# Quantum fidelity kernel with a trapped-ion simulation platform

Rodrigo Martínez-Peña<EMAIL_ADDRESS>Instituto de Física
Interdisciplinar y Sistemas Complejos (IFISC, UIB-CSIC), Campus Universitat de
les Illes Balears E-07122, Palma de Mallorca, Spain  Donostia International
Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
Miguel C. Soriano  Instituto de Física Interdisciplinar y Sistemas Complejos
(IFISC, UIB-CSIC), Campus Universitat de les Illes Balears E-07122, Palma de
Mallorca, Spain  Roberta Zambrini  Instituto de Física Interdisciplinar y
Sistemas Complejos (IFISC, UIB-CSIC), Campus Universitat de les Illes Balears
E-07122, Palma de Mallorca, Spain

###### Abstract

Quantum kernel methods leverage a kernel function computed by embedding input
information into the Hilbert space of a quantum system. However, large Hilbert
spaces can hinder generalization capability, and the scalability of quantum
kernels becomes an issue. To overcome these challenges, various strategies
under the concept of inductive bias have been proposed. Bandwidth optimization
is a promising approach that can be implemented using quantum simulation
platforms. We propose trapped-ion simulation platforms as a means to compute
quantum kernels and demonstrate their effectiveness for binary classification
tasks. We compare the performance of the proposed method with an optimized
classical kernel and evaluate the robustness of the quantum kernel against
noise. The results show that ion trap platforms are well-suited for quantum
kernel computation and can achieve high accuracy with only a few qubits.

Suggested keywords

##  I Introduction

The availability of noisy intermediate-scale quantum (NISQ) devices [1] has
catalyzed a surge in several research areas such as machine learning, many-
body physics, chemistry, and combinatorial optimization [2]. In particular,
quantum machine learning (QML) is a promising route toward quantum advantage,
where some theoretical studies [3, 4, 5, 6] and experimental evidence [7, 8,
9] have already indicated positive progress in this direction. While circuit-
based NISQ computing implementations are developed worldwide [2], an
alternative approach exploits NISQ analog or hybrid designs, with examples
ranging from quantum annealing [10], boson sampling [11], and QML itself [12,
13]. In this work we will focus on the QML implementation of a kernel method
in an analog simulation platform.

Kernel methods refer to classification or regression algorithms that leverage
a kernel function, denoted as $K(\bm{x},\bm{y})$, with
$\bm{x},\bm{y}\in\mathcal{X}$, being $\mathcal{X}$ the input space. The kernel
function transforms data points residing in the input space into a higher-
dimensional space, called feature space. This transformation usually
facilitates the distinction between different classes of data when they are
not linearly separable. A quantum kernel method is generally understood as the
case where a kernel function is computed by embedding the input information
into the Hilbert space of a quantum system (the feature space), offloading the
optimization process for the ML algorithm to a classical computer [14]. The
most common approach to computing quantum kernels is based on the quantum
state fidelity, i.e., the inner product of input-dependent quantum states.
Quantum advantage has been claimed as possible in realistic problems if the
embedding of inputs into these quantum states is classically intractable [14,
15].

However, finding a kernel function that could provide a quantum advantage
remains an open problem. One plausible strategy that could come to our minds
is to propose a very large quantum Hilbert space as the feature space since,
in principle, it would be much harder to simulate classically. But very large
Hilbert spaces can hinder the generalization capability of quantum models
[16], performing similarly or even worse than classical ML models [4]. Indeed,
training a kernel-based model ensures the discovery of the optimal model
parameters as the training landscape is convex. However, this assurance is
based on the assumption that the quantum kernel can be efficiently obtained
from quantum hardware. With an increasing number of qubits, the number of
measurements required to evaluate the kernels to sufficiently high precision
might scale exponentially, hindering trainability [16, 17, 18].

Several alternatives have been proposed to avoid this exponential scaling, all
of them under the notion of what is called inductive bias [16, 8]. An
inductive bias is a restriction over the set of functions a given model can
reproduce. They can be implemented in very diverse ways, such as projected
quantum kernels [4], quantum Fisher kernels [18], exploiting structured-data
encoding [19], and bandwidth optimization [20, 21]. While the former inductive
biases can be problem-dependent, bandwidth optimization is a general strategy
in which we simply tune the hyperparameters of our model. In particular, for
classical data the decrease in performance with respect to the number of
qubits can be compensated by the tuning of hyperparameters [20, 21], although
it might be possible to find classical kernels that perform as well as them in
some cases [22].

A promising choice for bandwidth optimization is quantum simulation platforms.
Quantum simulations are believed to be a classically hard problem together
with instantaneous quantum polynomial (IQP) random circuits or boson sampling
[23]. Some analog simulation platforms have already been experimentally
exploited for quantum kernels, such as Rydberg atoms [19] and nuclear magnetic
resonance (NMR) platforms [24], while more platforms have been numerically
studied, such as driven-dissipative spin chains [2], a single Kerr nonlinear
mode [25] and quantum annealers [26]. In this line of research, we propose to
compute quantum kernels through trapped-ion simulation platforms. Trapped and
laser-cooled atomic ions offer an excellent benchmark for simulating quantum
spin models with interactions [27, 28, 29]. Optical fields can control the
ions’ Coulomb interaction, enabling customizable, long-range spin-spin
interactions [30]. Ion trap quantum computers with logical quantum gates have
already been used for classification tasks [31, 32], even with quantum kernel
methods [33, 34].

