# The role of data embedding in equivariant quantum convolutional neural
networks

Sreetama Das1,2,3, Stefano Martina2,3, Filippo Caruso1,2,3 1Istituto Nazionale
di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO), I-50019 Sesto
Fiorentino, Italy  2Department of Physics and Astronomy, University of
Florence, Via Sansone 1, Sesto Fiorentino, I-50019, Italy  3European
Laboratory for Non-Linear Spectroscopy (LENS), University of Florence, Via
Nello Carrara 1, Sesto Fiorentino, I-50019, Italy

###### Abstract

Geometric deep learning refers to the scenario in which the symmetries of a
dataset are used to constrain the parameter space of a neural network and
thus, improve their trainability and generalization. Recently this idea has
been incorporated into the field of quantum machine learning, which has given
rise to equivariant quantum neural networks (EQNNs). In this work, we
investigate the role of classical-to-quantum embedding on the performance of
equivariant quantum convolutional neural networks (EQCNNs) for the
classification of images. We discuss the connection between the data embedding
method and the resulting representation of a symmetry group and analyze how
changing representation affects the expressibility of an EQCNN. We numerically
compare the classification accuracy of EQCNNs with three different basis-
permuted amplitude embeddings to the one obtained from a non-equivariant
quantum convolutional neural network (QCNN). Our results show a clear
dependence of classification accuracy on the underlying embedding, especially
for initial training iterations. The improvement in classification accuracy of
EQCNN over non-equivariant QCNN may be present or absent depending on the
particular embedding and dataset used. It is expected that the results of this
work can be useful to the community for a better understanding of the
importance of data embedding choice in the context of geometric quantum
machine learning.

##  I Introduction

Quantum computing holds the promise to surpass classical supercomputers in
achieving polynomial and exponential speed-up when performing certain tasks
[1, 2]. The recent breakthrough [3] of realizing such speed-up with state-of-
the-art noisy intermediate-scale quantum (NISQ) devices [4] has drawn
widespread attention and research interest to this field. One of the most
active research lines in recent times is to connect quantum computation with
classical machine learning (ML). In this context there exist two approaches,
the first one is to study how classical ML tools can facilitate quantum
information processing tasks [5, 6, 7, 8, 9]. The other is the more frequently
studied direction, i.e. to use a quantum system itself to build machine
learning models [10, 11] and in particular quantum neural networks (QNNs) [12,
13, 14, 15, 16, 17, 18, 19]. The central component in a prototypical QNN is a
quantum circuit with single and multiple qubit gates with trainable
parameters. The latter is called parametric quantum circuit (PQC) or
variational quantum circuit (VQC) [20, 21, 22, 23, 24]. PQC-based QNNs have
been used to design quantum analogs of well-known classical ML networks, e.g.,
quantum autoencoder [22], quantum convolutional neural networks (QCNNs) [25,
26, 27], quantum generative adversarial networks (QGANs) [28, 29, 30, 31, 32,
33], quantum generative diffusion models [17, 18, 19], etc. It is important to
note here that the cost function obtained from the above networks is optimized
using a classical optimization routine. In this sense, the quantum machine
learning networks are hybrid quantum-classical networks. Despite the potential
speed-up and signatures of success of quantum machine learning over classical
ML models [34, 35, 36, 37, 38], there exist several limitations for QNNs, the
barren plateau problem being the principal among them. The latter means that
for sufficiently deep quantum neural networks the gradient of the cost
function vanishes exponentially with the size of the PQC [39, 40, 41, 42]. On
one hand, an arbitrary PQC ideally should have high expressibility [43] to
ensure that the solution of the optimization problem is close enough to the
actual solution. On the other hand, PQCs with higher expressibility are more
prone to exhibiting barren plateau [39, 40]. It is therefore a crucial task to
mitigate barren plateau for a practical application of quantum machine
learning algorithms.

One way to improve the trainability and generalization in machine learning
algorithms is to introduce inductive bias in the network, i.e. to use some
prior known information about the dataset to build a problem-specific model
and constrain the optimization space of the network. Particularly, geometric
machine learning refers to a scheme in which the known symmetries of the
dataset are used to construct a network which respects those symmetries [44].
Such a network will contain a sufficiently good solution, yet explores a
smaller parameter space while training. For instance, the convolutional neural
network (CNN), which is greatly successful for image recognition, is
constructed by leveraging the translational symmetry of 2D images. Inspired
from this, a number of recent works study the role of symmetry in improving
QNN architectures [45, 46, 47, 48, 49, 50, 51]. The results show that a linear
map representing the QNN can adapt to a symmetric dataset if its action on the
data points commute with the action of the symmetry group (two operators $A$
and $B$ commute if $AB=BA$ or equivalently if $[A,B]\equiv AB-BA=0$). These
QNNs are called equivariant quantum neural networks (EQNNs), in analogy to
equivariant neural networks in classical geometric learning. EQNNs have less
number of trainable parameters and reduced expressibility, but they are
expected to show a faster training time and improved generalization compared
to a general QNN with high expressibility. Indeed, studies show that some
classes of symmetry-respecting QNNs are devoid of the barren plateau problem
[52, 53, 54]. Moreover, the performance of an EQNN shows improvement over a
non-equivariant QNN for pattern recognition and image classification [48, 55,
56, 54].

A set of symmetry operations form an abstract group $\mathcal{G}$. A
representation of a group $\mathcal{G}$ is a mapping $R:\mathcal{G}\rightarrow
GL$ where $GL$ is the space of invertible linear operators that preserve the
group structure of $\mathcal{G}$. Let us suppose we are given a classical data
point with a certain symmetry $\mathcal{G}$. If we encode the data point using
an $N$-dimensional quantum state, then the representations of the group
elements are $N\times N$ unitary matrices. However more than one unitary
representations for the same group is possible. Change of representation is
induced by the change in the quantum embedding of the data. In EQNN, the
action of the network on the input state must commute with representation
$R(g)$ for all group elements $g\in\mathcal{G}$. As different representations
have different commutator space, the choice of representation decides which
PQC and measurements are to be used in the EQNN. This has been discussed in
Ref. [57] where the authors conceptualize a network in which the
representation of the input state is altered at the intermediate layers by
applying a linear transformation on the quantum state. Also in Refs. [55, 54]
the authors encode classical images using an altered-basis amplitude embedding
in a way so that the resulting group representation simplifies the
construction of EQNN. However, whether the resulting EQNN has a similar
performance as before the basis-change is not clear. In this work, our
objective is to attain a better understanding of how a change in the data
embedding affects the performance of EQCNN. We present our theoretical
observations about the role of embedding and the resulting representation on
the construction of equivariant convolutional and pooling layers in EQCNNs.
For our numerical study, we choose standard amplitude embedding of images
along with other permuted-basis amplitude embedding, similar to Refs. [55,
56]. We then consider datasets in which the class labels are symmetric with
respect to reflection and $180^{\circ}$ rotation. We compare the
classification accuracy of a number of EQCNNs with different permuted basis
embeddings to a general non-equivariant QCNN for image classification. These
results show that the choice of embedding substantially affects the
performance of the EQCNN, and can be crucial for obtaining an improvement over
non-equivariant QCNNs.

