# Permutation-equivariant quantum convolutional neural networks

Sreetama Das1,3, 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

The Symmetric group $S_{n}$ manifests itself in large classes of quantum
systems as the invariance of certain characteristics of a quantum state with
respect to permuting the qubits. The subgroups of $S_{n}$ arise, among many
other contexts, to describe label symmetry of classical images with respect to
spatial transformations, e.g. reflection or rotation. Equipped with the
formalism of geometric quantum machine learning, in this work we propose the
architectures of equivariant quantum convolutional neural networks (EQCNNs)
adherent to $S_{n}$ and its subgroups. We demonstrate that a careful choice of
pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for
small subgroups of $S_{n}$. Our novel EQCNN architecture corresponding to the
full permutation group $S_{n}$ is built by applying all possible QCNNs with
equal probability, which can also be conceptualized as a dropout strategy in
quantum neural networks. For subgroups of $S_{n}$, our numerical results using
MNIST datasets show better classification accuracy than non-equivariant QCNNs.
The $S_{n}$-equivariant QCNN architecture shows significantly improved
training and test performance than non-equivariant QCNN for classification of
connected and non-connected graphs. When trained with sufficiently large
number of data, the $S_{n}$-equivariant QCNN shows better average performance
compared to $S_{n}$-equivariant QNN . These results contribute towards
building powerful quantum machine learning architectures in permutation-
symmetric systems.

##  I Introduction

The field at the interface of quantum technology and machine learning has been
subject to extensive research in recent times. A significant fraction of these
are aimed at building machine learning models using quantum systems. Of
particular interest are quantum neural networks (QNNs) [1, 2, 3, 4, 5],
analogous to classical neural networks in deep learning. The central component
in a prototypical QNN is a quantum circuit with single and multiple qubit
gates with trainable parameters. It has been widely known as parametric
quantum circuit (PQC) or variational quantum circuit (VQC) [6, 7, 8, 9, 10].
PQC-based QNNs have been used to design quantum analogs of well-known
classical machine learning architectures, e.g. quantum autoencoder [8],
quantum convolutional neural networks (QCNNs) [11, 12, 13], quantum generative
adversarial networks (QGANs) [14, 15, 16, 17, 18, 19], quantum generative
diffusion models [20, 21, 22] etc. The cost function obtained from the above
networks is optimized using a classical optimization routine, thus they are
hybrid quantum-classical networks. They are designed to function with both
classical and quantum data. Despite the potential speed-up and signatures of
success of quantum machine learning over classical ML models [23, 24, 25, 26,
27], there exist several limitations for QNNs, the barren plateau problem
being one of them [28, 29, 30, 31]. On one hand, an arbitrary hardware-
efficient PQC ansatze is expected to have high expressibility [32] 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 increasingly
prone to exhibiting barren plateau [28, 29]. It is therefore a crucial task to
mitigate barren plateau for a practical application of quantum neural
networks.

One way to improve the trainability and generalization in machine learning
algorithms is to introduce inductive bias in the learning model, 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
[33]. In past few years, the theory of geometric quantum machine learning has
been developed by a number of works [34, 35, 36, 37, 38, 39, 40]. The
performance of symmetry-enhanced QNNs, also called Equivariant QNNs (EQNNs),
has been investigated for a number of commonly encountered symmetries in
classical datasets as well as quantum systems. These studies show that some
classes of EQNNs are devoid of the barren plateau problem [41, 42, 43].
Moreover, the performance of an EQNN shows improvement over a non-equivariant
QNN for pattern recognition and image classification [37, 44, 45, 43, 46].

Among all the symmetry groups, the group of all permutations of $n$ objects,
which is also called Symmetric group ($S_{n}$), is a crucial one. It is known
that every group is isomorphic to some subgroup of a Symmetric group.
Symmetric group frequently arises in physical scenarios where certain
properties of a quantum state remain invariant under arbitrary permutation of
the qubits. To exemplify, the genuine multiparty entangled quantum states
remain genuinely entangled when the subsystems are permuted, or the set of
connected graphs remain connected under permutation of the vertices. There
also exist scenarios for which the permutation symmetry in a system
corresponds to a subgroup of $S_{n}$. For example, the states of a many-body
quantum system in presence of periodic boundary condition remain unchanged
under cyclic permutation of qubits, which corresponds to cyclic group $Z_{n}$.
Another example is the label symmetry of classical images, i.e. the class
labels of images remain unchanged under spatial transformations like
reflection or rotation, which corresponds to permuting the pixels in a certain
way. The $S_{n}$-equivariant QNN has been studied in [42], with a theoretical
proof of absence of barren plateau. For small subgroups of $S_{n}$ relevant to
classical images, EQNN can be easily built by visualizing the qubits as pixels
in a 2D plane [37].