In this work, we will introduce an inductive bias (i.e. a change in
expressivity) in a transverse-field Ising model by applying bandwidth
optimization, i.e., by tuning some of the model’s hyperparameters. We show
that this approach is suitable for solving binary classification tasks,
comparing the performance with an optimized classical kernel. Our goal is to
show via numerical simulations that an already available experimental platform
can be harnessed for quantum kernel methods, demonstrating its robustness
against noise. We introduce noise into the system in two different ways:
decoherence with a depolarizing channel after the dynamics, and statistical
noise with white noise at the entries of the kernel. For this, we require to
use an error mitigation technique to keep the kernel as a positive semi-
definite matrix, applying the shift method (see Sect. II.2).

Our results show that ion trap platforms are perfectly suited for the
computation of quantum kernels, being robust against different sources of
noise. We evaluate the classification tasks for an increasing Hilbert space
size, showing that only a few qubits are needed to obtain the best possible
accuracy.

##  II Methods

###  II.1 Kernel methods and support vector machines

In machine learning, kernel methods are algorithms used for classification or
regression. They utilize a kernel function, denoted as $K(\bm{x},\bm{y})$,
where $\bm{x},\bm{y}\in\mathcal{X}$ and $\mathcal{X}$ represents the input
space, that transforms data points $\mathcal{X}$ to a higher-dimensional
space, i.e. the feature space, usually facilitating the distinction between
different classes of data. A function
$K:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ is defined as a kernel
function if it fulfills two conditions: 1) it is symmetric, such that
$K(\bm{x},\bm{y})=K(\bm{y},\bm{x})$; and 2) it is a positive-semidefinite
function. The latter is known as the Mercer condition [35], which can be
expressed as:

$\sum^{M}_{i=1}\sum^{M}_{j=1}K(\bm{x}_{i},\bm{x}_{j})c_{i}c_{j}\geq 0,$ (1)

where $M$ is the number of training samples and $\\{c_{i}\\}$ is the set of
all possible real coefficients. A kernel function evaluated over pairs of data
points defines a symmetric and positive semidefinite matrix whose elements are
given by $K_{ij}:=K(\bm{x}_{i},\bm{x}_{j})$.

Here we will consider Support Vector Machines (SVM) as our kernel method [36].
Our starting point will be to introduce the simplest case: linear binary
classification. This problem consists on drawing a straight line that
separates two classes by assigning two different labels to them, like $y=\pm
1$, on opposite sites of the line. The decision function, i.e. the function
that assigns a label for new data, can be determined as:

$D(\bm{x})=\bm{w}^{\top}\cdot\bm{x}+b,$ (2)

where $\bm{w}\in\mathbb{R}^{m}$ is a vector with the same dimension as the
input and $b$ is an offset. The labeling is introduced by using the sign
function:

$y(\bm{x})=\text{sgn}(\bm{w}^{\top}\cdot\bm{x}+b).$ (3)

Let us assume we have $M$ training inputs $\bm{x}_{i}$ belonging either to
Class 1 with $y_{i}=1$ or Class 2 with $y_{i}=-1$. Since we assume the
training data are linearly separable, the following inequality is fulfilled:

$y_{i}D(\bm{x})=y_{i}(\bm{w}^{\top}\cdot\bm{x}_{i}+b)\geq 1,\quad
i=1,\dots,M.$ (4)

That is, the decision function yields $D(\bm{x}_{i})\geq 1$ for $y_{i}=1$ and
$D(\bm{x}_{i})\leq-1$ for $y_{i}=-1$, separating both classes by the
hyperplanes $D(\bm{x})=1$ and $D(\bm{x})=-1$. We can define a separating
hyperplane between these two: $\bm{w}^{\top}\cdot\bm{x}+b=c$, for $-1<c<1$.
The distance between the separating hyperplane and the training data sample
nearest to the hyperplane is called the margin. The generalization ability of
this model depends on the location of the separating hyperplane, and the
hyperplane with the maximum margin is denoted as the optimal one.

In fact, $\bm{w}$ represents the orthogonal vector of the optimal separating
hyperplane. If input data is bi-dimensional $\bm{w}$ defines a separation
line, and in higher dimensions, it defines a separating hyperplane. However, a
linear classifier will not suffice if the classification problem is nonlinear.
Here comes the concept of the feature map $\phi(\bm{x})$, which embeds data
points into a higher dimensional space (feature space) such that the modified
inputs may become linearly separable. The new decision function becomes

$D(\bm{x})=\bm{w^{\prime}}^{\top}\cdot\phi(\bm{x})+b,$ (5)

where $\bm{w^{\prime}}$ is now the vector of the optimal separating hyperplane
in the new feature space. The representer theorem [37] states that the vector
$\bm{w^{\prime}}$ can be written as a sum over a subset $S$ of the training
data points with real coefficients $\alpha_{i}$:

$\bm{w^{\prime}}=\sum_{i\in S}\alpha_{i}y_{i}\phi(\bm{x}_{i}).$ (6)

In the case of SVM, the subset $S$ corresponds to the support vectors, that
is, only the closest points to the decision boundary contribute. We write
again the decision function with this new information,

$D(\bm{x})=\sum_{i\in
S}\alpha_{i}y_{i}\phi(\bm{x}_{i})^{\top}\cdot\phi(\bm{x})+b,$ (7)

finally arriving at the introduction of the kernel, defined as an inner
product between feature vectors:

$K(\bm{x},\bm{x}^{\prime})=\phi(\bm{x})^{\top}\cdot\phi(\bm{x}^{\prime}).$ (8)

Under this definition, the kernel fulfills all the requirements exposed above.
The kernel trick consists then in substituting the product of feature vectors
by the kernel function, exploiting the fact that it might be easier to compute
the inner product than the vectors themselves. This is especially relevant
when the embedding happens in large feature spaces, as it usually happens with
quantum kernels. We will use this fact later to define our quantum kernel.

