# New model of non-Euclidean plane

Piotr Błaszczyk, Anna Petiurenko

###### Abstract.

We present a new model of a non-Euclidean plane, in which angles in a triangle
sum up to $\pi$. It is a subspace of the Cartesian plane over the field of
hyperreal numbers ${\mathbb{R}}^{*}$. The model enables one to represent the
negation of equivalent versions of the parallel axiom, such as the existence
of the circumcircle of a triangle, and Wallis’ or Lagendre’s axioms, as well
as the difference between non-Euclidean and hyperbolic planes.

The model has unique educational advantages as expounding its crucial ideas
requires only the basics of Cartesian geometry and non-Archimedean fields.

Keywords: Parallel postulate, semi-Euclidean plane, hyperbolic geometry,
hyperreal numbers

###### Contents

  1. 1 Introduction
  2. 2 Hilbert axioms for Euclidean and hyperbolic geometry
  3. 3 Semi-Euclidean plane
    1. 3.1 The Cartesian plane over the field of hyperreal numbers
    2. 3.2 Circumcircle of a triangle, Wallis’ and Lagendre’s axioms
    3. 3.3 Trigonometry
  4. 4 Limiting parallel rays

##  1\. Introduction

Figure 1. Straight lines in Klein (left) and Poincaré (right) disk.

Klein and Poincaré disks are classical models of non-Euclidean geometry. Both
consist of a fixed circle in the Euclidean plane, say $\Gamma$, representing
the plane. In the Klein disk, chords of $\Gamma$ are straight lines; in the
Poincaré disk, straight lines are diameters of $\Gamma$ or arcs of circles
orthogonal to $\Gamma$ (Fig. 1).

In the Poincaré model, an angle between intersecting circles is the Euclidean
angle between tangents to these lines drawn at their intersection point (Fig.
2). In the Klein disc, an angle between intersecting straight lines is
retrieved from the Poincaré model as presented in Fig. 3: for lines $n,m$, we
draw circles orthogonal to $\Gamma$ and determine the angle between them.

Figure 2. Angles in the Poincaré disc. Figure 3. Angles in the Klein disc.

Standard models of non-Euclidean planes, thus, involve a non-Euclidean
representation of straight lines (Poincaré) or angles (Klein). We present a
model in which both straight lines and angles are Euclidean as it takes the
concept of a straight line and angle from the Cartesian plane over an ordered
field.

A Euclidean plane is a Hilbert plane satisfying the parallel axiom and circle-
circle intersection axiom. The Cartesian plane over an ordered field closed
under the square root operation (i.e., over a Euclidean field) provides a
model of the Euclidean plane [7, 153]. Section 3 of our paper introduces a
non-Archimedean field of hyperreals $({\mathbb{R}}^{*},+,\cdot,0,1,<)$. Since
it is a Euclidean field, the Cartesian plane
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$ is a model of Euclidean geometry. We
show that its subspace $\mathbb{L}\times\mathbb{L}$, where
$(\mathbb{L},+,\cdot,0,1)$ is the ring of limited hyperreal numbers, is a
semi-Euclidean plane, that is, it satisfies Hilbert’s axioms of the so-called
absolute geometry, does not satisfy the parallel and Archimedean axioms, but
angles in any triangle sum up to $\pi$ (i.e., two right angles).

##  2\. Hilbert axioms for Euclidean and hyperbolic geometry

Hilbert’s Grundlagen der Geometrie, from [8] to [10] got eleven editions. [7]
includes its modern version adjusted to educational practice. Hilbert axioms,
as presented therein, differ from the original only in applying modern
symbols.111[6, 597–602] provides a concise account of Hilbert axioms; see also
[3].

Hilbert grouped his axioms due to primitive concepts, point, straight line,
plane, and the relation of betweenness, congruence of line segments, and
angles. The paper will discuss three of them.

Archimedes’ axiom

Given line segments $AB$ and $CD$, there is a natural number $n$ such that $n$
copies of $AB$ added together will be greater than $CD$.

Parallel axiom

For each point $A$ and each line $l$, there is at most one line containing $A$
that is parallel to $l$.

Hyperbolic axiom of parallels

If $l$ is any line and $A$ a point not on the line, then there exist through
$A$ two rays $AX^{\rightarrow}$ and $AY^{\rightarrow}$ which do not form one
and the same line and do not intersect the line $l$, while every ray emanating
from $A$ that lies in the angle formed by $AX^{\rightarrow}$ and
$AY^{\rightarrow}$ does intersect $l$.222See [9, 136], [6, 259], [7, 374].

Absolute (neutral) geometry consists of axioms of incidence, betweenness,
congruence of line segments and angles. One can also view it as a set of
propositions Elements I.1–28.

Generally, parallel lines are not intersecting lines. In the neutral geometry,
the transportation of angles enables one to construct a line through $A$ that
does not meet $l$. Hilbert’s parallel axiom translates, thus, into the claim:
There is exactly one line containing $A$ that is parallel to $l$.333[3, 75–77]
discusses a relationship between Hilbert’s and Euclid’s parallel axioms. In
hyperbolic geometry, next to not intersecting lines, one considers limiting
rays. Thus, Euclid’s parallel axiom requires one line through $A$, not meeting
$l$, its negation – at least two lines through $A$ not meeting $l$, the
hyperbolic axiom of parallels requires exactly two limiting rays originating
in $A$, not meeting $l$. In what follows, we will present a non-Euclidean
plane that is not hyperbolic.

##  3\. Semi-Euclidean plane

In this section, we present a model of a semi-Euclidean plane, i.e., a plane
in which angles in a triangle sum up to $\pi$ yet the parallel postulate
fails. [7, 311], introduces that term, but the very idea originates in [4, §
9]. Dehn built such a model owing to a non-Archimedean Pythagorean field
introduced in [8, § 12]; yet, it was a non-Euclidean field.444See also [7, §
18]. Example 18.4.3 expounds on Dehn’s model. We employ the Euclidean field of
hyperreal numbers. To elaborate, let us start by introducing these numbers.

###  3.1. The Cartesian plane over the field of hyperreal numbers

An ordered field $({\mathbb{F}},+,\cdot,0,1,<)$ is a commutative field
together with a total order that is compatible with sums and products. In such
a field, one can define the following subsets of ${\mathbb{F}}$:

  1. $\mathbb{L}=\\{x\in{\mathbb{F}}:(\exists n\in{\mathbb{N}})(|x|<n)\\}$,

  2. $\Psi=\\{x\in{\mathbb{F}}:(\forall n\in{\mathbb{N}})(|x|>n)\\}$,

  3. $\Omega=\\{x\in{\mathbb{F}}:(\forall n\in{\mathbb{N}})(|x|<\tfrac{1}{n})\\}$.

They are called limited, infinite, and infinitely small numbers, respectively.
Here are some relationships helpful to pursue our arguments.

  1. $(\forall x,y\in\Omega)(x+y\in\Omega,xy\in\Omega)$,

  2. $(\forall x,y\in\mathbb{L})(x+y\in\mathbb{L},xy\in\mathbb{L})$,

  3. $(\forall x\in\Omega)(\forall y\in\mathbb{L})(xy\in\Omega)$,

  4. $(\forall x\neq 0)(x\in\Omega\Leftrightarrow\ x^{-1}\in\Psi)$.

We can also encode these rules by terms such as $\Omega+\Omega\subset\Omega$,
or $\Omega\Omega\subset\Omega$, etc.

To clarify our account, let us observe that the following equality
$\Omega=\\{0\\}$ is a version of the well-known Archimedean axiom.

Since real numbers form the biggest Archimedean field, every field extension
of $({\mathbb{R}},+,\cdot,0,1,<)$ includes positive infinitesimals. The
standard way of extending real numbers employs formal power series, below we
sketch another method.

Let $\mathcal{U}$ be a non-principal ultrafilter on ${\mathbb{N}}$. The set of
hyperreals is defined as a reduced product
${\mathbb{R}}^{*}={\mathbb{R}}^{{\mathbb{N}}}/{\mathcal{U}}$. Sums, products,
and the order are introduced pointwise. The field of hyperreals
$({\mathbb{R}}^{*},+,\cdot,0,1,<)$ extends real numbers, hence, includes
infinitesimals and infinite numbers [1], [2]. Fig. 4 represents in a
schematized way a relationship between ${\mathbb{R}}$ and ${\mathbb{R}}^{*}$,
as well as between $\mathbb{L}$, $\Psi$, and $\Omega$.

Figure 4. The line of real numbers and its extension to hyperreals.

To get an algebraic insight, let us note that limited numbers form an ordered
ring, while infinitesimals are its maximal ideal. Due to the so-called
standard part theorem, one can show that the quotient ring is isomorphic to
the field of real numbers. In consequence, the set $\mathbb{L}$ finds the
following representation

$\mathbb{L}=\bigcup_{r\in{\mathbb{R}}}r+\Omega.$

Although the below diagrams spot on the origin of the coordinate system, one
can reiterate our arguments taking into account any point $(r,s)$, where
$r,s\in{\mathbb{R}}$, and its infinitesimal neighborhood, i.e.,
$(r,s)+\Omega\times\Omega$, in short $(r,s)+\Omega^{2}$.

In section § 3.3 below, we show how to introduce trigonometric functions and
other counterparts of real maps, such as the square root. Thus, the field of
hyperreals is a Euclidean field.