While QNN broadly refers to a general PQC architecture, in this work, we focus
our attention to a specific type of QNN, namely quantum convolutional neural
networks. We propose an architecture of equivariant quantum convolutional
neural network (EQCNN) for Symmetric group and its subgroups, which we jointly
refer to as permutation group. Our architecture is based on the well-known
model of QCNN introduced in Ref. [11], which shows a considerably good
performance for classifying topological phases of many-body quantum systems
[11, 47, 48, 49, 50] as well as classical images [51]. A few works have
studied EQCNN for classical images [45, 46], all of them using the quantum
amplitude embedding. For reflection or rotation symmetry of images, amplitude
embedding results in a representation of permutation group that acts locally
on each qubit. However, when using qubit embedding (also known as angle
embedding), the resulting representations correspond to subgroups of $S_{n}$
and act non-locally on multiple qubits, as we will elaborate in the next
section. In addition to that, the QCNN proposed in Ref. [11] is subject to
some structural restrictions, i.e. the translational invariance of the
convolution layers and the qubit-reducing feature of pooling layers. The main
challenge in designing EQCNN is to comply with these constraints under non-
local action of the unitary representations in general. In this work, we try
to resolve this issue. We demonstrate that for subgroups of $S_{n}$, it is
important to choose a particular order of pixel-to-qubit mapping which enables
one to build the EQCNN ansatze in a translationally-invariant manner. Equipped
with this, we put forward EQCNN architectures for reflection symmetry and
$\pi/2$-rotational symmetry of images, which shows improved performance
compared to non-equivariant QCNNs for classifying MNIST datasets. For full
permutation symmetry of $S_{n}$, our EQCNN is built by applying all QCNNs at
random with equal probability. Such a probabilistic picture of QNN, although
mentioned briefly in Ref. [38], has not been explored significantly before. We
build a quantum circuit to realize this stochastic action of the convolutional
and pooling layers. We test its performance for classification of connected
and non-connected graphs against non-equivariant QCNN as well as
$S_{n}$-equivariant QNN. While the EQCNN always performs better than the
former, for the latter it shows an improvement when trained with comparatively
large number of data-points.

##  II Methods

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

The QCNN architecture of ref. [11] is inspired from the multi-scale
entanglement renormalization ansatz (MERA) representation of quantum many-body
states [52]. Any QNN 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. The convolutional layer has a brick-like
structure on an 1D array of qubits, i.e. an $m$-qubit ($m<n$) trainable
convolutional ansatz is applied on all combinations of neighboring $m$ qubits,
mimicking the action of a kernel in CNN. It is a common practice to consider a
two-qubit convolutional ansatze. All ansatze within the same convolutional
layer have same trainable parameters. In the pooling layer, a subset of qubits
are measured and conditioned on each measurement outcome, a parametrized
rotation is applied on each qubit of the complementary subset. Similar to
convolutional layers the trainable parameters are shared within a pooling
layer. All the measured qubits are then traced out, thus reducing the
effective dimension of the system. In real quantum devices this is same as
ignoring the qubits in the subsequent stages and considering only the dynamics
on the remaining qubits. The sequence of convolutional and pooling layers 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 qubits reduction, the structure has a shallower depth of
$\mathcal{O}(\log n)$, which is preferable for implementation in NISQ devices.
This model of QCNN is also known to be free of barren plateau [41].

Figure 1: In the center, a $4\times 4$ image with pixel values
$\\{a,b,...,p\\}$ indicated in the top left corner of each pixel and
corresponding pixel positions in the bottom right corner. On the right, the
image after vertical reflection. On the left, the image after rotation by
angle $\pi/2$.