The parameters $\alpha_{i}$ can be efficiently computed by solving the
Lagrangian optimization problem for the soft margin SVM in its dual form:

$\arg\max_{\alpha_{i}}\left(\sum^{M}_{i=1}\alpha_{i}-\frac{1}{2}\sum^{M}_{i=1}\sum^{M}_{j=1}\alpha_{i}\alpha_{j}y_{i}y_{j}K(\bm{x}_{i},\bm{x}_{j})\right),$
(9)

subject to $\sum^{M}_{i=1}\alpha_{i}y_{i}=0$ with $0\leq\alpha_{i}\leq C$ for
$i=1,\dots,M$. The penalty term $C>0$ determines the balance between
minimizing the training error and maximizing the margin. When $C$ is small, it
results in a large margin but potentially high training error.

For this paper, we will make use of the soft margin SVM method from the
library Scikit-Learn [38]. In order to have a competitive benchmark to compare
with, we take the classical kernel known as the Radial Basis Function (RBF).
This kernel is defined as:

$K(\bm{x},\bm{y})=e^{-\gamma||\bm{x}-\bm{y}||^{2}},$ (10)

where $\gamma$ controls the precision of the kernel.

###  II.2 Quantum fidelity kernel

Once the SVM framework is introduced, we define the quantum kernel that we
will consider in this work, based on an analog implementation in an ion trap
platform. We start by specifying the dynamical system that will process the
input information. The Hamiltonian of our system is a Transverse-field Ising
model:

$H=\sum^{N}_{i>j=1}\frac{J}{|i-j|^{\alpha}}\sigma_{i}^{x}\sigma_{j}^{x}+\sum^{N}_{i=1}h_{i}\sigma_{i}^{z},$
(11)

where $N$ is the number of spins, $h_{i}$ is the onsite magnetic field for
each spin, $\sigma^{a}_{i}$ ($a=x,y,z$) are the Pauli matrices and $\alpha$ is
a hyperparameter that controls the decay of the spin-spin couplings, with
maximum strength $J$ for the nearest-neighbors in the chain. This model can be
generated using a combination of an effective magnetic field and Ising
interactions in ion trap platforms, with appropriate settings of the spin
phases [30]. The coupling decay is approximated by a power-law, where $\alpha$
can be tuned between $0$ and $3$ by varying parameters of the trap [39, 30].
Here we will fix $\alpha=1.13$ and $J=1$, following the values adopted by
experimentalists to obtain a medium-length interaction range [27]. Our
proposal is to use this ion trap to implement a kernel, by encoding the inputs
in the local magnetic fields. In particular, we introduce the input into the
external magnetic field of some or all the spins as $h_{i}=hx_{i}$, depending
on the dimension of the input vector. In general:

$H(\bm{x})=\sum^{N}_{i>j=1}\frac{J}{|i-j|^{\alpha}}\sigma_{i}^{x}\sigma_{j}^{x}+h\sum^{N}_{i=1}x_{i}\sigma_{i}^{z},$
(12)

where $h$ is the strength of the magnetic field for all spins and
$\bm{x}=\\{x_{1},x_{2},...\\}$ is the input vector for each input instance
with values scaled in the interval $x_{i}\in[-1,1]$. The chain length $N$ is
set to be equal or larger than the number of input features. For less features
than the chain length we will set the magnetic field in the remaining spins to
zero: for instance, for an input of only two features (and $N\geq 2$),
$\bm{x}=\\{x_{1},x_{2},0,...,0\\}$. If $N$ is a multiple of the number of
input features, one could redundantly encode the inputs in more than one spin.
In this input protocol, the number of qubits must scale linearly with respect
to the number of input features.

We will now move on to the definition of the quantum kernel. The system is
initialized with all spins in the state $\ket{0}$. Then, we apply the unitary
operator of the dynamics:

$\ket{\psi(\bm{x})}=e^{-iH(\bm{x})\Delta t}\ket{0},$ (13)

where $\Delta t$ indicates the time of the simulation. This operation of
encoding the input vector into a new Hilbert space is the feature mapping. To
construct the kernel, we need a dot product between two vectors in the new
feature space. This is expressed as

$K(\bm{x},\bm{y})=|\braket{\psi(\bm{y})}{\psi(\bm{x})}|^{2}=|\braket{0}{e^{iH(\bm{y})\Delta
t}e^{-iH(\bm{x})\Delta t}}{0}|^{2},$ (14)

where we introduced the squared absolute value for experimental purposes. In
order to compute the kernel matrix element for inputs $\bm{x}$ and $\bm{y}$,
we need to measure the overlap between states $\ket{\psi(\bm{x})}$ and
$\ket{\psi(\bm{y})}$. There are several ways of tackling this problem. The
most popular one with quantum computers is the quantum kernel estimation
method [15]. This method would require applying the following chain of
operations in our system: $1)$ initialize the system at $\ket{0}$ and apply
the unitary dynamics $e^{-iH(\bm{x})\Delta t}$; $2)$ a sudden quench has to be
done to apply the unitary dynamics $e^{iH(\bm{y})\Delta
t}=e^{-i(-H(\bm{y}))\Delta t}$; $3)$ measure the state of all qubits in the
$z$ direction; $4)$ repeat the whole process to estimate the frequency of
obtaining the state $\ket{0}$ in all the qubits after measuring.

