# Theoretical methods to design and test quantum simulators for the compact
Abelian Higgs model

Yannick Meurice1 1 Department of Physics and Astronomy, The University of
Iowa, Iowa City, IA 52242 USA

(August 30, 2024)

###### Abstract

The lattice compact Abelian Higgs model is a non-perturbative regularized
formulation of low-energy scalar quantum electrodynamics. In 1+1 dimensions,
this model can be quantum simulated using a ladder-shaped optical lattice with
Rydberg-dressed atoms Zhang  _et al._ (2018). In this setup, one spatial
dimension is used to carry the angular momentum of the quantum rotors. One can
use truncations corresponding to spin-2 and spin-1 to build local Hilbert
spaces associated with the links of the lattice. We argue that ladder-shaped
configurable arrays of Rydberg atoms can be used for the same purpose. We make
concrete proposals involving two and three Rydberg atoms to build one local
spin-1 space (a qutrit). We show that the building blocks of the Hamiltonian
calculations are models with one and two spins. We compare target and
simulators using perturbative and numerical methods. The two-atom setup
provides an easily controllable simulator of the one-spin model while the
three-atom setup involves solving nonlinear equations. We discuss approximate
methods to couple two spin-1 spaces. The article provides analytical and
numerical tools necessary to design and build the proposed simulators with
current technology.

##  I Introduction

There has been recent interest in using quantum simulations and quantum
computations to address problems with real-time and finite density in high-
energy physics Bañuls  _et al._ (2020); Klco  _et al._ (2021); Aidelsburger
_et al._ (2021); Wiese (2021); Gustafson  _et al._ (2021a); Lamm  _et al._
(2019); Gustafson and Lamm (2021); Brower  _et al._ (2020); Zohar  _et al._
(2016); Wiese (2013). One initial step is the simulation of Abelian gauge
theories Zohar and Reznik (2011); Tagliacozzo  _et al._ (2013). The Schwinger
model introduces fermions and can be studied with methods developed in many-
body physics Bañuls  _et al._ (2013); Buyens  _et al._ (2016); Bañuls  _et
al._ (2016); Funcke  _et al._ (2020). Quantum simulations Martinez  _et al._
(2016); Kasper  _et al._ (2017) and quantum computations Klco  _et al._
(2018); Kharzeev and Kikuchi (2020) have been performed for this model.

A bosonic variant is the compact Abelian Higgs model. The compactness allows
discrete character expansions formulations Bazavov  _et al._ (2015); Unmuth-
Yockey  _et al._ (2018); Meurice  _et al._ (2020) which are gauge-invariant
and solve Meurice (2020) the questions of gauge redundancy and the
implementation of Gauss’s law Unmuth-Yockey (2019); Bender and Zohar (2020).
The truncations do not break symmetries Meurice (2019, 2020); Meurice  _et
al._ (2020) but can affect the nature of phase transitions Zhang  _et al._
(2021); Hostetler  _et al._ (2021). The non-compact Brout-Englert-Higgs mode
is assumed to be decoupled hereafter. For a recent discussion of its effects
in the context of quantum simulations see Ref. Chanda  _et al._ (2021). An
Optical lattice implementation with spin-2 has been proposed Zhang  _et al._
(2018) in 1+1 dimensions. It is based on a ladder-shaped optical lattice with
Rydberg-dressed atoms. In this setup, one spatial dimension carries the
angular momentum of quantum rotors. Truncations corresponding to local Hilbert
spaces with various spin truncations associated with the links of the lattice
have been discussed Bazavov  _et al._ (2015); Unmuth-Yockey  _et al._ (2018);
Zhang  _et al._ (2018, 2021).

In the following, we argue that ladder-shaped configurable arrays of Rydberg
atoms Bernien  _et al._ (2017); Endres  _et al._ (2016); Keesling  _et al._
(2019); Cong  _et al._ (2021); Semeghini  _et al._ (2021), abbreviated CARA,
can be used for the same purpose. This platform has been used for other
lattice gauge theory models Surace  _et al._ (2020); Celi  _et al._ (2020);
Notarnicola  _et al._ (2020). See Ref. Wu  _et al._ (2021) for a review of the
use of Rydberg atoms. We make concrete proposals involving two and three
Rydberg atoms to build one local spin-1 space (a qutrit). We show that the
building blocks of the Hamiltonian calculations are simple models with one and
two spins. We compare target and simulators for these simple models using
perturbative and numerical methods. The two-atom setup provides an easily
controllable simulator of the one-spin model while the three-atom setup
involves nonlinear matching which could be tested with current technology. We
argue that near-term technology could be used to quantum simulate models with
two or more spins. More generally, programming with CARA amounts to a
geometrical assembling allowing a broad range of applications. The idea of
providing qutrits is very timely Ciavarella  _et al._ (2021); Gustafson
(2021).

The article is organized as follows. In Sec. II, we review the Lagrangian and
Hamiltonian formulation of the compact Abelian Higgs model with emphasis on
the meaning of the signs of the couplings. The general idea of ladder-shaped
CARA is introduced in Sec. III. The two and three atoms CARA for a single
spin-1 are discussed in Sec. IV. The coupling of two spins with an operator
that is the product of their respective angular momentum $L^{z}$’s is
discussed in Sec. V. We argue that the single spin-1 simulators can be tested
with current technology and that near-term technology could be used to quantum
simulate models with two or more spins. Implementations with universal quantum
computers are discussed in Sec. VI and the conclusions are provided in Sec.
VII.

##  II The lattice model

In this section we review the Lagrangian and Hamiltonian formulations of the
compact Abelian Higgs model with emphasis on the meaning of the signs of the
couplings. As we will see the sign question is important from the point of
view of quantum simulations.

###  II.1 Lagrangian formulation

We first review the Lagrangian path integral formulation of the Abelian Higgs
model at Euclidean time using most of the notations of Ref. Bazavov  _et al._
(2015) which should be consulted for more details. The partition function has
the form

$Z=\int D\phi^{\dagger}D\phi DUe^{-S}.$ (1)

The action is the sum of three terms

$S=S_{g}+S_{h}+S_{\lambda},$ (2)

where the gauge part is

$S_{g}=-\beta_{pl}\sum_{x}{\rm Re}\left[U_{pl.,x}\right],$ (3)

the hopping part

$\displaystyle S_{h}$ $\displaystyle=$
$\displaystyle-{\kappa_{\tau}}\sum_{x}\left[\phi_{x}^{\dagger}U_{x,\hat{\tau}}\phi_{x+\hat{\tau}}+\phi_{x+\hat{\tau}}^{\dagger}U^{\dagger}_{x,\hat{\tau}}\phi_{x}\right]$
(4) $\displaystyle-$
$\displaystyle{\kappa_{s}}\sum_{x}\left[\phi_{x}^{\dagger}U_{x,\hat{s}}\phi_{x+\hat{s}}+\phi_{x+\hat{s}}^{\dagger}U^{\dagger}_{x,\hat{s}}\phi_{x}\right],$

and the self-interaction

$S_{\lambda}=\lambda\sum_{x}\left(\phi_{x}^{\dagger}\phi_{x}-1\right)^{2}+\sum_{x}\phi_{x}^{\dagger}\phi_{x}.$
(5)

By writing

$\phi_{x}=|\phi_{x}|\exp(i\varphi_{x}),$ (6)

we can separate the compact and non-compact variables in $S_{h}$:

$\displaystyle S_{h}=$ $\displaystyle-$ $\displaystyle
2\kappa_{\tau}|\phi_{x}||\phi_{x+\hat{\tau}}|\sum\limits_{x}\cos(\varphi_{x+\hat{\tau}}-\varphi_{x}+A_{x,\hat{\tau}})$
(7) $\displaystyle-$ $\displaystyle
2\kappa_{s}|\phi_{x}||\phi_{x+\hat{s}}|\sum\limits_{x}\cos(\varphi_{x+\hat{s}}-\varphi_{x}+A_{x,\hat{s}}).$
(8)

It is assumed that $\kappa_{s}$ and $\kappa_{\tau}$ are positive as in
ferromagnetic interactions. This means that if we neglect the gauge fields and
the self-interactions, large values of $|\phi|$ favor the alignment of the
matter fields (configurations where all the angles $\varphi_{x}$ are equal),
as expected in the continuum limit of the free $O(2)$ scalar model.

