# Predicting Open-Hole Laminates Failure Using Support Vector Machines With
Classical and Quantum Kernels

Giorgio Tosti Balducci
Aerospace Structure and Materials
Delft University of Technology
Kluyverweg 1, 2629HS,
Delft, The Netherlands
&Boyang Chen
Aerospace Structure and Materials
Delft University of Technology
Kluyverweg 1, 2629HS,
Delft, The Netherlands
&Matthias Möller
Applied Mathematics
Delft University of Technology
Mekelweg 4, 2628CD,
Delft, The Netherlands
&Marc Gerritsma
Flow Physics and Technology
Delft University of Technology
Kluyverweg 1, 2629HS,
Delft, The Netherlands
&Roeland De Breuker
Aerospace Structure and Materials
Delft University of Technology
Kluyverweg 1, 2629HS,
Delft, The Netherlands
Corresponding author. Email<EMAIL_ADDRESS>
###### Abstract

Modeling open hole failure of composites is a complex task, consisting in a
highly nonlinear response with interacting failure modes. Numerical modeling
of this phenomenon has traditionally been based on the finite element method,
but requires to tradeoff between high fidelity and computational cost. To
mitigate this shortcoming, recent work has leveraged machine learning to
predict the strength of open hole composite specimens. Here, we also propose
using data-based models but to tackle open hole composite failure from a
classification point of view. More specifically, we show how to train
surrogate models to learn the ultimate failure envelope of an open hole
composite plate under in-plane loading. To achieve this, we solve the
classification problem via support vector machine (SVM) and test different
classifiers by changing the SVM kernel function. The flexibility of kernel-
based SVM also allows us to integrate the recently developed quantum kernels
in our algorithm and compare them with the standard radial basis function
(RBF) kernel. Finally, thanks to kernel-target alignment optimization, we tune
the free parameters of all kernels to best separate safe and failure-inducing
loading states. The results show classification accuracies higher than 90% for
RBF, especially after alignment, followed closely by the quantum kernel
classifiers.

_Keywords_ Composites $\cdot$ Support Vector Machines $\cdot$ Quantum Machine
Learning

##  1 Introduction

Modern aviation industry makes wide use of composite materials, thanks to
their lightweight and favorable mechanical properties. Frequently,
aeronautical structural elements are often not textbook flat composite panels,
but tailored components with complex mechanical responses. For instance,
composite panels often show cutouts in order to allow fastening or lightening
the structure or even for allowing the passage of wiring or cables. However,
the presence of holes in a composite plate induces stress concentrations that
can initiate damage which can propagate into intricate failure mechanisms
involving different modes.

Models for open hole composite failure have developed in different directions.
On the one hand, semi-empirical models were proposed to predict the allowables
of these structures, such as ultimate stength, and their statistical
distribution with respect to hole geometry, loading conditions, stacking
sequence, ply thickness, etc. Early attempts required experimental properties
from testing both the unnotched and notched laminate [1], while later models
removed the need of directly testing the open hole laminate [2, 3] or just
required the ply properties [4]. Despite being fast to evaluate and suitable
for preliminary design, semi-empirical models can make large errors when
extensive delaminations propagate from the notch, as it happens with ply-
scaled laminates.

Finite Element (FE) simulations allow for improved modeling of open hole
laminates failure. Open hole tension (OHT) has been extensively studied
numerically both for capturing the in-plane [5] and thickness size effects [6,
7, 8] on the ultimate strength and for reproducing the different failure modes
and their interactions [9, 10] with increasing detail. Furthermore, FE
simulations managed to quite accurately predict open-hole compression (OHC),
even though still struggling to predict the precise kink band formation[11,
12, 13]. However, the accuracy offered by FE models generally comes at the
price of high computational costs, possibly making them unfeasible when many
design iterations are required.

Therefore, there is a practical need for computationally efficient yet
accurate models that can simulate open hole composite laminates. A possibility
is offerred by machine learning surrogates, which have been employed in
composite design and optimization [14, 15, 16], constitutive law modeling and
multiscale analyses (see [17] for a comprehensive review) and damage
characterisation [18, 19]. Concerning open-hole composite failure, Furtado _et
al._ proposed a methodology to define allowables using four different machine
learning models [20]. Their methodology was applied to open-hole tensile
strength prediction for different dimensions, layups and material properties.
While their methods are demonstrated on data generated analytically [4], the
authors suggest using high fidelity finite element analyses for training,
potentially providing accurate data-based models.

Similarly, in this work we propose a machine learning surrogate for open hole
composites, which is accurate and efficient in inference. Differently from
[20] however, the approach we suggest is not to have a fast allowables
generator, but a classifier for ultimate failure of open hole composite
laminates. More precisely, our trained model takes a loading state as input,
such as the far field homogenized plane strain components and returns a binary
valuable ($\pm 1$) as output, depending on whether the load applied is lower
or higher than the notched laminate strength. In this sense, the surrogate
acts as a data-based generalized failure criterion which predicts at the
structural component level, rather than at the material level.