Let us now describe the different sources of noise in the system. We model the
experimental noise produced by decoherence with a depolarizing channel after
the unitary dynamics [40, 41]:

$\tilde{\rho}(\bm{x})=(1-p)\rho(\bm{x})+p\frac{I}{2^{N}},$ (15)

where $p$ is the probability of replacing the quantum state by the maximally
mixed state $I/2^{N}$ and $\rho(\bm{x})=\ket{\psi(\bm{x})}\bra{\psi(\bm{x})}$.
We assume that $p$ is the same for all inputs $\bm{x}$ to simplify the
setting, but this is not necessarily true in general. The quantum kernel is
given now by the Hilbert-Schmidt product of the feature vectors in the space
of density matrices:

$\tilde{K}(\bm{x},\bm{y})=\text{tr}(\tilde{\rho}(\bm{x})\tilde{\rho}(\bm{y}))=(1-p)^{2}K(\bm{x},\bm{y})+p(2-p)/2^{N}.$
(16)

Besides, we can add a Gaussian random number to the kernel matrix elements in
order to simulate statistical noise due to finite sampling (justified by the
central limit theorem):

$\tilde{K}^{\prime}(\bm{x},\bm{y})=\tilde{K}(\bm{x},\bm{y})+\xi(\bm{x},\bm{y}),$
(17)

where $\xi(\bm{x},\bm{y})$ is generated from a normal distribution of zero
mean and standard deviation $s$ for each pair of inputs $(\bm{x},\bm{y})$.
This can be related to the number of measurements $V$ through the relation
$s\sim 1/\sqrt{V}$. Since the kernel matrix has to be symmetric, we set
$\xi(\bm{x},\bm{y})=\xi(\bm{y},\bm{x})$. In this work, we consider
$p=0.01,0.1$ and $s=0.01,0.1$.

In order to use a kernel matrix for classification tasks, it must be a
positive-semidefinite matrix. The two sources of noise we previously
introduced might hinder this feature, so some mitigation technique should be
applied. We use the Tikhonov regularization [40], also known as the shifting
technique in this context [41]. It consists in shifting all the eigenvalues by
the minimum non-positive eigenvalue of $\tilde{K}^{\prime}$. This is
equivalent to subtracting it from the diagonal:

$\overline{K}=\left\\{\begin{array}[]{ll}\tilde{K}^{\prime}-\lambda_{\text{min}}I&\text{if
}\lambda_{\text{min}}<0\\\ \tilde{K}^{\prime}&\text{else}\end{array}\right.,$
(18)

where $\overline{K}$ is now a symmetric positive-semidefinite matrix.

###  II.3 Optimization

As was mentioned in the Introduction, we will introduce an inductive bias in
our models by applying bandwidth optimization. To this end, we perform a grid
search that will help to visualize the performance for all explored values of
hyperparameters. This can be numerically expensive for many hyperparameters
(and hard to visualize), so we limit ourselves to grid searches that vary two
hyperparameters simultaneously. For the classical case, we vary the scaling
factor of the RBF kernel $\gamma$ in Eq. (10) and the regularization parameter
$C$ (introduced after Eq. (9)). For the quantum case, we will vary $\Delta t$
and $h$ while fixing $C=1$ (recall we already fixed $J=1$ and $\alpha=1.13$).
We will also have a look at the performance as the number of spins increases
for $N=2,4,6$.

To measure the performance of our kernel methods, we use the classification
accuracy. It is defined as the number of correct predictions (CP) divided by
the total number of predictions (TP):

$A=\frac{\text{CP}}{\text{TP}}.$ (19)

This metric is appropriate since our dataset is well-balanced, with half of
each set from each classification class.

In order to find the optimal hyperparameters, we adopted a training,
validation, and testing strategy. Training is used to find the optimal
parameters of the SVM model for the training set (for a given value of the
hyperparameters). The validation set allows us to compute the performance of
the model (with the same hyperparameters) for new data that the trained model
has not seen. Based on the validation performance, we make the grid search in
order to find the optimal hyperparameters. Finally, at the test, we evaluate
the final performance of the model, with trained parameters and optimized
hyperparameters, for a new set of inputs.

Three binary classification tasks are evaluated. The Circles (C) and Moons (M)
tasks were generated from the library ScikitLearn [38]. These two tasks are
paradigmatic examples of benchmark classification tasks because they are easy
to generate, while not trivially solvable, and are very popular in the quantum
kernel literature [14, 42, 43, 44, 45, 26], even with experiments [46, 24].
The third task, called Ad Hoc (AH), was extracted from the Qiskit dataset
library in Python [47]. This dataset was originally designed to be exactly
separable using quantum kernels as proposed in [15]. We evaluate here the
capability of the classical RBF and quantum trapped ion kernels to solve these
three tasks.