In the following, we take the limit where $\lambda$ become large and positive.
The Brout-Englert-Higgs mode $|\phi|$ is then frozen to 1. The Nambu-Goldstone
mode $\varphi$ is compact. We call this model the compact Abelian Higgs model
. By shifting the integration variable $\varphi$ by $\pi$ on every other site
say in the spatial direction, we can flip the sign of $\kappa_{s}$ without
affecting the partition function. A similar reasoning can be applied for
$\kappa_{\tau}$ and the time direction. However, if observable are inserted in
the partition function, these changes of variable need to be performed for the
observables too. For instance, the magnetization becomes a staggered
magnetization. Similar considerations apply to the plaquette term and the sign
of $\beta_{pl.}$. This is discussed at length in Li and Meurice (2005);
Meurice (2009).

###  II.2 Hamiltonian and Hilbert space

Following Ref. Bazavov  _et al._ (2015); Zhang  _et al._ (2018); Unmuth-Yockey
_et al._ (2018) the continuous-time limit for the compact Abelian Higgs model
was taken in the field quantum number representation in the limit where the
Higgs quartic self-coupling goes to infinity. To take the time continuum
limit, one takes $\kappa_{\tau},\beta_{pl}\rightarrow\infty$ while
simultaneously taking $\kappa_{s}$, and the temporal lattice spacing, $a$, to
zero such that the combinations

$U\equiv\frac{1}{\beta_{pl}a}=\frac{g^{2}}{a},\quad
Y\equiv\frac{1}{2\kappa_{\tau}a},\quad X\equiv\frac{2\kappa_{s}}{a}$ (9)

are finite. These equations make clear that the signs of $U$, $X$ and $Y$ are
the same as $\beta_{pl.}$, $\kappa_{\tau}$ and $\kappa_{s}$ respectively. The
Hamiltonian for $N_{s}$ links reads

$\displaystyle H$
$\displaystyle=\frac{U}{2}\sum_{i=1}^{N_{s}}\left(L^{z}_{i}\right)^{2}$
$\displaystyle+\frac{Y}{2}{\sum_{i}}^{\prime}(L^{z}_{i+1}-L^{z}_{i})^{2}-X\sum_{i=1}^{N_{s}}U^{x}_{i}\
,$ (10)

where the sum, $\sum_{i}^{\prime}$, means that for open boundary conditions
(OBC) we need to include $(L^{z}_{1})^{2}+(L^{z}_{N_{s}})^{2}$. We used the
operator

$U^{x}\equiv\frac{1}{2}(U^{+}+U^{-}),$ (11)

with

$U^{\pm}\ket{m}=\ket{m\pm 1}.$ (12)

The quantum number $m$ corresponds to the Fourier modes in the character
expansion of the Lagrangian formulation Bazavov  _et al._ (2015). In practice,
we need to apply truncations. For a spin-$m_{max}$ truncation we have

$U^{\pm}\ket{\pm m_{max}}=0.$ (13)

In the following we mostly focus on the spin-1 truncation where $m=\pm 1,0$.
In this special case $U^{x}=L^{x}/\sqrt{2}$.

The truncations are compatible with the identities following from local or
global symmetries of several models with continuous Abelian symmetries Meurice
(2019, 2020); Meurice  _et al._ (2020), however they can affect the type of
phase transition present in the the model.

The physical interpretation of the three terms of the Hamiltonian is the same
as in conventional electrodynamics except for the fact that all the values
involved are discrete. The $U$-term represents the electric field energy. The
$Y$-term is associated with matter charges. They can be interpreted as charges
determined by Gauss’s law, in other words the difference between the two
plaquettes (electric field) on each side of a site in the Lagrangian form.
Finally the $X$-term is related to matter currents inducing temporal changes
in the electric field, again in the Lagrangian form. This corresponds to the
other inhomogeneous Maxwell equation involving the currents. In higher
dimensions, the discrete curl of a magnetic field appears as in the continuum
Meurice (2020).

###  II.3 Charge conjugation

As in standard quantum electrodynamics this Hamiltonian has a charge
conjugation symmetry. This will play an important role in the construction of
simulators because this property will translate into a global reflection
symmetry in the geometrical setup of the atoms. For this reason we remind the
basic equations associated with this symmetry. The charge conjugation
$\mathcal{C}$ is a unitary transformation which reverses the sign of $m$:

$\mathcal{C}\ket{m}=\ket{-m}$ (14)

It is clear that $\mathcal{C}^{2}=1$ and that

$\displaystyle\mathcal{C}L^{z}\mathcal{C}^{-1}=-L^{z},$ (15)
$\displaystyle\mathcal{C}U^{\pm}\mathcal{C}^{-1}=U^{\mp},$ (16)
$\displaystyle\mathcal{C}U^{x}\mathcal{C}^{-1}=U^{x}$ (17)

This implies that the Hamiltonian is invariant under charge conjugation:

$\mathcal{C}H\mathcal{C}^{-1}=H.$ (18)

###  II.4 The building blocks of the Hamiltonian formalism

In order to conduct actual experiments to quantum simulate the compact Abelian
Higgs model, we need to identify its building blocks. The first one is the
local spin-1 or higher spin Hilbert space where we need to implement the
operators $L^{z}$ and $U^{x}$. This will be discussed in Sec. IV. The second
is the coupling of two spins with an operator that is the product of their
respective $L^{z}$’s. This will be discussed in Sec. V.

##  III Rydberg atom simulators

###  III.1 A ladder-shaped optical lattice simulator

In Ref. Zhang  _et al._ (2018), a quantum simulator for the spin-2 truncation
of the Hamiltonian in Eq. (II.2) was proposed. The general idea is to use an
$5\times N_{s}$ optical lattice that one can visualize as a ladder with
$N_{s}$ rungs. There is only one atom per rung and the five sites on each rung
represent the 5 possible values for $m$, with $m=0$ at the center. Tunneling
in the direction orthogonal to the rung is not allowed. Tunneling along the
rung generates the $X$-term. The $U$-term is created by a parabolic potential.
The $Y$-term is mediated by the $1/r^{6}$ interactions of the Rydberg-dressed
atoms Zeiher  _et al._ (2016). These interactions were chosen to be attractive
and favoring ferromagnetism: neighbor atoms with the same $m$ are closer to
each other than atoms with different $m$’s. By taking the distance between the
rungs $a_{s}$ larger than the distance between the sites on the rungs $a_{r}$,
it is possible to do perturbation in $a_{r}/a_{s}$ in Pythagoras theorem and
approximately generate the quadratic $Y$-term. A more complete discussion and
illustrations can be found in Ref. Zhang  _et al._ (2018).

###  III.2  CARA simulators

In the following we discuss the possibility of adapting the idea of Ref. Zhang
_et al._ (2018) to configurable arrays of Rydberg atoms Bernien  _et al._
(2017); Endres  _et al._ (2016); Keesling  _et al._ (2019); Cong  _et al._
(2021); Semeghini  _et al._ (2021) denoted CARA. They can be configured by
positioning ${}^{87}Rb$ atoms separated by controllable (but not too small)
distances, homogeneously coupled to the excited Rydberg state $\ket{r}$ with a
detuning $\Delta$. The ground state is denoted $\ket{g}$ and the two possible
states $\ket{g}$ and $\ket{r}$ can be seen as a qubit. The Hamiltonian reads

$H=\frac{\Omega}{2}\sum_{i}(\ket{g_{i}}\bra{r_{i}}+\ket{r_{i}}\bra{g_{i}})-\Delta\sum_{i}n_{i}+\sum_{i<j}V_{ij}n_{i}n_{j},$
(19)

with

$V_{ij}=\Omega R_{b}^{6}/r_{ij}^{6},$ (20)

for a distance $r_{ij}$ between the atoms labelled as $i$ and $j$. This
repulsive interaction prevents two atoms close enough to each other to be in
the $\ket{r}$ state. This is the so-called blockade mechanism.

This setup has been successfully used to simulate the Kibble-Zurek mechanism
for chiral clock models Keesling  _et al._ (2019). It has been used to propose
simulators for other gauge theories Surace  _et al._ (2020); Celi  _et al._
(2020); Notarnicola  _et al._ (2020). Simulations with Rydberg atoms are
reviewed in Ref. Wu  _et al._ (2021).

In subsection III.1, we discussed a setup Zhang  _et al._ (2018) where one
direction of the optical lattice was used to carry the $2S+1$ spin degrees of
freedom on the sites of the rungs. We will now try to replace each rung by a
line of $2S+1$ Rydberg atoms close enough to each other to prevent more than
one atom to be in the $\ket{r}$ state. The spin-2 case is illustrated in Fig.
1. For instance, $m=2$ corresponds to $\ket{rgggg}$. Also notice that charge
conjugation is implemented by a reflection about the horizontal axis passing
by the $m=0$ states.

Figure 1:  The 5 spin-2 states with a 5 atom setup.