Due to the proposition [7, 16.2], the Cartesian plane over the field of
hyperreals is a model of Euclidean plane, with straight lines and circles
given by equations $ax+by+c=0$, $(x-a)^{2}+(y-b)^{2}=r^{2}$, where
$a,b,c,r\in{\mathbb{R}}^{*}$ and due to the equation of a straight line,
parameters $a,b$ have to satisfy condition $a^{2}+b^{2}\neq 0$; angles between
straight lines are defined as in the Cartesian plane over the field of real
numbers. Specifically, on the plane ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$,
angles in triangles sum up to $\pi$. Parallel lines are of the form $y=mx+b$
and $y=mx+c$, while a perpendicular to the line $y=mx+b$ is given by the
formula $y=-\frac{1}{m}x+d$.

Figure 5. Perpendicular lines with infinitesimal and infinitely large slopes.

Now, let us take a subspace $\mathbb{L}\times\mathbb{L}$ of the plane
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$. In that plane, circles are defined
by analogous formula, namely $(x-a)^{2}+(y-b)^{2}=r^{2}$, where
$a,b,c,r\in\mathbb{L}$, while every line in $\mathbb{L}\times\mathbb{L}$ is of
the form $l\cap\mathbb{L}\times\mathbb{L}$, where $l$ is a line in
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$. Since we want plane
$\mathbb{L}\times\mathbb{L}$ include lines such as $y_{1}=\varepsilon x$,
where $\varepsilon\in\Omega$, it also has to include the perpendicular
$y_{2}=\frac{-1}{\varepsilon}x$, but $\frac{-1}{\varepsilon}\notin\mathbb{L}$.
Formula $l\cap\mathbb{L}\times\mathbb{L}$, where $l=ax+by+c$ and
$a,b,c\in{\mathbb{R}}^{*}$ guarantees the existence of the straight line
$y_{2}$ in $\mathbb{L}\times\mathbb{L}$. Finally, the interpretation of an
angle is the same as in the model ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$.

Explicit checking shows that the model characterized above satisfies Hilbert
axioms of absolute plane geometry plus the circle-circle and line-circle
axioms; see [7, §14–17].

Figure 6. Non-Euclidean plane $\mathbb{L}\times\mathbb{L}$ vs. Euclidean plane
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$.

With regard to parallel lines, let us consider the horizontal line $y\equiv 1$
and two specific lines through $(0,0)$, namely $y_{1}=\varepsilon
x,y_{2}=\delta x$, where $\varepsilon,\delta\in\Omega$; see Fig. 6. Since
$\Omega\mathbb{L}\subset\Omega$, the following inclusions hold
$y_{1},y_{2}\subset\mathbb{L}\times\Omega$. In other words, values of maps
$y_{1}(x),y_{2}(x)$ are infinitesimals, given that $x\in\mathbb{L}$. The same
obtains for any line of the form $y=\mu x$, with $\mu\in\Omega$. Since there
are infinitely many infinitesimals, there are infinitely many lines through
$(0,0)$ not intersecting the horizontal line $y\equiv 1$.

Since every triangle in $\mathbb{L}\times\mathbb{L}$ is a triangle in
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$, it follows that angles in a triangle
on the plane $\mathbb{L}\times\mathbb{L}$ sum up to $\pi$ (Fig. 7).

Figure 7. Triangles in Euclidean plane
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$ and its subspace
$\mathbb{L}\times\mathbb{L}$.

###  3.2. Circumcircle of a triangle, Wallis’ and Lagendre’s axioms

In our model, one can easily find counterexamples for various versions of the
parallel axiom. Below, we discuss three of them.

In the Elements, Book IV, proposition 5, Euclid requires to circumscribe a
circle about a given triangle. To this end, he assumes that perpendicular
bisectors of two sides of a triangle meet. Indeed, the intersection point is
the center of the circle circumscribing a triangle. The existence of such a
circumcircle is an equivalent version of the parallel axiom. Below we show it
does not hold in the plane $\mathbb{L}\times\mathbb{L}$.

Let us take the line $y\equiv-\varepsilon$ and points $A=(-1,-\varepsilon)$,
$B=(1,-\varepsilon)$ on it. A line through the point $C=(0,0)$ and $A$ has the
equation $y=\varepsilon x$. Perpendicular bisectors of the sides $AB$ and $AC$
have the equations $x=0$ and $2x+2\varepsilon y+\varepsilon^{2}+1=0$,
respectively. They meet at the point
$\left(0,-\varepsilon-\frac{1}{\varepsilon}\right)$. However,
$\left(0,-\varepsilon-\frac{1}{\varepsilon}\right)\notin\mathbb{L}\times\mathbb{L}$.
Fig. 8 depicts three perpendiculars to the sides of the triangle $ABC$.

Figure 8. Triangle in $\mathbb{L}\times\mathbb{L}$ with no circumcircle (left)
and its counterpart in $\mathbb{R}^{*}\times\mathbb{R}^{*}$ (right).

The so-called Wallis axiom reads: Given any triangle $\triangle ABC$ and given
any segment $DE$, there exists a triangle $\triangle DEF$ having $DE$ as one
of his sides such that $\triangle ABC\sim\triangle DEF$ [6, 216].

(a) Wallis axiom.

(b) Legender’s Axiom.

Figure 9.

It is yet another equivalent version of the parallel axiom [6, 216–217].

Let $ABC$ be a triangle with vertices $O$, $(\varepsilon,0)$, $(1,0)$, and
take $DE=[0,1]$. Then, the triangle $DEF$ equiangular with the triangle $ABC$
has the vertex $F=(0,\tfrac{1}{\varepsilon})$ outside the plane
$\mathbb{L}\times\mathbb{L}$.

Finally, let us take Legender’s axiom that reads: For any angle and any point
$A$ in the interior of that angle, there exists a line through $A$ and not
through that angle vertex which intersects both sides of the angle [6, 223].

Lagendre’s axiom is equivalent to the parallel axiom, given the Archimedean
axiom [7, 324].

To show it fails in our model, take $A=(0,1)$, and an angle with arms made by
rays $y_{1}\equiv 0$ and $y_{2}=\tan(\pi-\varepsilon)x$ emanating from $O$.
Now, any line cutting $y_{1}$ does not meet $y_{2}$. Moreover, the straight
line $y_{3}\equiv 1$ does not meet any arm of the angle.

###  3.3. Trigonometry

Let $f$ be a real map, i.e. $f:{\mathbb{R}}\mapsto{\mathbb{R}}$. Its extension
to a map on ${\mathbb{R}}^{*},$
$f^{*}:{\mathbb{R}}^{*}\mapsto{\mathbb{R}}^{*}$, is defined by

(3.1) $f^{*}([(r_{n})])=[(f(r_{1}),f(r_{2}),...)].$

If $r\in{\mathbb{R}}$, then $f^{*}(r)=[(f(r),f(r),...)]$. Since we identify
real number $r$ with hyperreal $[(r,r,...)]$, the equality
$f^{*}([(r,r,...)])=f(r)$ obtains, meaning, $f^{*}$ extends $f$,
$f^{*}_{|{\mathbb{R}}}=f$.

Putting $f=\sqrt{\ \ }$ in definition (3.1), we get

$\sqrt{[(r_{n})]}^{*}=[(\sqrt{r_{n}})]=[(\sqrt{r_{1}},\sqrt{r_{2}},...)],\ \
\mbox{for}\ \ [(r_{j})]>0.$

Similarity, under the definition (3.1), we have

$\sin^{*}[(r_{n})]=[(\sin r_{1},\sin r_{2},...)],\ \ \cos^{*}[(r_{n})]=[(\cos
r_{1},\cos r_{2},...)].$

Since for every $n$ the identity $\sin^{2}r_{n}+\cos^{2}r_{n}=1$ holds, we
have

$(\sin^{*}x)^{2}+(\cos^{*}x)^{2}=1.$

Similarly, every trigonometric identity translates into an identity involving
the maps $\sin^{*}$ and $\cos^{*}$, $\tan^{*}$ and $\cot^{*}$. Interpreting
angles and rotations in ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$, one can
apply these ∗maps. Yet, that way of introducing trigonometry is hardly popular
among geometry scholars.

While dealing with angles in a Cartesian plane over a Pythagorean field, i.e.,
closed under the operation $x\mapsto\sqrt{1+x^{2}}$, Hilbert’s solution was
the following [8, §9]. Let points $O,E,C$ have coordinates $(0,0)$, $(1,0)$,
$(c_{x},c_{y})$. Then the rotation of the angle $\angle EOC$ about $O$ turns
point $(x,y)$ into $(x^{\prime},y^{\prime})$ defined as follows

$x^{\prime}=\frac{c_{x}}{\sqrt{c_{x}^{2}+c_{y}^{2}}}x-\frac{c_{y}}{\sqrt{c_{x}^{2}+c_{y}^{2}}}y,$
$y^{\prime}=\frac{c_{y}}{\sqrt{c_{x}^{2}+c_{y}^{2}}}x+\frac{c_{x}}{\sqrt{c_{x}^{2}+c_{y}^{2}}}y.$

Thus, terms $\frac{c_{x}}{\sqrt{c_{x}^{2}+c_{y}^{2}}}x$ and
$\frac{c_{y}}{\sqrt{c_{x}^{2}+c_{y}^{2}}}y$ are to mimic the cosine and sine
of the angle $\angle COE$, nonetheless, they make sense in any Pythagorean
field.