For the Circles and Moons tasks, Gaussian noise is introduced during the
generation of the datasets (with standard deviation equal to 0.2 and 0.3
respectively), in such a way that points belonging to one class can enter into
the area that corresponds to the other class (See Figs. 1 (a) and (b)). For
the Ad Hoc task (Fig. 1 (c)), we set the separation gap as $\Delta=0.3$ (see
[15] for more details). Note that the input data noise is the only noise
source we considered for the classical kernel.

The datasets are equally divided into three sets (train, validation, and test)
with 333 points per set. We remark that given this number of data points and
the resolution of the grid search (we used 100 points per hyperparameter), we
may find more than one equal-valued minima in the grid search. We need to
select one of them to compute the test accuracy. Our choice is to take the
middle element of the ordered list of minima, having checked that the other
points work equally well.

##  III Numerical results

###  III.1 Classical kernel

We begin by presenting the results of the classical RBF kernel. Table 1 shows
the values of the optimal hyperparameters for the three different tasks,
together with the validation accuracy $A_{\text{val}}$ and test accuracy
$A_{\text{test}}$. We observe that the values of validation accuracy are
larger than those of test accuracy, as expected. The fact that we do not
surpass the value $0.9$ of test accuracy in any of the tasks is due to the
presence of noise during the generation of the datasets, as explained in the
previous section.

Task | $\gamma$ | $C$ | $A_{\text{val}}$ | $A_{\text{test}}$
---|---|---|---|---
C | $\numprint{2.783e-3}$ | $\numprint{2.984e7}$ | $\numprint{0.910}$ | $\numprint{0.889}$
M | $\numprint{4.977e-2}$ | $\numprint{8.111e6}$ | $\numprint{0.937}$ | $\numprint{0.892}$
AH | $\numprint{2.984e0}$ | $\numprint{2.105e4}$ | $\numprint{0.853}$ | $\numprint{0.841}$
Table 1: Optimal hyperparameters for the classical RBF kernel. The tasks are
represented as C (Circles), M (Moons), and AH (Ad Hoc). The values of
$A_{\text{val}}$ and $A_{\text{test}}$ represent the accuracy with the optimal
hyperparameters for the validation and test sets respectively. Figure 1:
Decision function ((a)-(c)) and optimization landscape ((d)-(f)) of the RBF
classical kernel model. Figures (a)-(c) were obtained for the test sets by
extracting the optimal hyperparameters of figures (d)-(f). The three different
tasks are referred to as Circles ((a) and (d)), Moons ((b) and (e)), and Ad
Hoc ((c) and (f)).

Figures 1 (d)-(f) represent the classification accuracy during the validation
procedure of the classical kernel over the hyperparameter space. The optimal
hyperparameters of Table 1 were obtained by selecting the values that maximize
the validation accuracy, as described in Section II.3. For the Circles and
Moon tasks (Figs. 1 (d) and (e)), we observe that a broad combination of
hyperparameters allows us to solve them. However, the Ad Hoc task (Fig. 1 (f))
seems to require a higher nonlinear response from the kernel. Using the kernel
with a small scale factor $\gamma$ tends to make the SVM behave like a linear
classifier. Conversely, a large scale factor $\gamma$ makes the output
classifier highly sensitive to small input changes, which could lead to
overfitting even with margin maximization. Figure 1 (f) shows how this task is
best solved with the higher values of $\gamma$ (higher nonlinearity) but
without overfitting, as the test accuracy of Table 1 demonstrates.

###  III.2 Quantum kernel

Now we introduce the results of the quantum kernel. Table 2 shows the optimal
hyperparameters and classification accuracy of the quantum kernel with the
Hamiltonian given in Eq. (12) for the same tasks. This table contains only the
noiseless case, i.e., the kernel given by Eq. (14). As in the classical case,
we compute the validation and test accuracy for the optimal hyperparameters
($h$ and $\Delta t$), while exploring the number of qubits $N$ as well.