This paper also aims at comparing classical and quantum computation for a
classification problem in composite mechanics. To do this, we train the
machine learning surrogate using _kernel_ -based support vector machines
(SVMs) [21], where the kernel function can be computed both in classical and
quantum logic. As it will be clear in the next sections, quantum computation
offers a way to encode information into exponentially large Hilbert spaces and
to define an inner product in this spaces, effectively generating a kernel.
This allows to explore the generalization potential of quantum machine
learning, while leaving the SVM optimization to well-established classical
quadratic optimization algorithms.

The rest of the paper is structured as follows. Section 2 describes the
machine learning problem, by defining the input, the data sampling strategy
and the labeling criterion. Section 3 briefly introduces the SVM dual problem,
the RBF kernels and the quantum kernels. More details about these methods are
available in the appendices following the main body of the manuscript.
Finally, Section 4 presents the classification results for all kernels and
Section 5 outlines conclusions and future work.

All data and code used in this work are made publicly available (see [22],
[23] respectively).

##  2 Machine learning problem

Figure 1: Geometry of the open hole composite plate and homogenous strain
loads components.

Our method was applied to predict failure of an open hole composite specimen
similar in geometry and material properties to the one experimentally tested
in [24]. The specimen was modeled and meshed with the Abaqus finite element
code [25] and it was loaded with different combinations of axial and shear
strains and constrained with periodic boundary conditions. All the details of
the specimen properties and of the finite element analyses are left to
Appendix A.

The input of our surrogate models are homogenized far field strains
$\bm{\varepsilon}=\left[\varepsilon_{11},\,\varepsilon_{22},\,\gamma_{12}\right]^{\top}$,
which derive from enforcing periodic boundary conditions on opposite faces of
the plate. The displacements of the left/right and top/bottom faces
respectively can be linked through some reference degrees of freedom

$\displaystyle U_{1}$ $\displaystyle=u_{1}^{R}-u_{1}^{L}$ (1) $\displaystyle
U_{2}$ $\displaystyle=u_{2}^{T}-u_{2}^{B}$ $\displaystyle U_{3}$
$\displaystyle=u_{2}^{R}-u_{2}^{L}$ $\displaystyle U_{4}$
$\displaystyle=u_{1}^{T}-u_{1}^{B},$

where directions $1$ and $2$ are the horizontal and vertical directions in
Figure 1. The homogenized strains are then obtained as

$\displaystyle\varepsilon_{11}$ $\displaystyle=\frac{U_{1}}{D_{1}}$ (2)
$\displaystyle\varepsilon_{22}$ $\displaystyle=\frac{U_{2}}{D_{2}}$
$\displaystyle\gamma_{12}$
$\displaystyle=\frac{U_{3}}{D_{1}}+\frac{U_{4}}{D_{2}},$

where $D_{1}$ and $D_{2}$ are the planar dimensions of the plate.

Figure 2: (a) Radial strategy for sampling the far-field strain space. The
bigger dots on the edge of the
$\left(\varepsilon_{11},\varepsilon_{22}\right)$ volume represent the final
applied load in different nonlinear FE incremental-iterative analyses. The
arrow represents a loading path with the load increments unevenly distributed.
(b) Load-displacement curve corresponding to the loading path in (a).

As mentioned, the input space was sampled through nonlinear incremental-
iterative finite element analyses. Figure 2 illustrates the sampling strategy
used in this work in the simplified case of two-dimensional input. We refer to
this technique as _radial sampling_ , due to the fact that the design of
experiments (DoE) does not directly affect all the points in this input space,
but only the ones on the boundary. On the other hand, all the intermediate
points are generated internally by the FE solver and they correspond to the
homogenized strain values at every time increment111Of course, the user
maintains a certain control on the inner samples values, by the choice of
initial, minimum and maximum time steps.. For this work, we chose the sampling
space to be the hypercube $\left[-10^{-2},\,10^{-2}\right]^{\otimes 3}$ in
$\mathbb{R}^{3}$, meaning that all three components of the applied strains
vector have the same bounds.

###  2.1 Labeling criterion

Each strain sample was assigned a label based on an ultimate failure
criterion. In particular, we defined failure by the loss of stiffness of the
laminate for given a user-defined threshold.

From the results of the FE analyses with periodic boundary conditions, one
obtains the reaction forces $F_{1}$, $F_{2}$, $F_{3}$ and $F_{4}$ conjugate to
the degrees of freedom in Equation 1. These provide the homogenized stresses,
which can then be derived via the Hill-Mandel principle of energy balance as

$\displaystyle\sigma_{11}$ $\displaystyle=\frac{F_{1}}{tD_{2}}$ (3)
$\displaystyle\sigma_{22}$ $\displaystyle=\frac{F_{2}}{tD_{1}}$
$\displaystyle\sigma_{12}$
$\displaystyle=\frac{F_{3}U_{3}+F_{4}U_{4}}{\gamma_{12}tD_{1}D_{2}},$

where $t$ is the thickness of the plate.