The manuscript is arranged as follows. In Sec. II, we start with an
introduction to QCNN and EQCNN, followed by discussing the role of
representation in constructing an EQCNN. In Sec. III we describe the
reflection symmetry group and its different representations used in this work,
alongside presenting the unitary ansatze used in building the EQCNNs and the
non-equivariant QCNN. We present our findings in Sec. IV and conclude in Sec.
V.

Figure 1: The structure of QCNN for 10 qubits. The orange and the cyan boxes
represent respectively the parametrized convolutional and pooling ansatze. The
input quantum state is $|\psi\rangle$ and the final measurement is $M$.

##  II Equivariant quantum convolutional neural network

###  II.1 Quantum Convolutional Neural Network (QCNN)

Convolutional neural networks (CNNs) are classical ML models extensively used
for image classification, speech recognition etc [58, 59]. They consist of a
sequence of convolutional and pooling layers followed by a fully-connected
layer at the end. The convolution operation can be visualized as a $d\times d$
matrix with trainable weights, known as kernel, traversing along the height
and width of the input image. At each position of the kernel, the dot product
between it and the $d\times d$ block of the input image over which it is
placed is calculated to get a feature map of the input image. The trainable
weights of the kernel are the same within a convolutional layer. It is common
to apply a nonlinear activation function after each convolutional layer. In
the pooling layer, the dimension is reduced by aggregating over areas of the
feature map by taking the average or the maximum value. After application of a
number of convolution and pooling layers, the feature maps are flattened to 1D
and used as the input layer of a fully-connected neural network that performs
the prediction. The cost function is calculated using the output nodes and the
network is trained. One important aspect of CNN is that it can recognize a
particular feature of an image irrespective of the translation of that feature
in the image plane. Thus, CNNs respect the translational symmetry of images.

In Ref. [25], the authors proposed a quantum analogue of CNN. We show this
architecture in Fig. 1. The architecture is inspired from the multiscale
entanglement renormalization ansatz (MERA) representation of quantum many-body
states [60]. Any quantum neural network has three components– an $n$-qubit
quantum register encoding the input quantum state $|\Psi\rangle$, a PQC
$\mathcal{U}_{\theta}$ acting on $|\Psi\rangle$, and lastly a measurement $M$
on some (or all) of the qubits. In case of QCNN, $\mathcal{U}_{\theta}$
consists of a series of convolutional and pooling layers. In the
$i^{\mathrm{th}}$ convolutional layer, an $m$-qubit ($m<n$) trainable
convolutional ansatz $U_{i}(\Theta_{i})$ is applied on all combinations of
neighboring $m$ qubits, mimicking the action of a kernel in CNN. Here
$\Theta_{i}=\\{\theta_{i}^{1},\theta_{i}^{2},..,\theta_{i}^{k}\\}$ is a set of
$k$ trainable parameters which is the same for all $U_{i}(\Theta_{i})$ in the
$i^{\mathrm{th}}$ convolutional layer. In the pooling layer, from each pair of
neighboring qubits one qubit is measured in a particular basis, and
conditioned on the measurement outcomes a set of parametrized rotations are
applied on the other qubit. These actions constitute the pooling ansatz
$V_{i}(\Phi_{i})$. Similar to convolution layers the parameters
$\Phi_{i}=\\{\phi_{i}^{1},\phi_{i}^{2},..,\phi_{i}^{l}\\}$ are shared within a
pooling layer. Note that one can also relax the translational invariance
condition by using convolutional and pooling ansatze with different parameters
within a layer [61]. Following the pooling layer, all the measured qubits are
traced out, thus reducing the effective dimension of the system. In real
quantum devices this is synonymous with ignoring the qubits in the subsequent
stages after the measurement, and considering only the dynamics on the
remaining qubits. The sequence of a convolutional layer followed by a pooling
layer is then repeated until a small fraction of qubits are left. In analogy
to the fully-connected part of a CNN, one can then apply a general PQC on the
remaining qubits at the end and finally measure them to obtain the prediction.
The network is then trained using a suitable loss function and optimization
algorithm. Compared to a generic deep quantum neural network, in this case due
to progressive qubit reduction, the structure has a shallower depth of
$\mathcal{O}(\log n)$, which is preferable for training a QNN. In [25] the
authors demonstrated the utility of QCNN for topological phase recognition of
quantum many-body states as well as optimization of error correcting codes.
Later, this architecture of QCNN was as well studied for classical image
classification in [62]. In this case, there is an additional step in which the
classical images are encoded as quantum states and used as the input to the
network. In this regard, two widely used embedding methods are amplitude
embedding and qubit embedding– the former encode each classical information
value in one of the quantum state amplitudes while the latter employs one
qubit to encode one value. For large images amplitude embedding is
advantageous as we need logarithmically less qubits compared to number of
pixels to encode them. For smaller images, qubit embedding is preferred since
the information about each pixel is more easily encoded in the single qubits.
In [62], the convolution and pooling layers were applied until only a single
qubit remained, on which Pauli-$Z$ measurement is performed. This structure is
suitable for binary classification since the expectation value of the
measurement operator on a qubit can be associated with the two classes to be
distinguished. Overall, good classification accuracy for simple datasets
(e.g., MNIST and Fashion MNIST) can be reached. However, one must note that
the state-of-the-art QCNN is still too immature to learn the complex features
of general complex image datasets. Thus, in general CNNs can achieve higher
classification accuracies than QCNNs.

Figure 2: A $2\times 2$ image and its transformations under reflection and
rotation by $180^{\circ}$. The table shows group representations of these two
symmetries with changing embedding of the image as a quantum state.
$\mathcal{N}$ is the quantum state normalization factor. Figure 3: The
relation between quantum-encoded image state $|\Psi\rangle$, basis-permuting
matrix $A$, and the representations $R(g)$ and $R^{\prime}(g)$ before and
after the basis permutation, respectively.

###  II.2 Equivariant QCNN with label-symmetry

The architecture of CNNs respect the translational symmetry of 2D images.
However, there can exist additional symmetries in a dataset. Particularly, for
classical images one of the most commonly occurring symmetry is label symmetry
in which the class labels of the images remain unchanged under a set of
operations. For example, in MNIST dataset the labels of digits 1, 8 and 0 are
reflection invariant. Let us consider a binary-classification task for a
dataset $\mathcal{X}$ with images $x_{i}$ and corresponding class labels
$y_{i}\in\mathcal{Y}$ where $\mathcal{Y}=\\{0,1\\}$. There is a function
$f:\mathcal{X}\rightarrow\mathcal{Y}$ that maps the images to their labels.
The task of the QCNN is to prepare a quantum circuit $f^{\prime}_{\theta}$
that closely approximates $f$. A set of operations $\mathcal{G}$ form a label
symmetry group for $\mathcal{X}$ if for applications of $g\in\mathcal{G}$ to
the images, the assigned labels remain unchanged, i.e.,

$f(g(x_{i}))=y_{i}=f(x_{i})\quad\forall x_{i}\in\mathcal{X},\quad\forall
g\in\mathcal{G}.$ (1)

Note that the data points themselves may not be invariant under the group
action, i.e., $g(x_{i})\neq x_{i}$ in general. The QCNN $f^{\prime}_{\theta}$
used to classify $\mathcal{X}$ should not predict different labels for inputs
related by the symmetry operations. The QCNNs that satisfy this condition are
called EQCNNs. Recent works explore EQNN and its improvements over a general
non-equivariant QNN for image classification [55, 56]. In particular, we use
the result from [55] which shows that a PQC $\mathcal{U}_{\theta}$ along with
a measurement $M$ will construct an EQNN $f^{\prime}_{\theta}$ if the
following condition holds,

$\big{[}R(g),\mathcal{U}_{\theta}^{\dagger}M\mathcal{U}_{\theta}\big{]}=0\quad\forall
g\in\mathcal{G},$ (2)

where $R(g)$ is the unitary representation of $g\in\mathcal{G}$. It was also
proved in [49, 57] that a map $\mathcal{U}_{\theta}$ and a measurement $M$
will form an EQNN if and only if

$[R(g),\mathcal{U}_{\theta}]=0\quad\mbox{and}\quad[R(g),M]=0,\quad\forall g$
(3)

Thus if we define the commutator space of $R(g)$ to be the space of all
operators that commute with $R(g)$, then $\mathcal{U}_{\theta}$ and $M$ must
belong to that commutator space. It is easy to see that Eq. (2) is satisfied
when Eq. (3) is true.