No noise
---
Task |  N | $h$ | $\Delta t$ | $A_{\text{val}}$ | $A_{\text{test}}$
C |  2 | $\numprint{0.404}$ | $\numprint{2.105}$ | $\numprint{0.763}$ | $\numprint{0.715}$
C |  4 | $\numprint{0.368}$ | $\numprint{16.298}$ | $\numprint{0.907}$ | $\numprint{0.889}$
C |  6 | $\numprint{1.874}$ | $\numprint{2.310}$ | $\numprint{0.907}$ | $\numprint{0.874}$
M |  2 | $\numprint{0.423}$ | $\numprint{49.770}$ | $\numprint{0.637}$ | $\numprint{0.568}$
M |  4 | $\numprint{2.257}$ | $\numprint{2.783}$ | $\numprint{0.931}$ | $\numprint{0.898}$
M |  6 | $\numprint{2.984}$ | $\numprint{2.105}$ | $\numprint{0.931}$ | $\numprint{0.898}$
AH |  2 | $\numprint{1.630}$ | $\numprint{2.205}$ | $\numprint{0.742}$ | $\numprint{0.724}$
AH |  4 | $\numprint{1.556}$ | $\numprint{6.734}$ | $\numprint{0.835}$ | $\numprint{0.778}$
AH |  6 | $\numprint{1.556}$ | $\numprint{6.734}$ | $\numprint{0.865}$ | $\numprint{0.823}$
Table 2: Optimal hyperparameters for the quantum kernel model without noise. The tasks are represented as C (Circles), M (Moons) and AH (Ad Hoc). $N$ represents the number of qubits, $h$ is the strength of the external magnetic field and $\Delta t$ is the unitary evolution’s time scale. The values of $A_{\text{val}}$ and $A_{\text{test}}$ represent the accuracy with the optimal hyperparameters for the validation and test sets respectively. |  $s=0.01$ and $p=0.01$ |  $s=0.1$ and $p=0.01$ |  $s=0.01$ and $p=0.1$ |  $s=0.1$ with $p=0.1$
---|---|---|---|---
Task |  N | $h$ | $\Delta t$ | $A_{\text{val}}$ | $A_{\text{test}}$ | $h$ | $\Delta t$ | $A_{\text{val}}$ | $A_{\text{test}}$ | $h$ | $\Delta t$ | $A_{\text{val}}$ | $A_{\text{test}}$ | $h$ | $\Delta t$ | $A_{\text{val}}$ | $A_{\text{test}}$
C |  2 | $\numprint{0.183}$ | $\numprint{19.630}$ | $\numprint{0.781}$ | $\numprint{0.721}$ | $\numprint{0.705}$ | $\numprint{7.391}$ | $\numprint{0.763}$ | $\numprint{0.706}$ | $\numprint{0.486}$ | $\numprint{1.918}$ | $\numprint{0.775}$ | $\numprint{0.706}$ | $\numprint{0.705}$ | $\numprint{7.391}$ | $\numprint{0.757}$ | $\numprint{0.706}$
C |  4 | $\numprint{1.024}$ | $\numprint{2.535}$ | $\numprint{0.907}$ | $\numprint{0.883}$ | $\numprint{0.278}$ | $\numprint{29.836}$ | $\numprint{0.898}$ | $\numprint{0.883}$ | $\numprint{0.890}$ | $\numprint{2.915}$ | $\numprint{0.907}$ | $\numprint{0.889}$ | $\numprint{0.423}$ | $\numprint{25.950}$ | $\numprint{0.904}$ | $\numprint{0.883}$
C |  6 | $\numprint{0.559}$ | $\numprint{6.734}$ | $\numprint{0.907}$ | $\numprint{0.886}$ | $\numprint{0.739}$ | $\numprint{4.642}$ | $\numprint{0.901}$ | $\numprint{0.883}$ | $\numprint{0.774}$ | $\numprint{5.094}$ | $\numprint{0.907}$ | $\numprint{0.880}$ | $\numprint{0.368}$ | $\numprint{59.948}$ | $\numprint{0.901}$ | $\numprint{0.892}$
M |  2 | $\numprint{1.417}$ | $\numprint{2.310}$ | $\numprint{0.637}$ | $\numprint{0.583}$ | $\numprint{1.874}$ | $\numprint{2.535}$ | $\numprint{0.616}$ | $\numprint{0.574}$ | $\numprint{1.556}$ | $\numprint{2.105}$ | $\numprint{0.628}$ | $\numprint{0.598}$ | $\numprint{1.292}$ | $\numprint{3.199}$ | $\numprint{0.628}$ | $\numprint{0.544}$
M |  4 | $\numprint{1.177}$ | $\numprint{8.902}$ | $\numprint{0.931}$ | $\numprint{0.874}$ | $\numprint{1.789}$ | $\numprint{3.054}$ | $\numprint{0.910}$ | $\numprint{0.898}$ | $\numprint{1.963}$ | $\numprint{2.915}$ | $\numprint{0.931}$ | $\numprint{0.898}$ | $\numprint{2.257}$ | $\numprint{2.105}$ | $\numprint{0.913}$ | $\numprint{0.892}$
M |  6 | $\numprint{2.848}$ | $\numprint{2.205}$ | $\numprint{0.931}$ | $\numprint{0.895}$ | $\numprint{1.874}$ | $\numprint{3.352}$ | $\numprint{0.922}$ | $\numprint{0.901}$ | $\numprint{2.848}$ | $\numprint{2.205}$ | $\numprint{0.931}$ | $\numprint{0.895}$ | $\numprint{1.707}$ | $\numprint{3.199}$ | $\numprint{0.910}$ | $\numprint{0.898}$
AH |  2 | $\numprint{1.789}$ | $\numprint{2.009}$ | $\numprint{0.739}$ | $\numprint{0.703}$ | $\numprint{3.765}$ | $\numprint{1.000}$ | $\numprint{0.676}$ | $\numprint{0.676}$ | $\numprint{1.789}$ | $\numprint{2.009}$ | $\numprint{0.736}$ | $\numprint{0.703}$ | $\numprint{1.789}$ | $\numprint{2.009}$ | $\numprint{0.682}$ | $\numprint{0.703}$
AH |  4 | $\numprint{1.417}$ | $\numprint{7.055}$ | $\numprint{0.823}$ | $\numprint{0.778}$ | $\numprint{2.364}$ | $\numprint{8.498}$ | $\numprint{0.724}$ | $\numprint{0.763}$ | $\numprint{1.556}$ | $\numprint{6.734}$ | $\numprint{0.814}$ | $\numprint{0.778}$ | $\numprint{1.417}$ | $\numprint{7.391}$ | $\numprint{0.715}$ | $\numprint{0.799}$
AH |  6 | $\numprint{1.292}$ | $\numprint{8.498}$ | $\numprint{0.868}$ | $\numprint{0.820}$ | $\numprint{1.177}$ | $\numprint{10.235}$ | $\numprint{0.757}$ | $\numprint{0.802}$ | $\numprint{1.556}$ | $\numprint{18.738}$ | $\numprint{0.859}$ | $\numprint{0.859}$ | $\numprint{1.417}$ | $\numprint{7.391}$ | $\numprint{0.751}$ | $\numprint{0.820}$
Table 3: Optimal hyperparameters for the quantum kernel model with noise. The
tasks are represented as C (Circles), M (Moons) and AH (Ad Hoc). $N$
represents the number of qubits, $h$ is the strength of the external magnetic
field and $\Delta t$ is the unitary evolution’s time scale. The values of
$A_{\text{val}}$ and $A_{\text{test}}$ represent the accuracy with the optimal
hyperparameters for the validation and test sets respectively.