With current technology, it is difficult to get 5 atoms on a line close enough
to make the blockade mechanism efficient. However it seems possible to do it
with three atoms as in a spin-1 truncation. In the following we will
concentrate on this simpler realization.

###  III.3 Spin-1 CARA implementations

For spin-1, we propose the correspondence

$\displaystyle\ket{1}\rightarrow\ket{rgg},$
$\displaystyle\ket{0}\rightarrow\ket{grg},$ (21)
$\displaystyle\ket{-1}\rightarrow\ket{ggr}.$

This is illustrated in Fig. 2. Again, charge conjugation is implemented as a
reflection with respect to the horizontal axis.

Figure 2:  Three spin-1 states with the three atom setup, the 5 other possible
states are not displayed.

An even simpler setup consists in using only two Rydberg atoms and having
$\ket{m=0}$ to be a state without Rydberg states, namely

$\displaystyle\ket{1}\rightarrow\ket{rg},$
$\displaystyle\ket{0}\rightarrow\ket{gg},$ (22)
$\displaystyle\ket{-1}\rightarrow\ket{gr}.$

This is illustrated in Fig. 3.

Figure 3:  Spin-1 states with the two atom setup.

Since this second possibility is the simplest, it will be the starting point
of the presentation of the next sections.

##  IV One spin system

In this section, we discuss the one spin-1 Hilbert space and the local
Hamiltonian for the target model and the CARA implementations with two and
three atoms.

###  IV.1 Target model

The local part of the target Hamiltonian is

$H^{1T}=\frac{U}{2}(L^{z})^{2}-XU^{x}.$ (23)

We will discuss the spectrum of this Hamiltonian with emphasis on the
symmetries in order to build matching simulators. Since $H^{1T}$ is invariant
under charge conjugation, we introduce the $\mathcal{C}$ eigenstates

$\ket{\pm}\equiv\frac{1}{\sqrt{2}}(\ket{1}\pm\ket{-1}).$ (24)

with $\mathcal{C}$-eigenvalues $\pm 1$:

$\mathcal{C}\ket{\pm}=\pm\ket{\pm}.$ (25)

They are also eigenstates of $(L^{z})^{2}$ with eigenvalue 1. Note that

$L^{z}\ket{\pm}=\ket{\mp}.$ (26)

In addition $\mathcal{C}\ket{0}=\ket{0}$.

There is only one $\mathcal{C}$-odd state which is $\ket{-}$. It is
annihilated by $U^{x}$

$U^{x}\ket{-}=0.$ (27)

Consequently,

$H^{1T}\ket{-}=\frac{U}{2}\ket{-},$ (28)

for any value of $X$.

In the $\mathcal{C}$-even sector, we have

$\displaystyle U^{x}\ket{0}=\frac{1}{\sqrt{2}}\ket{+},$ (29) $\displaystyle
U^{x}\ket{+}=\frac{1}{\sqrt{2}}\ket{0},$ (30)

and the eigenvalues are obtained from the even matrix in the $\ket{0},\
\ket{+}$ basis:

$H^{1T}_{even}=\begin{pmatrix}0&-\frac{X}{\sqrt{2}}\\\
-\frac{X}{\sqrt{2}}&\frac{U}{2}\end{pmatrix}.$ (31)

The two eigenstates are

$\ket{0}_{X}=\cos\phi\ket{0}+\sin\phi\ket{+},$ (32)

with eigenvalue

$E_{0}(X)=\frac{1}{4}(U-\sqrt{U^{2}+8X^{2}}\ ),$ (33)

and

$\ket{+}_{X}=\cos\phi\ket{+}-\sin\phi\ket{0},$ (34)

with eigenvalue

$E_{+}(X)=\frac{1}{4}(U+\sqrt{U^{2}+8X^{2}}\ ).$ (35)

The mixing angle obeys the equation

$\tan\phi=-\sqrt{2\ }\frac{E_{0}(X)}{X}.$ (36)

If we treat $X$ as a perturbation, we obtain that at the lowest nontrivial
order

$\displaystyle E_{0}(X)$ $\displaystyle\simeq-\frac{X^{2}}{U},$ (37)
$\displaystyle E_{+}(X)$ $\displaystyle\simeq\frac{U}{2}+\frac{X^{2}}{U},$
(38) $\displaystyle\phi$ $\displaystyle\simeq\sqrt{2}\frac{X}{U}.$ (39)

These results can also be derived using standard perturbative formulas Sakurai
(1994). The perturbation has no diagonal element and the energy corrections
occur at second order.

###  IV.2 Two Rydberg atom implementation

The two atom setup discussed in Sec. III.3 provides a very simple
implementation of a single spin-1 system. Introducing $\pm 1$ labels for the
top and bottom atoms, we call $n_{\pm 1}$ the occupation of their $\ket{r}$
state. The list of the possible states and their occupations are given in
Table 1.

Setup | Ket | $n_{+1}$ | $n_{-1}$ | Short | Energy ($\Omega=0)$
---|---|---|---|---|---
$\begin{matrix}\bullet\\\ \circ\end{matrix}$ | $\ket{rg}$ | 1 | 0 | $\ket{1}$ | $-\Delta$
$\begin{matrix}\circ\\\ \circ\end{matrix}$ | $\ket{gg}$ | 0 | 0 | $\ket{0}$ | 0
$\begin{matrix}\circ\\\ \bullet\end{matrix}$ | $\ket{gr}$ | 0 | 1 | $\ket{-1}$ | $-\Delta$
$\begin{matrix}\bullet\\\ \bullet\end{matrix}$ | $\ket{rr}$ | 1 | 1 | $\ket{2}$ | $-2\Delta+V_{0}$
Table 1: Graphical representation of the two atoms in space, the symbol
$\circ$ represents the ground state $\ket{g}$ and $\bullet$ the Rydberg state
$\ket{r}$, ket notation, occupations, short notation and energy for
$\Omega=0.$

The Hamiltonian for the physical two-atom system is

$\displaystyle H^{2R}=$
$\displaystyle-\Delta(n_{+1}+n_{-1})+V_{0}n_{+1}n_{-1}$ (40)
$\displaystyle+\frac{\Omega}{2}\sum_{\pm 1}(\ket{g_{\pm 1}}\bra{r_{\pm
1}}+\ket{r_{\pm 1}}\bra{g_{\pm 1}}).$ (41)

We want to match this Hamiltonian with the target $H^{1T}$. We first consider
the first term of the target Hamiltonian. The splitting between $\ket{0}$ and
$\ket{\pm 1}$ is $\frac{U}{2}$ when $\Omega=0$, and can be implemented by
setting

$\Delta=-\frac{U}{2}.$ (42)

In addition we want to suppress transitions to the $\ket{rr}$ state by
introducing a large enough energy $V_{0}$. This is the blockade mechanism.
This can be achieved by positioning the two atoms close enough to each other.

In order to determine $\Omega$, we compare the action of

$H_{1}^{2R}=\frac{1}{2}\sum_{\pm 1}(\ket{g_{\pm 1}}\bra{r_{\pm 1}}+\ket{r_{\pm
1}}\bra{g_{\pm 1}})$ (43)

on the simulator Hilbert space to the action of $U^{x}$ on the target Hilbert
space.

$\displaystyle H_{1}^{2R}\ket{gg}$
$\displaystyle=\frac{1}{2}(\ket{rg}+\ket{gr}),$ (44) $\displaystyle
H_{1}^{2R}\frac{1}{\sqrt{2}}(\ket{rg}+\ket{gr})$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{gg}+\ket{rr}),$ (45) $\displaystyle
H_{1}^{2R}\frac{1}{\sqrt{2}}(\ket{rg}-\ket{gr})$ $\displaystyle=0.$ (46)

Comparing with the action of $U^{x}$ on the three states of the target Hilbert
space, we see that, except for the heavy state $\ket{rr}$, they are identical.
Consequently, we can set

$\Omega=-X.$ (47)

Except for possible transitions to $\ket{rr}$, the correspondence is exact and
the linear formula applies for arbitrary values of $X$. The good matching for
$U=1$ with $X=0.5$ and $U=1$ with $X=1.5$ is demonstrated in Fig. 4.

Figure 4: $|\bra{m}U(t)\ket{m=1}|^{2}$, one site with exact Hamiltonian $U=1,\
X=0.5$ (solid lines) and Rydberg Hamiltonian with $\Omega=-0.5$, $\Delta=-0.5$
and $V_{0}=64|\Omega|=32$ (empty symbols) (top), and $U=1,\ X=1.5$ (solid
lines) and Rydberg Hamiltonian $H^{2R}$ with $\Omega=-1.5$, $\Delta=-0.5$ and
$V_{0}=64|\Omega|=96$ (empty symbols) (bottom).