Considering angles and rotations in a Cartesian plane, Hartshorne mimics the
real function $tan$.555See [7, §16–17]. Greenberg excludes the Poincare length
in a non-Archimedean, Euclidean field.666See [6, 320]. Max Dehn introduced
semi-Euclidean planes over non-Archimedean Pythagorean fields. And indeed, for
a long time, non-Archimedean, Euclidean fields have not been known, and it
could be a reason why Hilbert did not introduce the circle-circle axiom into
his system of axioms. The only non-Archimedean fields known at that time were
Pythagorean fields.

##  4\. Limiting parallel rays

In this section we show that the plane $\mathbb{L}\times\mathbb{L}$ is not
hyperbolic by considering limiting parallel rays.

Figure 10. Limiting parallel rays

Take the point $A=(0,\varepsilon)$, for some positive infinitesimal
$\varepsilon$. Rays $y_{1}=\varepsilon^{2}x+\varepsilon$ and
$y_{2}=-\varepsilon^{2}x+\varepsilon$ contain $A$ and do not meet $y_{3}\equiv
0$. Suppose $y=\delta x+\varepsilon$ is the limiting ray, that is, such that
every line inside the angle with arms $y$ and $AO$ meets $y_{3}$. Slope
$\delta$ has to satisfy the condition $\tfrac{-\varepsilon}{\delta}\in\Psi$.
However, $y_{5}=\sqrt{\delta}x+\varepsilon$ lies inside the angle with arms
$y$ and $AO$, contains $A$ and does not meet $y_{3}$. Therefore,
$\mathbb{L}\times\mathbb{L}$ is not a hyperbolic plane.

Usually, non-Euclidean and not hyperbolic planes are modeled through some
tricks concerning the algebraic properties of a field. It could be, for
example, a Poincaré disc over a Pythagorean field that is not Euclidean. Our
plane $\mathbb{L}\times\mathbb{L}$ is not hyperbolic because it is semi-
Euclidean. Nevertheless, one can also observe that both Euclidean and
hyperbolic parallel axioms fail because straight lines in that plane are too
‘short’.

## References

  * [1] Błaszczyk, P.: 2016, A Purely Algebraic Proof of the Fundamental Theorem of Algebra. AUPC 8, 6–22; https://didacticammath.up.krakow.pl/article/view/3638
  * [2] Błaszczyk, P.: 2021, Galileo’s paradox and numerosities. Zagadnienia Filozoficzne w Nauce 70, 73–107.
  * [3] Błaszczyk, P., Petiurenko, A.: 2021, Commentary to Book I of the Elements. AUPC 13, 43–93; https://didacticammath.up.krakow.pl/article/view/9386
  * [4] Dehn, M.: 1900, Legendre’schen Sätze über die Winkelsumme im Dreieck. Mathematische Annalen 53(3), 404–439.
  * [5] Fitzpatrick, R.: 2007, Euclid’s Elements of Geometry translated by R. Fiztpatrick; http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
  * [6] Greenberg, M.: 2008, Euclidean and Non-Euclidean Geometries, Freeman & Co., New York.
  * [7] Hartshorne, R.: 2000, Geometry: Euclid and Beyond, Springer, New York.
  * [8] Hilbert, D.: 1899: Grundlagen der Geometrie. Festschrift Zur Feier Der Enthüllung Des Gauss-Weber-Denkmals in Göttingen. Teubner, Leipzig 1899, 1–92. In: K. Volkert (Hrsg.), David Hilbert, Grundalgen der Geometrie (Festschrift 1899), Springer, Berlin, 2015.
  * [9] Hilbert, D.: 1971: Foundations of Geometry. Translated by L. Unger. Open Court, La Salle, Illinois.
  * [10] Hilbert, D.: 1972, Grundlagen der Geometrie. Teubner, Stuttgart.
  * [11] Petiurenko, A.: 2022, Foundations of Geometry for Secondary Schools and Prospective Teachers. PhD thesis. Pedagogical University of Cracow, Kraków.


# The doubly charmed strange tetraquark

Marek Karliner<EMAIL_ADDRESS>School of Physics and Astronomy
Tel Aviv University, Tel Aviv 69978, Israel  Jonathan L. Rosner
<EMAIL_ADDRESS>Enrico Fermi Institute and Department of Physics
University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA

(January 5, 2022)

###### Abstract

The LHCb experiment at CERN has discovered a doubly charmed isoscalar
tetraquark $T_{cc}$ with the quantum numbers of $cc\bar{u}\bar{d}$ and mass of
about 3875 MeV/$c^{2}$, decaying to $D^{0}D^{0}\pi^{+}$ through the
intermediate channel $D^{*+}D^{0}$. We present a study of its strange
companions with the quantum numbers of $cc\bar{q}\bar{s}$, where $q=u,d$ and
isospin violation is neglected.

###### pacs:

14.20.Lq, 14.20.Mr, 12.40.Yx

The LHCb Experiment at CERN has discovered an exotic meson
$T_{cc}=cc\bar{u}\bar{d}$ decaying to $D^{0}D^{0}\pi^{+}$ through the channel
$D^{*+}D^{0}$, with a mass around 3875 MeV/$c^{2}$ Aaij  _et al._ (2021a, b).
This mass is just below that for which the pion can form a $D^{*+}$ resonance
with either $D^{0}$. It is also within 7 MeV/$c^{2}$ of the value of $3882\pm
12$ MeV/$c^{2}$ predicted in Ref. Karliner and Rosner (2017) using the same
quark-model parameters that led to the successful prediction Karliner and
Rosner (2014) of the mass of the doubly charmed baryon $\Xi_{cc}^{++}$ Aaij
_et al._ (2017). We use the methods of Ref. Karliner and Rosner (2017) to
predict the masses of the strange companions of the $T_{cc}$, and propose a
study of the $D^{0}D^{0}K^{+}$ system to search for excited companions of the
doubly charmed strange tetraquark. We comment on production and decay of the
$T_{cc,s}$ states.

A generalization of Table II in Ref. Karliner and Rosner (2017) gives a spin-1
state at 4106 MeV which can decay to $D^{*}D_{s}$ or $DD_{s}^{*}$, composed of
a $cc$ diquark with spin 1 and a $\bar{q}\bar{s}~{}(q=u,d)$ antidiquark with
spin zero. We reproduce the table in Ref. Karliner and Rosner (2017) which led
to the successful prediction of the $T_{cc}$ mass (Table I) and then indicate
the changes that apply to a $T_{cc;s}=cc\bar{q}\bar{s}$ (Table II). Here we
have taken $m^{b}_{q}=363$ MeV, $m^{b}_{s}=538$ MeV, and the superscript $b$
denotes values suitable for baryons (and tetraquarks).

There are also states with spin 0, 1, and 2 composed of a $cc$ diquark with
spin 1 and a $\bar{q}\bar{s}$ with spin 1. The spin-weighted average mass
$\bar{M}$ of the corresponding hyperfine multiplet differs from 4106 MeV only
in the mass of the light $\bar{q}\bar{s}$ diquark. For a nonstrange diquark
this “bad”$-$“good” difference has long been known to be 200
MeV=$(2/3)[M(\Delta)-M(N)]$, as in Karliner and Rosner (2014) and earlier
references therein. It is encouraging that a recent lattice QCD calculation
Francis  _et al._ (2021) finds this difference to be 198(4) MeV. The
corresponding difference for a strange diquark is $(m^{b}_{q}/m^{b}_{s})\cdot
200$ MeV = 135 MeV, so the spin-weighted average mass of the hyperfine
multiplet with two spin-1 $cc$ and $\bar{q}\bar{s}$ diquarks is obtained by
adding the “bad”$-$“good” $\bar{q}\bar{s}$ diquark mass difference to the
state in Table II, $\bar{M}=4106+135=4241$ MeV.

The two spin-1 $cc$ and $\bar{q}\bar{s}$ diquarks can form $S$-wave states
with with spin $(0,1,2)$. The mass splittings between the members of this
multiplet result only from hyperfine, i.e., spin-dependent interaction between
the two diquarks. They scale like $\Delta M=(-2x,-x,x)$ for spin $(0,1,2)$,
respectively. The $D^{0}D^{0}K^{+}$ threshold is at 4223 MeV. Depending on the
sign of $x$, at least either the spin-zero or spin-2 three-body resonance
should be able to decay to $D^{0}D^{0}K^{+}$. Indeed, we expect the spin-spin
interaction to be repulsive, i.e¡ $x>0$. In quark-model calculations, whether
it is quark-quark or quark-antiquark, we always have that spin-spin
interaction is antiferromagnetic, i.e., spin 1 has higher energy than spin
zero. This is likely to carry over into interaction of two spin-1 diquarks,
since such an interaction presumably can be viewed as the sum of interactions
of quarks in one diquark with quarks in the other diquark.

