$ refers to
the almost contact structure. Every plane on the distribution is generated by
some vector field $X$ on it and $\phi X$, which results from the action of
$\phi$ on $X$. The point is that manifolds such as Nil and
$\widetilde{{\rm{SL}}(2,\mathbb{R})}$ do not have constant sectional curvature
but do have constant $\phi$-holomorphic sectional curvature.

Let us mention that every Sasakian manifold is $\varepsilon$-contact and
$K$-contact. Thus, by our previous results Flores-Alfonso  _et al._ (2021),
the $\eta$-Einstein geometries described above are vacuum solutions of NMG.
Let us discuss this in further detail starting with the Nil geometry which is
described by the metric

$g=\frac{\ell^{2}}{4}\bigg{(}{\rm d}x\otimes{\rm d}x+{\rm d}y\otimes{\rm
d}y+\eta\otimes\eta\bigg{)},$ (9)

where

$\eta=x{\rm d}y-{\rm d}z.$ (10)

In this case, the sectional curvature of planes in the contact distribution is
$k=-3\ell^{-2}$. In NMG, the length scale $\ell$ is set by the cosmological
constant by

$\ell^{2}=-\frac{1}{2\Lambda},$ (11)

and the NMG parameters must obey

$m^{2}=21\Lambda<0,$ (12)

in order for (9) to satisfy the equations of motion.

We now turn to the universal cover of ${\rm{SL}}(2,\mathbb{R})$, which is
described by the metric

$g=\ell^{2}\big{(}{\rm d}r\otimes{\rm d}r+\sinh^{2}r\cosh^{2}r{\rm
d}\theta\otimes{\rm d}\theta+\eta\otimes\eta\big{)},$ (13)

with

$\eta={\rm d}\psi+\sinh^{2}r{\rm d}\theta.$ (14)

The Sasakian structure on $\widetilde{{\rm{SL}}(2,\mathbb{R})}$ includes the
(1,1) tensor

$\displaystyle\phi=$ $\displaystyle-\sinh r\cosh r\frac{\partial}{\partial
r}\otimes{\rm d}\theta$ $\displaystyle+\frac{1}{\sinh r\cosh
r}\frac{\partial}{\partial\theta}\otimes{\rm d}r-\frac{\sinh r}{\cosh
r}\frac{\partial}{\partial\psi}\otimes{\rm d}r.$ (15)

Additionally, let us mention that the $\phi$-holomorphic sectional curvature
of the metric is $k=-7\ell^{-2}$. Moreover, inserting Eq. (13) into the NMG
equations of motion, Eq. (1), yields

$\ell^{2}=-\frac{109}{50\Lambda},\quad\text{and}\quad
m^{2}=\frac{625}{109}\Lambda<0,$ (16)

which tell us how the cosmological constant fixes the curvature radius and how
the coupling constants are constrained so that the geometry of
$\widetilde{{\rm{SL}}(2,\mathbb{R})}$ is a vacuum solution.

In sum, except for Sol, all Thurston geometries are either a space with a
product structure, constant sectional curvature or constant $\phi$-holomorphic
sectional curvature. For the Sol geometry we write its metric as

$g=\ell^{2}\big{(}e^{-2z}{\rm d}x\otimes{\rm d}x+e^{2z}{\rm d}y\otimes{\rm
d}y+{\rm d}z\otimes{\rm d}z\big{)}.$ (17)

The isometry algebra of this metric is solvable.

Considering now the NMG equations, we find that the conformal scale is fixed
by

$\ell^{2}=-\frac{1}{2\Lambda},$ (18)

telling us that $\Lambda$ must be negative and, what is more,

$m^{2}=5\Lambda.$ (19)

Anticipating the results provided below let us note that the Jordan normal
form of the Cotton-York tensor for the Sol geometry is

$Y=2l^{3}\left(\begin{array}[]{ccc}-1&0&0\\\ 0&0&0\\\
0&0&1\end{array}\right),$ (20)

which tells us that it has two independent real eigenvalues. Thus, the Sol
geometry is algebraically general in the classification of García  _et al._
(2004).

_Lorentzian Signature._ —Since NMG is a gravity theory, obviously, one is
interested in Lorentzian geometries. Some Lorentzian metrics are obtained from
Euclidean ones by a Wick rotation Wick (1954). For instance Minkowski from
$E^{3}$, de Sitter from $S^{3}$, and anti-de Sitter (AdS) from $H^{3}$ or
$\widetilde{{\rm{SL}}(2,\mathbb{R})}$. However, since there is no
distinguishable direction over which to perform the rotation other
inequivalent metrics arise from the same Euclidean geometry. The canonical
Lorentzian metric on the three-sphere is an example

$g=\frac{\ell^{2}}{4}\bigg{(}-\eta\otimes\eta+{\rm d}\Theta\otimes{\rm
d}\Theta+\sin^{2}\Theta{\rm d}\Phi\otimes{\rm d}\Phi\bigg{)},$ (21)

where

$\eta={\rm d}\Psi+\cos\Theta{\rm d}\Phi.$ (22)

This metric is not diffeomorphic to de Sitter spacetime but both are obtained
from Wick rotations applied to the Thurston geometry $S^{3}$.