###  IV.3 Three Rydberg atom simulator

In the three atom setup, we have an additional atom which is associated with
the $m=0$ state. We label its $\ket{r}$ occupation $n_{0}$. The Hamiltonian
reads

$\displaystyle H^{3R}=$ $\displaystyle-\Delta_{0}n_{0}-\Delta\sum_{j=0,\pm
1}n_{j}$ (48)
$\displaystyle+V_{0}(n_{0}n_{+1}+n_{0}n_{-1})+V_{0}^{\prime}n_{+1}n_{-1}$ (49)
$\displaystyle+\frac{\Omega}{2}\sum_{j=0,\pm
1}(\ket{g_{j}}\bra{r_{j}}+\ket{r_{j}}\bra{g_{j}}),$ (50)

with

$V^{\prime}_{0}=\frac{V_{0}}{64}$ (51)

when the three atoms are located equidistantly on a line as in Fig . 2. The
spin-1 sector is shown in Table 2.

Setup | Ket | Energy ($\Omega=0$)
---|---|---
$\begin{matrix}\bullet\\\ \circ\\\ \circ\end{matrix}$ | $\ket{1}$ | $-\Delta$
$\begin{matrix}\circ\\\ \bullet\\\ \circ\end{matrix}$ | $\ket{0}$ | $-\Delta-\Delta_{0}$
$\begin{matrix}\circ\\\ \circ\\\ \bullet\end{matrix}$ | $\ket{-1}$ | $-\Delta$
Table 2:  Spin-1 states for the three atom simulator.

The auxiliary sector has five states shown in Table 3.

Setup | Ket | Energy ($\Omega=0$)
---|---|---
$\begin{matrix}\circ\\\ \circ\\\ \circ\end{matrix}$ | $\ket{0^{\prime\prime}}$ | 0
$\begin{matrix}\bullet\\\ \bullet\\\ \circ\end{matrix}$ | $\ket{1^{\prime}}$ | $-2\Delta-\Delta_{0}+V_{0}$
$\begin{matrix}\bullet\\\ \circ\\\ \bullet\end{matrix}$ | $\ket{0^{\prime}}$ | $-2\Delta+\frac{V_{0}}{64}$
$\begin{matrix}\circ\\\ \bullet\\\ \bullet\end{matrix}$ | $\ket{-1^{\prime}}$ | $-2\Delta-\Delta_{0}+V_{0}$
$\begin{matrix}\bullet\\\ \bullet\\\ \bullet\end{matrix}$ | $\ket{3}$ | $-3\Delta-\Delta_{0}+2V_{0}+\frac{V_{0}}{64}$
Table 3:  Auxiliary states for the three atom simulator.

Following previous notation, we also define

$\ket{\pm^{\prime}}\equiv\frac{1}{\sqrt{2}}(\ket{1^{\prime}}\pm\ket{-1^{\prime}}),$
(52)

and

$H^{3R}_{1}=\frac{1}{2}\sum_{j=0,\pm
1}(\ket{g_{j}}\bra{r_{j}}+\ket{r_{j}}\bra{g_{j}}),$ (53)

Unlike the previous situation with two atoms, $H^{3R}_{1}$ only connects the
spin-1 sector with the auxiliary sector

$\displaystyle H^{3R}_{1}\ket{0}$
$\displaystyle=\frac{1}{2}\ket{0^{\prime\prime}}+\frac{1}{\sqrt{2}}\ket{+^{\prime}},$
(54) $\displaystyle H^{3R}_{1}\ket{+}$
$\displaystyle=\frac{1}{2}\ket{+^{\prime}}+\frac{1}{\sqrt{2}}\ket{0^{\prime}}+\frac{1}{\sqrt{2}}\ket{0^{\prime\prime}},$
(55) $\displaystyle H^{3R}_{1}\ket{-}$
$\displaystyle=\frac{1}{2}\ket{-^{\prime}}.$ (56)

Using the corresponding matrix elements together with standard perturbation
theory Sakurai (1994), we obtain the perturbative matching equation for the
energy differences:

$\displaystyle\frac{X^{2}}{U}$
$\displaystyle=\frac{\Omega^{2}}{2}(\frac{1}{\Delta-\frac{V_{0}}{64}}-\frac{1}{\Delta}),$
(57) $\displaystyle\frac{U}{2}+\frac{X^{2}}{U}$
$\displaystyle=\Delta_{0}+\frac{\Omega^{2}}{4}(\frac{1}{\Delta+\Delta_{0}}+\frac{2}{V_{0}-\Delta}-\frac{1}{V_{0}-\Delta-\Delta_{0}}).$

Similarly we can try to match perturbative expressions for the mixing angle
$\phi$ defined in Eq. (36). However, $\phi$ contributions appear at first
order in $X$ in the target model (because $U^{x}$ connects $\ket{0}$ and
$\ket{+}$), but only at second order in $\Omega$ in the three-atom simulator.
More explicitly,

$\sqrt{2}\frac{X}{U}=\frac{\Omega^{2}}{2\sqrt{2}\Delta_{0}}(\frac{1}{\Delta}+\frac{1}{V_{0}-\Delta-\Delta_{0}}).$
(58)

This apparently contradicts the idea that $X$ and $\Omega$ should be
proportional for small values of $X$. If we are given $U$ and $X$, we have
three nonlinear equations for $\Omega$, $\Delta$ , $\Delta_{0}$ and $V_{0}$
and generically, we expect one-parameter families of solutions. By fixing one
of the unknowns, one can look for solutions using Newton’s method. More
generally, it is easy to find accurate numerical solutions for the energies
and mixing of the simulators and it seems possible to attack the matching
problem non perturbatively. In order to pursue such effort, it would be useful
to know the range of values corresponding to feasible experiments.

With today’s technology, it seems difficult to create non homogeneous detuning
and we should consider the limit where $\Delta_{0}$ is zero. In this limit,
$\ket{0}$ and $\ket{+}$ are degenerate when $\Omega$ is set to zero. In the
$\ket{0}$ and $\ket{+}$ basis, the energy matrix up to second order in
$\Omega$ has the form

$-\Delta\mathbb{1}-\frac{\Omega^{2}}{4}\mathbb{M},$ (59)

with

$\mathbb{M}=\begin{pmatrix}\frac{1}{\Delta}+\frac{2}{V_{0}-\Delta}&\frac{\sqrt{2}}{\Delta}+\frac{\sqrt{2}}{V_{0}-\Delta}\\\
\frac{\sqrt{2}}{\Delta}+\frac{\sqrt{2}}{V_{0}-\Delta}&\frac{2}{\Delta}+\frac{1}{V_{0}-\Delta}-\frac{2}{\Delta-\frac{V_{0}}{64}}\end{pmatrix}.$
(60)

The matrix $\mathbb{M}$ determines the mixing angle and the eigenvalues.

###  IV.4 An example of approximate solution with three Rydberg atoms

The three-atom simulator leads to nonlinear equations. However, it is not
difficult to find approximate solutions when $\Delta$ and $V_{0}$ are in a
specific ratio. As a simple example, we picked

$\Delta_{0}=0\ {\rm and}\ V_{0}=2\Delta.$ (61)

If we neglect the $\frac{V_{0}}{64}$ term, we have

$\mathbb{M}\simeq\frac{1}{\Delta}\begin{pmatrix}3&2\sqrt{2}\\\ 2\sqrt{2}\
&1\end{pmatrix}.$ (62)

Given that there is an overall minus sign in front of $\mathbb{M}$, its
largest eigenvalue corresponds to the lowest energy state and the mixing angle
approximately satisfies the equation

$\tan\phi\simeq\frac{1}{\sqrt{2}}$ (63)

Comparing with Eq. (36), we find that this angle corresponds to the situation
$X=U$. Note that for this significant value $X/U=1$, the angle is not small
and the linear approximation of Eq. (36) given in Eq. (39) is not accurate.
Instead, we used the exact value of $E_{0}$ given in Eq. (33). For the
simulator, this large mixing angle is not controlled by $\Omega^{2}$. This is
a feature of degenerate perturbation theory. This procedure is justified by
the good quality of the agreement between target and simulator shown in Fig.
5.