###  II.2 Label symmetry

For a classical or quantum dataset, a label symmetry exists if the class
labels of all the data-points remain unchanged under a set of operations on
those data-points. For example, in MNIST dataset the labels of digits
$\\{0,1,8\\}$ are unchanged when the images are reflected, or the labels of a
quantum state being entangled or non-entangled are invariant under local
unitary transformations on a qubit. Let us consider a binary-classification
task for a dataset $\mathcal{X}$ with data-points $x_{i}$ and corresponding
class labels $y_{i}\in\mathcal{Y}$ where $\mathcal{Y}=\\{0,1\\}$. The function
that maps the data-points to their labels is
$f:\mathcal{X}\rightarrow\mathcal{Y}$. The task of QNN is to realize a quantum
circuit $f_{\theta}$ that closely approximates $f$. A set of operations
$\mathcal{G}$ form a label symmetry group of $\mathcal{X}$ if for all group
elements $g\in\mathcal{G}$, the assigned labels remain unchanged, i.e.,

$f(g(x_{i}))=y_{i}=f(x_{i}),\quad\forall x_{i}\in\mathcal{X},\;\forall
g\in\mathcal{G}.$ (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. Thus, the parametric function
$f_{\theta}$ used to classify $\mathcal{X}$ will be consistent with
$\mathcal{G}$ if it predicts same label for inputs related by the symmetry
operations. The QNNs respecting this constraint are called Equivariant QNNs
(EQNNs). It was proved in [38, 53] that a PQC $\mathcal{U}_{\theta}$ and a
measurement $M$ will form an EQNN if and only if

$[R(g),\mathcal{U}_{\theta}]=0\quad\mbox{and}\quad[R(g),M]=0.$ (2)

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. In the context of EQCNN,
$\mathcal{U}_{\theta}$ is composed of a series of convolutional and pooling
layers. Thus, equivariance of $\mathcal{U}_{\theta}$ can be obtained by making
sure that each convolution and pooling layer is equivariant with respect to
$R(g)$.

###  II.3 EQCNN with permutation symmetry

The elements of a symmetry group can act either as inner symmetry or outer
symmetry [38]. For the former, $R(g)$ can be decomposed as a tensor product of
locally acting unitary operators. A well known example is the local unitary
group $\mathrm{SU}(2)$ acting on a system of $n$ qubits, and its
representation $R(g)=g^{\otimes n}$ which acts locally on each qubit, where
$g\in\mathrm{SU}(2)$. Another commonly occurring instance of inner symmetry is
when classical images are encoded as quantum states using amplitude embedding,
and the representation $R(g)$ of reflection or rotation group symmetry is a
tensor product of Pauli-$X$ matrices and qubit identity operators acting
locally on different qubits. The architecture and performance of equivariant
quantum neural networks for inner symmetries have been investigated in a
number of works [37, 38, 44, 45, 46].

On the other hand, $R(g)$ for outer symmetry groups cannot be decomposed into
locally acting components, an example being the Symmetric group $S_{n}$ which
manifests itself as a symmetry with respect to permuting the qubits. The
physical operation of permuting two qubits is expressed mathematically using
SWAP gate. One can show that, for every element $g$ of $S_{n}$, $R_{g}$ can be
written as a product of SWAP gates, as we will exemplify in the next section.
When a system of $n$ qubits has full permutation symmetry, each qubit can be
swapped with any other qubit. The fulfillment of equivariance condition Eq.
(2) then requires that $\mathcal{U}_{\theta}$ must remain unchanged when the
qubits are permuted in an arbitrary way. When using layers of single-qubit
gates and two-qubit entangling gates to construct $\mathcal{U}_{\theta}$,
permutation equivariance can be achieved if one uses single qubit gates with
same parameters on all qubits, and entangling gates to connect all possible
${n}\choose{2}$ combinations of two qubits. In reference Ref. [42], the
authors show that this architecture of $S_{n}$-equivariant QNN is devoid of
barren plateau, and benchmark its performance for classification of connected
and non-connected graph states.