The metric in Eq. (21) does not have constant sectional curvature. However, if
we restrict ourselves to planes in the contact distribution then the sectional
curvature of (21) is constant. This is

$\kappa(p)=\frac{7}{\ell^{2}},$ (23)

for $p$ in the contact distribution generated by $\eta$. The metric is
$\eta$-Einstein and $K$-contact. Moreover Eq. (23) is highly suggestive it
relates to $\widetilde{{\rm{SL}}(2,\mathbb{R})}$. Indeed, requiring the metric
in Eq. (21) to satisfy the NMG equations of motion leads to

$\ell^{2}=\frac{109}{50\Lambda},\quad\text{and}\quad
m^{2}=\frac{625}{109}\Lambda>0,$ (24)

which should be compared with Eqs. (16). Perhaps these results are not so
surprising when one keeps in mind that SU(1,1) is isomorphic to
${\rm{SL}}(2,\mathbb{R})$.

The metric in Eq. (21) is a special case of the only SU(2) left-invariant
family of Lorentzian geometries Boucetta, Mohamed and Chakkar, Abdelmounaim
(2019). Except for $S^{2}\times E^{1}$, all Thurston geometries are left-
invariant metrics. The para-Sasakian Nil geometry mentioned above belongs to
one of two nonflat families of left-invariant Lorentzian Nil geometries. The
other one is

$g=\frac{\ell^{2}}{4}\bigg{(}{\rm d}x\otimes{\rm d}x+{\rm d}y\otimes{\rm
d}y-\eta\otimes\eta\bigg{)},$ (25)

where $\eta$ is given by Eq. (10). This metric has been studied in the physics
literature before, see, e.g., Ref. Mukhopadhyay  _et al._ (2014) where the
metric is studied in TMG and the Cotton-York tensor is interpreted as the
energy-momentum tensor of a perfect fluid. This carries over to the case at
hand as follows. The $\varepsilon$-contact, $\eta$-Einstein, and $K$-contact
properties of the metric and the contact form imply Flores-Alfonso  _et al._
(2021)

$K=ag+b\eta\otimes\eta,$ (26)

for some constants $a$ and $b$. A direct calculation shows Eq. (25) is a
solution of NMG. In fact, the restrictions on the parameter space are exactly
as in Eq. (32). By defining a unit timelike form

$u=\frac{l}{2}\eta,$ (27)

we now have

$\frac{1}{2m^{2}}K=pg+(\rho+p)u\otimes u.$ (28)

In other words, on shell, the NMG equations are effectively those of Einstein
gravity coupled to a perfect fluid, where the isotropic pressure and energy
density are given by

$p=3\Lambda,\quad\text{and}\quad\rho=5\Lambda,$ (29)

respectively.

The geometry in Eq. (25) and other left-invariant metrics on Nil, SU(2), and
SU(1,1) have been studied before in the context of NMG, as Lorentzian Bianchi
type metrics Ahmedov and Aliev (2011a, b). What is more, except for $H^{3}$
and $E^{1}\times H^{2}$, one or more Lorentzian analogs of the Thurston
geometries have been studied in these works.

_The Dumitrescu-Zeghib metrics._ —Thus far, we have seen that some Lorentzian
left-invariant metrics sharing isometry group with the Thurston metrics are
NMG vacua. Some also lead to applications in physics. Recalling that there is
no Lorentzian counterpart to the correspondence between Riemannian metrics and
reductions of the structure group of the frame bundle (cf. Theorem 5.8 in
Isham (1999)), the Lorentzian counterpart to the geometrization theorem
remains an open problem.

Nevertheless, amongst the various possible Lorentzian analogs of the Thurston
metrics, in this Letter we study those found in Dumitrescu and Zeghib (2010),
where it was established that there are exactly four left-invariant metrics
which are Lorentzian and at the same time are preserved by the canonical
action of the isometry group. This is compelling because it is exhaustive and
provides only a handful of metrics. More precisely, it was established that
any compact locally homogeneous Lorentz threefold $M$ is isometric to a
quotient of a Lorentz homogeneous space $G/I$ by a discrete subgroup $\Gamma$
of $G$ acting properly and freely on $G/I$. A famous, closely related,
_noncompact_ spacetime is the Bañados-Teitelboim-Zanelli (BTZ) black hole
Bañados  _et al._ (1992). It is locally homogeneous and is isometric to a
quotient of a Lorentz homogeneous space by a discrete subgroup Bañados  _et
al._ (1993). Specifically, it is AdS spacetime with additional (nonobvious)
identifications; for further details we recommend Ayón-Beato  _et al._ (2004).
A further result of Dumitrescu and Zeghib (2010) is that when $I$ is
noncompact then $G/I$ is isometric to a Lie group $L$ endowed with a
Lorentzian left invariant metric, where $L$ is isomorphic to $E^{3}$,
$\widetilde{{\rm{SL}}(2,\mathbb{R})}$, Nil or Sol.