Furthermore, we can compute the energy spectrum in this simple example. The
eigenvalues of $\mathbb{M}$ are approximately $5/\Delta$ and $-1/\Delta$. In
addition, we have up to second order in $\Omega$:

$E_{-}=-\Delta-\frac{\Omega^{2}}{4}\frac{1}{V_{0}-\Delta}.$ (64)

Consequently, in this simple example, we have

$\displaystyle E_{+}-E_{0}$
$\displaystyle=\frac{3}{2}\frac{\Omega^{2}}{\Delta},$ (65) $\displaystyle
E_{-}-E_{0}$ $\displaystyle=\frac{\Omega^{2}}{\Delta}.$ (66)

In the target model with $X=U$, we have

$\displaystyle E_{+}-E_{0}$ $\displaystyle=\frac{3}{2}U,$ (67) $\displaystyle
E_{-}-E_{0}$ $\displaystyle=U.$ (68)

The ratio of differences are both 3/2 and we can match the scale for
$\Omega^{2}/\Delta=U$. Given that the matrix has been approximated, we should
look for matching in the region $X\simeq U$. As a numerical example, we used
$\Omega=1$, $\Delta=15$ and $V_{0}=30$ and found good matching for $U=0.064,\
X=0.067$ both close to $1/15=0.0666...$ This is illustrated in Fig. 5. Note
that the time scale in $1/\Omega$ units is significantly larger than in the
two atom case. This is due to the extra $\Omega/\Delta$ factor in the energy
scale.

Figure 5: $|\bra{m}U(t)\ket{m=1}|^{2}$ for one site with exact Hamiltonian
$U=0.064,\ X=0.067$ (solid lines), Rydberg Hamiltonian with $\Omega=1$,
$\Delta=15$ and $V_{0}=30$ (empty symbols).

##  V Two-spin system

In this section, we follow the same sequence as in Sec. IV for a two-spin
system motivated by the Hamiltonian of Eq. (II.2).

###  V.1 Target model

For the target model we consider two spins called left (L) and right (R)
connected by a $Y$-term. The target Hamiltonian for the two-spin system is
chosen to be

$\displaystyle H^{2T}=$ $\displaystyle
H^{1T}_{L}+H^{1T}_{R}+\frac{Y}{2}(L_{L}^{z}-L_{R}^{z})^{2}$ (69)
$\displaystyle=$
$\displaystyle\frac{U}{2}((L_{L}^{z})^{2}+(L_{R}^{z})^{2})-X(U_{R}^{x}+U_{L}^{x})+\frac{Y}{2}(L_{L}^{z}-L_{R}^{z})^{2}$

For the time evolution we consider an initial state $\ket{0,0}$, and calculate
the probability to stay in that state or to be in the state

$\ket{S}\equiv\frac{1}{2}(\ket{0,1}+\ket{0,-1}+\ket{1,0}+\ket{-1,0}).$ (70)

obtained by applying the $X$-term on the initial state. This guarantees a
significant overlap when $X$ is not too small. For the comparison of the four
and six atom simulators we picked a special target situation where solutions
of the one-spin problems are available, more specifically we picked $U=1$,
$X=1.2$ and $Y=0.2$.

###  V.2 Four Rydberg atom simulator

For a simulator with four atoms, we use the two-atom setup for two pairs and
include the additional $V_{ij}$ appearing in Eq. (19). The Hamiltonian reads

$\displaystyle H^{4R}=$ $\displaystyle H^{2R}_{L}+H^{2R}_{R}$
$\displaystyle+V_{1}(n_{+1L}n_{-1R}+n_{-1L}n_{+1R})$ (71)
$\displaystyle+V_{2}(n_{+1L}n_{+1R}+n_{-1L}n_{-1R})$

When the $V_{i}$ are positive, we need to have the opposite signs closer
(bipartite charge conjugations on alternate sites). In other words, if the
interactions are repulsive and decreasing like $1/r^{6}$, we need to put atoms
with the same $m$ farther apart. This is illustrated in Fig. 6.

Figure 6:  Rydberg interactions for the four atom simulator.

For the standard $1/r^{6}$ Rydberg interactions as in Eq. (20), and using
$a_{r}$ and $a_{s}$ as the vertical (as in one spin) and horizontal (coupling
the two spins) lattice spacings respectively, and their ratio

$\rho\equiv\frac{a_{r}}{a_{s}},$ (72)

we have

$\displaystyle V_{1}=$ $\displaystyle V_{0}\rho^{6},$ (73) $\displaystyle
V_{2}=$ $\displaystyle V_{0}(\rho/\sqrt{1+\rho^{2}}\ )^{6}.$ (74)

The matching condition when $X=\Omega=0$ reads

$\displaystyle\Delta$ $\displaystyle=-\frac{U}{2}-\frac{Y}{2},$ (75)
$\displaystyle V_{1}$ $\displaystyle=Y,$ (76) $\displaystyle V_{2}$
$\displaystyle=-Y.$ (77)

The second equation can be solved using

$\rho=(\frac{Y}{V_{0}})^{1/6}.$ (78)

The third equation has no solutions for positive $V_{2}$. In Eq. (73) $V_{2}$
is smaller than $V_{1}$ but nevertheles positive. This implies that it is not
possible to exactly match the energy of the states $\ket{+1,-1}$ and
$\ket{-1,+1}$. The possible solutions to this problem are: 1) ask
experimentalists to adjust the couplings locally, 2) consider
$Y$-perturbations over situations with non-zero $\Omega$. On the other hand,
the phase structure and dynamical features of the simulator are worth
exploring even if the matching with the target is not perfect. Note that for
the six-atom setup discussed below, we will see that the matching in the limit
$\Omega=X=0$ is approximately possible following the mechanism invoked in Ref.
Zhang  _et al._ (2018).

In order to give an idea of the size of the effects discussed above for the
four atom system, we have considered the problem mentioned in the target
section, first with the futuristic $V_{2}=-Y$ (the agreement is excellent) and
then with $V_{2}$ is as in Eq. (73) (deviations from exact are quite visible
for $t\gtrsim 3$). The results are displayed in Fig. 7.

Figure 7: $|\bra{0,0}U(t)\ket{0,0}|^{2}$ and $|\bra{S}U(t)\ket{0,0}|^{2}$,
with $\ket{S}$ defined in Eq. (70), for the target two-spin Hamiltonian with
$U=1$, $X=1.2$ and $Y=0.2$ (solid lines) and the four atom simulator with
$\Delta$=-0.6, $\Omega$=-1.2, $V_{0}=64$, $V_{1}=0.2$, with $V_{2}=-0.2$ (top)
and $V_{2}=0.13$ (bottom).

###  V.3 Six Rydberg atom implementation

We now consider a six-atom setup with two three-atom spin-1 setups. The
Hamiltonian reads

$\displaystyle H^{6R}=$ $\displaystyle H^{3R}_{L}+H^{3R}_{R}$
$\displaystyle+V_{1}(n_{+1L}n_{-1R}+n_{-1L}n_{+1R})$
$\displaystyle+V_{2}(n_{0L}(n_{+1R}+n_{-1R}))$ (79)
$\displaystyle+V_{2}((n_{+1L}+n_{-1L})n_{0R})$
$\displaystyle+V_{3}(n_{+1L}n_{+1R}+n_{-1L}n_{-1R})$

When the $V_{i}$ are positive, we again need to have the atoms representing
spins of opposite signs closer in space. This is illustrated in Fig. 8.

Figure 8:  Rydberg interactions for the six atom simulator.

For the standard Rydberg interactions of Eq. (20), we have $V_{1}$ and $V_{2}$
as in Eq. (73) and

$V_{3}=V_{0}(\rho/\sqrt{1+4\rho^{2}}\ )^{6}$ (80)

Matching condition when $X=\Omega=0$ are

$\displaystyle\Delta_{0}$ $\displaystyle=\frac{U}{2}+Y$ $\displaystyle
V_{1}-V_{2}$ $\displaystyle=\frac{Y}{2}$ (81) $\displaystyle V_{1}-V_{3}$
$\displaystyle=2Y,$

These equations have approximate solutions with a tuned $\rho$ when $Y$ is not
too large. In Fig. 9, we used $\Delta$=15, $\Omega$=1, $V_{0}=30$, and
empirically tuned $\rho$ to a value 0.326. A rescaling $K=0.05464$ has been
applied to the simulator time to match the target evolution. In Fig. 9, we
show the case $\Delta$=15, $\Omega$=1, $V_{0}=30$, as in the one spin-1, with
$\rho$=0.326.

Figure 9: $|\bra{0,0}U(t)\ket{0,0}|^{2}$ and $|\bra{S}U(t)\ket{0,0}|^{2}$,
with $\ket{S}$ defined in Eq. (70), with the target two-spin Hamiltonian with
$U=1$, $X=1.2$ and $Y=0.2$ (solid lines) and the six atom simulator with
$\Delta$=15, $\Omega$=1, $V_{0}=30$, $\rho$=0.326. A rescaling $K=0.05464$ has
been applied to the simulator time to match the target evolution.