Table I: Contributions to the mass of the lightest tetraquark $T(cc\bar{u}\bar{d})$ with two charmed quarks and $J^{P}=1^{+}$.  Contribution | Value (MeV)
---|---
$2m^{b}_{c}$ | 3421.0
$2m^{b}_{q}$ | 726.0
$a_{cc}/(m^{b}_{c})^{2}$ | 14.2
${-}3a/(m^{b}_{q})^{2}$ | ${-}150.0$
$cc$ binding | ${-}129.0$
Total |  $3882.2\pm 12$
Table II: Contributions to the mass of the lightest tetraquark $T(cc\bar{q}\bar{s})$ with two charmed quarks, $\bar{q}\bar{s}$ in a state of spin zero, and $J^{P}=1^{+}$.  Contribution | Value (MeV)
---|---
$2m^{b}_{c}$ | 3421.0
$m^{b}_{q}+m_{s}$ | 901.0
$a_{cc}/(m^{b}_{c})^{2}$ | 14.2
${-}3a/(m^{b}_{q}m^{b}_{s})$ | ${-}101.2$
$cc$ binding | ${-}129.0$
Total |  $4106\pm 12$

The identification of states decaying to $D_{s}$ or $D_{s}^{*}$ poses
significant challenges. The largest exclusive branching fraction of the
$D_{s}$ is to $K^{+}K^{-}\pi^{+}$, with ${\cal B}=(5.39\pm 0.15)\%$ Zyla  _et
al._ (2020). Furthermore, the soft photon in $D_{s}^{*}\to\gamma D_{s}$ will
be very difficult to identify, preventing full reconstruction of the
$D_{s}^{*}$ decay. Table III lists some prospective final states of the
predicted $T_{cc,s}(4106)$. The most promising decay is $T_{cc,s}^{++}\to
D^{*+}D_{s}$, where $D^{*+}\to D^{0}\pi^{+}$ (giving an identifiable soft
pion) and $D_{s}\to K^{+}K^{-}\pi^{+}$ (giving a fully reconstructed final
state).

Table III:  Decay modes of the ground-state singly and doubly charged $T_{cc,s}$ with spin 1 and mass 4106 MeV.  Decaying | Quark | Prospective
---|---|---
state | content | final state
$T_{cc,s}^{+}$ | $cc{\bar{u}\bar{s}}$ | $D^{0}D_{s}^{*+}$
|  | $D^{*0}D_{s}^{+}$
$T_{cc,s}^{++}$ | $cc{\bar{d}\bar{s}}$ | $D^{+}D_{s}^{*+}$
|  | $D^{*+}D_{s}^{+}$

Other calculations of $M(T_{cc,s})$ include, e.g., 3975, 3979 MeV for the
singly and doubly charged state Ren  _et al._ (2021) based on a molecular
picture, and 4156 MeV Eichten and Quigg (2017), based on heavy-quark symmetry
(giving 3978 MeV for the nonstrange state). A comprehensive list of
theoretical mass predictions for the $T_{cc}$ states can be found in Refs.
Aaij  _et al._ (2021a, b).

The predictions for $T_{cc}$ and $T_{cc,s}$ masses in Tables I and II are
based on the same approach. Therefore, if it turns out that the mass of the
lightest doubly-charmed strange tetraquark $M(T_{cc,s})$ is significantly
different from 4106 MeV, it will imply that LHCb’s $T_{cc}$ candidate reported
in Aaij  _et al._ (2021a, b) is unlikely to be the state predicted in Ref.
Karliner and Rosner (2017). If so, the most probable interpretation will be a
molecular state, but one also needs to examine the possibility that it is a
kinematic effect, as discussed below.

The LHCb analysis of the $D^{0}D^{0}\pi^{+}$ system via a unitarized Breit-
Wigner formalism gives rise to a resonance at a mass of $361\pm 40$ keV below
$D^{*+}D^{0}$ threshold, or at approximately $M_{0}=3874.7$ MeV. (We shall use
units in which $c=\hbar=1$.) We show the boundary of the $D^{0}D^{0}\pi^{+}$
Dalitz plot along with the maximum of the two-dimensional distribution in Fig.
1. The proximity of this maximum to the intersection of the two $M(D^{*+})$
dashed straight lines is a cautionary signal of a possible kinematic
enhancement.

Figure 1: Solid curve: boundary of $D^{0}D^{0}\pi^{+}$ Dalitz plot for a
center-of-mass (c.m.) energy of $M_{0}=3874.7$ MeV. Dashed straight lines
denote central values of $m(D^{*+})=2010.26$ MeV. The cross marks the
approximate maximum of the two-dimensional distribution. Axis units are in
GeV2.

The lowest-lying $D^{0}K^{+}$ resonant subsystem in the three-body
$D^{0}D^{0}K^{+}$ system is called $D_{s1}^{*+}(2700)$ in Ref. Zyla  _et al._
(2020). Its mass is $2714\pm 5$ MeV and its width is $122\pm 10$ MeV.
Henceforth we shall refer to this resonance as $D_{s}(2714)$. With its spin-
parity $1^{-}$ and its mass about 600 MeV above the $D^{*}_{s}(2112)$ it is a
candidate for a 2S radial excitation of that state.

Figure 2: Boundary of $D^{0}D^{0}K^{+}$ Dalitz plot for center-of-mass (c.m.)
energy of 4588 MeV. Axes denote squared effective masses of the two
$D^{0}K^{+}$ combinations. Dashed straight lines correspond to
$m(D^{0}K^{+})=2714$ MeV. Dot-dashed ovals correspond to c. m. energy 4527 and
4649 MeV. Dot-dashed straight lines correspond to $M(D_{s1})=2714\pm 61$ MeV.

The boundary of the Dalitz plot in Fig. 2 is for a value of
$M(D^{0}D^{0}K^{+})=4588$ MeV which makes it just tangent to the $D^{0}K^{+}$
resonance band at 2714 MeV. One is then invited to look for a peak near 4588
MeV in the distribution of $M(D^{0}D^{0}K^{+})$. If one is seen, it could
indicate that the tangency condition helps to generate a three-body resonance
with quark content $cc\bar{q}\bar{s}$, though probably a broad one in view of
the large width of the $D_{s1}(2714)$. The dot-dashed ovals and straight lines
correspond to displacing $M(D_{s1}$ by $\pm\Gamma/2$ from its central value.

Figure 3: Relations among processes involving production of states containing
two charmed quarks. Columns correspond to nonstrange (left) and strange
(right) particles. Rows correspond to fragmentation of the $cc$ diquark into
an antidiquark (top) and a quark (bottom). Processes in the left hand column
have been observed.  Figure 4: Relations among processes involving production
of states containing one bottom quark. Columns correspond to nonstrange (left)
and strange (right) particles. Rows correspond to fragmentation of the $b$
quark into a diquark (top) and an antiquark (bottom). All processes have been
observed.  Figure 5: Relations among processes involving production of states
containing one charmed quark. Columns correspond to nonstrange (left) and
strange (right) particles. Rows correspond to fragmentation of the $c$ quark
into a diquark (top) and an antiquark (bottom). All processes have been
observed.

We gain some insight into the possible production rate of a $T_{cc,s}$ state
by comparison with $T_{cc}$ production. This corresponds to the top $SU(3)$
relation in Fig. 3. One can get a rough idea about the relevant relative
fragmentation probabilities of a color antitriplet $cc$ diquark with mass
$\sim 3.4$ GeV by looking at the corresponding processes for a $b$ quark with
mass $\sim 4.2$ GeV Zyla  _et al._ (2020). These are described in Fig. 4; the
corresponding relations involving a charmed quark are shown for comparison in
Fig. 5. The fragmentation of a $b$ quark into a strange quark accounts for
roughly 1/8 of $b$ fragmentation into a nonstrange quark Aaltonen  _et al._
(2008); Aaij  _et al._ (2019, 2021c).

In more detail: For $b$ quark fragmentation at 13 TeV, Ref. Aaij  _et al._
(2019) quotes

$\displaystyle f_{s}/(f_{u}+f_{d})$ $\displaystyle=$ $\displaystyle 0.122\pm
0.006~{},$ (1) $\displaystyle f_{\Lambda_{b}}/(f_{u}+f_{d})$ $\displaystyle=$
$\displaystyle 0.259\pm 0.018~{},$ (2)

where $f_{q}$ are the fragmentation functions of a $b$ quark to a $B_{q}$
hadron, and similarly for the $\Lambda_{b}$. The latter relation can be
interpreted as describing the relative probability of a heavy color
(anti)triplet fragmenting into $[ud]$ (a “good” diquark) vs. the sum of
probabilities of fragmentation into $u$ or $d$ quarks. So in this case we need

$f_{[ud]}/f_{q}=2\times 0.259=0.518~{}(q=u,~{}d)~{}.$ (3)

Further, from $f_{s}/(f_{u}+f_{d})$ we have

$f_{s}/f_{q}=2\times 0.122=0.244~{}.$ (4)

We then assume $SU(3)$ breaking is the same in quarks and diquarks:

$f_{qs}/f_{[ud]}=f_{s}/f_{q}=0.244~{}.$ (5) $f_{qs}/f_{q}=f_{ud}/f_{q}\times
f_{qs}/f_{ud}=0.518\times 0.244=0.126~{}.$ (6)

The estimate of diquark-quark symmetry could be verified by comparing the
production of $T_{cc}$ with that of $\Xi_{cc}^{++}$, the former being 0.518 of
the latter. LHCb can already provide an estimate for this ratio.

One must then take account of differences in branching fractions into
observable final states: $K^{-}\pi^{+}$ and $K^{-}\pi^{+}\pi^{-}\pi^{+}$
(about 10%) for $D^{0}$ and $K^{+}K^{-}\pi^{+}$ (about 5%) for $D_{s}$. All
told, one might expect the signal for the lightest strange doubly charmed
tetraquark to be about 1/16 of that for the $T_{cc}$. Background might be
higher because the $D_{s}$ signal is weaker in the $T_{cc,s}$ than the second
$D^{0}$ signal in the $T_{cc}$.