For $E^{3}$ and $\widetilde{{\rm{SL}}(2,\mathbb{R})}$ these are the Minkowski
and AdS metrics, respectively. The Minkowski metric is one of only three left-
invariant Lorentzian metrics on Nil Rahmani (1992), however, it is not
preserved by the canonical action of Nil. From Ref. Boucetta, Mohamed and
Chakkar, Abdelmounaim (2019) we know that there are in total seven left-
invariant Lorentzian metrics on $\widetilde{{\rm{SL}}(2,\mathbb{R})}$.
However, only the AdS metric is the one which is preserved by its canonical
action.

The Minkowski metric is flat and thus the only requirement for it to be a NMG
vacuum solution is $\Lambda=0$. Turning to AdS we see that it is analogous to
two Thurston geometries. On the one hand, it has negative constant sectional
curvature like $H^{3}$. This implies that for it to satisfy the NMG equations
it must satisfy Eq. (7) with the negative sign. On the other hand, AdS can
also be obtained from Eq. (13), i.e.,

$g=\ell^{2}\big{(}{\rm d}r\otimes{\rm d}r+\sinh^{2}r\cosh^{2}r{\rm
d}\theta\otimes{\rm d}\theta-\eta\otimes\eta\big{)},$ (30)

where $\eta$ is the same as in Eq. (14). This means that AdS spacetime is also
analogous to the geometry of $\widetilde{{\rm{SL}}(2,\mathbb{R})}$.

The Nil group is also called the Heisenberg group because of its relation to
quantum mechanics. In Ref. Dumitrescu and Zeghib (2010) Lorentz-Heisenberg
(Lorentz-Sol) is the name given to the unique left-invariant Lorentzian
geometry which is preserved by the canonical action of the Nil (Sol) group. We
use the same nomenclature in what follows.

We write the Lorentz-Heisenberg metric as

$g=\frac{\ell^{2}}{4}\bigg{(}-{\rm d}x\otimes{\rm d}x+{\rm d}y\otimes{\rm
d}y+\eta\otimes\eta\bigg{)},$ (31)

where $\eta$ is as in Eq. (10). This geometry parallels its Euclidean
counterpart. That is, it is para-Sasakian, $\eta$-Einstein and has constant
$\varphi$-holomorphic sectional curvature $c=3\ell^{-2}$. Here $\varphi$ is
the (1,1) tensor in the almost para-contact structure of the Nil group. It is
a NMG vacuum under the constraints

$m^{2}=21\Lambda>0,$ (32)

and the cosmological constant is positive, as

$\ell^{2}=\frac{1}{2\Lambda}.$ (33)

Unlike the Lorentzian Nil geometries, there are Lorentzian left-invariant Sol
metrics which cannot be obtained by (obvious) Wick rotations of Eq. (17).
However, it has been proven that only seven such metrics exist Boucetta,
Mohamed and Chakkar, Abdelmounaim (2019). The Lorentz-Sol metric is the only
one among them which is conformally flat (“Petrov” type $O$). We write the
metric as

$g=\ell^{2}\big{(}2e^{-z}{\rm d}x\otimes_{s}{\rm d}z+e^{2z}{\rm d}y\otimes{\rm
d}y\big{)},$ (34)

where $A\otimes_{s}B=1/2\,(A\otimes B+B\otimes A)$. This metric is _not_ a
solution of Eq. (1) since

${\rm Ric}(g)=-2{\rm d}z\otimes{\rm d}z,$ (35)

and the NMG tensor $K$ defined in Eq. (2) vanishes. However, this means it is
actually a solution of pure new massive gravity, where we have used the same
nomenclature as in Pure Lovelock Gravity Kastor and Mann (2006); Cai and Ohta
(2006). Hence, the configuration requires $\Lambda=0$ and leaves $m^{2}$
unspecified.

Although other left-invariant Sol metrics have been studied in NMG Ahmedov and
Aliev (2011a), the metric in Eq. (34) is a new solution. This motivated us to
inspect the other Lorentzian Sol metrics. It was found that there are only
three other metrics which satisfy the NMG equations of motion and we discuss
them below.

_New Sol geometries._ —We start with metric

$g_{I}=\ell^{2}\big{(}e^{-2z}{\rm d}x\otimes{\rm d}x+e^{2z}{\rm d}y\otimes{\rm
d}y-{\rm d}z\otimes{\rm d}z\big{)}.$ (36)

whose Cotton-York tensor in Jordan form is given by Eq. (20) which makes it
algebraically general. The theory fixes the characteristic length scale $\ell$
by

$\ell^{2}=\frac{1}{2\Lambda}.$ (37)

Moreover, the theory’s coupling constants are constrained by

$m^{2}=5\Lambda>0,$ (38)

which should be compared to Eqs. (18) and (19).

Similarly, another algebraically general Sol geometry is given by

$g_{I^{\prime}}=\ell^{2}\big{(}-e^{-2z}{\rm d}x\otimes{\rm d}x+e^{2z}{\rm
d}y\otimes{\rm d}y+{\rm d}z\otimes{\rm d}z\big{)}.$ (39)

This time, the Jordan form of the Cotton-York tensor has the form

$Y=2\ell^{3}\left(\begin{array}[]{ccc}0&1&0\\\ -1&0&0\\\
0&0&0\end{array}\right).$ (40)

Note that the difference between Eqs. (36) and (39) is that the corresponding
eigenvalues of Eqs. (20) and (40) are real and purely imaginary, respectively.
Much like Minkowski and AdS spacetimes, this Lorentzian geometry obeys NMG
dynamics under the same conditions as its Euclidean counterpart, i.e., Eqs.
(18) and (19).