In the context of EQCNN, $\mathcal{U}_{\theta}$ can be obtained by assuring
that each convolution and pooling layer is equivariant with respect to the
representation in the output of the previous layer. Since in our case these
layers are composed of two-qubit local variational ansatze, it is sufficient
to make these ansatze equivariant with respect to the two-qubit local symmetry
representations $r(g)$. In detail, for the $i^{\mathrm{th}}$ layer,

$[r(g),U_{i}(\Theta_{i})]=0\quad\mathrm{and}\quad[r(g),V_{i}(\Phi_{i})]=0\quad\forall
r(g).$ (4)

In particular, $U_{i}$ and $V_{i}$ are constituted of parameterized single-
qubit and two-qubit rotational gates which can be expressed as $e^{-i\theta
H}$, where $H$ is a Hermitian operator and is called the _generator_ of that
gate. In this case, Eq. (4) is satisfied if

$[r(g),H]=0$ (5)

holds for all the generators of the single and two qubit gates used in the
construction of $U_{i}$ and $V_{i}$. Given a symmetry representation $R(g)$,
these generators can be found using a variety of methods as discussed in Refs.
[49, 48, 57].

###  II.3 Role of data embedding

If an image has symmetries, then the representation of that symmetry group
depends on the particular way the image is encoded using qubits. In this work,
we will consider amplitude embedding (AE) in which $N$ pixel values are
encoded as the amplitudes of the basis states of $n=\log N$ qubits. Let us
consider two simple label symmetries– reflection with respect to the vertical
axis and rotation by $180^{\circ}$. In standard AE the pixel values of the
classical image are encoded row-wise into the amplitudes of the quantum state.
Let us consider a small image for which $N=4$ and $n=2$. For standard AE the
representation of reflection group is $\\{\mathbb{I}^{\otimes
2},\mathbb{I}\otimes X\\}$ and that for the $180^{\circ}$ rotation group is
$\\{\mathbb{I}^{\otimes 2},X\otimes X\\}$, where $\mathbb{I}$ is the qubit
identity operator and $X$ is the Pauli-$X$ operator. However, if one modifies
the order in which the pixels are encoded, in a way presented in the table in
Fig. 2, then the representation for the reflection symmetry group becomes
$\\{\mathbb{I}^{\otimes 2},X\otimes\mathbb{I}\\}$ and that for the
$180^{\circ}$ rotation symmetry group becomes $\\{\mathbb{I}^{\otimes
2},\mathbb{I}\otimes X\\}$.

In a more general scenario, let us suppose $A$ is a $2^{n}\times 2^{n}$ matrix
that acts on the standard amplitude-encoded input state $|\Psi\rangle$ and
alters the order in which pixel values are encoded. Thus $A$ has to be a
permutation matrix which permutes the coefficients of the canonical basis
states of $\mathcal{H}^{n}$, or equivalently permutes the basis states. For
input state $|\Psi\rangle$ with standard AE the group representation is
$R(g)$, while for input state $|\Psi^{\prime}\rangle$ with basis-permuted AE,
it is $R^{\prime}(g)$. If $|\Phi\rangle$ and $|\Phi^{\prime}\rangle$ are the
states obtained after applying the corresponding symmetry transformations,
then the relation between them can be summarized as that in Fig. 3. From this
we can write the following–

$\displaystyle R^{\prime}(g)A|\Psi\rangle=AR(g)|\Psi\rangle$
$\displaystyle\Rightarrow$ $\displaystyle R^{\prime}(g)=AR(g)A^{-1}.$ (6)

In other words, $R(g)$ and $R^{\prime}(g)$ are related by a basis permutation
and they are two equivalent representations of the same group. In the same
way, the unitary operators in the commutator space of $R(g)$ are related by a
basis permutation $A$ to the unitary operators in the commutator space of
$R^{\prime}(g)$.

However, the architecture of EQCNNs is a special case of QCNN differing in the
following aspects from a general QNN.

  1. 1.

$\mathcal{U}_{\theta}$ is composed of $m$-qubit local unitary ansatze which
are equivariant with respect to the locally acting components of the full
symmetry representation. Whether these locally equivariant ansatze can realize
the full set of globally equivariant ansatze depends on the particular group
and its representations [63].

  2. 2.

Due to translational symmetry, in one particular layer of EQCNN all local
ansatze must be the same. However, the local symmetry representations may not
be same for all groups of $m$ qubits. Let us suppose that in the
$i^{\mathrm{th}}$ layer there are $s$ distinct local representations
$r_{i}^{j}(g)$ ($j=1,2,...,s$) and the corresponding sets of equivariant
generators are $\sigma_{i}^{j}$. Then the local ansatze in this layer must be
generated from the elements which are common to all $\sigma_{i}^{j}$. Thus,
the translation symmetry hinders the use of the full set of equivariant
generators with respect to the local symmetries and reduces the expressibility
of the network. In other words, the EQCNN is capable of taking advantage of
only some of the local symmetries that exist in groups of $m$ qubits and not
the global or more-than-$m$ qubit symmetries.

  3. 3.

This dependence on local symmetries becomes more significant after each
pooling layer when some of the qubits are traced out. It means the local
ansatze are not applied on all the local groups of qubits.

From the discussion above, it is evident that EQCNN ansatze depend on the
local symmetries and the reduced symmetries in the subsequent layers. Thus, a
change in representation $R(g)$ is expected to vary the expressibility. We
note here again that the changes in representation and the expressibility is
accompanied with a basis permutation of the input state itself. Though the set
of input states are different for different representations, the correlations
between images related by a symmetry operation remain unchanged.

There are a few earlier works where particular basis-permuted embedding was
used in order to facilitate construction of the ansatze that satisfy Eq. (3)
[55, 54]. In this work, our aim is to compare the non-equivariant QCNN with
standard AE to EQCNNs with a number of basis-permuted AEs. In order to do
this, we first choose a particular $R^{\prime}(g)$ and then we find the
corresponding $A$ to be applied to the standard amplitude encoded state
$|\Psi\rangle$ to get the basis-permuted embedding $|\Psi^{\prime}\rangle$.
Observing that the matrices that are related by a similarity transformation
have the same eigenvalues, we can write

$\displaystyle M^{-1}R(g)M=N^{-1}R^{\prime}(g)N$ $\displaystyle\Rightarrow$
$\displaystyle R^{\prime}(g)=NM^{-1}R(g)MN^{-1},$ (7)

where $M$ and $N$ are matrices constituted of eigenvectors of respectively
$R(g)$ and $R^{\prime}(g)$. Thus, the desired value of $A$ is

$A=NM^{-1}.$ (8)

In the next section we discuss in detail the architecture of our EQCNN and
non-equivariant QCNN for different embeddings.