The first thing that calls our attention from the results presented in Table 2
is that the performance significantly improves when increasing the number of
qubits from $N=2$ to $N=4$ in all tasks. This transition is directly related
to the size of the quantum feature space where the inputs are introduced,
increasing the expressivity of the model. However, the Ad Hoc task is the only
one that also displays a significant change when increasing the number of
spins from $N=4$ to $N=6$. This suggests that the task at hand determines the
optimal size of the Hilbert space, depending on the expressiveness required.

Regarding the effect of statistical (s) and depolarization (p) noise, we
analyze four different situations where the quantum kernel is given by Eqs.
(16)-(18), summarized in Table 3. In the first case, we added a small amount
of both statistical and depolarizing noise, with $s=0.01$ and $p=0.01$. Here
the test accuracy is slightly decreased in almost all the cases, but we also
observe that it can be also slightly increased, as in the Circles task with
$N=6$. Then, we increase the statistical noise to $s=0.1$, finding a small
negative effect over validation accuracy that was not present with $s=0.01$.
Instead, when increasing the depolarizing noise to $p=0.1$ while keeping
$s=0.01$, both validation and test accuracy return to values similar to the
noiseless case, with even higher performance in the Ad Hoc task for $N=6$.
Finally, the case of larger statistical and depolarizing noise ($s=0.1$ and
$p=0.1$) shows that test performance is almost identical to the noiseless
situation, even larger in some cases. However, the validation accuracy
decreases again as in the case of $s=0.1$ and $p=0.01$, meaning that this
effect is produced by the statistical noise.

To visualize the change in performance with noise, Fig. 2 represents the
performance of the quantum kernel for the Ad Hoc task in all the studied
situations. The $x$ axis describes the noise parameters and the $y$ axis
represents the test accuracy. The color of each bar corresponds to a given
number of qubits. Figure 2 illustrates our main findings, namely that test
accuracy increases with system size and that statistical noise can be
detrimental, although an appropriate amount of depolarizing noise can
compensate for this negative effect, even finding the best performance for the
Ad Hoc task for $N=6$ with $s=0.01$ and $p=0.1$.

Figure 2: Test accuracy of the quantum kernel model for the Ad Hoc task. The
color of each bar corresponds to a given number of qubits and each group of
bars corresponds to a set of noise parameters.

When comparing the optimal quantum kernels with the classical kernel, the
quantum models for $N=4$ and $N=6$ obtain a very similar performance to the
classical case, even under the effect of noise. For the Ad Hoc task, we can
also find one situation where the test accuracy of the quantum kernel is
larger than in the classical case ($N=6$ for $s=0.01$ and $p=0.1$). On the one
hand, the fact that statistical noise (less measurement precision) tends to
decrease the performance is an evident negative effect that can have severe
consequences, such as the loss of any possible generalization advantage of
quantum kernels [41]. However, as shown in [5], quantum kernels with a
controlled additive sampling noise in SVM are robust, where the noisy
hyperplane is close to the noiseless hyperplane. On the other hand,
depolarizing noise corresponds to a loss of information about the quantum
state with probability $p$, so large values of $p$ would also hinder the
performance of quantum devices. But as we observe here, it can act as a
regularization parameter for small values. We remark that in any case,
techniques can be carried out to mitigate the effect of both sources of noise
[41, 40, 33].

We now proceed to visualize the relationship between the optimal
hyperparameters and task performance. Figure 3 shows the validation accuracy
in terms of the hyperparameters $h$ and $\Delta t$ for the systems associated
with $s=0.01$ and $p=0.01$ in Table 3. We notice that the optimization
landscape is qualitatively similar in all the studied situations, so Fig. 3 is
a representative example. The left and right columns correspond to $N=2$ and
$N=6$ respectively, while each row corresponds to a different task. As already
observed in Table 3, the validation accuracy reaches higher values for $N=6$
than for $N=2$ due to the larger Hilbert space. But Fig. 3 also shows that the
region of hyperparameters that could solve the tasks may be broad. For the
Circles task (Fig. 3 (d)), we observe a linear tendency $h\Delta
t\sim\text{cte}$ for the best hyperparameter’s area, while the best
hyperparameters are more concentrated for the Ad Hoc task (Figs. 3 (f)). The
Moons task (Figs. 3 (e)) is an intermediate case where the best
hyperparameters seem to follow a linear tendency but over a broad region.