The laminate stiffness in the two axial directions and in shear can thus be
defined at every timestep $t$ as

$\displaystyle E_{1}^{(t)}$
$\displaystyle=\frac{\sigma_{11}^{(t)}}{\varepsilon_{11}^{(t)}}$ (4)
$\displaystyle E_{2}^{(t)}$
$\displaystyle=\frac{\sigma_{22}^{(t)}}{\varepsilon_{22}^{(t)}}$
$\displaystyle G_{12}^{(t)}$
$\displaystyle=\frac{\sigma_{12}^{(t)}}{\varepsilon_{12}^{(t)}}.$

The stiffness degradation $d_{S}$ is defined as the minimum ratio between the
instantaneous stiffness and the corresponding stiffness measure in the linear
elastic region,

$d_{S}^{(t)}=\min\left\\{\frac{E_{1}^{(t)}}{E_{1}^{(0)}},\,\frac{E_{2}^{(t)}}{E_{2}^{(0)}},\,\frac{G_{12}^{(t)}}{G_{12}^{(0)}}\right\\}$
(5)

Therefore, given $M$ the total number of samples, every sample
$\bm{\varepsilon}^{(m)}$ ($m=1,\dots,M$) is assinged a label $y^{(m)}=-1$ if
$d^{(m)}<\bar{d}_{S}$ and $y^{(m)}=+1$ otherwise.

##  3 Methodology

Figure 3: The methodology used in this work. The dataset is generated by
nonlinear finite element analyses, then labeled and split into training and
testing sets. The training set is used first to train the kernel, by
optimizing the kernel-target alignment (KTA), then in a grid search cross-
validation to find the best slack penalty $C$ of the SVM. With all the
hyperparameters fixed, the SVM is trained for increasing dataset sizes and the
classification accuracy is evaluated on the testing set.

As already mentioned, we solve the ultimate failure binary classification
problem using the SVM algorithm [21]. This consists in the following quadratic
optimization problem in dual form

$\displaystyle\max_{\bm{\alpha}}$
$\displaystyle\quad\sum_{m=1}^{M}\alpha^{(m)}-\frac{1}{2}\sum_{m,m^{\prime}=1}^{M}y^{(m)}y^{(m^{\prime})}\alpha^{(m)}\alpha^{(m^{\prime})}\kappa\left(\bm{x}^{(m)},\,\bm{x}^{(m^{\prime})}\right)$
(6) $\displaystyle\mathrm{s.t.}$ $\displaystyle\quad 0\leq\alpha^{(m)}\leq
C,\quad m=1,\,\dots,\,M$
$\displaystyle\quad\sum_{m=1}^{M}\alpha^{(m)}y^{(m)}=0,$

where $\bm{x}^{(m)}=\bm{\varepsilon}^{(m)}$, $y^{(m)}=\pm 1$ are the labels,
respectively non-failed and failed, $\alpha^{(m)}$ are the Lagrange
multipliers and $C$ is the slack penalty. The kernel function
$\kappa\left(\cdot,\,\cdot\right)$ is a similarity metric between two samples
in a higher-dimensional feature space. More details on the SVM algorithm are
left to Appendix B.

The performance of the dual SVM depends on the choice of its hyperparameters,
namely the kernel function $\kappa$ and slack penalty $C$. To restrict the
search space, the kernel function is generally parametrized via one or more
parameters $\bm{\theta}$ and the standard practice is to perform a grid-search
cross-validation procedure in the $\left(\bm{\theta},C\right)$ space. In this
work, we use instead a mixed procedure, where the kernel function is
determined by optimizing the kernel-target alignment (KTA) [26] and the slack
penalty is found by grid search cross-validation. The overall methodology is
illustrated in Figure 3, where we refer to the two steps as _kernel training_
and _SVM selection_. Once the SVM has been fully determined, it can be trained
by solving Equation 6 and its learning ability can be measured as the accuracy
on unseen test data, for different training dataset sizes.

We compare one classical and two quantum kernels. The classical kernel is the
radial basis function (RBF) kernel, defined as

$\kappa_{\text{RBF}}\left(\bm{x}^{(m)},\,\bm{x}^{(m^{\prime})}\right)=\exp\left(-\gamma\|\bm{x}^{(m)}-\bm{x}^{(m^{\prime})}\|^{2}\right).$
(7)

RBF is a powerful kernel which corresponds to a feature map in an infinite-
dimensional feature space [27]. It induces a Gaussian similarity function,
whose width is controlled by the hyperparameter $\gamma$.

On the other hand, the quantum kernel is defined via a _quantum embedding_ ,
which is constructed via data-depending unitary transformations
$U\left(\bm{x}\right)$ that prepare the quantum state

$\ket{\psi\left(\bm{x}\right)}=U\left(\bm{x}\right)\ket{0}.$ (8)

Given two samples $\bm{x}^{(m)}$ and $\bm{x}^{(m^{\prime})}$, the quantum
kernel is simply the inner product

$\kappa_{\text{Q}}\left(\bm{x}^{(m)},\,\bm{x}^{(m^{\prime})}\right)=\absolutevalue{\langle\psi\left(\bm{x}^{(m)}\right),\,\psi\left(\bm{x}^{(m^{\prime})}\right)\rangle}^{2}.$
(9)

Figure 4 shows the generic quantum embedding and the two specific ones used in
this work, which are the hardware efficient embedding (HE2) [28] and the
instantaneous quantum polynomial (IQP) [29] one. To have more expressive
feature mapping, either the _width_ or the _depth_ of the quantum embedding
can be increased. The first one is the number of qubits, which can be even
higher than the number of features in the dataset, by cyclically re-encoding
the features to generate a highly nonlinear and potentially better separable
feature space. Meanwhile, the embedding’s depth can be increased by repeating
a base data-encoding block, such as IQP and HE2. Even in this case, re-
encoding of the features may lead to a higher expressivity of the overall
feature map [30]. For a short summary of relevant quantum computing concepts,
we refer the reader to Appendix C.