The third Sol geometry under consideration is

$g_{N}=\ell^{2}\big{(}-e^{2z}{\rm d}y\otimes{\rm d}y-2{\rm d}x\otimes_{s}{\rm
d}y+{\rm d}z\otimes{\rm d}z\big{)},$ (41)

for which

$Y=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&0\\\ 0&0&0\end{array}\right),$ (42)

indicating that it is algebraically type $N$. At a first glance, the metric in
Eq. (41) can be seen to admit a null vector field

$V=\frac{\partial}{\partial x}.$ (43)

This field is covariantly constant, hence, the metric describes a pp-wave
spacetime. The NMG tensor is

$K(g_{N})=\frac{8}{\ell^{2}}{\rm Ric}(g_{N})=\frac{16}{\ell^{2}}e^{2z}{\rm
d}y\otimes{\rm d}y,$ (44)

so for this pp wave to be a vacuum solution it requires

$\Lambda=0,\quad\text{and}\quad m^{2}>0,$ (45)

as

$\ell^{2}=\frac{4}{m^{2}}.$ (46)

Since this solution is type $N$, it must belong the general family found in
Ahmedov and Aliev (2010). However, as a specific geometry, it represents a new
solution of NMG. Moreover, it is the only pp-wave vacuum solution which has
Sol isometry.

To close this section, let us remark that Eqs. (34), (39) and (41), which are
new, together with Eq. (36) which was discussed in Ahmedov and Aliev (2011a,
b), all have different Petrov type.

_Closing Remarks._ —We conclude by mentioning that early work on Thurston
geometries as Euclidean gravitational configurations came from string theory
Gegenberg  _et al._ (2002). A stringy 3D gravity theory was constructed as a
way to shed light on to the Thurston conjecture Gegenberg and Kunstatter
(2004). A key feature of that work is that the gravity sector is given by
general relativity. Since most of the Thurston geometries are not Einstein
manifolds, then in previous analyses a unifying dynamics required matter
content. In this Letter, unifying dynamics comes from modifying Einstein
gravity. In other words, we do not consider matter fields. This establishes
that all eight Thurston geometries are NMG vacua. To the best of our
knowledge, NMG is the only 3D gravity theory where this happens.

We thank Eloy Ayón-Beato for suggesting to us we explore Thurston geometries
in new massive gravity. We are also grateful to Mokhtar Hassaïne for helpful
discussions. D. F.-A. acknowledges financial support from CONACYT through a
postdoctoral research grant. This work has been partially funded by CONACYT
Grant No. A1-S-11548.

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††thanks: The author’s work is supported by National Science and Technology
Council, Taiwan (NSTC 110-2115-M-003-019-MY3).

# Generating quantum feature maps for SVM classifier

Bang-Shien Chen<EMAIL_ADDRESS>Department of Mathematics, National Taiwan
Normal University, Taipei 11677, Taiwan  Jann-Long Chern
<EMAIL_ADDRESS>Department of Mathematics, National Taiwan Normal
University, Taipei 11677, Taiwan

###### Abstract

We present and compare two methods of generating quantum feature maps for
quantum-enhanced support vector machine, a classifier based on kernel method,
by which we can access high dimensional Hilbert space efficiently. The first
method is a genetic algorithm with multi-objective fitness function using
penalty method, which incorporates maximizing the accuracy of classification
and minimizing the gate cost of quantum feature map’s circuit. The second
method uses variational quantum circuit, focusing on how to contruct the
ansatz based on unitary matrix decomposition. Numerical results and
comparisons are presented to demonstrate how the fitness fuction reduces gate
cost while remaining high accuracy and conducting circuit through unitary
matrix obtains even better performance. In particular, we propose some
thoughts on reducing and optimizing the gate cost of a circuit while remaining
perfect accuracy.

support vector machine; quantum feature map; genetic algorithm

##  I Introduction

Quantum computing is a type of calculation using quantum mechanical
properties, such as superposition, interference, and entanglement. It is well
known that quantum computing is capable of solving certain computational
problems faster than classical computers. Applying quantum mechanics on
machine learning [4, 27] is expected to have a performance on speeding up
calculations. As for the classifier, quantum kernel method via support vector
machine [10], among other development by different research groups [5, 9, 16,
18, 20, 22, 24, 28], has been attested a powerful mean of using high
dimensional quantum state space. These studies show the possibility of machine
learning using quantum computers, which may have a boost in the future.