One may try to explain the previous results by establishing a connection
between the value of the optimal hyperparameters and the underlying dynamical
features of the model. However, it seems more complicated than that. Notice
that the kernel of Eq. (14) has three hyperparameters related to the dynamics,
$J$, $h$, and $\Delta t$, but in fact, only two of them are independent (when
combined as $J\Delta t$ and $h\Delta t$ in the unitary operator). Then the
coefficient $J/h$ indicates the relative weight of each of the two terms of
the Hamiltonian in Eq. 12. Recalling that we previously fixed $J=1$, our
results suggest that the relative weight of the external magnetic field term
to the coupling term would only become a relevant factor in solving some
specific tasks. For instance, the Moons task does not strongly depend on the
rate $J/h$: in Fig. 3 (e) we observe that values of the hyperparameters such
as $0.3<h<3$ with $\Delta t\sim 10$ provide a similar performance. That is,
the rates $1/3<J/h<3$ work equally well. On the contrary, the Ad Hoc task
requires a very specific ratio $J/h$: Fig. 3 (f) shows a clear optimal region
of hyperparameters in the center of the figure, around $\Delta t\sim 10$ and
$h\sim 1$.

Figure 3: Optimization landscape for the quantum kernel model with the
validation datasets. The statistical noise strength is given by $s=0.01$ and
the depolarizing noise parameter is $p=0.01$. The left column corresponds to
$N=2$ while the right column corresponds to $N=6$. The top row is the Circles
task, the middle column is the Moons task, and the bottom column is the Ad Hoc
task.

##  IV Discussion

Kernel methods are a promising direction in the search for applications of
quantum machine learning techniques. In particular, classically intractable
quantum simulation models can be exploited to explore quantum advantages. Our
work proposes ion trap platforms as an analog experimental platform to compute
quantum kernels, demonstrating their viability through numerical simulations
of realistic scenarios with depolarizing and statistical noise.

The computation of quantum kernels proposed here for trapped ions builds on
the kernel estimation protocol of Ref. [15]. Computing the unitary dynamics
given by $-H(\bm{y})$ only requires the change of sign of the couplings and
the definition of new magnetic fields depending on the input $\bm{y}$. One
potential challenge is whether it is possible to make this sudden change
between $H(\bm{x})$ and $-H(\bm{y})$ on the fly. There are other possible
routes to measure the overlap between two quantum states that might be applied
to this quantum simulation platform. For example, the SWAP test and extensions
[48, 49] would allow evolving in parallel the states $\ket{\psi(\bm{x})}$ and
$\ket{\psi(\bm{y})}$ without the sudden quench. Classical machine learning
techniques can help here [50, 51], and full tomography allows us to obtain the
quantum kernel as well [52]. Finally, a promising direction might be to use
randomized measurements [53, 54], where the overlap of states generated by
independent Hamiltonians can be measured.

We evaluated the performance of the proposed quantum kernel in three different
binary classification tasks and compared the results with the well-known RBF
classical kernel. We made a grid search of optimal hyperparameters for both
classical and quantum models, also exploring the number of qubits in the
latter. The bandwidth optimization of the quantum model introduces an
inductive bias that restricts the type of functions that can be solved. We
find that both classical and quantum models reach a very similar performance,
closely saturating the maximum possible test accuracy for the given test sets
without overfitting. This means that the proposed quantum model is operational
for classification tasks, being robust under the presence of different sources
of noise.

The search for optimal quantum kernels was carried out for only two
hyperparameters with a very specific Hamiltonian (eq. (11)) and input scheme,
but more possibilities can be explored. The hyperparameters $\alpha$ and $C$
can be also tuned, the input can be redundantly injected (like
$\bm{x}=\\{x_{1},x_{2},x_{1},x_{2},...\\}$ in eq. (11)), or we can even add
new terms to the Hamiltonian such as a magnetic field in the $x$ axis. We
tested these possibilities for the presented tasks, reaching a similar or
worse performance. However, we do not discard that a more systematic
evaluation of these strategies could bring an improvement. In fact, the
quantitative performance in different tasks is expected to vary under
different interactions or driving in the Hamiltonian.

Future work will be devoted to exploring the effect of increasing the number
of qubits to solve tasks with a larger number of input features. On the one
hand, one can expand the number of features that can be codified in a single
qubit by feeding them over time with an input-dependent driving field, as
presented in Ref. [52]. On the other hand, projected quantum kernels can be
explored in this setting [4], limiting the effective Hilbert space dimension
to maximize the generalization ability of the model.

## ACKNOWLEDGMENTS

We acknowledge the Spanish State Research Agency, through the María de Maeztu
project CEX2021-001164-M funded by the MCIN/AEI/10.13039/501100011033 and
through the COQUSY project PID2022-140506NB-C21 and -C22 funded by
MCIN/AEI/10.13039/501100011033, MINECO, through the QUANTUM SPAIN project, and
EU through the RTRP - NextGenerationEU within the framework of the Digital
Spain 2025 Agenda. The CSIC Interdisciplinary Thematic Platform (PTI) on
Quantum Technologies in Spain is also acknowledged. RMP acknowledges the QCDI
project funded by the Spanish Government and part of this work was also funded
by MICINN/AEI/FEDER and the University of the Balearic Islands through a pre-
doctoral fellowship (Grant No. MDM-2017-0711-18-1).

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