Figure 4: Quantum embeddings. (a) Generic quantum embedding made of one- and
two-qubit gates. The _width_ of the embedding is the number of qubits, while
the _depth_ is the number of layers, which is a minimal block of gates. (b)
IQP quantum embedding layer, which is parameter free, but encodes products of
features as $ZZ$ interactions. (c) HE2 quantum embedding layer, parametrized
by $\left[\theta_{0},\theta_{1},\dots,\theta_{P-1}\right]$

##  4 Results

We tested our machine learning models on a dataset of 1960 labelled strain
vectors $\bm{\varepsilon}^{(m)}$, which we obtained by uniformly sampling the
homogeneous strain/stress pairs from the FE simulations of the open-hole
composite specimen. The input homogeneous strains in both normal and shear
directions were varied between ${1}\mathrm{e}{-4}$ and ${1}\mathrm{e}{4}$
microstrains and a stiffness degradation threshold of 0.9 was used to
discriminate non-failed and failed loading states.

Both classical- and quantum-kernel SVMs were implemented using different
Python libraries. We used PyTorch for training the RBF kernel and PennyLane
for the quantum kernels. These libraries implement automatic differentiation
(AD), which allows to optimize the KTA with gradient-based methods. We also
used JAX together with PennyLane to just-in-time compile the quantum kernel
functions. Concerning the classification problem, we employed the SVM and
grid-search cross validation routines available from the Scikit-Learn Python
package.

Figure 5: Training history of the RBF kernel’s KTA. Figure 6: KTA of quantum
kernels before and after training for 9 different quantum embeddings. The
basic layer for all embeddings is the HE2, which has trainable parameters. The
tag wXdY indicates width and depth of the embedding.

The KTA of both RBF and quantum kernels was maximized using stochastic
gradient descent and Adam parameters update [31]. Figure 5 shows the kernel
alignment training of the RBF kernel. Figure 6 presents instead the KTAs
before and after training for nine different quantum kernels with HE2
embedding. It can be seen that increasing width and depth of these kernels
generally improves their KTA. A higher number of qubits means that the strain
features are mapped in a higher dimensional space, which can favor
separability of the classes. On the other hand, increasing the depth benefits
the kernel alignment, since it results in more expressive feature maps. Also,
every additional layer of the HE2 embedding doubles the number of free
parameters, explaining why optimization of deeper kernels mostly leads to
higher gains in KTA. However, the advantage of increasing these quantum
encoding resources does not scale uniformly. Already with 6 qubits and 3 HE2
layers, the optimization only modestly improves the KTA, likely due to the
vanishing KTA gradients [32].

Figure 7: Grid-search cross validation for the RBF kernel. $\gamma=1.065$
corresponds to the kernel with highest kernel-target alignment.

To find the hyperparameter $C$ that guarantees the highest off-training
accuracy of the SVM algorithm, we used grid-search cross validation for the
kernels considered. The validation accuracy values are reported in Figure 7
for multiple values of $C$ and $\gamma$. We observe that kernels with
$\gamma\leq 10$ achieve the higest scores, with the highest-KTA $\gamma$
scoring first for the whole range of $C$ values. Furthermore, the accuracy of
the maximally-aligned RBF kernel increases monotonically with $C$, which
suggests the usefulness of maximizing the KTA, but also that the class
boundary in this feature space is densely populated and still requires a tight
margin.

Figure 8: Summary of the grid-search cross validation results for different
quantum kernels. The three figures correspond respectively to the IQP
embeddding, the HE2 kernel with untrained parameter and HE2 with trained
parameters. Different shades of the same color correspond to different depths
and widths. From lighter to darker, the points correspond to embeddings of
increasing widths and of increasing depth per fixed width.

The same analysis was performed for all the quantum kernels considered, where
we wanted to take into account the effect on accuracy of different embeddings
and of maximizing the kernel-target alignment. The results are reported in
Figure 8, which shows accuracies roughly between 67% and 87% for all
embeddings with different values of $C$. Except for IQP case, increasing $C$
leads to higher accuracies, hinting to the the need of a tight bound when
mapping with these embeddings, similar to the RBF kernel. Unfortunately, for
$C\geq{1}\mathrm{e}{4}$, the optimization of the dual SVM failed to converge
for the quantum kernels, likely due to numerical ill-conditioning, presumably
preventing from reaching higher accuracies. In fact, we observe that the
accuracies of both untrained and trained HE2 kernels monotonically increase
with $C$. For the trained HE2 case, this is true regardless of the number of
qubits and depth. Furthermore, Figure 8 also shows that increasing the
embedding resources, especially the number of qubits, pays off more when also
optimizing the KTA.