Figure 4: (a) The convolutional ansatze and (b) the pooling ansatze used in
this work. The specific use case of each ansatz is discussed in Sec. III.

Dataset |  # Qubits |  Embedding | Ansatz |  M
---|---|---|---|---
|  |  |  U1 |  V1 |  U2 |  V2 |  U3 |  V3 |  U4 |  V4 |
Fashion MNIST |  8 |  AE | 1 | 6 | 1 | 6 | 4 | 7 | - |  - |  $\sigma_{Z}$
|  |  AE 1 | 1 | 6 | 2 | 6 | 3 | 6 | - |  - |  $\sigma_{X}$
|  |  AE 2 | 1 | 7 | 5 | 9 | 5 | 9 | - |  - |  $\sigma_{Z}$
Cifar10/ Blood MNIST |  10 |  AE/AE1 | 1 | 6 | 1 | 7 | 1 | 9 | 4 |  8 |  $\sigma_{Z}$
|  |  AE 1/AE | 1 | 6 | 2 | 6 | 3 | 6 | 3 |  6 |  $\sigma_{X}$
|  |  AE 2 | 1 | 7 | 5 | 9 | 5 | 9 | 5 |  9 |  $\sigma_{Z}$

Table 1: The datasets, their quantum embeddings, corresponding circuit indices
of convolutional and pooling ansatze from Fig. 4 and measurements $M$ used in
the construction of EQCNN. Here we use AE, AE 1 and AE 2 to indicate
respectively standard AE, basis-permuted AE 1 and basis-permuted AE 2

.

##  III EQCNN with different data embeddings

We consider classification of images whose labels remain invariant under a
reflection about the vertical axis and $180^{\circ}$ rotation. In both cases,
the underlying groups have only two group elements– the identity operation
which keeps the image unchanged, and the operation corresponding to reflection
or rotation. They form the abstract group $\mathbb{Z}_{2}$. In our
experiments, for the reflection-symmetric images we choose to employ the
classes 0 (tshirt/top) and 1 (trouser) of Fashion MNIST dataset, and classes 1
(car) and 2 (bird) of Cifar10 dataset. For rotationally symmetric images, we
perform classification between classes 1 and 6, as well as between classes 4
and 5 of Blood MNIST dataset. The Fashion MNIST dataset is a collection of
70000 greyscale images of different clothing articles and accessories. Each
image has dimension $28\times 28$ pixels and the dataset has 10 classes in
total. The Cifar10 dataset has 60000 RGB images of animals and vehicles
divided into 10 classes, each image having dimension $32\times 32$ pixels. The
Blood MNIST dataset is a collection of $32\times 32$ pixels RGB images of
human Blood cells divided into 8 classes. Each class denotes a cell type and
contains a few hundred to about a thousand of images. We downsize the Fashion
MNIST images to $16\times 16$ pixels and encode them using 8 qubits. Regarding
the Cifar10 and Blood MNIST images, we keep the dimension unchanged but they
are transformed from RGB to greyscale and we use 10 qubits for the embedding.
In all cases, we use two-qubit convolutional ansatze and assume periodicity in
the qubit register i.e. the first and the last qubits are nearest neighbours.

For the non-equivariant QCNN we use standard amplitude embedding. For
constructing $\mathcal{U}_{\theta}$, it is possible to choose from a large
number of non-equivariant convolutional and pooling ansatze, which may affect
the overall performance of the QCNN. In this work, we build the two-qubit
convolutional ansatze by applying a parametrized arbitrary rotation on each
qubit and a $CNOT$ gate to entangle them, as shown in circuit 3 in Fig. 4,
where
$\mathrm{Rot}(\theta_{1},\theta_{2},\theta_{3})=R_{z}(\theta_{3})R_{y}(\theta_{2})R_{z}(\theta_{1})$.
As the pooling ansatze we use parametrized controlled $z$ and controlled $x$
rotations on the target qubit, when the control qubit is in computational
basis states $|1\rangle$ and $|0\rangle$ respectively. This is presented as
circuit 9 in Fig. 4. In the last step we measure the remaining qubit 1 in $z$
basis.

For the EQCNN, we use standard AE along with two different basis-permuted
embeddings, namely basis-permuted _AE 1_ and _basis-permuted AE 2_.

####  III.0.1 Standard AE

In standard amplitude embedding of 2D square images using $n$ qubits, the
representation of the reflection group is

$\mathcal{G}_{\mathrm{ref}}\equiv\\{\mathbb{I}^{n},\mathbb{I}_{1}\otimes\mathbb{I}_{2}\otimes...\otimes\mathbb{I}_{n/2}\otimes
X_{(n/2)+1}\otimes...\otimes X_{n}\\},$ (9)

where $\mathbb{I}^{n}$ is the $n$-qubit identity operator, and
$\mathbb{I}_{i}$ and $X_{i}$ denote respectively the identity operator and
Pauli-$X$ operator applied on $i^{\mathrm{th}}$ qubit. The second element of
$\mathcal{G}$ corresponding to reflection operation, leaves the first half
qubits unchanged and flips the second half. This is because, in the standard
AE of square images, the amplitudes related to a single row span all the bases
with fixed values for the first half qubits and all the possible combinations
of $0$ and $1$ for the second half qubits. For a rectangular $2^{n_{1}}\times
2^{n_{2}}$ image, the group representation becomes,

$\mathcal{G}_{\mathrm{ref}}\equiv\\{\mathbb{I}^{n},\mathbb{I}_{1}\otimes\mathbb{I}_{2}\otimes...\otimes\mathbb{I}_{n_{1}}\otimes
X_{n_{1}+1}\otimes...\otimes X_{n_{1}+n_{2}}\\}.$ (10)

Similarly, one can check that the representation of $180^{\circ}$ rotation
group for standard AE is,

$\mathcal{G}_{\mathrm{rot}}\equiv\\{\mathbb{I}^{n},X_{1}\otimes
X_{2}\otimes...\otimes X_{n}\\}.$ (11)

####  III.0.2 Basis-permuted AE 1

Basis-permuted AE 1 swaps the representations of reflection and rotation
groups in Eq. (9) and Eq. (11). We choose a basis-permuting matrix $A$ such
that the image is reflected when $X$ is applied on all the qubits. Thus the
representation of reflection group becomes,

$\mathcal{G}_{\mathrm{ref}}\equiv\\{\mathbb{I}^{n},X_{1}\otimes
X_{2}\otimes...\otimes X_{n}\\}.$ (12)

For images with rotational symmetry, we apply $A^{-1}$ to permute the basis
vectors so that the representation of the $180^{\circ}$ rotation group
becomes,

$\mathcal{G}_{\mathrm{rot}}\equiv\\{\mathbb{I}^{n},\mathbb{I}_{1}\otimes\mathbb{I}_{2}\otimes...\otimes\mathbb{I}_{n/2}\otimes
X_{(n/2)+1}\otimes...\otimes X_{n}\\}.$ (13)

####  III.0.3 Basis-permuted AE 2

We permute the basis vectors in a way such that the representation of both the
reflection and rotation group becomes

$\mathcal{G}_{\mathrm{ref/rot}}\equiv\\{\mathbb{I}^{n},\mathbb{I}_{1}\otimes
X_{2}\otimes\mathbb{I}_{3}\otimes
X_{4}\otimes...\otimes\mathbb{I}_{n-1}\otimes X_{n}\\},$ (14)

i.e., the identity operator and Pauli-X operator are applied alternatively on
the qubits. The matrix $A$, however, is different for the two groups in this
case.