Another estimate of $T_{cc,s}$ production compares the fragmentation of $cc$
into $\bar{q}\bar{s}$ with $cc$ fragmentation into $u$, which gives rise to
the observed $\Xi_{cc}^{++}$ Aaij  _et al._ (2017). This comparison can
involve measurement of the intermediate fragmentation of $cc$ into $s$ and
hence observation of the $\Omega_{cc}=ccs$. We have noted prospects for
detecting the $\Omega_{cc}$ in Ref. Karliner and Rosner (2014). As in the
preceding discussion, a rough estimate of the relevant relative fragmentation
probabilities can be obtained by comparing the fragmentation of a $b$ quark
into $\,\Xi_{b}\,(=bqs)$ and to a $B^{-}$ meson $(=b\bar{u})$.

The observation by LHCb of a tetraquark $T_{cc}$ implies a rich spectrum of
its strange counterpart with quark content $cc\bar{q}\bar{s}$. If the observed
$D^{*+}D^{0}$ enhancement is merely due to kinematics, one might expect the
prediction of $M(T_{cc,s})=4106$ MeV to fail. A corresponding effect in
$M(D^{0}D^{0}K^{+})$ around threshold would occur at
$M(D_{s}(2714)D^{0})=4588$ MeV. If, on the other hand, one generalizes the
approach of Ref. Karliner and Rosner (2017) to systems with quark content
$cc\bar{q}\bar{s}$, one expects several states decaying to a pair of charmed
mesons, one nonstrange and one strange, including one around 4100 MeV decaying
to $D^{*}D_{s}$ and $DD_{s}^{*}$ and one not far from threshold decaying to
$D^{0}D^{0}K^{+}$.

## ACKNOWLEDGMENTS

We thank Vanya Belyaev and Tomasz Skwarnicki for correspondence on LHCb
$b$-quark fragmentation measurements, and an anonymous referee for
constructive comments.. The research of M.K. was supported in part by NSFC-ISF
grant No. 3423/19.

## References

  * Aaij  _et al._ (2021a) Roel Aaij _et al._ (LHCb), “Observation of an exotic narrow doubly charmed tetraquark,” (2021a), arXiv:2109.01038 [hep-ex] .
  * Aaij  _et al._ (2021b) Roel Aaij _et al._ (LHCb), “Study of the doubly charmed tetraquark $T_{cc}^{+}$,” (2021b), arXiv:2109.01056 [hep-ex] .
  * Karliner and Rosner (2017) Marek Karliner and Jonathan L. Rosner, “Discovery of doubly-charmed $\Xi_{cc}$ baryon implies a stable ($bb\bar{u}\bar{d}$) tetraquark,” Phys. Rev. Lett. 119, 202001 (2017), arXiv:1707.07666 [hep-ph] .
  * Karliner and Rosner (2014) Marek Karliner and Jonathan L. Rosner, “Baryons with two heavy quarks: Masses, production, decays, and detection,” Phys. Rev. D 90, 094007 (2014), arXiv:1408.5877 [hep-ph] .
  * Aaij  _et al._ (2017) Roel Aaij _et al._ (LHCb), “Observation of the doubly charmed baryon $\Xi_{cc}^{++}$,” Phys. Rev. Lett. 119, 112001 (2017), arXiv:1707.01621 [hep-ex] .
  * Francis  _et al._ (2021) Anthony Francis, Philippe de Forcrand, Randy Lewis, and Kim Maltman, “Diquark properties from full QCD lattice simulations,” (2021), arXiv:2106.09080 [hep-lat] .
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), “Review of Particle Physics,” PTEP 2020, 083C01 (2020).
  * Ren  _et al._ (2021) Huimin Ren, Fan Wu, and Ruilin Zhu, “Hadronic molecule interpretation of $T^{+}_{cc}$ and its beauty-partners,” (2021), arXiv:2109.02531 [hep-ph] .
  * Eichten and Quigg (2017) Estia J. Eichten and Chris Quigg, “Heavy-quark symmetry implies stable heavy tetraquark mesons $Q_{i}Q_{j}\bar{q}_{k}\bar{q}_{l}$,” Phys. Rev. Lett. 119, 202002 (2017), arXiv:1707.09575 [hep-ph] .
  * Aaltonen  _et al._ (2008) T. Aaltonen _et al._ (CDF), “Measurement of Ratios of Fragmentation Fractions for Bottom Hadrons in $p\bar{p}$ Collisions at $\sqrt{s}$ = 1.96-TeV,” Phys. Rev. D 77, 072003 (2008), arXiv:0801.4375 [hep-ex] .
  * Aaij  _et al._ (2019) Roel Aaij _et al._ (LHCb), “Measurement of $b$ hadron fractions in 13 TeV $pp$ collisions,” Phys. Rev. D 100, 031102 (2019), arXiv:1902.06794 [hep-ex] .
  * Aaij  _et al._ (2021c) Roel Aaij _et al._ (LHCb), “Precise measurement of the $f_{s}/f_{d}$ ratio of fragmentation fractions and of $B^{0}_{s}$ decay branching fractions,” Phys. Rev. D 104, 032005 (2021c), arXiv:2103.06810 [hep-ex] .


# Quantum transport on networks for supervised classification

Shmuel Lorber, Oded Zimron, Inbal Lorena Zak, Anat Milo and Yonatan Dubi
Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva,
84105, Israel

###### Abstract

Classification, the computational process of categorizing an input into pre-
existing classes, is now a cornerstone in modern computation in the era of
machine learning. Here we propose a new type of quantum classifier, based on
quantum transport of particles in a trained quantum network. The classifier is
based on sending a quantum particle into a network and measuring the
particle’s exit point, which serves as a ”class” and can be determined by
changing the network parameters. Using this scheme, we demonstrate three
examples of classification; in the first, wave functions are classified
according to their overlap with predetermined (random) groups. In the second,
we classify wave-functions according to their level of localization. Both
examples use small training sets and achieve over 90% precision and recall.
The third classification scheme is a ”real-world problem”, concerning
classification of catalytic aromatic-aldehyde substrates according to their
reactivity. Using experimental data, the quantum classifier reaches an average
86% classification accuracy. We show that the quantum classifier outperforms
its classical counterpart for these examples, thus demonstrating quantum
advantage, especially in the regime of ”small data”. These results pave the
way for a novel classification scheme, which can be implemented as an
algorithm, and potentially realized experimentally on quantum hardware such as
photonic networks.

Recent years have seen a huge advance in computational capabilities which are
based on machine-learning (ML) algorithms. Classification, pattern
recognition, complex predictions and adaptive imitation, among others, are
typical problems where ML shows incredible advantage over regular search or
inference algorithms [1, 2, 3, 4]. However, although ML algorithms work very
well and have found a large variety of applications with unprecedented
success, they are limited in a very basic way - they are emulators, meaning
that they require a digital component (e.g. a transistor) to act as a non-
linear device (e.g. to mimic the inherent non-linearity of a neuron). Efforts
have been devoted to finding hardware elements which directly show non-linear
functionality (for example memristors [5, 6, 7, 8, 9] or other physical
implementations [10, 11, 12, 13, 14]).

In light of recent advances in quantum computing (and its advantages over
classical computation for certain problems [15]), various generalizations of
ML algorithms to quantum computation have been suggested, [16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 4,
40, 41, 33, 42, 43] and were applied to various tasks such as quantum
encoding, simulating many-body systems, quantum state discrimination, learning
from experiments [22, 28, 29, 30, 31, 32, 33, 44], and supervised
classification [45]. Importantly, these approaches to quantum machine learning
consider the ”standard” quantum computer architecture, namely qubits and
gates.

Here we take a very different approach, and propose a quantum approach to
Supervised classification (SC) , based on quantum transport of particles
through a ”quantum classification network” (QCN), which in principle can be
implemented in current experiments [46].

SC is one of the most basic computational tasks performed by machine-learning
algorithms [2]. In SC (in its most simplistic form) the algorithm is supplied
with a training set which contains a series of tagged inputs (for instance a
series of images, each tagged to label the content of the image, e.g. ”cat”,
”dog”, etc.). The algorithm uses these images to ”train” the system, i.e. to
optimize a set of internal parameters. Then, when a test input is supplied,
the algorithm can categorize it into its appropriate class.

We provide three fundamental examples for QCN. The first is based on
classifying wave-functions according to their overlap with predetermined
groups, the second is based on classifying wave functions according to their
level of localization. The third example, based on experimentally measured
data, is based on classification of substrates according to their reactivity
properties. We show that within these examples, the quantum-network based SC
outperforms standard machine-learning algorithms under certain conditions,
especially for small data sets, thus providing a direct example for quantum
advantage. Our proposal can be implemented experimentally on various systems
such as networks of quantum dots or quantum wave-guide arrays, thus paving the
way for a new design for hardware-based quantum computations.

Setup and formulation
Classification protocol.– We start by describing the general classification
protocol using quantum transport, namely the problem of assigning an input
vector $\Psi$ to a specific class $C[\Psi]$. The vectors belong to a Hilbert
space of size $L$ (e.g. $L=2$ for qubits), and there are $N$ classes to choose
from. Our classification protocol is defined as follows. A network is
constructed which has $L$ entry nodes, a ”hidden layer” of $M$ nodes and a
layer of $N_{c}$ exit nodes, corresponding to $N_{c}$ classes (see Fig. 1).
The network is excited such that the excitation is defined by the state $\Psi$
(i.e. a quantum particle enters the network through the source sites in a way
which is defined by the state $\Psi$, as described in the following section).
As a result, current flows through the network and exits through the drain
sites. From the currents, the class of $\Psi$, $C[\Psi]$ is determined by
simply evaluating the currents from the different drain sites, and assigning
the class to the node from which current is maximal.