Figure 9: Test data classification accuracy of classical and quantum kernels
for increasing training set size. The RBF kernel-SVMs were trained with both
$C={1}\mathrm{e}{7}$ and $C={1}\mathrm{e}{4}$, as the latter is the highest
value of $C$ for which the quantum-kernel SVMs could still be fitted. Quantum-
kernel classifiers were trained instead with the embedding arhitecture and $C$
value ensuring highest accuracy during grid-search cross validation

Classical and quantum kernels are finally compared in Figure 9, which shows
how 5 different models classify a test set of strain loading data when fitted
on progressively larger training sets. A similar comparison on additional
classification metrics can be found in Appendix D. The RBF kernel achieves 80%
accuracy with just 10% the total training set size, and with
$C={1}\mathrm{e}{7}$ it reaches over 90% with just half the training points.
In comparison, all quantum kernel classifiers are at least 5% less accurate
than the best RBF-kernel SVM. However, especially for HE2 embeddings, the
scores are similar to the $C={1}\mathrm{e}{4}$ RBF case, suggesting that RBF
and HE2 kernels separate the non-failed and failed classes to a similar
extent. Changing the embedding from HE2 to IQP, there is a drop in accuracy
for small training set sizes, while the performance is similar when more than
half the training set is used. On the other hand, the effect of training the
kernel is less visible at this stage, reflecting the fact that the accuracies
obtained during grid-search cross validation are alike for untrained and
trained HE2.

##  5 Conclusion

In this paper, we proposed a methodology to build a binary classifier from
finite element analyses data for the particular case of an open hole composite
specimen. We studied the case of in-plane strain loading of the specimen where
the objective is to correctly label strain combinations that lead to ultimate
failure.

From a design of experiment point of view, we demonstrated a radial sampling
strategy technique, where the choice of which simulations to make to cover the
input space takes into account the incremental-iterative nature of the
nonlinear FE method. We then proposed a labelling criterion of homoegenized
strain-stress pairs based on residual in-plane stiffness.

For classification of the labelled data, we used the kernel-based SVMs, which
also allowed us to compare the performance of the recently proposed quantum
kernels against the more traditional RBF. Furthermore, we employed kernel-
target alignment to improve class separbility of both RBF and the HE2
embedding kernel.

For all the kernel examined, the corresponding SVMs separate non-failed and
failed loading states with good accuracy. The RBF-based model classify more
accurately than its quantum counterparts, although this likely happens due to
numerical ill-conditioning in the current quantum SVM implementation. These
numerical issues can likely be fixed by studying the dual SVM problem for the
problematic instances, which will be the subject of future work.

Regarding kernel alignment, optimizing the KTA is shown to be powerful for
RBF, since the SVM for the trained kernel outperforms the other RBF-based
models in terms of accuracy. Aligning quantum kernels for this dataset also
helps them to better separate the two classes, but for simple architectures
the improvement is moderate, while more complex embeddings only reach the
scores of the more simple ones after they have been aligned. Furthermore, one
should remember that optimizing quantum kernels is almost always more
computationally involved than for RBF, as the formers can have highly
parametrized embeddings, while RBF is completely defined by the single
parameter $\gamma$.

Extensions of this work can go in many directions. From the point of view of
the problem, it would be interesting to increase the number of degrees of
freedom, by allowing the notch radius or the lamination sequence to also
change. The latter could be written in terms of lamination parameters [33] to
have a continuos representation.

In terms of algorithms, both classical and quantum kernels can be explored
further. RBF is the most popular choice for classical kernels, but certainly
not the only one. Due to Mercer’s condition, any function which defines a
positive semi-definite kernel matrix is a valid kernel function [27].
Obviously, the design space is vast, but automated procedures help reduce the
search for instance by exploring combinations of only a fixed set of standard
kernel functions.

On the other hand, the freedom of designing and parametrizing quantum
embedding circuit also makes the choice of a quantum kernel nontrivial. Within
the limits of classical simulation of quantum circuits, one could experiment
with increasing number of qubits or different layering strategies, for
instance the one proposed in [34] for the task of satellite image
classification. From an optimization point of view, a recent technique has
been proposed to maximize the quantum kernel alignment KTA and solving the SVM
in a single optmization loop [35], which would of course greatly reduce the
computational cost. Nevertheless, to truly understand a potential
competitiveness of quantum kernels, it is probably most important to remove
layers of simulation and study the effects of statistical and hardware noise
on SVM convergence and accuracy.

##  Appendix A Open hole specimen features and finite element model details

###  A.1 Geometry and material properties

Figure 10: Finite element mesh of the composite plate.

The plate’s hole has a 6 mm diameter and the in-plane dimensions are both 5
times the hole diameter. The ply material is IM7/8552 prepreg (carbon fibres
and epoxy matrix) and each ply has $t=0.125$ mm thickness. We considered the
lamination sequence $\left[45/90/-45/0\right]_{S}$ for a total of 8 plies and
1 mm laminate thickness.

###  A.2 Details of the FE models

All finite element models were done using the Abaqus finite element code [25]
and Python scripting was used to automatically generate a different FE models
for each of the strain loading combinations [36, 37].

The meshed part is illustrated in Figure 10, which shows that a radial mesh
was obtained by seeding the hole edge 4 times as much as the outer edges.
Since no delaminations were expected due to the absence of ply blocks, the
elements were chosen to be S4 shells elements of the Abaqus Standard Element
Library [38], whose in-plane and bending behaviour are described by the
classical lamination theory (CLT), once the stacking sequence and ply
thicknesses are specified.