##  VI Digital implementations

So far we have discussed an analog simulator with a computational basis given
by the states $\ket{g}$ and $\ket{r}$ for each atom. However, it is also
possible to calculate the time evolution corresponding to the Rydberg atom
Hamiltonians $H^{nR}$ using digital methods, for instance a universal quantum
computer with qubits. Before going farther, it should be mentioned that a
large $V_{0}$ requires small Trotter steps (of order $1/V_{0}$) in order to
suppress the contributions with multiple $\ket{r}$ states by fast
oscillations. This is not optimal with NISQ machines and it would be
profitable to find a native way to implement the blockade. However, we will
discuss the results to give an idea of the time scales involved.

As a simple example, we considered the two atom Hamiltonian $H^{2R}$ with the
values $\Omega=-1.5$, $\Delta=-0.5$ as in Fig. 4, but with a lower value
$V_{0}=10$. We used a Trotter step $\delta t=0.1=1/V_{0}$. We used Qiskit with
the classical simulator called with the instruction
`Aer.get_backend(’qasm_simulator’)` and which sample the so-called statevector
Anis (2021). This simulator does not include a realistic noise model
corresponding to current hardware such as, for instance, `ibmq_lima`. The
circuit is shown in Fig. 10 with Qiskit notations. We used 1,000 shots for
each of the times.

Figure 10: Qiskit circuit for a Trotter form of the evolution operator for
$H^{2R}$ described in the text .

In Fig. 10, $P$ is the phase gate

$P(\phi)\equiv\begin{pmatrix}1&0\\\ 0&\exp(i\phi)\end{pmatrix},$ (82)

and we have the usual definition

$RX(\lambda)\equiv\exp(-i\frac{\lambda}{2}X).$ (83)

The results of the Qiskit simulations are shown in Fig. 11. The Trotter errors
become quite significant when $t\gtrsim 2$. Based on recent runs on IBMQ
Gustafson  _et al._ (2021b), we would expect that computations on actual IBMQ
hardware with a slightly larger Trotter step would lead to reasonable results
for $t\lesssim 1$. The two spin system can be implemented with universal
quantum computers provided that an all-to-all connectivity is available.
Hardware considerations will be discussed in a separate paper.

Figure 11: $|\bra{m}U(t)\ket{m=1}|^{2}$, one site with exact Hamiltonian
$U=1,\ X=1.5$ (solid lines) and Rydberg Hamiltonian $H^{2R}$ with
$\Omega=-1.5$, $\Delta=-0.5$ and $V_{0}=10$, with the circuit of Fig. 10
calculated with the Qiskit simulator (empty symbols).

##  VII Conclusions

In summary, we have proposed a ladder-shaped CARA with two and three atoms for
a single spin-1 and four and six atoms for two coupled spins. In one spatial
dimension this is all we need to control in order to study larger systems. We
compared target and simulators using perturbative and numerical methods. The
two-atom setup provides an easily controllable simulator of the one-spin model
while the three-atom setup involves a nonlinear matching. However when two
spins are coupled, the six-atom simulator provides solutions closer to the
target than the four-atom simulator for small $\Omega$. Approximate
implementations of the two-spin model appear to be possible with near term
technology. Extensions to spin-2 and implementations in 2+1 dimensions are
under investigation.

Acknowledgments. This work is supported in part by the U.S. Department of
Energy (DoE) under Award Number DE-SC0019139. Special thanks to Alex Keesling
who enthusiastically supported the idea using the Rabi driven term to replace
the tunneling in the orginal formulation and made very interesting
suggestions. We also thank Shan-Wen Tsai, Jin Zhang, Johannes Zeiher, James
Corona, Erik Gustafson, Denis Candido, Michael Flatte, Craig Pryor, Nate
Gemelke, Tout Wang, Shangtao Wang and the members of QuLAT for suggestions and
comments.

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# Individual Packet Features are a Risk to Model Generalisation in ML-Based
Intrusion Detection

Kahraman Kostas , Mike Just , and Michael A. Lones K. Kostas, M. Just, and M.
A. Lones are with the Department of Computer Science, Heriot-Watt University,
Edinburgh EH14 4AS, UK, e-mail: kk97, m.just<EMAIL_ADDRESS>Kostas
supported by Republic of Turkey Ministry of National Education.
0000-0002-4696-1857 0000-0002-9669-5067 0000-0002-2745-9896

###### Abstract

Machine learning is increasingly used for intrusion detection in IoT networks.
This paper explores the effectiveness of using individual packet features
(IPF), which are attributes extracted from a single network packet, such as
timing, size, and source-destination information. Through literature review
and experiments, we identify the limitations of IPF, showing they can produce
misleadingly high detection rates. Our findings emphasize the need for
approaches that consider packet interactions for robust intrusion detection.
Additionally, we demonstrate that models based on IPF often fail to generalize
across datasets, compromising their reliability in diverse IoT environments.

###### Index Terms:

IoT security, Network Security, Intrusion detection, Machine learning, Attack
Detection.

##  I Introduction

The continually growing number of connected devices, coupled with the
heterogeneity of hardware and software designs across manufacturers, presents
a significant challenge for securing the Internet of Things (IoT) ecosystem
[1]. The inherent variety in these devices can lead to complex security
vulnerabilities, potentially compromising network security. For example,
weakly secured IoT devices can be exploited by malicious actors, turning them
into “botnet zombies” used in large-scale cyberattacks against critical
infrastructure [2].

The multifaceted nature of IoT security challenges has attracted considerable
scholarly attention. Notably, intrusion detection employing machine learning
(ML) techniques has emerged as a particularly active area. Although specific
ML methods have evolved over time, the fundamental reliance on data remains
constant, underscoring the indispensability of robust datasets in data-driven
approaches.

Current intrusion detection studies with ML primarily rely on three feature
types [3]: flow-based features (analyzing network traffic statistics) [4, 5],
window-based features (focusing on packet variations within a timeframe) [6,
3], and individual network packet features [7, 8].

The third approach is the simplest — it uses features extracted from
individual packets, and does not take into account interactions between
packets — yet despite this, previous studies have reported high detection
accuracy using this approach.

In this study, we critically examine the efficacy of IPF in ML-based intrusion
detection systems, revealing the approach to have significant limitations,
particularly in terms of the ability of models to generalise. We demonstrate,
through extensive experiments on various public datasets, that reliance on IPF
can lead to vulnerabilities and reduced robustness in intrusion detection. Our
findings underscore the importance of adopting more informative feature sets,
such as flow-based and window-based features, to enhance the security of IoT
ecosystems. Notably, this study is the first to provide empirical evidence of
the risks associated with IPF.

##  II Why IPF Does Not Tell the Whole Story

Detecting an attack using IPF alone is challenging. In this section, we’ll
elaborate on the reasons behind this difficulty. In the subsequent sections,
we support this notion by presenting examples from existing literature and our
own experimentals. For reproducibility, our scripts are publicly
available111Source code available at:github.com/kahramankostas/IPF.

This challenge can be explained with a simple example: SYN Flood [9], which is
a Denial-of-Service (DoS) attack based on the exploit of 3-way handshake in
the TCP protocol. During the 3-way handshake process, the client that wants to
establish a connection sends a SYN packet to the server. The server receiving
this packet sends a SYN-ACK packet to the client. Finally, the client responds
with the ACK packet, and the TCP connection is established. (see Figure 1).

Figure 1: Legitimate and malicious use of 3-way handshake

(a) The legitimage usage.

(b) SYN Flood attack.

During an attack, the server is sent a larger number of SYN packets for which
a SYN-ACK reply is sent, but the ACK reply is never received. Server resources
are impacted, and subsequent legitimate connection requests could be dropped.
The distinguishing individual packet features in this case are typically those
such as IP or MAC addresses (or other network-specific features) which are
identifying within one dataset, but which do not generalise to other datasets
or network environments.

This phenomenon extends beyond SYN Flood attacks; vari- ous attack scenarios
causing time, size, or protocol anomalies, such as DoS, distributed DoS (DDos)
and man-in-the-middle (MitM) attacks, display similar characteristics. To
efficiently detect and mitigate an attack, it is essential to adopt an
approach that considers the context and interaction of packets, rather than
solely relying on individual packet characteristics.

There are some exceptions, such as for single-packet attacks that exploit
malformed packets. These attacks typically involve sending malformed packets
that the device cannot handle, leading to malfunctions or crashes in the
receiving device [10]. However, addressing these attacks usually involves
packet filtering or firewalls, not ML behavioural analysis, since they can be
effectively managed using signature/rule-based approaches.

##  III IPF Usage in the Literature

The use of individual packets in attack detection is common. Our literature
review, conducted in [3], reveals that 10 out of 68 studies from 2019 to 2023
incorporate IPFs. Table I summarises key information from these studies, such
as the dataset, data types, and metrics, and Table II lists the attacks
detected.