To give a concrete example (as in Fig. 1), consider states from an $L=3$
Hilbert space, which can be classified into two classes. A specific state
$\Psi$ is encoded into the injection of particles to the network from the
source (blue circles in Fig. 1), and current leaves from the two drain sites
(orange circles in Fig. 1). If most of the current comes out of site $1$ (say
the top site), then the state $\Psi$ will be classified as belonging to class
$1$ (i.e. $C[\Psi]=1$) and vice versa. This scheme is substantially different
from the one proposed in Ref. [45], which involves a local measurement after a
stochastic quantum random walk, and requires local control of both Hamiltonian
parameters and local dephasing.

Figure 1:  Schematic representation of the quantum classification network. A
wave function $\Psi$ of length $L$ ($L=3$ in this example) is encoded into an
input excitation of particles into the network, who’s parameters are
determined following a training protocol. The particles exit the network
through one $N_{c}$ sites, representing the $N_{c}$ classes. The class of
$\Psi$ is then determined by the exit site from which the current is maximal.

The quantum network– We model the QCN using a tight-binding Hamiltonian of the
form $\mathcal{H}=\sum_{i,j}h_{ij}c^{\dagger}_{i}c_{j}+h.c.$, where
$c^{\dagger}_{i}(c_{i})$ creates (annihilates) a particle on the network node
$i$. This is a general description which can describe many quantum transport
networks, including (but not limited to) electron transport in quantum dots
[47], exciton transport in bio-networks [48] and photons in wave-guides [49,
50]. We distinguish between nodes on the entry sites (indexed
$i_{in}=1,...,L$), on the network (”hidden”) layer (indexed $i=1,...,M$) and
exit sites (indexed $i_{out}=1,...,N_{c}$). The network structure is such that
entry sites are coupled to all network sites but not to each other and not
directly to the exit sites (and similarly the exit sites are only connected to
the network sites). The network sites may be interconnected between
themselves.

Current propagation through the network, as well as the connection of the
network to the environment (i.e. input and output nodes) is modeled using the
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) quantum master equation [51, 52]
for the density matrix of the system, $\rho$,

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-i[H,\rho]+\Sigma_{k}(V_{k}^{\dagger}\rho
V_{k}-\frac{1}{2}(V_{k}^{\dagger}V_{k}\rho+\rho V_{k}^{\dagger}V_{k}))$
$\displaystyle=$ $\displaystyle-i[H,\rho]+\mathcal{L}[\rho]$

where $[\cdot,\cdot]$ is the commutator, and $V_{k}$ are the so called
Lindblad operators. In the above setup, k=[1,2] for $V_{in}$ ($V_{out}$)
describing the source and drain terms.

The drain term, $V_{out}$, is defined by a set of $N_{c}$ $V$\- operators
describing the extraction of current from the network, of the form
$V_{out,n}=\gamma^{1/2}c_{r}$ [53], where $c_{r}$ annihilates a particle on
the site $r$, where $r=1,...,N_{c}$ are indices of the exit sites. The source
term $V_{in}$ is used to encode the wave-functions that are classified. If a
vector $\Psi$ is considered, the encoding is performed by setting
$V_{in}=\gamma^{1/2}_{in}\sum^{L}_{i=1}\Psi(i)c^{\dagger}_{i}$, where the
index $i=1,...,L$ runs over the source sites.

Once the Hamiltonian is defined (see next section for the training protocol)
and the source and drain Lindbladians are provided, we proceed by evaluating
the current from each exit node [53] (see Methods section for details)

Network training and validation.– In order to succeed in classification, the
network parameters - in this case, the tight-binding amplitudes in the
Hamiltonian - need to be determined according to a given training set. A
Training set

$\displaystyle TS=\left\\{\varphi_{n},C_{n}\right\\},~{}n=1,...,N_{TS}~{}~{}$

is a set of input vectors $\varphi_{n}$ along with their classification
$C_{n}$, where $C_{n}$ is an integer from 1 to $N_{c}$, referring to the class
of the state $\varphi_{n}$, and $n=1,...,N_{TS}$ is an index running on all
the vector states in the training set.

Once the training set is chosen, the next step is the system training, i.e.
finding a Hamiltonian that can achieve the classification described above.
According to the classification protocol described above, we define
$J_{r}[\varphi]$ as the current output from the drain site $r=1,..,N_{c}$ when
the input vector is $\varphi$. The class of $\varphi$ is thus defined as

$\mathcal{C}[\varphi]=\mathrm{Index}\left(\mathrm{Max}\left(J_{r}[\varphi],~{}r=1,...,N_{c}\right)\right)$
(2)

, i.e. the site index of the drain site with maximal current.

Training the Hamiltonian thus amounts to minimizing the cost function

$CF(\mathcal{H},TS)=\sum^{N_{TS}}_{n=1}\left(\mathcal{C}[\varphi_{n}]-C_{n}\right)^{2}~{}.$
(3)

For the optimization we typically use particle-swarm optimization (PSO)
algorithms [54], in combination with standard gradient-descent algorithms,
which were found to yield the fastest and most stable convergence.

Once the training is performed, the QCN is validated by applying the algorithm
to $N_{v}$ validation vectors, $\psi_{n}(n=1,...,N_{v})$, which do not belong
to the training set but their ”true” class, $\mathcal{C}_{n}$ can be
calculated (or is known externally). Then, by comparing $\mathcal{C}_{n}$ to
the output class of the algorithm $\mathcal{C}[\psi_{n}]$, the performance of
the QCN can characterized by the standard measures of classification, namely
precision, $P$, and recall $R$ [2] (perfect precision $P=1$ means that in the
classification process there were no false positives, and perfect recall $R=1$
means there were no false negatives).

We note on the fly that ”training”, in the way described above, is not the
same as the training defined in modern machine-learning algorithms. Indeed, in
our case, the full set of data is apriori required, and ”training” implies
optimizing the cost function with respect to the given data (rather than
updating the system parameters when new data arrives, as in modern machine
learning algorithms). In that sense, our protocol is closer to Hebbian
learning of a Hopfield network [55].

Results

Example I : group overlap classification.– Our first example entails SC
according to the overlap of an input wave-function with different sets of pre-
determined groups of known wave-functions. Consider $G$ groups of wave-
functions, each group containing $N_{G}$ wave-functions
$\varphi^{g}_{n},~{}g=1,...,G,~{}n=1,...,N_{G}$. The class of an input vector
$\psi$ will be determined according to its overlap with the wave-functions of
the group $n$. Put in mathematical form, we define the average overlap of
$\psi$ with the group $g$,

$\eta_{g}=\frac{1}{N_{G}}\sum^{N_{G}}_{n=1}|\langle\varphi^{g}_{n}|\psi\rangle|^{2}$
(4)

(we consider for simplicity only real-valued wave-functions, otherwise, in the
case of complex-valued wave-functions one can take the absolute value). The
classification of $\psi$ is defined as the index of the group of maximal
overlap,
$\mathcal{C}[\psi]=\mathrm{Index}[\mathrm{Max}\\{\eta_{1},\eta_{2},...\eta_{G}\\}]$.
The quantum network classification is obtained by setting the network to have
$G$ exit nodes, and training the network such that when a vector
$\varphi^{g}_{n}$ is input, most of the current will leave through exit node
$g$, i.e. by minimizing the cost function

$CF(\mathcal{H},TS)=\sum^{G}_{g}\sum^{N_{G}}_{n}\sum^{G}_{i}(J_{i}[\varphi^{g}_{n}]-\delta_{i,g})^{2}$
(5)

, where $i$ is the index of the exit node and $\delta$ is the Kronecker delta
function.

We start our demonstration of QCN by examining the simplest possible case of
$L=2,G=2,N_{G}=2$, namely by classifying inputs (of length $2$, i.e. qubits)
according to their overlap with only two vectors. The training set of the two
vectors, $\varphi_{1,2}$, is given by

$\displaystyle
TS=\begin{cases}\varphi_{1}=\cos(x)|0\rangle+\sin(x)|1\rangle,C_{1}=1\\\
\varphi_{2}=\sin(x)|0\rangle+\cos(x)|1\rangle,C_{2}=2\end{cases}$ (6)

where we chose $x=0.3$ (other values give similar results). A network of 2-4-2
was chosen (namely, 4 nodes in the ”hidden layer”, $M=4$), and trained (by
varying the hopping matrix elements such that the cost function is minimized)
using PSO as mentioned above.

Once the training is complete, we tested the classification using validation
set containing 1000 random states $\psi_{n}$, each assigned a class according
to its overlap in the following way. Defining the overlap
$\eta^{(n)}_{g}=\langle\psi_{n}|\varphi_{g}\rangle$, the class of $\psi_{n}$
is $\mathcal{C}_{n}=1$ (or 2) if $\eta^{(n)}_{1}>\eta^{(n)}_{2}$ (or
$\eta^{(n)}_{1}<\eta^{(n)}_{2}$). The class $\mathcal{C}[\psi_{n}]$ defined by
the QCN is the index of the exit site which carries the most current when
$\varphi_{n}$ is inserted into the network.