Figure 5: Test set accuracy obtained from equivariant and non-equivariant
QCNNs when using maximum possible set of equivariant generators. The vertical
axis shows the average accuracy obtained over 10 randomly initialized runs and
the standard deviation is indicated with the shaded area. (a) Classification
of classes $0$ and $1$ of Fashion MNIST dataset. The inset shows a magnified
part of the same with the horizontal axis spanning a few hundreds of
iterations. (b) Classification of classes $1$ and $2$ of Cifar10 dataset.

Now we describe in detail the construction of the EQCNNs in these three cases.
We try to use the largest possible set of equivariant generators in each case
for constructing the equivariant convolutional and pooling ansatze, this is
summarized in Table 1. However, we also investigate a scenario in which a
subset of equivariant generators is used. All the convolutional and pooling
ansatze used in this work are presented in Fig. 4.

Let us consider the standard AE for reflection group for which the
representation is given by Eq. (9). In the first layer of the EQCNN,
equivariant convolutional and pooling layers can be built if $U_{1}$ and
$V_{1}$ commutes with all the possible two-qubit local group representations
within $\mathcal{G}$ which are
$\\{\mathbb{I}\otimes\mathbb{I},\mathbb{I}\otimes
X,X\otimes\mathbb{I},X\otimes X\\}$. To satisfy the commuting condition, we
use circuit 1 in Fig. 4(a) for $U_{1}$ and the circuit 6 in Fig. 4(b) for
$V_{1}$ (note that the control qubit is in the $x$ basis). One can check that
these are the only ansatze that commutes with all local representations in
this layer. In circuit 6, we have used two $R_{x}$ gates with different
parameters to draw an analogy with the other pooling circuits, however one can
merge them into a single parametrized $R_{x}$ gate to be applied when the
control qubit is either in $|0\rangle$ or $|1\rangle$. Afterwards, we trace
out the even-indexed qubits leaving the reduced representation
$\\{\mathbb{I}^{5},\mathbb{I}_{1}\otimes\mathbb{I}_{3}\otimes\mathbb{I}_{5}\otimes
X_{7}\otimes X_{9}\\}$ for $n=10$ when considering the Cifar10 dataset, and
$\\{\mathbb{I}^{4},\mathbb{I}_{1}\otimes\mathbb{I}_{3}\otimes X_{5}\otimes
X_{7}\\}$ for $n=8$ when considering the Fashion MNIST dataset. The two-qubit
local representations for the convolutional ansatze remain unchanged, thus in
the second layer we still use circuit 1 for $U_{2}$. In the second pooling
layer, we will trace out qubits 3 and 7 and apply conditional rotations on
qubits 1 and 5. The local representations are
$\\{\mathbb{I}_{1}\otimes\mathbb{I}_{3},\mathbb{I}_{5}\otimes X_{7}\\}$ for
$n=10$, and $\\{\mathbb{I}_{1}\otimes\mathbb{I}_{3},X_{5}\otimes X_{7}\\}$ for
$n=8$. For the former, both circuit 7 and circuit 8 are equivariant
generators, however since we can use only one ansatze in a pooling layer, we
choose circuit 7 without loss of generality. For $n=8$ we use circuit 6 as
$V_{2}$. In the third layer, the reduced representations are
$\\{\mathbb{I}^{3},\mathbb{I}_{1}\otimes\mathbb{I}_{5}\otimes X_{9}\\}$ for
$n=10$ and $\\{\mathbb{I}^{2},\mathbb{I}_{1}\otimes X_{5}\\}$ for $n=8$. For
the former, we use the circuit 1 as $U_{3}$. For pooling, we trace out qubit 5
by applying conditional rotation on qubit 1, thus $V_{3}$ should commute with
$\\{\mathbb{I}_{1}\otimes\mathbb{I}_{5}\\}$. Here, we are free to use any
pooling ansatze, so we choose to use the same one as in the non-equivariant
QCNN, i.e. circuit 9. For $n=8$, $U_{3}$ should commute with
$\\{\mathbb{I}_{1}\otimes X_{5}\\}$, thus both circuit 1 and circuit 4 are
equivariant ansatze. Since we have already used circuit 1 in previous layers,
we use circuit 4 for $U_{3}$ and the circuit 7 for $V_{3}$. At this point for
$n=8$ we measure the remaining qubit 1 in $z$ basis to get the result. For
$n=10$ instead, we still have two qubits left with the reduced representation
being $\\{\mathbb{I}^{2},\mathbb{I}_{1}\otimes X_{5}\\}$. We apply a further
layer with circuit 4 as $U_{4}$. For the pooling layer, this time we choose
circuit 8 as $V_{4}$. We measure the remaining qubit 1 in $z$ basis.

Figure 6: Test set accuracy of Blood MNIST images obtained from equivariant
and non-equivariant QCNNs when using maximum possible set of equivariant
generators. The vertical axis shows the average accuracy obtained over 10
randomly initialized runs and the standard deviation is indicated with the
shaded area.

For $180^{\circ}$ rotation group with standard AE, at every layer the reduced
representation is a tensor product of Pauli-X matrices. Thus, $U_{i}$ and
$V_{i}$ must commute with $\\{\mathbb{I}^{\otimes 2},X\otimes X\\}$ for all
layers. The convolutional ansatze that satisfies the latter are circuit 1,
circuit 2 and circuit 3 in Fig. 4(a). Therefore, we use circuit 1 for $U_{1}$,
circuit 2 for $U_{2}$, and circuit 3 for $U_{3}$ and $U_{4}$. For pooling
ansatze, we are left with the unique choice of circuit 6 for $V_{i}$ at every
layer. At the end, to obtain the prediction, we measure the remaining qubit 1
in the $x$ basis.

For basis-permuted AE 1, we just swap the EQCNN structure for reflection and
rotation group discussed above.

For basis-permuted AE 2, in the first layer, $U_{1}$ must commute with
$\\{\mathbb{I}\otimes X,X\otimes\mathbb{I}\\}$ and $V_{1}$ must commute with
$\\{\mathbb{I}\otimes X\\}$, thus we use respectively circuit 1 and circuit 7.
In all the subsequent layers, the reduced group representation is simply
$\\{\mathbb{I}^{\otimes k}\\}$, $k$ being the number of qubits in each layer.
Since no equivariance constraint is imposed, we use the circuit 5 for $U_{i}$
and the circuit 9 for $V_{i}$ for $i>1$. In the end, we measure the remaining
qubit in the $z$ basis to get the result.