However, a suitable feature map, corresponding to a suitable kernel, plays a
big role in the kernel method. A quantum circuit is employed to produce a
feature map and takes the dataset from its original low dimensional real space
to high dimensional quantum state space, i.e., the Hilbert space [25]. In
general, the quantum circuit should be designed to increase its capacity to
explore the Hilbert space and encode probability distribution more efficient
but also avoid expensive gate cost and circuit size to decrease quantum noise.
Taking this into account, we first modified the method proposed in [3], which
automatically generates quantum feature maps by using multi-objetive genetic
algorithm [7, 15] both to maximize the accuracy and to minimize the size of
the circuits. Then, we proposed another method learning through parameterized
variational circuit [12] with different ansatz constructed by unitaty
decomposition [11, 13, 23, 29].

The paper is organized as follows. A brief review of support vector machine
and quantum kernel method is presented in Section II. Section III describes
the genetic algorithm in more details, including how we encode the problem and
how we optimize the multi-objective fitness function. Section IV explains the
variational method we proposed, using hardware efficient ansatz and unitary
decomposition ansatz. Section V presents the results of applying the algorithm
to three datasets: moonshape data, ad hoc data and a dataset generated by a
specific group structure. Finally, we draw our conclusions and summarize our
learned lessons in Section VI.

##  II Quantum Kernels

###  II.1 Support vector machine

Support vector machine (SVM) is a widely used classical supervised
classification model in machine learning. Suppose we have a set of data points
$\\{(x_{i},y_{i})\,|\,x_{i}\in\mathbb{R}^{n},\,y_{i}=\pm 1\\}$, SVM finds a
pair of parallel hyperplane $w\cdot x+b=\pm 1$ that divides the data points
into two classes, where the decision boundary corresponds to the hyperplane
$w\cdot x+b=0$ with orientation controlled by $w$ and offset controlled by
$b$. The goal is to differ the data points by classes as much as possible,
i.e., maximize the margin (distance) between the parallel hyperplane, where we
can express the problem as $\min\frac{1}{2}\|w\|_{2}^{2}\ \text{s.t.}\
y_{i}(w\cdot x_{i}+b)\geq 1$. The optimization problem for the primal problem
can be formulated into Lagrange dual problem:

$\begin{array}[]{cl}\max&L(w,b,\alpha)=\displaystyle\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y_{j}x_{i}x_{j}\\\
\text{s.t.}&\displaystyle\sum_{i=1}^{n}\alpha_{i}y_{i}=0,\ \alpha_{i}\geq
0\end{array}$ (1)

When the data points are not linearly separable, we use a feature map $\phi$
to map the data into high dimensional space for more features or to low
dimensional space to eliminate unimportant features. The kernel trick is to
offer a more efficient and less expensive way to transform data into higher
dimensions, where the data can be linearly seperated in the higher dimensional
space. The kernel function, denoted as
$K(x_{i},x_{j})=\phi(x_{i})\phi(x_{j})$, allows us to perform SVM by just
knowing how to calculate the inner product of $\phi(x_{i})$ and $\phi(x_{j})$
instead of knowing what the feature map $\phi$ is. In particular, the RBF
kernel, i.e., the Gaussian kernel is often used. Applying the kernel trick, we
can rewrite the Lagrange dual problem:

$\begin{array}[]{cl}\max&L(w,b,\alpha)=\displaystyle\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y_{j}K(x_{i},x_{j})\\\
\text{s.t.}&\displaystyle\sum_{i=1}^{n}\alpha_{i}y_{i}=0,\ \alpha_{i}\geq
0\end{array}$ (2)

###  II.2 Quantum kernel method

A big challenge in quantum machine learning is how we encode the classical
information to quantum states for quantum computing. The kernel method can be
implemented to quantum computing by considering the quantum circuit as an
encoding function $\Phi$ where the quantum feature map is defined as
$|\Phi(x)\rangle=U(x)|0\rangle^{n}$ and the kernel is naturally defined as
$K(x_{i},x_{j})=|\langle\Phi(x_{i})|\Phi(x_{j})\rangle|^{2}$. This provides
the data points to be mapped into quantum Hilbert space [25], where using
quantum computer will have a big advantage. Using quantum circuits, a data
$x\in\mathbb{R}^{n}$ is mapped to an n-qubit quantum feature state
$\Phi(x)=U(x)|0^{n}\rangle\langle 0^{n}|U^{\dagger}(x)$ through a unitary
circuit $U(x)$, and we can employ the circuit to evaluate the kernel by:

$K(x_{i},x_{j})=|\langle\Phi(x_{i})|\Phi(x_{j})\rangle|^{2}=|\langle
0^{n}|U^{\dagger}(x_{i})U(x_{j})|0^{n}\rangle|^{2}$ (3)

The kernel can be estimated on a quantum computer by evolving the initial
state $|0^{n}\rangle$ with $U^{\dagger}(x_{i})U(x_{j})$ and counting the
frequency of the $0^{n}$ outcome. As long as the qauntum kernel
$K(x_{i},x_{j})$ can be computed effciently, we can do optimization in the
Hilbert space while other optimization steps can be performed on classical
computer. For more details about quantum kernel estimation, please refer to
[10]. Note that the ZZFeatureMap they provided is an unitary circuit defined
as $U=\widetilde{U}_{\Phi(x)}H^{\otimes 2}\widetilde{U}_{\Phi(x)}H^{\otimes
2}$, where
$\widetilde{U}_{\Phi(x)}=\exp(i\Phi_{1}(x)ZI+i\Phi_{2}(x)IZ+i\Phi_{1,2}(x)ZZ)$,
is only suitable for a specific dataset. The main challenge is to seek
suitable kernel function for different datasets, which led to researches on
automatically learning kernels [3, 8].