This structure of $S_{n}$-equivariant QNN is translationally symmetric on an
array of qubits, thus can be used as the first convolutional layer of a
permutation-equivariant QCNN. However, it is not straightforward to construct
pooling layers in this case, as choosing a particular set of qubits to trace
out inevitably breaks the full permutation symmetry. Additionally, the reduced
representation on the remaining qubits after the first pooling layer is not a
faithful representation. It was pointed out in Ref. [38] that one possible
technique is to randomly trace out all combinations of $n\choose n/2$ qubits
with equal probability to retain full permutation symmetry. In this work, we
have used this strategy to construct equivariant pooling layers. Moreover, we
show how to construct equivariant measurements for a probabilistic ansatze.

There exist other practically useful scenarios when the relevant labels of
quantum states remain unchanged under the action of a subgroup of $S_{n}$. For
example, when each pixel value of a classical image is encoded using a qubit
(angle embedding), the operation of reflecting or rotating the image
corresponds to permuting the qubits in a certain way, eliminating full
freedom. In a way, this is an inner symmetry for which the group
representation acts locally on a set of qubits which are allowed to permute
within that set. In such cases, equivariant $\mathcal{U}_{\theta}$ can be
achieved using less than ${n}\choose{2}$ two-qubit gates and a lower-depth
circuit, as we will describe in the following subsections.

####  II.3.1 Reflection-equivariant QCNN

Let us consider classification of images whose labels remain invariant under a
reflection about the vertical axis. There are two group elements here– the
identity operation $e$ which keeps the image unchanged, and the reflection
operation $r$. They form the abstract group $\mathbb{Z}_{2}\equiv\\{e,r\\}$.
For angle embedding of classical images into quantum states, the symmetry
representation of reflection group acts non-locally on certain groups of
qubits through SWAP gates. For example, for a $4\times 4$ image as in Fig. 1,
the representation of the reflection group is,

$R^{\mathrm{ref}}=\Big{\\{}\mathbb{I}^{\otimes
16},\;\mathrm{SWAP}_{1,4}\mathrm{SWAP}_{2,3}\mathrm{SWAP}_{5,8}\mathrm{SWAP}_{6,7}\mathrm{SWAP}_{9,12}\mathrm{SWAP}_{10,11}\mathrm{SWAP}_{13,16}\mathrm{SWAP}_{14,15}\Big{\\}}$
(3)

where $\mathrm{SWAP}_{i,j}$ denotes swap operation between pixel values at
positions $i$ and $j$ of the image before reflection. Here, qubit pairs
$\\{\\{1,4\\}$, $\\{2,3\\}..\\}$ constitute subsystems on each of which a SWAP
gate acts locally.

Now, for an arbitrary order of embedding the pixel values in 1D array of
qubits, one can check that applying convolutional ansatze in the typical
brick-like structure is not equivariant with respect to the representation in
Eq. (3). We demonstrate that by adhering to a particular pixel-to-qubit
embedding order, one can build equivariant convolutional ansatze for all
layers by having the brick-like arrangement of two-qubit ansatze. By pixel-to-
qubit embedding order, we imply a particular one-to-one correspondence between
pixel values $\\{a,b,c,...\\}$ and the qubits indices $\\{1,2,3,..\\}$ which
encodes them. This pixel-to-qubit embedding order and the resulting EQCNN have
been presented in Fig. LABEL:eqcnn_refl for a $4\times 4$ image. Here the
qubit indices increase from 1 to 16 when moving from top to bottom of the 1D
array. Due to this altered embedding order, the group representation changes
to,

$R^{\mathrm{ref}}=\Big{\\{}\mathbb{I}^{\otimes
16},\;\mathrm{SWAP}_{1,16}\mathrm{SWAP}_{2,15}\mathrm{SWAP}_{3,14}\mathrm{SWAP}_{4,13}\mathrm{SWAP}_{5,12}\mathrm{SWAP}_{6,11}\mathrm{SWAP}_{7,10}\mathrm{SWAP}_{8,9}\Big{\\}}.$
(4)

In the pooling layer, we trace out half of the qubits in a way so that the
action of the reflection group has a faithful representation on the remaining
qubits, which is the following after the first pooling layer,

$R^{\mathrm{ref}}=\Big{\\{}\mathbb{I}^{\otimes
8},\;\mathrm{SWAP}_{5,12}\mathrm{SWAP}_{6,11}\mathrm{SWAP}_{7,10}\mathrm{SWAP}_{8,9}\Big{\\}}.$
(5)

Similarly, after the second pooling layer, the reduced representation is

$R^{\mathrm{ref}}=\Big{\\{}\mathbb{I}^{\otimes
4},\;\mathrm{SWAP}_{6,11}\mathrm{SWAP}_{7,10}\Big{\\}},$ (6)

and that after the third pooling layer is

$R^{\mathrm{ref}}=\Big{\\{}\mathbb{I}^{\otimes
2},\;\mathrm{SWAP}_{7,10}\Big{\\}}.$ (7)

The measurement is performed on the remaining two qubits $\\{7,10\\}$.