Damage initiation was modeled with the Hashin criterion, while damage
evolution was represented in a smeared crack fashion. For this purpose, the
cohesive law available in Abaqus [38] was employed to model the stiffness
degradation due to matrix and fiber tensile and compressive failure.

##  Appendix B Support vector machines, kernel methods and kernel-target
alignment

###  B.1 Primal SVM

The SVM is the linear decision model

$y=\bm{w}^{\top}\bm{x}+b$ (10)

which assigns labels through the sign function

$\mathrm{sgn}(y)=\mathrm{sgn}\left(\bm{w}^{\top}\bm{x}+b\right).$ (11)

In Equations 10 and 11, $\bm{w}$ is the vector normal to the decision
hyperplane and $b$ is the intercept.

The optimal hyperplane is found by maximizing the _geometric margin_ of the
dataset, which can be proved to be

$\gamma^{*}=\frac{1}{\|\bm{w}\|}.$ (12)

By minimizing the squared norm $\|\bm{w}\|^{2}$ one obtaines the primal
optimization problem of the SVM,

$\displaystyle\min_{\bm{w},\,b}$ $\displaystyle\quad\frac{1}{2}\|\bm{w}\|^{2}$
(13) $\displaystyle\mathrm{s.t.}$ $\displaystyle\quad
y^{(m)}\left(\bm{w}^{\top}\bm{x}^{(m)}+b\right)\geq 1\quad m=1,\,\dots,\,M,$

where $m$ identifies the sample and $M$ is the total number of training
samples.

Equation 13 enforces exact separability, which can lead to overfitting. A way
to improve generalization is the so-called _soft margin SVM_ , which modifies
Equation 13 by introducing the constraints slack variables $\xi^{(m)}$ and the
penalty constant $C$,

$\displaystyle\min_{\bm{w},\,b}$
$\displaystyle\quad\frac{1}{2}\|\bm{w}\|^{2}+C\sum_{m=1}^{M}\xi^{(m)}$ (14)
$\displaystyle\mathrm{s.t.}$ $\displaystyle\quad
y^{(m)}\left(\bm{w}^{\top}\bm{\varepsilon}^{(m)}+b\right)\geq 1-\xi^{(m)}\quad
m=1,\,\dots,\,M$ $\displaystyle\quad\xi^{(m)}\geq 0.$

###  B.2 Dual SVM and kernels

By introducing the Lagrange multipliers $\alpha^{(m)}$ and $\beta^{(m)}$, one
can write the Lagrangian of the SVM optimization problem,

$\displaystyle\mathcal{L}\left(\bm{w},b,\bm{\xi},\bm{\alpha}\right)=$
$\displaystyle\frac{1}{2}\langle\bm{w},\,\bm{w}\rangle+C\sum_{m=1}^{M}\xi^{(m)}$
(15)
$\displaystyle-\sum_{m=1}^{M}\alpha^{(m)}\left(y^{(m)}\left(\bm{w}^{\top}\bm{x}^{(m)}+b\right)-1+\xi^{(m)}\right)-\sum_{m=1}^{M}\beta^{(m)}\xi^{(m)}.$

The _dual_ soft-margin SVM is obtained by setting all the derivatives of the
Lagrangian in Equation 15 equal to zero,

$\displaystyle\max_{\bm{\alpha}}$
$\displaystyle\quad\sum_{m=1}^{M}\alpha^{(m)}-\frac{1}{2}\sum_{m,m^{\prime}=1}^{M}y^{(m)}y^{(m^{\prime})}\alpha^{(m)}\alpha^{(m^{\prime})}\langle\bm{x}^{(m)},\,\bm{x}^{(m^{\prime})}\rangle$
(16) $\displaystyle\mathrm{s.t.}$ $\displaystyle\quad
0\leq\alpha^{(m)}+\beta^{(m)}\leq C,\quad m=1,\,\dots,\,M$
$\displaystyle\quad\sum_{m=1}^{M}\alpha^{(m)}y^{(m)}=0.$

Equation 16 is still a linear model in the original feature space. However, by
introducing a feature map

$\phi:\bm{x}\longrightarrow\phi(\bm{x})$ (17)

we can map the features nonlinearly and potentially to a manifold where they
are more easily separable. Furthermore, replacing $\bm{x}$ with $\phi(\bm{x})$
in Equation 16, we see that the mapped features only appear in the inner
product

$\kappa\left(\bm{x}^{(m)},\,\bm{x}^{(m^{\prime})}\right)=\langle\phi(\bm{x}^{(m)}),\,\phi(\bm{x}^{(m^{\prime})})\rangle,$
(18)

which is known as the _kernel_ of the feature map. The advantage of having
only inner product of features (_kernel trick_) is the possibility of
classifying in nonlinear feature spaces without having to compute the feature
map explicitly.