This reveals numerous studies with reported high success rates in attack
detection using IPF. Given that we have posited that IPFs are an ineffective
basis for attack detection, it is important to understand this discrepancy.

TABLE I: Studies in the literature using IPF, their datasets and results. The
dataset specified as Private* is the same in both studies.

Study | Dataset | Accuracy | Recall | Precision | F1 Score
---|---|---|---|---|---
[11] |  Edge-IIoTset(CSV) | 100.00 | 100.00 | 100.00 | 100.00
[12] |  Edge-IIoTset(CSV) | 100.00 | 89.00 | 95.00 | 87.00
[13] |  MQTT-IoT-IDS2020(CSV) | 99.98 | 99.98 | 99.98 | 99.98
[14] |  MQTTset(CSV) | 91.00 | 91.00 | 77.00 | 80.00
[8] |  AB-TRAP (CSV) |  |  |  | 100.00
[15] |  AWID2 (RAW) | 99.96 | 99.99 | 99.95 | 99.97
[16] |  ISCXIDS2012 (RAW) | 99.42 | 99.41 | 99.34 | 99.37
|  CICIDS2017 (RAW) | 97.87 | 98.16 | 97.59 | 97.83
[17] | Private | 100.00 | 100.00 | 100.00 | 100.00
[7] | Private* |  | 98.00 | 99.00 | 99.00
[18] | Private* |  | 100.00 | 100.00 | 100.00

TABLE II: List of studies and the attacks they include Study | Attacks
---|---
[11] | DoS/DDoS, Scanning, MitM, Injection, Malware
[12] | DoS/DDoS, Scanning, MitM, Injection, Malware
[13] | Brute-Force, Scanning,
[14] | DoS/DDoS, MitM, Injection, Packet Manipulation
[8] | Scanning
[15] | DoS, Injection, Authentication , Wireless Attack
[16] | DoS/DDoS, Brute-Force, Infiltration, Scanning, Botnet, Web
[17] | Wrong setup, DDoS, Probing, Scanning, MitM
[7] | DoS, MitM, Scanning, IoT-toolkit
[18] | DoS

Most studies in Table I use open access datasets, and those used in [8, 11,
12, 13, 14] have a similar nature in that all contain raw data as well as
individual packet specifications in pre-extracted CSV format. All of the
studies used these pre-extracted versions. Another feature of these datasets
is that they contain a single file for each attack, with no information about
sessions. Therefore, in all these studies, it seems likely that the training
and test data were created from a single file.

This promotes identifying features, such as IP addresses, across training and
test sets, causing information leakage. Such leakage can inflate performance
metrics by granting the model access to information during testing that it
would not have during deployment. Consequently, these features serve as hidden
variables, offering a shortcut to class identification. The model may then
prioritize them over learning true underlying patterns, as illustrated in
Figure 2. For example, in [11, 13, 14], although the reporting of features
used in these studies is unclear, there is no evidence that these identifying
features have been removed or censored. In this respect, it is quite possible
that identifying features were used in their models.

Nevertheless, the inclusion of source and destination-based identifiers is
generally considered a significant flaw, and most studies tend to remove such
features from their data. For instance, in [12], IP addresses were removed.
However, this study suffers from a different type of information leak, with
the dataset containing features such as ports, IDs, synchronisation, and
acknowledgement numbers. Despite these features being extracted from
individual packets, they provide non-generalizable insights into the
interrelation of packets. For instance, when a flow is established between two
ports, all individual packets within that flow bear the corresponding port
numbers (see Figure 2 as an example, focusing on port and ID numbers). In a
train/test split that does not consider the flows, even with more robust
evaluation methods like cross-validation (CV), it is highly likely that
packets representing the same flow will occur in both the training data and
the test data. Moreover, the fact that a single file with no distinct sessions
is used to represent each attack makes it very challenging to split in a
manner that prevents information leakage. So, in this study, it can be seen
that the features that stand out for many attacks are sequence and
acknowledgement numbers, which are session-based identifiers, and it is quite
possible that they are subject to information leakage.

In none of these studies [8, 11, 12, 13, 14] did we see any indication that
flows were taken into account in the training and test split, or that these
session-based identifiers were removed (Although session-based features are
mostly removed in Bertoli et al.’s work [8], the low complexity in the
structure of the dataset still leads to generalisation problems. this will be
examined in Section IV-B) . Therefore information leakage between the training
and test data is distinctly possible. While the results published in these
studies may appear promising, models relying on identifying features due to
information leakage will not generalise to other datasets, and consequently
the results are likely to be misleading.

Two studies [15, 16] used raw pcap files rather than pre-extracted features.
However, the use of raw data can be quite dangerous because these files also
contain identifiers such as IP and MAC addresses. In one of these studies
[15], there is no indication that these identifying features were
deleted/discarded, so it is quite likely that they were used. In the other
study[16], identifying features such as IP addresses and frame number were
removed. Despite this precaution, we suspect that some identifying information
might have inadvertently leaked. For example, in their study, the internet
checksum has been used, which is an identifying feature as it stores a hashed
version of the source and destination IP addresses. In addition, both studies
had no mechanism to prevent sessions from being split across train and test
sets, providing a potential route for information leakage.

The other three studies [17, 7, 18] listed in Table I did not use publicly
available data. Many of the features used in [17] are identifying features
(such as frame number/time, eth source/destination, source/destination IP).

In two studies [7, 18], identifying features such as IP, MAC, and frame number
were removed, but session-based segregation was not applied, likely causing
session characteristics to leak between training and test sets. Notably, [7]
claims the testing phase was isolated from training data, although the process
for achieving this is not clear.

Figure 2: An 80/20 split for a hypothetical dataset containing 40 samples (4
session, 40 network packets), showcasing the division into 80% training data
and 20% testing data for model development and evaluation, respectively. This
example illustrates how information leakage takes place. In the scenario
involving the source IP attribute, information leakage occurs because this
address uniquely identifies both the attacker and the benign source, remaining
constant across both the training and test sets. If we examine other features
such as source port numbers and ID numbers, they vary across different
sessions but exhibit correlations within each session (port numbers remain
constant, ID numbers increment sequentially). Consequently, if packets from
the same session are found in both the training and test sets, it can result
in information leakage. Benign: Malicious:

##  IV Experimental Case Studies

In the previous section, we highlighted the role that identifying features may
play in giving an incorrect impression that IPF-based approaches are
effective. Next, we strengthen this claim using experimental case studies. We
also consider another potentially misleading factor, low data complexity.

###  IV-A Identifying Features

Figure 3: Comparison of CV and isolated data performance of features on some
attacks in the IoT-NID dataset with DT.

(a) IoT-NID dataset ARP Spoofing attack. (b) IoT-NID dataset Telnet Bruteforce
attack.

(c) IoT-NID dataset HTTP flooding attack. (d) IoT-NID dataset UDP Flooding
attack.

In SectionII and III we argued that IPF-based attack detection yielded
unrealistic results based on information leakage. To substantiate our claim,
we conducted experiments using the IoT-NID [19] dataset, which offers multiple
sessions for each attack. This dataset records each attack execution as a
separate session, providing an opportunity to explore the identifiability of
session-based features. We utilised examples from this dataset to demonstrate
the effectiveness of session-based identifiers. Our approach involved merging
the first and second sessions of the attacks to create a new dataset. We then
performed a 10-times 10-fold CV on this merged dataset, employing one of the
session-based identifiers at each iteration to distinguish between benign and
attack samples. In the subsequent step, we executed the separation by
utilising the first session as the training set and the second session as the
testing set. Figure 3d illustrates the distribution of these features for four
attacks.

The results of this analysis reveal a notable pattern: in the majority of
cases, the identifying features demonstrate better success in the CV scenario,
while in isolation, they exhibit limited effectiveness. Figure 3d visually
depicts this trend. In this context, these results support the hypothesis that
these features lead to information leakage.

However, upon a more comprehensive examination of the figures, we observed
that certain features consistently display exceptional performance in both
scenarios. Dport feature in Telnet Brute-force (Figure 3b) and HTTP Flooding
(Figure 3d) attacks are prime examples of this consistent success across the
two scenarios. If we examine the models associated with the features that
achieve this unexpected success, as shown in Figure 4, we see that these
models have an unusually simple structure. This brings us to the second
explanation for achieving high performance with individual packet features:
low data complexity.

Figure 4: Visualization of high-achieving decision tree models. (a) HTTP Flood
model using dport feature, (b) Brute-Force model using dport feature.