##  III Genetic Algorithm

Genetic Algorithm is a evolutionary algorithm inspired by the process of
natural selection, which has extensively been used to solve optimization
problems. The evolution starts from a population of randomly generated
chromosomes, with the population in each iteration called generation. The
fitness of every chromosome in the population is calculated, selected from the
current population, and each chromosome’s gene is modified to form a new
generation for the next iteration of the algorithm.

###  III.1 Chromosome and Population

A chromosome is a potential solution to the problem being solved, consists of
a set of genes/numbers. Note that encoding problems to individual genes is
often the hardest, a careful analysis between the circuit and numbers is
necessary. We modified the encoding method [3] as follows, the first three
bits determine whether the gate is a Rotation gate, Hadamard gate, CNOT gate
or identity gate, where any unitary operation can be approximated to Rotation
gate, Hadamard gate and CNOT gate. If the first three bits determine to be a
Rotation gate, the last three bits determine a coefficient between
$[\pi/8,\pi]$ for the rotation parameter. As for the circuit size, we use
2-qubits in order to compare with ZZFeatureMap later. The $i$th gate acts on
$i$ (mod 2) qubit, CNOT gate acts on $i$ (mod 2) and $i$+1 (mod 2)
specifically. We added more bits to determine the rotation angle for more
precisely results. The model starts by generating a binary string with the
size $M\times N\times 6$, where $M$ is the number of chromosomes and $N$ is
the number of genes.

Figure 1: Modified encoding example based on [3].

###  III.2 Fitness and Matting pool

Fitness is also considered as objective function or cost function. We pass a
solution to this function and it returns the fitness of that solution. In the
below multi-objective fitness function (5), we aim to both maximize the
accuracy and minimize the gate cost. A common strategy for multi-objective
problem is to find the pareto front [1] by choosing high domination points
with crowd distance techniques, however, pareto front method does not suit our
situation because high-accuracy performance is usually hard to find. If we
don’t give more weight to the accuracy factor, the model will be influenced
more by gate cost, thus it tends to find low-cost circuit rather than high-
accuracy circuits. The accuracy is simply the mean accuracy of the given test
data and label, with the QSVM model generated by a quantum circuit. We use the
gate cost proposed in [14], by counting the sequence of basic physical
operations required for implementation on a quantum computer:

$\text{GateCost}=\text{Rgate}+2\text{Hgate}+5\text{CNOTgate}$ (4)

With the two ingredients, we can define the multi-objective fitness function.
Our main problem is to minimize the gate cost with a constraint of maximizing
the accuracy. We consinder the reciprocal of square of accuracy as the penalty
function and replace the problem as follows, where $w$ is a positive constant
(penalty weight) that depends on different datasets.

$\text{Fitness}=\text{GateCost}+w/\text{Accuracy}^{2}$ (5)

The matting pool chooses the parents for the next generation. The simplest
method is to first determine a pool size and then choose the best solutions
based on the pool size. Note that the same chromosome’s fitness should be
calculated in every generation, due to the fact that the accuracy are encoded
by probability distribution, computing in every generation guarantees the
stability of the outcome by a chromosome. For the mutation operation, we set a
probability at $80\%$ to randomize a chromosome and simply do a bit flip.

##  IV Variational Ansatz

In quantum mechanics, every quantum circuit is equivalent to an unitary
matrix. Because that the feature map is calculated by a quantum circuit, it is
also calculated by a specific unitary matrix, which our goal is to find the
unitary matrix corresponding to a feature map that suits the dataset. We
propose two ways to construct a variational quantum circuit with learnable
parameters in order to find the unitary matrix.

###  IV.1 Hardware Efficient Ansatz

We use the hardware efficient ansatz, which is spanned by special unitaries
and entangle layers, proposed in [12]. This ansatz is often used as the
parameterized circuit for variational quantum eigensolver [19], consisting of
a RY gate layer, RZ gate layer, then a CNOT gate layer. One can simply add the
depths by repeating the layers where the ansatz usually ends with the rotaion
layer. We encode the linear combination of the data points as the parameter of
each rotation gate. An one depth 2-qubit feature map can be formulated as
follow:

$\begin{array}[]{c}(e^{I\phi_{1}Z}\otimes
e^{I\phi_{2}Z})(e^{I\phi_{3}Y}\otimes e^{I\phi_{4}Y})\\\ \\\\[-6.00006pt]
\phi_{j}=a_{j}x_{1}+b_{j}x_{2}\\\ \end{array}$ (6)

Figure 2: Hardware efficient ansatz.

After the circuit is constructed, we set the accuracy of the QSVM model as the
cost function and update the parameters through classical optimizer COBYLA
(constraint optimization by linear approximation). Note that COBYLA only
performs one objective function evaluation per optimization iteration.