The kernels mostly used in machine learning are the polynomial, Gaussian and
sigmoid kernels

$\kappa\left(\bm{x},\,\bm{x}^{\prime}\right)=\begin{cases}(\gamma\bm{x}^{\top}\bm{x}^{\prime}+c_{0})^{d}&\quad\mathrm{(polynomial)}\\\
\exp\left(-\gamma\|\bm{x}-\bm{x}^{\prime}\|^{2}\right)&\quad\mathrm{(Gaussian)}\\\
\tanh\left(\gamma\bm{x}^{\top}\bm{x}^{\prime}+c_{0}\right)&\quad\mathrm{(sigmoid)}\end{cases}$
(19)

###  B.3 Kernel-target alignment

The _alignment_ between two kernels is defined as

$\displaystyle A\left(K^{(1)},\,K^{(2)}\right)$ $\displaystyle=\frac{\langle
K^{(1)},\,K^{(2)}\rangle_{F}}{\sqrt{\langle
K^{(1)},\,K^{(1)}\rangle_{F}\langle K^{(2)},\,K^{(2)}\rangle_{F}}}$ (20)
$\displaystyle=\frac{\langle
K^{(1)},\,K^{(2)}\rangle_{F}}{\|K^{(1)}\|_{F}\|K^{(2)}\|_{F}},$

where $K$ is the _kernel matrix_ , obtained by taking the kernel of all pairs
of features, and

$\langle K^{(1)},\,K^{(2)}\rangle_{F}=\tr\left(K^{(1)\top}K^{(2)}\right).$

The alignment between two kernels is always lesser or equal to 1, where 1
corresponds to perfect alignment.

Assume a kernel $\kappa_{\bm{\theta}}$, parametrized by $\bm{\theta}$ and
define the _target kernel_ matrix as

$K^{*}=\mathbf{y}\mathbf{y}^{\top}.$ (21)

The _kernel-target alignment_ (KTA) of $\kappa_{\bm{\theta}}$ is the alignment
between the chosen kernel and the target,

$\displaystyle A\left(K_{\bm{\theta}},\,K^{*}\right)$
$\displaystyle=\frac{\langle
K_{\bm{\theta}},\,K^{*}\rangle_{F}}{\|K_{\bm{\theta}}\|_{F}\|K^{*}\|_{F}}$
(22) $\displaystyle=\frac{\langle
K_{\bm{\theta}},\,K^{*}\rangle_{F}}{M\|K_{\bm{\theta}}\|_{F}},$

where $K_{\bm{\theta}}$ is the kernel matrix of $\kappa_{\bm{\theta}}$.

The KTA enjoys theoretical properties such as concentration around its
expected value and generalisation [26] and therefore it is indicative of the
ability of a kernel to separate classes of data.

##  Appendix C Quantum computing notions

###  C.1 Quantum states

The basic logical unit in quantum computing is the _qubit_. Mathematically
speaking, this is a unit-norm vector in the complex 2-dimensional space
$\mathbb{C}^{2}$ defined as a linear combination of two orthogonal basis
states, $\ket{0}$ and $\ket{1}$, i.e.

$\ket{\psi}=\psi_{0}\ket{0}+\psi_{1}\ket{1},\quad\psi_{0},\,\psi_{1}\in\mathbb{C},$
(23)

where the $\ket{\cdot}$ notation is used to indicate unit vectors.

As opposed to classical bits, Equation 23 shows that a single qubit can be in
any complex superposition of the two basis states. However, reading of a
quantum state can only happen through a measurement, which will make the qubit
collapse to one of the two basis states, $\ket{0}$ or $\ket{1}$. More
specificallly, the qubit is measured as $\ket{0}$ with probability
$p_{0}=\psi_{0}^{2}$ and as $\ket{1}$ with probability $p_{1}=\psi_{1}^{2}$.
Since these are the only two possible outcomes, it must be that
$p_{0}+p_{1}=\psi_{0}^{2}+\psi_{1}^{2}=1$, which explains the unitary norm of
the qubit.

Similarly, a states of $n$ qubits is defined as a superpositions of basis
states that correspond to bitstrings, that is

$\ket{\psi}=\psi_{0}\ket{0\dots 0}+\psi_{1}\ket{0\dots
1}+\dots+\psi_{N-1}\ket{1\dots
1}\quad\psi_{0},\,\dots,\,\psi_{N-1}\in\mathbb{C},$ (24)

where $N=2^{n}$.

The exponential relation between the number of qubits and the number of
possible bitstrings speaks for the potential advantage of _quantum
superposition_ , which allows multiple classical information states to be
processed simultaneously through a quantum algorithm. Quantum superposition is
at the heart of fundamental algorithms with proved complexity improvement such
as quantum integer factoring [39] and quantum database search [40].

Nevertheless, the quantum state is inaccessible as readable information and
measurement will collapse the wavefunction to only one of the $2^{n}$ basis
states. Similarly to the single-qubit case, the basis state $\ket{i}$ has
probability $p_{i}=\psi_{i}^{2}$ of being measured and

$\sum_{i=0}^{N-1}p_{i}=\sum_{i=0}^{N-1}\psi_{i}^{2}=1.$ (25)

Quantum states can be prepared by applying unitary transformations to a
reference state, such as the all-zero state,

$\ket{\psi}=U\ket{0}^{\otimes n},$ (26)

where $U$ is the generic unitary transformation.

Figure 11: As an example of quantum circuit, the GHZ state can be prepared by
first creating a uniform superpositon of all qubits through the Hadamard gate
$H$ and then introducing entanglement with CNOT gates in cascade.