This structure of EQCNN can be conveniently embedded on a quantum register
with 2D square lattice architecture. We demonstrate this in Fig.
LABEL:eqcnn_refl(b). An advantage of 2D is that all the pooling ansatze are
applied on nearest-neighbour qubits. However, to retain full translation
symmetry of the convolutional layer, one needs to apply the convolutional
ansatze on two qubits $\\{a,d\\}$ which are not nearest neighbours in this
case. Square lattice architectures have been realized in NISQ devices such as
Rigetti quantum hardware ANKAA-2 and Pasqal neutral atom quantum processing
units.

Figure 3: EQCNN for reflection and $\pi/2$-rotation symmetry for 16 qubits.
For non-nearest-neighbour applications of two-qubit convolutional ansatze in
the first and third layer, we use same integers to clearly indicate qubits on
which they apply within a sub-layer .

####  II.3.2 Reflection and $\frac{\pi}{2}$-rotation equivariant QCNN

Next we consider images having label symmetry with respect to both reflection
and a rotation by angle $\pi/2$. We present the corresponding EQCNN in Fig. 3.
There are two symmetry groups at play– the reflection symmetry group
$\\{e,r\\}$, and the rotational symmetry group $\\{e,r^{\prime}\\}$ where
$r^{\prime}$ corresponds to $\pi/2$-rotation. For the natural order of pixel-
to-qubit embedding, the group representation for the latter is

$R^{\mathrm{rot}}=\Big{\\{}\mathbb{I}^{\otimes
16},\;\mathrm{SWAP}_{1,16}\mathrm{SWAP}_{2,15}\mathrm{SWAP}_{3,14}\mathrm{SWAP}_{4,13}\mathrm{SWAP}_{5,12}\mathrm{SWAP}_{6,11}\mathrm{SWAP}_{7,10}\mathrm{SWAP}_{8,9}\Big{\\}},$
(8)

as can be understood from Fig. 1. Note that Eq. (8) is the same as the
representation in Eq. (4). Hence in this case, rotational equivariance is
obtained from the brick-like structure of convolutional layer for the natural
order of pixel-to-qubit embedding. However, to ensure reflection equivariance
for this embedding, an additional layer of convolutional ansatze needs to be
applied but in a different fashion. In detail, the two-qubit convolutional
ansatz is applied on every qubit and and its $7^{\mathrm{th}}$ nearest
neighbour when counting the qubits from top to bottom in Fig. 3. The reasoning
behind this construction can be understood by observing how the nearest
neighbour qubits are rearranged under reflection. The construction of first
pooling layer ensures that the representations of both the groups on the
remaining qubits are faithful. These reduced representations are

$\displaystyle R^{\mathrm{ref}}=\Big{\\{}\mathbb{I}^{\otimes
8},\;\mathrm{SWAP}_{5,8}\mathrm{SWAP}_{6,7}\mathrm{SWAP}_{9,12}\mathrm{SWAP}_{10,11}\Big{\\}},$
(9) $\displaystyle R^{\mathrm{rot}}=\Big{\\{}\mathbb{I}^{\otimes
8},\;\mathrm{SWAP}_{5,12}\mathrm{SWAP}_{6,11}\mathrm{SWAP}_{7,10}\mathrm{SWAP}_{8,9}\Big{\\}}.$
(10)

In the second convolutional layer, the brick-like structure alone is
sufficient to ensure equivariance with respect to both the symmetries. The
reduced representations after the second pooling layer are,