Note that, one can obtain the same EQCNN for symmetry representations in Eq.
(9) and Eq. (14) by changing the qubits on which the first pooling layer acts.
To elaborate, one can choose to trace out qubits $\\{6,7,8,9,10\\}$ in the
first pooling layer when the representation is that in Eq. (9). However, we
consider a more realistic scenario in which the two-qubit gate connectivity
between the qubits is constrained due to the underlying architecture of the
quantum hardware. In our case, we implicitly assume that the gate connectivity
is exactly the one shown in Fig. 1. Therefore, our convolutional and pooling
layers act on different set of qubits for these two representations.

Figure 7: Test set accuracy of (a) Fashion MNIST and (b) Cifar10 images
obtained from equivariant and non-equivariant QCNNs when using a subset of
equivariant generators. The other details remain same as that in the caption
of Fig. 5. Figure 8: Test set accuracy obtained from equivariant and non-
equivariant QCNNs for Blood MNIST dataset when using a subset of equivariant
generators. All other details remain same as in the caption of Fig. 6.

##  IV Results

We have used Pennylane [64] quantum simulator to implement the networks. In
this work we consider only binary classification. For each dataset we choose a
batch size $p$, which is the number of randomly sampled images used for one
training iteration. As loss function, we use mean squared error (MSE) which is
defined as

$C=\frac{1}{p}\sum\limits_{i=1}^{p}(f^{\prime}_{\theta}(x_{i})-y_{i})^{2},$
(15)

where $f^{\prime}_{\theta}(x_{i})$ is the expectation value of the measurement
operator for the $i^{\mathrm{th}}$ input quantum state in the batch. Since the
expectation values of Pauli operators lies in the range $(-1,1)$, the class
labels corresponding to the two classes are mapped to $y_{i}\in\\{-1,1\\}$ for
all the datasets. We use Nesterov moment optimizer for training with a
learning rate 0.01. We train the network for 2000 or 3000 iterations depending
on the dataset. This scheme is run for 10 instances in parallel, for each of
which the parameters are randomly initialized. We calculate the average test
accuracies and standard deviations over these parallel runs after each 10
iterations.

First, we discuss the results when the maximum possible set of equivariant
generators are used. In the following paragraphs, we use ‘AE 1’ and ‘AE 2’ to
imply respectively basis-permuted AE 1 and basis-permute AE 2. For the Fashion
MNIST dataset, the training is performed for $2000$ iterations of batches
composed of 32 randomly-sampled images. We plot the average test set
accuracies and standard deviations for the different equivariant and non-
equivariant models with increasing number of iterations in Fig. 5(a). Even
though Fashion MNIST is a relatively simple greyscale dataset, the advantage
of using an EQCNN over non-equivariant QCNN is clear from the plots. Standard
AE and AE 2 achieve a higher accuracy in a much shorter time. The performance
of AE 1 lags behind and is comparable to non-equivariant QCNN. All EQCNNs have
significantly lower standard deviation compared to the non-equivariant QCNN.

In Fig. 5(b) we present the test set accuracies for Cifar10 images. In this
case, the network is trained for $3000$ iterations with batch size 64. We
again observe a very fast convergence and higher accuracy for standard AE and
AE 2 for lower number of iterations, as well as very low standard deviation.
Compared to that, AE 1 shows a slower convergence, lower accuracy and high
standard deviation. Overall, non-equivariant QCNN achieves highest accuracy
for longer training iterations.

The results for Blood MNIST dataset are presented in Fig. 6. We train the
network for 2000 iterations with batch size 32 and 64 respectively for
classifying between classes $\\{4,5\\}$ and classes $\\{1,6\\}$. Remember that
in this case the representations corresponding to standard AE and AE 1 are
swapped. Let us consider the classification of classes 4 and 5. The test
accuracies show a similar behaviour to those of reflection-symmetric datasets
wherein AE 1 and AE 2 performs better than the others for lower iterations.
For longer iterations, all equivariant and non-equivariant QCNNs performs
equally well. For classification of classes 1 and 6, in contrast to all other
results, AE 1 has significantly lower accuracy for low number of iterations,
while all other embeddings performs equally well.

Let us now see how the above trends in accuracy change when using a subset of
the equivariant generators. For this, we replace all use-cases of circuit 2
and circuit 3 by circuit 1. Thus, for reflection-symmetric images, the EQCNN
corresponding AE 1 is now generated only from $R_{x}$ and $R_{xx}$ gates. The
same applies for rotationally-equivariant EQCNN corresponding to standard AE.
We also replace all use-cases of circuit 7 by circuit 8, i.e. all pooling
layers are generated from $R_{z}$ and $R_{x}$ gates. The resulting behaviour
is presented in Fig. 7 and Fig. 8. For reflection-symmetric images, this
improves the accuracy obtained from AE 1, which either surpasses or matches
closely to that obtained from non-equivariant QCNN. For both cases of Blood
MNIST dataset, the accuracy of AE 1 significantly decreases.

We note that the non-equivariant convolutional ansatze has six trainable
parameters compared to three trainable parameters in all the equivariant
convolutional ansatze. To compare these two QCNNs with an equal number of
trainable parameters, we append circuit 2 to circuit 1 to build a new
convolutional ansatze with six trainable parameters, and use it in all the
layers of EQCNN with AE 1 for classification of Cifar10 dataset. In Fig. 9, we
present the resulting behaviour which shows a higher accuracy for EQCNN
compared to the non-equivariant QCNN.

Overall, we observe that the performance of an EQCNN varies depending the
classical-to-quantum embedding. In particular, when the group representation
is a tensor product of Pauli-X matrices acting on all the qubits, the EQCNN
has a lower accuracy. On the other hand, when the representation is a tensor
product of Pauli-X matrices acting on half of the qubits and the identity
operator acting on the rest half, the EQCNNs have a faster convergence and
higher accuracy. For the latter, the pooling choice can make a little
difference in the initial training regime, i.e. the EQCNNs show slightly
different performance when the pooling layers act on different set of qubits.
We note, however, that this behaviour may not show for every dataset, as
evident from Fig. 5.

Figure 9: Test set accuracy of Cifar10 for non-equivariant QCNN and EQCNN with
6 trainable parameters in the convolutional ansatze.

##  V Conclusion

Quantum machine learning, though currently is at its inception, has already
been useful in designing novel ML-based algorithms and also has shown some
advantages over classical ML. However, there are still a number of open
problems when it comes to understanding how to ensure a sufficiently good
performance from a QNN. Equivariant QNNs are promising candidates to improve
the training and generalization of quantum machine learning algorithms. A
typical application of quantum machine learning is image classification which
is an ubiquitous task in many daily-life scenarios. For this task, it is
possible to construct equivariant QCNN compatible with the label symmetry of
images additionally respecting the general translational symmetry of 2D
images. In this work, we have explored the connections between the classical-
to-quantum embedding of images, the resulting representation of a symmetry
group, and the structure of the EQCNN respecting that symmetry. We considered
datasets of images characterized by reflection and rotation symmetry and
different amplitude embeddings of these images obtained by basis-permutation.
Our theoretical observations ascertained that the local representations play a
crucial role in deciding the equivariant ansatze to be used and hence the
expressibility of the EQCNN. Our numerical results support this by showing a
largely varying test set classification accuracy corresponding to different
embeddings. It will be interesting to explore if instead of using 2-qubit
local ansatze, an $m$-qubit ansatz with $m>2$ can reduce the dependency of the
EQCNN on local symmetries. It is also possible to compare amplitude embedding
with other kinds of embedding, e.g. qubit embedding and dense qubit embedding
[62], to investigate their effect on the EQCNN performances. Finally, one can
also run the QCNN circuits in the real quantum hardwares, for example by using
the corresponding plugins provided by Pennylane. In this case, it will be
interesting to see if EQCNNs have better noise robustness compared to non-
equivariant QCNNs due to reduced number of parametrized gates used in the
former.