###  IV.2 Unitary Decomposition Ansatz

Due to the fact that quantum circuits are equivalent to unitary matrices, we
use the unitary decomposition ansatz proposed in [23] in order to produce
arbitrary unitaries. Given an even size unitary matrix $M$, which the matrix
we want to encode to a quantum circuit is always the size of $2^{k}$, it can
be decomposed to smaller unitaries $L_{1},L_{2},R_{1},R_{2}$ and diagonal
matrices $C,S$ with the size of $2^{k-1}$ by Cosine-Sine Decomposition such
that $C^{2}+S^{2}=I_{2^{k-1}}$, as below:

$\begin{array}[]{l}M=\begin{bmatrix}L_{1}&0\\\
0&L_{2}\end{bmatrix}\begin{bmatrix}C&-S\\\
S&C\end{bmatrix}\begin{bmatrix}R_{1}&0\\\ 0&R_{2}\end{bmatrix}\end{array}$ (7)

The cosine-sine matrix is in the form of multiplexor gate, while the left and
right factor can further be decomposed. Given an unitary matrix $X$ with the
form of $X_{1}\oplus X_{2}$, it can be decomposed to $(I\otimes V)(D\oplus
D^{\dagger})(I\otimes W)$ where $D^{2},V$ are the eigenvalues and eigenvectors
of the marix $X_{1}X_{2}^{\dagger}$, that is,

$\begin{array}[]{l}\begin{bmatrix}X_{1}&0\\\
0&X_{2}\end{bmatrix}=\begin{bmatrix}V&0\\\
0&V\end{bmatrix}\begin{bmatrix}D&0\\\
0&D^{{\dagger}}\end{bmatrix}\begin{bmatrix}W&0\\\ 0&W\end{bmatrix}\end{array}$
(8)

$V,W$ are now smaller unitary matrices and the diagonal matrix is also in the
form of multiplexor gate. The whole process is called the Quantum Shannon
Decomposition, please refer to [23] for more deatails. At last, the single
qubit unitary matrices can be decomposed by Euler Decomposition (ZYZ
Decomposition). Again, We encode the linear combination of the data points as
the parameter of every rotation gates. A 2-qubit feature map can be formulated
as follow:

$\begin{array}[]{c}U_{\phi_{1,2,3}}M_{\text{R}_{\text{Z}}\phi_{4,5}}U_{\phi_{6,7,8}}M_{\text{R}_{\text{Y}}\phi_{9,10}}U_{\phi_{11,12,13}}M_{\text{R}_{\text{Z}}\phi_{14,15}}U_{\phi_{16,17,18}}\\\
\\\\[-3.99994pt]
U_{\phi_{j,j+1,j+2}}=\exp({i\phi_{j}\text{IZ}+i\phi_{j+1}\text{IY}+i\phi_{j+2}\text{IZ}})\\\
\\\\[-6.00006pt] \phi_{j}=a_{j}x_{1}+b_{j}x_{2}\end{array}$ (9)

Figure 3: Unitary decomposition ansatz.

##  V Results

###  V.1 Moonshape dataset

First, we consider the moonshape dataset generated by Sklearn [21]. By using
genetic algorithm, we choose the weight $w$ = 20 in equation (5) because the
data here is rather simple, which we do not put much emphasis on the accuracy
factor. The genetic algorithm produces the simple uncorrelated circuit with
low gate cost and perfect accuracy as illustrated in Figure 4. If we choose
lower weight, the algorithm will produce some circuit with only one gate cost
but a lower accuracy, due to the fact that this particular circuit’s fitness
is lower than any other circuits with perfect accuracy. If we choose higher
weight, the algorithm will eventually find the optimal solution but with more
time cost. Note that the circuit can be optimized if the gates share the same
rotation axis and parameter, and can be combined to a single rotation gate
with additional angles.

Figure 4: Moonshape data using genetic algorithm.

By using variational circuits, both hardware efficient ansatz and unitary
decomposition produce perfect accuracy. For hardware efficient ansatz, we
choose to use only one depth such that the circuit’s gate cost is only 4,
again the data here is rather simple. A disadvantage of unitary decomposition
ansatz is its high gate cost, unlike hardware efficient ansatz, the circuit is
fixed.

Figure 5: Moonshape data using hardware efficient ansatz. Figure 6: Moonshape
data using unitary decomposition ansatz.

###  V.2 Adhoc dataset

Secondly, we consider the ad hoc data generated by Qiskit [2]. For genetic
algorithm, we choose the weight $w$ = 80 in equation 4 because the data here
is more complex, which we need to put more emphasize on the accuracy factor.
We obtain 90$\%$ accuracy and we compare the circuit with ZZFeatureMap [10],
though ZZFeatureMap will always get perfect accuracy because the ad hoc
dataset is generated by ZZFeatureMap. We have an advantage by gate cost is 23
in compare with ZZFeatureMap’s gate cost is 34 as illustrated in Figure 7.
From lots of experiments, we observe that with high accuracy circuits, more
gate cost circuit makes the QSVM model more over fitting.

Figure 7: Ad hoc dataset using genetic algorithm (left) and ZZFeatureMap
(right).