$\displaystyle R^{\mathrm{ref}}=\Big{\\{}\mathbb{I}^{\otimes
4},\;\mathrm{SWAP}_{5,8}\mathrm{SWAP}_{9,12}\Big{\\}},$ (11) $\displaystyle
R^{\mathrm{rot}}=\Big{\\{}\mathbb{I}^{\otimes
4},\;\mathrm{SWAP}_{5,12}\mathrm{SWAP}_{8,9}\Big{\\}}.$ (12)

The third convolutional layer is constructed in the same way as the second
convolutional layer. In the third pooling layer, it is impossible to trace out
any two qubits, maintaining equivariance with respect to both the symmetry
groups. Hence, we use a pooling layer for which the rotational equivariance is
broken but the reflection equivariance is protected. Lastly, the remaining two
qubits $\\{5,8\\}$ are measured.

Note that, one can obtain the EQCNN with respect to $\pi/2$-rotation symmetry
alone by removing the additional convolutional layer in the first layer, and
changing the last pooling ansatze in a way to protect rotational equivariance.
As a result, the final measurement will be on a different set of qubits.

Figure 5: The circuit for realizing EQCNN for symmetric group $S_{4}$ on the
input state $\rho$. In the controlled gates, the circles with no fillings and
black fillings represent states $|0\rangle$ and $|1\rangle$ respectively.
Figure 6: Samples of Fashion MNIST images corresponding to class 0 and class 8
sized down to $4\times 4$ pixels. Figure 7: (a) The convolutional ansatze and
(b) the pooling ansatze used to construct EQCNN and non-equivariant QCNN for
reflection and $\pi/2$-rotation symmetry. (c) The convolutional ansatze used
in $S_{n}$-equivariant QCNN.

####  II.3.3 $S_{n}$-equivariant QCNN

When a system has full permutation symmetry, we use the $S_{n}$-equivariant
QNN with all-to-all two-qubit gate connections as the first convolutional
layer. For the pooling operation, let us consider a scenario in which no
explicit pooling ansatz is applied, but half of the qubits are traced out
successively at each pooling layer. In the first pooling layer, we randomly
trace out any $n/2$ qubits. This can be done in $n\choose n/2$ possible ways,
each occurring with equal probability to retain the permutation equivariance.
The second convolutional layer has the same structure as the first
convolutional layer, however it is applied on the remaining $n/2$ qubits. In
the second pooling layer, we randomly trace out $n/4$ qubits from $n/2$ qubits
in $n/2\choose n/4$ possible ways, and apply the third convolutional layer on
the remaining $n/4$ qubits. We keep adding convolutional and pooling layers
until only one qubit is left. Another way to envision this is to think that
there are in total

$P=\binom{n}{n/2}\times\binom{n/2}{n/4}\times\binom{n/4}{n/8}\times...\times\binom{2}{1}$
(13)

number of possible QCNNs, which are applied on the input state with equal
probability. For each of these QCNNs, the last remaining qubit is measured to
obtain the loss function. We schematically depict the idea discussed above in
Fig. LABEL:sn_conv_example for a simple case of $n=4$.

Now we build a quantum circuit that can exactly realize the probabilistic
nature of the EQCNN described above. We show this in Fig. 5. Additional to the
target register encoding the input state $\rho$, there is an auxiliary quantum
register $A_{1}$ with $\lceil\log P\rceil=4$ qubits, which is initialized in
an equal superposition $|\Psi_{A_{1}}\rangle$ of the first $P=12$ orthogonal
basis states. We apply the first convolutional layer $C_{1}$ on $\rho$. Since
no trainable pooling ansatz is present, tracing out $n/2$ randomly chosen
qubits in the first pooling layer is equivalent to applying the second
convolutional layer $C_{2}$ on all possible $\binom{4}{2}=6$ pairs of qubits.
To ensure that they are applied at random with equal probability, we apply
them controlled on the states of the first three qubits in $A_{1}$. In the
next and final pooling layer, randomly tracing out any one qubit from a qubit
pair is equivalent to randomly choosing a qubit and measuring it. For this, we
add another single-qubit auxiliary register $A_{2}$ initialized in state
$|0\rangle$. Now for each qubit pair in the target register, controlled on the
last qubit of $A_{1}$ being in state $|0\rangle$ or $|1\rangle$, we swap
either of the qubits with $A_{2}$. Finally, we measure Pauli-Z operator on
$A_{2}$ to obtain the loss function. A detailed mathematical description of
how this circuit works is discussed in the Appendix.