###### Acknowledgements.

This work was supported by the European Commission’s Horizon Europe Framework
Programme under the Research and Innovation Action GA n. 101070546–MUQUABIS,
by the European Union’s Horizon 2020 research and innovation programme under
FET-OPEN GA n. 828946–PATHOS, by the European Defence Agency under the project
Q-LAMPS Contract No B PRJ- RT-989, and by the MUR Progetti di Ricerca di
Rilevante Interesse Nazionale (PRIN) Bando 2022 - project n. 20227HSE83 –
ThAI-MIA funded by the European Union - Next Generation EU. S.M. acknowledges
financial support from PNRR MUR project PE0000023-NQSTI. S.D. and S.M. thank
Paolo Braccia for useful discussions regarding some of the initial ideas.

## References

  * Shor [1997] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing 26, 1484 (1997), https://doi.org/10.1137/S0097539795293172 .
  * Grover [1996] L. K. Grover, A fast quantum mechanical algorithm for database search, in  _Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing_, STOC ’96 (Association for Computing Machinery, New York, NY, USA, 1996) p. 212–219.
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Preskill [2018] J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  * Huang  _et al._ [2022a] H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill, Provably efficient machine learning for quantum many-body problems, Science 377, eabk3333 (2022a), https://www.science.org/doi/pdf/10.1126/science.abk3333 .
  * Martina  _et al._ [2022] S. Martina, L. Buffoni, S. Gherardini, and F. Caruso, Learning the noise fingerprint of quantum devices, Quantum Machine Intelligence 4, 8 (2022).
  * Martina  _et al._ [2023a] S. Martina, S. Gherardini, and F. Caruso, Machine learning classification of non-markovian noise disturbing quantum dynamics, Physica Scripta 98, 035104 (2023a).
  * Martina  _et al._ [2023b] S. Martina, S. Hernández-Gómez, S. Gherardini, F. Caruso, and N. Fabbri, Deep learning enhanced noise spectroscopy of a spin qubit environment, Machine Learning: Science and Technology 4, 02LT01 (2023b).
  * [9] E. Canonici, S. Martina, R. Mengoni, D. Ottaviani, and F. Caruso, Machine learning based noise characterization and correction on neutral atoms nisq devices, Advanced Quantum Technologies n/a, 2300192, https://onlinelibrary.wiley.com/doi/pdf/10.1002/qute.202300192 .
  * Dalla Pozza  _et al._ [2022] N. Dalla Pozza, L. Buffoni, S. Martina, and F. Caruso, Quantum reinforcement learning: the maze problem, Quantum Machine Intelligence 4, 11 (2022).
  * Das  _et al._ [2023] S. Das, J. Zhang, S. Martina, D. Suter, and F. Caruso, Quantum pattern recognition on real quantum processing units, Quantum Machine Intelligence 5, 16 (2023).
  * Schuld  _et al._ [2015] M. Schuld, I. Sinayskiy, and F. Petruccione, An introduction to quantum machine learning, Contemporary Physics 56, 172 (2015), https://doi.org/10.1080/00107514.2014.964942 .
  * Biamonte  _et al._ [2017] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature 549, 195 (2017).
  * Benedetti  _et al._ [2019] M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Science and Technology 4, 043001 (2019).
  * Perdomo-Ortiz  _et al._ [2018] A. Perdomo-Ortiz, M. Benedetti, J. Realpe-Gómez, and R. Biswas, Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers, Quantum Science and Technology 3, 030502 (2018).
  * Schuld and Killoran [2022] M. Schuld and N. Killoran, Is quantum advantage the right goal for quantum machine learning?, PRX Quantum 3, 030101 (2022).
  * Parigi  _et al._ [2023] M. Parigi, S. Martina, and F. Caruso, Quantum-noise-driven generative diffusion models (2023), arXiv:2308.12013 [quant-ph] .
  * Zhang  _et al._ [2023] B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Generative quantum machine learning via denoising diffusion probabilistic models (2023), arXiv:2310.05866 [quant-ph] .
  * Cacioppo  _et al._ [2023] A. Cacioppo, L. Colantonio, S. Bordoni, and S. Giagu, Quantum diffusion models (2023), arXiv:2311.15444 [quant-ph] .
  * Peruzzo  _et al._ [2014] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a photonic quantum processor, Nature Communications 5, 4213 (2014).
  * McClean  _et al._ [2016] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New Journal of Physics 18, 023023 (2016).
  * Romero  _et al._ [2017] J. Romero, J. P. Olson, and A. Aspuru-Guzik, Quantum autoencoders for efficient compression of quantum data, Quantum Science and Technology 2, 045001 (2017).
  * Mitarai  _et al._ [2018] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  * Schuld  _et al._ [2020] M. Schuld, A. Bocharov, K. M. Svore, and N. Wiebe, Circuit-centric quantum classifiers, Phys. Rev. A 101, 032308 (2020).
  * Cong  _et al._ [2019] I. Cong, S. Choi, and M. D. Lukin, Quantum convolutional neural networks, Nature Physics 15, 1273 (2019).
  * Li  _et al._ [2020] Y. Li, R.-G. Zhou, R. Xu, J. Luo, and W. Hu, A quantum deep convolutional neural network for image recognition, Quantum Science and Technology 5, 044003 (2020).
  * Henderson  _et al._ [2020] M. Henderson, S. Shakya, S. Pradhan, and T. Cook, Quanvolutional neural networks: powering image recognition with quantum circuits, Quantum Machine Intelligence 2, 2 (2020).
  * Braccia  _et al._ [2021] P. Braccia, F. Caruso, and L. Banchi, How to enhance quantum generative adversarial learning of noisy information, New Journal of Physics 23, 053024 (2021).
  * Braccia  _et al._ [2022] P. Braccia, L. Banchi, and F. Caruso, Quantum noise sensing by generating fake noise, Phys. Rev. Appl. 17, 024002 (2022).
  * Rudolph  _et al._ [2022] M. S. Rudolph, N. B. Toussaint, A. Katabarwa, S. Johri, B. Peropadre, and A. Perdomo-Ortiz, Generation of high-resolution handwritten digits with an ion-trap quantum computer (2022), arXiv:2012.03924 [quant-ph] .
  * Boyle and Nikandish [2023] A. O. Boyle and R. Nikandish, A hybrid quantum-classical generative adversarial network for near-term quantum processors (2023), arXiv:2307.03269 [quant-ph] .
  * Tsang  _et al._ [2023] S. L. Tsang, M. T. West, S. M. Erfani, and M. Usman, Hybrid quantum-classical generative adversarial network for high resolution image generation (2023), arXiv:2212.11614 [quant-ph] .
  * Zhou  _et al._ [2023] N.-R. Zhou, T.-F. Zhang, X.-W. Xie, and J.-Y. Wu, Hybrid quantum–classical generative adversarial networks for image generation via learning discrete distribution, Signal Processing: Image Communication 110, 116891 (2023).