Figure 11 shows a unitary operation as a quantum circuit, i.e. a sequence of
single- and two-qubit operations. Here, Hadamard gates $H$ are first applied
to every qubit, where

$\displaystyle H\ket{0}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right)$ (27)
$\displaystyle H\ket{1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{0}-\ket{1}\right).$

This first layer of Hadamard gates creates the uniform superposition state

$\frac{1}{\sqrt{2^{n}}}\left(\ket{0\dots 0},\,\ket{0\dots
1},\,\dots,\,\ket{1\dots 1}\right),$ (28)

where each basis state can be sampled with the equal probability $1/2^{n}$.
This is often a starting point state in many quantum algorithms.

Following, CNOT gates act between neighbouring couples of qubits as

$\displaystyle\mathrm{CNOT}\ket{00}$ $\displaystyle=\ket{00}$ (29)
$\displaystyle\mathrm{CNOT}\ket{01}$ $\displaystyle=\ket{01}$
$\displaystyle\mathrm{CNOT}\ket{10}$ $\displaystyle=\ket{11}$
$\displaystyle\mathrm{CNOT}\ket{11}$ $\displaystyle=\ket{10}.$

CNOT gates are used to set qubits in an _entangled_ state, a condition in
which any operation on any of the qubits affects also the rest of the state.
In particular, the series of CNOT gates creates one of the maximally entangled
Greenberger-Horne-Zeilinger (GHZ) states [41], specifically

$\ket{\mathrm{GHZ}}=\frac{1}{\sqrt{2}}\left(\ket{0000}+\ket{1111}\right).$
(30)

###  C.2 Quantum embedding

State preparation can be used to embed classical data into quantum states, by
mapping the features to a unitary transformation.

$\ket{\phi\left(\bm{x}\right)}=U(\bm{x})\ket{0},$ (31)

A complete review of the different types of quantum embeddings is beyond the
current scope and the interested reader is pointed to [42] for a critical
overeview.

###  C.3 Quantum kernels

Quantum embeddings are effectively feature maps in the Hilbert space
$\mathcal{H}\subseteq\mathbb{C}^{2^{n}}$. The kernel associated with it
computes the overlap between quantum feature vectors in $\mathcal{H}$, that is

$\kappa\left(\bm{\varepsilon},\bm{\varepsilon}^{\prime}\right)=\absolutevalue{\innerproduct{\phi\left(\bm{\varepsilon}\right)}{\phi\left(\bm{\varepsilon}^{\prime}\right)}}^{2},$
(32)

where the _braket_ notation $\innerproduct{\cdot}{\cdot}$ indicates the inner
product between two vectors in $\mathcal{H}$.

By introducing Equation 31 in Equation 32, the quantum kernel can be rewritten
as

$\kappa\left(\bm{x},\,\bm{x}^{\prime}\right)=\matrixelement{0}{U^{\dagger}\left(\bm{x}^{\prime}\right)U\left(\bm{x}\right)}{0},$
(33)

which shows that the quantum kernel can be computed as the probability of the
all-zeros state, after applying the direct embedding for $\bm{x}$ and the
reversed embedding for $\bm{x}^{\prime}$.

##  Appendix D Classical and quantum SVM comparison on different
classification metrics

$N_{\mathrm{train}}$ | Accuracy | Jaccard Index | Precision | Recall | Specificity
---|---|---|---|---|---
RBF kernel
156 | 0.694 | 0.627 | 0.784 | 0.759 | 0.795
313 | 0.750 | 0.708 | 0.835 | 0.824 | 0.838
470 | 0.788 | 0.751 | 0.886 | 0.832 | 0.895
627 | 0.788 | 0.750 | 0.893 | 0.825 | 0.903
784 | 0.838 | 0.813 | 0.939 | 0.859 | 0.944
940 | 0.824 | 0.797 | 0.915 | 0.861 | 0.921
1097 | 0.848 | 0.827 | 0.938 | 0.874 | 0.943
1254 | 0.878 | 0.866 | 0.943 | 0.913 | 0.945
1411 | 0.873 | 0.860 | 0.937 | 0.912 | 0.939
1568 | 0.882 | 0.869 | 0.955 | 0.906 | 0.958
HE2W6D3 kernel
156 | 0.699 | 0.626 | 0.804 | 0.739 | 0.823
313 | 0.731 | 0.675 | 0.835 | 0.780 | 0.846
470 | 0.756 | 0.708 | 0.858 | 0.802 | 0.869
627 | 0.779 | 0.741 | 0.878 | 0.826 | 0.887
784 | 0.795 | 0.762 | 0.892 | 0.839 | 0.900
940 | 0.794 | 0.757 | 0.897 | 0.829 | 0.907
1097 | 0.805 | 0.773 | 0.902 | 0.844 | 0.910
1254 | 0.797 | 0.765 | 0.884 | 0.849 | 0.891
1411 | 0.814 | 0.785 | 0.911 | 0.851 | 0.917
1568 | 0.818 | 0.790 | 0.912 | 0.855 | 0.919
Table 1: Comparison of five different classification scores between the RBF-
SVM and the trained HE2W6D3 quantum kernel SVM. The classifiers were trained
with increasing fractions of the training dataset. Notice that the RBF-SVM
problem used $C={1}\mathrm{e}{7}$, while the HE2W6D3 quantum kernel SVM used
$C={1}\mathrm{e}{4}$, which is the highest $C$ values before the occurrence of
convergence issues.

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