For this example, we find $P_{2}=R_{2}=1$, i.e. perfect classification. Put
simply, there are no validation vectors $\psi_{n}$ which belonged to class 1
and, when injected to the trained network, had most current come out of node 2
(and vice versa). This is depicted in Fig. 2a. For simplicity, we define for
each $\psi_{n}$ an overlap balance
$\tilde{\eta}_{n}=\frac{\eta^{(n)}_{1}-\eta^{(n)}_{2}}{\eta^{(n)}_{1}+\eta^{(n)}_{2}}$
and a current balance
$\tilde{J}_{n}=\frac{J_{1}[\psi_{n}]-J_{2}[\psi_{n}]}{J_{1}[\psi_{n}]+J_{2}[\psi_{n}]}$.
Both $\tilde{\eta}_{n}$ and $\tilde{J}_{n}$ vary between $\pm\frac{1}{2}$, and
perfect classification means that they have the same sign for every
$\psi_{n}$. In Fig. 2a we plot $\tilde{J}_{n}$ vs $\tilde{\eta}_{n}$ for all
the validation vectors. Indeed, for all $\psi_{n}$’s, the sign of $\tilde{J}$
and $\tilde{\eta}$ is the same.

Of course, this is an extremely simple example. To generalize it, we have
extended the study to the cases of $N_{G}=20$ and $N_{G}=40$ (still with
$G=2$, i.e. two classes each containing $N_{G}$ vectors). The states were of
the same form, $\varphi_{i}=\cos(x_{i})|0\rangle+\sin(x_{i})|1\rangle$, where
$x_{i}$ were chosen randomly from a uniform distribution. Using the same
training protocol as for the $N_{G}=2$, we find
$P_{20}=0.9961,~{}R_{20}=0.9644$ and $P_{40}=0.96767,~{}R_{40}=0.9984$, i.e.
almost perfect classification. The current balance $\tilde{J}$ vs overlap
balance $\tilde{\eta}$ for $N_{G}=20$ is shown in Fig. 2b. The outliers,
namely validation states which give a false positive or negative, are such
that have an extremely similar overlap between the two training sets, and thus
lie close to the $\tilde{\eta}=0$ point, as shown in the inset to Fig. 2b. The
specific shape of the plot arises due to the specific choice of vectors in the
overlap classes, Eq. 6. Similar tests for other random choices of vectors give
similar results.

Figure 2:  Performance of the QCN protocol for group overlap classification.
a: Current balance $\tilde{J}$ vs overlap balance $\tilde{\eta}$ for
$N_{v}=1000$ validation vectors for $G=2,~{}N_{G}=2$. No points reside on the
2nd and 4th quarters, indicating that $\tilde{J}$ and $\tilde{\eta}$ always
have the same sign. This means perfect classification for this case, with
$P=R=1$. b: Same as a, for $G=2,~{}N_{G}=40$ (a training set consisting of 2
groups with 20 vectors each). We find $P=0.9914$ and $R=0.9745$, almost
perfect classification. The outliers are due to validation vectors which have
very close overlap to the two classes, and thus lie close to the
$\tilde{\eta}=0$ point (see enlargement of this area in the inset).

To see how the QCN compares to a classical classification network (CCN), we
ran the same classification problem with a standard classical neural network
(see Methods section for details). While for $N_{G}=2$ the classical algorithm
performed perfectly, it provided rather poor classification for $N_{G}=20,40$.
In Fig. 3 we plot $P$ and $R$ for $N_{G}=2,20,40$ for the QCN (stars) and
classical algorithm (circles). Solid and dashed lines are guides to the eye.
It is clearly visible that the QCN substantially outperforms the CCN in this
regime. However, as we go to much larger training sets (e.g., $N_{G}=600$, not
shown) the classical algorithm gives $P=0.9853,R=1$. This affirms that the
quantum advantage we observe is limited to small training sets.

To further elaborate on the quantum advantage of the QCN, in Fig. 3b the cost
function is plotted vs the number of iterations (so-called calculation epoch)
of the training and validation sets for the QCN (red solid and dashed lines,
respectively) and the CCN (blue solid and dashed lines, respectively), for
$N_{G}=40$. For the QCN the validation set follows the training set behavior,
which is what is expected from a well-performing classification network. For
the classical network, on the other hand, the validation set begins to
diverge, leading to poor classification. This behavior is a hallmark for the
so-called ”overfitting” problem [56, 57]. It thus seems that the QCN overcomes
the overfitting problem for small data sets. We raise here a conjecture that
for this problem, finding the minimum of the cost function on a (quantum)
Hilbert space is a convex optimization problem, but becomes non-convex for the
classical problem, which leads to the overfitting. Proof of this conjecture is
left to future studies.

Figure 3: Quantum advantage demonstrated on overlap-based classification. (a)
Precision and recall with varying sizes of training sets of the QCN (stars)
and CCN (circles). As can be seen, except for $N_{TS}=2$, the QCN algorithm
shows advantage over the classical algorithm. (b) Cost function vs training
epoch of the training and validation sets (solid and dashed lines,
respectively) for the QCN and CCN (red and blue, respectively), for an
$N_{G}=40$ overlap classification. While for the quantum network the
validation set cost function follows the training set cost function, the
classical validation cost function diverges, a clear indication of
”overfitting” due to the small training set.

As a final example for overlap classification, we performed the calculation
for $G=4$, namely 4 classification classes. We use a 3-5-4 network. The
results for the precision and recall (averaged over the 4 classes) is given in
table 1 below. Comparing to the classical algorithm, we again find substantial
quantum advantage for the QCN.

| $N_{G}=4$ | $N_{G}=20$ | $N_{G}=40$
---|---|---|---
Precision QCN | 0.9309 | 0.9116 | 0.9383
Recall QCN | 0.9000 | 0.8888 | 0.8646
Precision CCN | 0.5002 | 0.6537 | 0.9100
Recall CCN | 0.2874 | 0.6848 | 0.6533
Table 1: Classification success values for $G=4$

We point out that this example, although simple to understand, is a somewhat
atypical classification problem. The reason is that in this example, the
classes and the training set are defined by the same group of input vectors.
We thus proceed with a more standard example in the next section.

Example II : classification by level of localization.– The next classification
assignment the QCN was trained to achieve is classification based on level of
localization. We measure the level of localization by the Inverse
Participation Ratio (IPR) [58, 59, 60], which is a well-known measure of
localization defined by

$\displaystyle\mathcal{I}[\psi]=\left(\sum^{L}_{i=1}|\psi(i)|^{4}\right)^{-1}~{}~{}.$
(7)

The IPR ranges from $\mathcal{I}=1$ for a completely localized wave-function
to $\mathcal{I}=L$ for a fully delocalized wave-function (where the weight is
equal over all basis state). Using different measures of localization (e.g.
weave-function entropy) yields similar results.

As a first example, we set $L=5$ and randomly chose $N_{TS}=10$ states,
divided into $G=2$ groups of ”localized states” with $\mathcal{I}<2$ and
”extended states” with $\mathcal{I}>3$ (states with $2<\mathcal{I}<3$ were
discarded). After training a 5-8-2 network, 1000 validation wave-functions
were tested.

In Fig. 4 we plot the current from exit node 1 (red circles) and exit node 2
(blue circles) as a function of the IPR of the test wave-function. It can be
clearly seen that for validation states with $\mathcal{I}<3$ most of the
current flows out of node 2 and vice versa, as indeed required for this
classification. Precision and recall for this case study were found to be
$P=0.9409,~{}R=0.9330$.

As for the case of overlap classification, we proceed to compare the
performance of the QCN vs the classical algorithm for different number of
training set vectors $N_{TS}$. As seen in Fig. 4(b), where the precision and
recall are plotted as a function of $N_{TS}$ for the QCN (stars) and classical
algorithm (circles), the QCN consistently outperforms the classical algorithm
over the entire range of examined $N_{TS}$.

Figure 4: Performance and quantum advantage of localization-based
classification and quantum advantage. (a) Currents from exit node 1 (red
circles) and 2 (blue circles) for 1000 wave-functions, plotted against the
wave-functions IPR, $\mathcal{I}$. A clear ability to classify between
localized ($\mathcal{I}<2$) and delocalized $\mathcal{I}>3$ wave functions is
seen - for localized states most of the currents exit from node 1, and for
delocalized states most of the current exists from site 2. (b) Comparison of
precision and recall of quantum and classical network classification based on
wave-function level of localization (via the IPR, see text) with varying sizes
of $N_{TS}$. Data clearly shows the advantage of the QCN

Robustness against dephasing.– Dephasing, or the general loss of wave-function
coherence, may destroy the ability of the network to perform classification,
since the classification depends on the coherent transport of the excitations
through the trained network. To explore the effect of dephasing, one can ad a
Zeno-type local dephasing terms to the Lindbladian [61] with the $V-$operators
$V_{i}=(\Gamma_{\mathrm{dep}})^{1/2}c^{\dagger}_{i}c_{i}$, where $i$ runs over
all network sites and $\Gamma_{\mathrm{dep}}$ is the dephasing rate. In Fig. 5
we plot the precision and recall for a group IPR classification for
$N_{TS}=30$ (same calculation as in Fig. 2) as a function of dephasing rate
$\Gamma$ (where $\Gamma$ is taken in units of the average hopping matrix
element of the Hamiltonian $\overline{t}$). We find that while the precision
actually increases with dephasing rate, the recall decreases substantially as
$\Gamma\sim 100\overline{t}$. One can evaluate this rate for realistic
systems. Noting that $\overline{t}\sim 1~{}\mu$eV for quantum dots [47] and
$\overline{t}\sim 1$ eV for photonic mazes [50], it follows that the dephasing
time should be longer than $\sim 50$ ns for quantum dots and $\sim 50$ fs for
photonic mazes, which is well within reach in current experimental setups.
Similar calculations with different parameters (e.g. different $G$ and
$N_{TS}$) gave similar results.