  * Rebentrost  _et al._ [2014] P. Rebentrost, M. Mohseni, and S. Lloyd, Quantum support vector machine for big data classification, Phys. Rev. Lett. 113, 130503 (2014).
  * Liu  _et al._ [2021] Y. Liu, S. Arunachalam, and K. Temme, A rigorous and robust quantum speed-up in supervised machine learning, Nature Physics 17, 1013 (2021).
  * Abbas  _et al._ [2021] A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, The power of quantum neural networks, Nature Computational Science 1, 403 (2021).
  * Huang  _et al._ [2022b] H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, and J. R. McClean, Quantum advantage in learning from experiments, Science 376, 1182 (2022b), https://www.science.org/doi/pdf/10.1126/science.abn7293 .
  * Caro  _et al._ [2022] M. C. Caro, H.-Y. Huang, M. Cerezo, K. Sharma, A. Sornborger, L. Cincio, and P. J. Coles, Generalization in quantum machine learning from few training data, Nature Communications 13, 4919 (2022).
  * McClean  _et al._ [2018] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications 9, 4812 (2018).
  * Holmes  _et al._ [2022] Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  * Cerezo  _et al._ [2021] M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nature Communications 12, 1791 (2021).
  * Wang  _et al._ [2021] S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, Nature Communications 12, 6961 (2021).
  * Sim  _et al._ [2019] S. Sim, P. D. Johnson, and A. Aspuru-Guzik, Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms, Advanced Quantum Technologies 2, 1900070 (2019), https://onlinelibrary.wiley.com/doi/pdf/10.1002/qute.201900070 .
  * Bronstein  _et al._ [2021] M. M. Bronstein, J. Bruna, T. Cohen, and P. Veličković, Geometric deep learning: Grids, groups, graphs, geodesics, and gauges (2021), arXiv:2104.13478 [cs.LG] .
  * Larocca  _et al._ [2022] M. Larocca, F. Sauvage, F. M. Sbahi, G. Verdon, P. J. Coles, and M. Cerezo, Group-invariant quantum machine learning, PRX Quantum 3, 030341 (2022).
  * Mernyei  _et al._ [2022] P. Mernyei, K. Meichanetzidis, and İsmail İlkan Ceylan, Equivariant quantum graph circuits (2022), arXiv:2112.05261 [cs.LG] .
  * Skolik  _et al._ [2023] A. Skolik, M. Cattelan, S. Yarkoni, T. Bäck, and V. Dunjko, Equivariant quantum circuits for learning on weighted graphs (2023), arXiv:2205.06109 [quant-ph] .
  * Meyer  _et al._ [2023] J. J. Meyer, M. Mularski, E. Gil-Fuster, A. A. Mele, F. Arzani, A. Wilms, and J. Eisert, Exploiting symmetry in variational quantum machine learning, PRX Quantum 4, 010328 (2023).
  * Nguyen  _et al._ [2022] Q. T. Nguyen, L. Schatzki, P. Braccia, M. Ragone, P. J. Coles, F. Sauvage, M. Larocca, and M. Cerezo, Theory for equivariant quantum neural networks (2022), arXiv:2210.08566 [quant-ph] .
  * Zheng  _et al._ [2023] H. Zheng, Z. Li, J. Liu, S. Strelchuk, and R. Kondor, Speeding up learning quantum states through group equivariant convolutional quantum ansätze, PRX Quantum 4, 020327 (2023).
  * East  _et al._ [2023] R. D. P. East, G. Alonso-Linaje, and C.-Y. Park, All you need is spin: Su(2) equivariant variational quantum circuits based on spin networks (2023), arXiv:2309.07250 [quant-ph] .
  * Pesah  _et al._ [2021] A. Pesah, M. Cerezo, S. Wang, T. Volkoff, A. T. Sornborger, and P. J. Coles, Absence of barren plateaus in quantum convolutional neural networks, Phys. Rev. X 11, 041011 (2021).
  * Schatzki  _et al._ [2022] L. Schatzki, M. Larocca, Q. T. Nguyen, F. Sauvage, and M. Cerezo, Theoretical guarantees for permutation-equivariant quantum neural networks (2022), arXiv:2210.09974 [quant-ph] .
  * West  _et al._ [2023a] M. T. West, J. Heredge, M. Sevior, and M. Usman, Provably trainable rotationally equivariant quantum machine learning (2023a), arXiv:2311.05873 [quant-ph] .
  * West  _et al._ [2023b] M. T. West, M. Sevior, and M. Usman, Reflection equivariant quantum neural networks for enhanced image classification, Machine Learning: Science and Technology 4, 035027 (2023b).
  * Chang  _et al._ [2023] S. Y. Chang, M. Grossi, B. L. Saux, and S. Vallecorsa, Approximately equivariant quantum neural network for $p4m$ group symmetries in images (2023), arXiv:2310.02323 [quant-ph] .
  * Ragone  _et al._ [2023] M. Ragone, P. Braccia, Q. T. Nguyen, L. Schatzki, P. J. Coles, F. Sauvage, M. Larocca, and M. Cerezo, Representation theory for geometric quantum machine learning (2023), arXiv:2210.07980 [quant-ph] .
  * LeCun  _et al._ [2015] Y. LeCun, Y. Bengio, and G. Hinton, Deep learning, Nature 521, 436 (2015).
  * Schmidhuber [2015] J. Schmidhuber, Deep learning in neural networks: An overview, Neural Networks 61, 85 (2015).
  * Vidal [2008] G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101, 110501 (2008).
  * Liu  _et al._ [2023] Y.-J. Liu, A. Smith, M. Knap, and F. Pollmann, Model-independent learning of quantum phases of matter with quantum convolutional neural networks, Phys. Rev. Lett. 130, 220603 (2023).
  * Hur  _et al._ [2022] T. Hur, L. Kim, and D. K. Park, Quantum convolutional neural network for classical data classification, Quantum Machine Intelligence 4, 3 (2022).
  * Marvian [2022] I. Marvian, Restrictions on realizable unitary operations imposed by symmetry and locality, Nature Physics 18, 283 (2022).
  * Bergholm  _et al._ [2022] V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, S. Ahmed, V. Ajith, M. S. Alam, G. Alonso-Linaje, B. AkashNarayanan, A. Asadi, J. M. Arrazola, U. Azad, S. Banning, C. Blank, T. R. Bromley, B. A. Cordier, J. Ceroni, A. Delgado, O. D. Matteo, A. Dusko, T. Garg, D. Guala, A. Hayes, R. Hill, A. Ijaz, T. Isacsson, D. Ittah, S. Jahangiri, P. Jain, E. Jiang, A. Khandelwal, K. Kottmann, R. A. Lang, C. Lee, T. Loke, A. Lowe, K. McKiernan, J. J. Meyer, J. A. Montañez-Barrera, R. Moyard, Z. Niu, L. J. O’Riordan, S. Oud, A. Panigrahi, C.-Y. Park, D. Polatajko, N. Quesada, C. Roberts, N. Sá, I. Schoch, B. Shi, S. Shu, S. Sim, A. Singh, I. Strandberg, J. Soni, A. Száva, S. Thabet, R. A. Vargas-Hernández, T. Vincent, N. Vitucci, M. Weber, D. Wierichs, R. Wiersema, M. Willmann, V. Wong, S. Zhang, and N. Killoran, Pennylane: Automatic differentiation of hybrid quantum-classical computations (2022), arXiv:1811.04968 [quant-ph] .