For variational circuits, hardware efficiency ansatz has produced an accuracy
of 92.5$\%$, while unitary decomposition ansatz has reached almost perfect
accuracy with 97.5$\%$. We choose to use four depths of hardware efficient
ansatz in order to have similar numbers of parameters with unitary
decomposition ansatz, and it turns out that with more complex datasets,
unitary decomposition ansatz have better performance in general.

Figure 8: Ad hoc dataset using hardware efficient ansatz. Figure 9: Ad hoc
dataset using unitary decomposition ansatz.

We compared different kernels and quantum feature map generating methods
together, the RBF (Guassian) kernel can already obtain well performance with
simple datasets and using ZZFeatureMap for the quantum kernel is only suitable
for the ad hoc dataset. Comparing genetic algorithm (GA), hardware efficient
ansatz (HE) and unitary decomposition ansatz (UD), genetic algorithm works
well with simple datasets and can efficiently decrease the gate cost, but with
more complex datasets, it is hard to find global solution and consumes a lot
of time. Hardware efficient ansatz is flexible with gate cost by choosing
suitable depths, and have slightly better accuracy than genetic algorithm.
Unitary decomposition ansatz has the best accuracy overall, but has the
disadvantage of high fixed gate cost.

Table 1: Comparing classification methods. Method | Moonshape | Ad hoc
---|---|---
acc | gate | time | acc | gate | time
SVM-linear | 80.0$\%$ |  |  | 60.0$\%$ |  |
SVM-RBF | 100$\%$ | $(\gamma=0.5)$ |  | 80.0$\%$ | $(\gamma=8)$ |
QSVM-ZZ | 50.0$\%$ | 34 |  | 100$\%$ | 34 |
QSVM-GA | 100$\%$ | 3 | 66.30s | 90.0$\%$ | 23 | 2067.96s
QSVM-HE | 100$\%$ | 4 | 33.51s | 92.5$\%$ | 31 | 197.68s
QSVM-UD | 100$\%$ | 38 | 209.61s | 97.5$\%$ | 38 | 220.06s

###  V.3 Covariant quantum kernel dataset

Thirdly, we compare our genetic algorithm model with quantum kernel training
[9], which is also designed to find a proper kernel via trainable parameters.
Because our dataset does not fit the group structure with the circuit they
designed, we try to compare by implementing our model to their dataset [2]
with 14 features generated by specific group cosets.

Table 2: Comparison with covariant kernel. | Quantum covariant kernel | QSVM multi-fitness
---|---|---
Accuracy | 100$\%$ | 100$\%$
Gate cost | 51 | 41$\rightarrow$3
Figure 10: Comparison with Covariant kernel.

After experiments, we found a circuit with the gate cost of 41 and perfect
accuracy, the rotation gates on the left side of the barrier are identity
gates and can be ignored. It is obvious that the 4 Hadamard gates at the last
qubit does not effect and can be removed to decrease the gate cost to 33. The
circuit has already outperformed covariant quantum kernel’s circuit, due to
the fact that their circuit is with the gate cost of 51 (21 R gates and 6 CZ
gates). Moreover, the circuit can be optimized as shown in Figure 11, we test
the necessity of each rotation gate in order to check if every feature is
useful, and it turns out that only 4 features are essential here. Again, it is
trivial to remove qubits without feature parameters involved, the last qubit
without any gates and the two qubits with only a Hadamard gate and CNOT gate
can be removed. We repeat the steps before, as we found that the two Hadamard
gates and CNOT gates can also be removed, that leaves 4 rotation gates. Again,
we test the necessity of each rotation gate where only 3 are essential, 2
among them can be added on the same qubit in order to decrease the size of
circuit. As a result, we optimized the circuit to a simple uncorrelated
circuit with the gate cost of 3 and perfect classification accuracy. More
details are available at my Github [30].

Figure 11: Optimizing circuit.

##  VI Conclusion

In this work, we optimize the quantum feature map of QSVM using genetic
algorithm and quantum variational circuit. A quantum circuit is generated as a
feature map that maps the data points into Hilbert space, and then classified
with quantum kernel method applied QSVM. The genetic algorithm modified in
this study [3], aims to maximize the accuracy of QSVM while minimizing the
gate cost of quantum circuit with a multi-objective fitness function. The
variational circuit learns through paremeters with a given ansatz, based on
special unitary groups and unitary matrix decomposition. We test our models on
three different datasets, a simpler moonshape dataset, a more complex ad hoc
dataset and a specifically generated dataset for comparison. We all found high
accuracy of classification, where more complex dataset is needed to increase
the penalty weight $w$ to decrease the influence of gate cost or increase the
circuit depth in order to have more learnable parameters.

There are many more things to explore, including testing the circuit by
variational quantum classifier (VQC), quantum neural network (QNN) or quantum
distance-based classifier. Adding trainable parameters $\theta_{i}$ to the
circuit that can be optimized using SPSA is also another strategy. We believe
it is possible to find suitable feature maps that classification can be well-
performed in the Hilbert space.

##  VII Acknowledgement

Thank the authors of [3] for helpful suggestions and communications, which
help improve the presentation of this paper.

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