Figure 5: Robustness against dephasing. Precision and recall of group
localization classification for $N_{TS}=30$ as a function of dephasing rate.

Implementation of QCN on chemical data

The two previous examples, while representative, are somewhat artificial -
they were created to test the QCN algorithm. Next, we demonstrate the action
of the QCN on real experimental data. For this, we turn to the world of
chemistry and catalysis, where the use in machine-learning algorithms is
beginning to emerge, yet many problems are characterized by having a small
data set, i.e. ”Small Data” problems [62, 63].

Specifically, we consider the classification of aromatic aldehyde substrates,
based on their preferred reaction conditions to a deuteration reaction. The
aldehydes are classified on a scaling reactivity, into 3 broad categories: (1)
those that undergo full deuteration with a catalyst named SIPr, but due to
their reactivity tend to proceed and form benzoin. Suppressing this further
reaction demands adding also boronic acid (BA) to the reaction. (2) Aromatic
aldehydes that undergo full deuteration with SIPr alone, and (3) those that
require a triazolium catalyst (TAC) for their full deuteration. The first
group contains the most reactive substrates, while the 3rd group contains the
least reactive ones. The substrate reactivity depends on various properties,
including steric hindrance, dipole charges, partial charges obtained from
natural population analysis (NPA) [64], etc. Since no formal method for
determining of specific reactivity conditions is known, using machine learning
algorithms may prove useful.

In the experiment, 60 substrates (i.e. molecules) were tested, and
experimentally classified into one of the three classes described above. We
use these data for the classification scheme as follows. Each substrate is
characterized by a list of $10$ numerical values corresponding to its physical
characteristics (some stated above, see SI for the full data), which display a
broad distribution over the substrates. The values are then normalized such
that they are distributed in the range $[-1,1]$, and then constructed into a
”normalized wave function”, namely a list of normalized parameter values,
normalized by itself to unity. With $3$ classes and $10$ numerical
characteristics, the network thus has $10$ input nodes (representing the
numerical characteristics of the substrate) and $G=3$ output nodes
(representing their reactivity).

We note that since we are working with a very small data set, the typical
procedure for classification protocols, namely dividing the data set into a
training set and validation set, and determining the performance of the
protocol from the chosen training set, may be inadequate. We therefore proceed
with a slightly more detailed analysis. We focus on accuracy, $A$, defined as
the number of correct predictions divided by the total number of predictions
[2], rather than precision and recall, because of the small data set size. For
an estimation of the protocol accuracy, we thus repeat this procedure $10$
times, randomly choosing a validation set and a training set at each
repetition, and evaluate the accuracy of the protocol for each realization.
This gives us an average accuracy and a standard deviation, which is a better
statistical description for the protocol performance.

The first step is to determine the optimal network size. We select a training
set of size $N_{TS}=10$ (and a validation set with $10$ vectors), and repeat
the calculation 10 times. In Fig. 6a we plot the accuracy (mean and standard
deviation) as a function of the size of the internal layer (e.g. black circles
in Fig. 1). We find that a network with 7 sites gives the best accuracy, $\sim
86\%$.

We then turn to determine how many vectors the training set requires to obtain
good classification accuracy. Using the optimal network size found above, Fig.
6b shows the accuracy as a function of training set size (for each size, 10
different realizations were chosen and the accuracy mean and standard
deviation were evaluated). Surprisingly, we find that the best accuracy was
actually reached for a rather small training set size $N_{TS}=10$, and that
increasing the training set size actually reduced the accuracy on average.
Similar results were obtained for the precision and recall (see SI).

The results of Fig. 6 show that the QCN protocol averages at around an
accuracy of $A=0.86$, and for some validation sets reaches up very close to
$100\%$ accuracy (similar values were obtained for the precision and recall).
These values are comparable to (and even higher from) the accuracy of similar
calculations performed with classical algorithms [62, 65, 66, 67], yet were
obtained with a much smaller training set (of only $\sim 10$ vectors). This
demonstrates the possibility of quantum advantage in the QCN protocol in the
regime of small data.

Figure 6: Performance of the QCN protocol on real-world data. The
classification protocol (see text for details) is used for randomly chosen
training and validation sets, from which the average and standard deviation of
the accuracy are evaluated. (a) Average classification accuracy for a network
of constant $N_{TS}=10$, as a function of the size $M$ of the intermediate
(hidden) layer (see Fig. 1). The network that presents the best performances
is a $10-7-3$ network. (b) Mean accuracy of the QCN classification for
different numbers of training set vectors, $N_{TS}$. Best result was obtained
for $N_{TS}=10$, with accuracy $A=0.86$, and reaching up to $A=1.0$ for some
validation sets. The fact that the best $N_{TS}$ and $M$ are so small shows
the potential of the QCN protocol for small data problems. Error bars mark the
standard deviation

Discussion

Admittedly, the QCN algorithm we present is computationally more expensive,
and for large system sizes may be more time consuming, than the classical
algorithms. However, it show a ”quantum advantage”, especially relevant for
the regime of small data [68], where available datasets are limited in size, a
regime which seems highly relevant in fields such as material design and
chemistry [69, 70, 71, 62]. This may become practical for situations where
calculation time is an unimportant factor but recall and precision are
important,

What is the origin of the quantum advantage of the QCN algorithm? one possible
conjecture is as follows. The cost-function of the QCN algorithm is based on
the relation between the currents and the hopping matrix elements in a
network, a relation which is typically of Lorentzian nature. This means that
the optimization landscape is fairly shallow, i.e. the local minima are not
”exponentially deep” but ”Lorentzianly deep”, which allows the algorithm to
effectively escape local minima, and reach the global minimum even for a small
training set. How come this works for such small training sets? these
questions are left for future investigations.

Importantly, the QCN scheme we present here is not only an algorithm to be
implemented on a computer. Rather, it is a scheme that can in principle be
implemented using current state-of-the-art experiments using photonic networks
(e.g., [72, 46]). Thus, this work paves the way for a quantum machine aimed at
performing specific machine-learning computational tasks, beyond the standard
paradigm of quantum computing.

Methods

Current from the Lindblad equation

As discussed in the Results section, the system is described by a general
tight-binding Hamiltonian and the Lindblad equation,

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-i[H,\rho]+\Sigma_{k}(V_{k}^{\dagger}\rho
V_{k}-\frac{1}{2}(V_{k}^{\dagger}V_{k}\rho+\rho V_{k}^{\dagger}V_{k}))$
$\displaystyle=$ $\displaystyle-i[H,\rho]+\mathcal{L}[\rho]~{}.$

We limit the calculation to a single-exciton manifold (we expect only minor
changes if one uses the full Fock space, as demonstrated in [53].

We solve the Lindblad equation and evaluate the current at the steady state.
i.e. $\dot{\rho}_{s}=0$ [73, 48, 74, 75]

Once $\rho_{s}$ is at hand, we evaluate the current, $J$, from the extraction
sites $J_{i}=\frac{d\langle n_{i}\rangle}{dt}$ where $\langle n_{i}\rangle$ is
the on-site density ($i$ is the site identifier) , via the relation $\langle
n_{i}\rangle=Tr(\hat{n_{i}}\rho)$. Substituting the expression for the on-site
density into $J_{i}$ gives an equation for the current,

$\displaystyle J_{ij}=\frac{d\langle
n_{i}\rangle}{dt}=\frac{d}{dt}Tr(\hat{n_{i}}\rho)$ (8)

Inserting the derivative into the trace and applying it on its contents yields

$\displaystyle J_{i}=Tr(\dot{\hat{n}}_{i}\rho_{s}+\hat{n}_{i}\dot{\rho}_{s})$
(9)

Considering only exit sites, this expression formally becomes

$\displaystyle
J_{ext}=Tr(\hat{n}_{ext}(-i[\mathcal{H},\rho_{s}]+\mathcal{L}_{ext}[\rho_{s}]))=0$
(10)

The expression above vanishes due to trivial current conservation at the
steady state (the two terms are identical in magnitude and opposite in sign).
In order to evaluate the exiting current, we thus evaluate only the term
relating to the exit sites (or more specifically, to the measuring apparatus),
e.g., the term

$\displaystyle J_{ext}=Tr(\hat{n}_{ext}\mathcal{L}_{ext}[\rho_{s}])~{}~{}.$
(11)

Training the classical classification network In order to compare the QCN and
the classical one, we use the following procedure. First, we take exactly the
same training set for both cases. Then, we train the classical network using
”Keras” (https://keras.io/), which is a convenient application programming
interface (API) for implementing machine learning models in Python. Activation
function of the CCN was ”Relu”, output layer activation was a ”sigmoid”, the
optimizer of the network was ”Adam”, and loss function used was
”crossentropy”.

Data availability
No datasets were generated or analysed during the current study. Please
contact the corresponding author(s) with questions or concerns.

Code availability
The custom code that was created during the work that led to the main results
of this article is published in a public GitHub repository:
https://github.com/ShmueLorber/Quantum-Classification

Author contributions
Y.D. conceived the project and supervised the calculations. S.L. designed and
performed the calculations. O.Z. performed some calculations and assist in
data analysis. I.L.Z. and A.M. supplied the chemical data and performed the
chemical experiments to generate the data. All authors participated in the
analysis of the data. S.L. and Y.D. wrote the manuscript, and all authors
commented on the manuscript.

Competing interests
The authors declare no competing interests.

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