(a)

(b)

###  IV-B Low Data Complexity

Most attacks follow a pattern and produce uniform outputs. For example, the
distribution of the size of the attack packets in the HTTP flood attack on
four different datasets [20, 21, 12, 19] is shown in Figure 5. Because of the
uniform nature of the attacks, if the complexity level of the dataset is low
in studies using individual packets, even very basic characteristics, such as
size or time, can become identifying features.

Figure 5: The distribution of packet size malicious data (HTTP Flood) in four
different datasets [20, 21, 12, 19].

Figure 6f presents compelling empirical evidence to support this assertion.
Specifically, in Figures 6a-LABEL:sub@fig:comp_model4, we depict the UDP
attacks observed in four distinct sessions. A detailed analysis of these
figures reveals notable similarities between sessions 1 (6a) and 3 (6c), as
well as between sessions 2 (6b) and 4 (6d). Moreover, it is noteworthy that
the size of attack packets remains constant even across these different
sessions.

When incorporating only packet size as a feature in this example, we achieve
perfect detection performance (see Table I). Even when testing the model
trained with the first session on the third session, where the model has not
encountered this data previously, we observe a notably high success rate. The
same holds true for sessions 2 and 4, indicating strong consistency in their
detection capabilities. However, this superior success is only a consequence
of the information leakage problem. The packet size in a UDP attack may change
in another attack, and this model will fail to catch these attacks, or it will
detect benign packets of this specific size as attacks.

Figure 6e provides another insightful example, showcasing the change of labels
over time in the Kitsune dataset [6], Mirai attack. In this graph, packets
before a specific time exhibit all benign characteristics, while those
occurring after the designated time are identified as attacks.

Figure 6: Distribution of labels in Kitsune and IoT-NID. (a), (b), (c), and
(d) show the size histograms of sessions 1, 2, 3, and 4 for UDP attack in the
IoT-NID dataset, respectively. (e) and (f) depict the distribution of labels
over time for Mirai and Video Injection attacks, respectively, in the Kitsune.

(a)  (b)

(c)  (d)

(e)

(f)

Additionally, we present yet another example, Figure 6f, in which all packets
up to a certain time demonstrate benign behaviour, while thereafter, a mixture
of attack and benign packets is evident. The graph illustrates a dense
concentration of data over a brief temporal window. However, focusing on a
smaller portion of this mixture (the last 2000 samples) reveals a rhythmic
pattern between attack and benign traffic over time. The models trained using
the time feature for these two attacks perform well on these datasets, but are
not capable of detecting the same attack on another dataset or in real-life.

Figure 7: Complexity analysis of UDP, Mirai, and Video injection attacks
according to time and size characteristics. (a) and (b) Mirai, (c) and (d)
UDP, (e) and (f) Video injection time and size complexity, respectively. The
central number depicted in the figures represents the comprehensive complexity
score, which is derived as the average of 22 distinct analytical methods. (The
complexity score ranges from 0 to 100, where higher values indicate increasing
complexity). These 22 methods stem from six different approaches to assessing
complexity, each distinctly colour-coded for clarity of categorisation [22]:
red for feature-based, orange for linearity-based, yellow for neighbourhood-
based, light green for network-based, dark green for dimensionality-based, and
blue for class imbalance-based.

(a)  (b)

(c)

(d)  (e)

(f)

To get a better understanding of the complexity of the data, we calculated the
complexity score [22] of the attack files. Figure 7 shows the computation of
the complexity scores of 3 different data using time and size characteristics.
When these figures are analysed, it can be seen that the UDP flood attack has
low size complexity and high time complexity. On the other hand, Mirai and
video injection attacks show low size complexity and high time complexity.

TABLE III: Near-perfect classification of 2 datasets afflicted by low data
complexity. In Kitsune Mirai and Video injection data, the labels follow a
certain time pattern. In the IoT-NID UDP data, attack packets have a specific
size. In the Cross-Validation part of the table, the results depict the mean
and standard deviation based on 10 repetitions of 10-fold cross-validation
(using DT). In the isolated part, the mean and standard deviation are
presented from 100 repetitions. Temporal information is not used in order to
keep the features at the simplest level. The “time” feature here refers to a
timestamp, not a temporal feature.

| Dataset | Subdataset | Feature | ML | Accuracy | Precsion | Recall | F1 Score | Kappa
---|---|---|---|---|---|---|---|---|---
Cross-validated | Kitsune | Video Inj | Size | DT | 0.959±0.000 | 0.479±0.000 | 0.500±0.000 | 0.489±0.000 | 0.000±0.000
Kitsune | Mirai | Size | DT | 0.899±0.001 | 0.804±0.002 | 0.918±0.001 | 0.842±0.002 | 0.688±0.003
IoT-NID | UDP-S2 | Size | DT | 1.000±0.000 | 1.000±0.000 | 1.000±0.000 | 1.000±0.000 | 0.999±0.001
Kitsune | Video Inj | Time | DT | 0.988±0.000 | 0.933±0.002 | 0.919±0.002 | 0.926±0.001 | 0.852±0.003
Kitsune | Mirai | Time | DT | 0.997±0.000 | 0.998±0.000 | 0.990±0.001 | 0.994±0.000 | 0.988±0.001
IoT-NID | UDP-S2 | Time | DT | 0.841±0.007 | 0.833±0.008 | 0.839±0.008 | 0.835±0.008 | 0.671±0.016
Isolated | IoT-NID | UDP-S2vsS3 | Size | DT | 0.032±0.000 | 0.016±0.000 | 0.500±0.000 | 0.031±0.000 | 0.000±0.000
IoT-NID | UDP-S2vsS4 | Size | DT | 1.000±0.000 | 1.000±0.000 | 1.000±0.000 | 1.000±0.000 | 0.999±0.000
IoT-NID | UDP-S2vsS3 | Time | DT | 0.526±0.000 | 0.508±0.000 | 0.566±0.000 | 0.379±0.000 | 0.017±0.000
IoT-NID | UDP-S2vsS4 | Time | DT | 0.866±0.000 | 0.858±0.000 | 0.865±0.000 | 0.861±0.000 | 0.722±0.000

Table III shows the results of the attack detection process using only one
feature for each of the three attacks with shared complexity values. Upon
examination of the CV (10-times, 10-fold) results in Table III, we observe
that the UDP attack, characterised by very low complexity in terms of size,
achieves flawless detection. Similarly, in the Mirai attack, very low time
complexity leads to nearly perfect detection rates. Conversely, the video
injection attack, which possesses a relatively higher size complexity but
lower time complexity, did not achieve substantial success in terms of size-
based detection but demonstrated significant success when considering time-
based detection.

In our second experiment, capitalising on the multi-session nature of the UDP
attack, We conducted tests involving the second session, pairing it with the
fourth session, which has very similar characteristics. Additionally, we
tested the second session with the third sessions, which are notably different
from it. The similarities and differences between these sessions can be better
understood by examining their distributions in Fig 6f. The results, shown
under “isolated" in Table III, indicate exceptional performance in the fourth
sessions that were similar, while not achieving notable results in the third
sessions exhibited substantial dissimilarities from the second session.

What all of these examples have in common is that high levels of success can
be achieved by using individual features that are not suitable for attack
detection. Although identifying features are not used here, the uniformity of
attack data enables even basic individual characteristics to serve as
identifying features.

To give more insight into the relationship between complexity and overfitting,
we have expanded this sample a little more. Figure 8 shows the results of
another experiment on complexity. In this experiment, a basic feature (packet
size) was used to separate all attack types in the IoT-NID dataset. From this
graph, we can observe the inverse correlation between complexity and success.
Especially, if the complexity is below the critical level (45), highly
successful discrimination is achieved. On the other hand, as complexity
increases, success seems to decrease.

Figure 8: Reflection of the change in the complexity value due to a single
feature on the model success (with DT).

##  V Conclusion

In summary, this study highlights the limitations of relying solely on
individual packet features (IPF) for intrusion detection within IoT
environments. While conducted within the IoT domain to emphasise security
considerations in IoT devices, the implications extend beyond this realm. Our
findings stress the necessity of holistic approaches considering contextual
features and packet interactions for effective intrusion detection in various
network security contexts.

Through literature review and experimental analysis, we’ve shown that although
IPF may show high detection rates, they suffer from inherent flaws like
information leakage and low data complexity. This underscores the importance
of prioritising robust ML-based intrusion detection systems that address the
multifaceted nature of security challenges in networked environments.

Moving forward, researchers and practitioners should prioritise comprehensive
approaches incorporating contextual features and packet interactions to
enhance IoT ecosystem resilience and networked environment security against
evolving cyber-threats. By addressing these limitations and embracing
sophisticated detection methodologies, we can strengthen defences and mitigate
risks posed by malicious actors.

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