A drawback of the above circuit is that with increasing $n$, $P$ increases
very fast and soon becomes exponential in $n$, resulting in an exponentially
deep circuit. This can be avoided if instead of tracing out half of the qubits
from previous layer, one chooses to trace out a constant number $m$ of qubits
at each pooling layer, and repeats this for $k<\frac{n}{m}$ times such that
$P$ is polynomial in $n$. At the end of $k^{\mathrm{th}}$ layer, measurements
are performed on all the remaining qubits. This strategy remains valid since
tracing out exactly half of the qubits is a convention rather than a strict
requirement for QCNN, although the logarithmic depth may not hold depending on
the relation between $m$ and $n$. We leave it for future research works to
investigate the performance of this polynomially deep EQCNN.

As mentioned in ref. [38], the technique of probabilistically applying
different unitary ansatze on the input states can also be used to design a
dropout mechanism in a general quantum neural network. A few works exist in
literature that proposes quantum dropout by randomly dropping single and two-
qubit gates [54, 55] as well as the qubits themselves [10], whereas our
strategy matches well with that discussed in ref. [56]. Particularly, ours is
a special case in which half of the qubits from the previous layer are
randomly chosen and dropped along with all gates acting on them on subsequent
layers. However, it also implies dropping half of the trainable parameters at
each layer, which greatly reduces the expressibility and can result in poor
trainability of the network. Instead, one may choose to drop a small constant
number of $m$ qubits at each successive layer, as discussed also in the
previous paragraph.

Figure 9: An instance of initial parameters for which $S_{n}$-EQCNN performs
better than $S_{n}$-EQNN when the train-test split has $5:1$ ratio.

##  III Results

We compare the classification accuracy of equivariant and non-equivariant
QCNNs for the symmetry groups discussed in the last section. We build and
train all the networks using Pennylane quantum simulator [57].

For reflection symmetry alone, we use class 0 (tshirt) and class 8 (handbag)
of Fashion MNIST dataset, whereas for images symmetric with respect to
reflection and $\pi/2$-rotation, we use class 0 and 1 of MNIST dataset. The
original images are $28\times 28$ pixels, we downsize them to $4\times 4$
pixels to embed the pixel values into 16 qubits using $R_{Y}$ rotation. Some
samples of the downsized images for Fashion MNIST dataset are presented in
Fig. 6. As the convolutional and pooling ansatze, we use the circuits in Fig.
7(a) and 7(b) respectively, where
$\mathrm{Rot}(\theta_{1},\theta_{2},\theta_{3})=R_{Z}(\theta_{3})R_{Y}(\theta_{2})R_{Z}(\theta_{1})$.
One can choose any other set of single-qubit and two-qubit gates without loss
of generality. For building EQCNNs we follow the embeddings and architectures
in Fig. LABEL:eqcnn_refl and Fig. 3, while for non-equivariant QCNNs we use
standard pixel-to-qubit embedding order and the architectures in Ref. [51]. We
train the models using Nesterov moment optimizer with learning rate 0.005. As
loss function, we use binary cross-entropy. We obtain the average test-set
classification accuracy over $10$ identical runs, each with random initial
parameters and a batch size of $32$. The behaviour of mean test-set accuracy
with increasing number of training iterations is presented in Fig.
LABEL:compare_accuracy. For fashion MNIST, EQCNN has a higher classification
accuracy than non-equivariant QCNN. For MNIST images, considering the
$\pi/2$-rotational symmetry alone results in a higher accuracy than the non-
equivariant QCNN. However, taking into account both reflection and
$\pi/2$-rotational symmetry reduces the accuracy significantly. This can be
due to the highly constrained structure of EQCNN under both the symmetries,
for which the expressibility degrades significantly.

We apply the $S_{n}$-equivariant QCNN for classification of connected and non-
connected graphs with $4$ vertices. The graphs are randomly generated using
Erdős-Rényi model with an edge probability of $45\%$. The quantum graph states
are generated from the graphs using the standard method from [58, 59], and are
embedded into a four-qubit quantum register using amplitude embedding. The
property of being a connected or non-connected graph remains unchanged under
arbitrary permutations of the vertices, in this case the qubits. From all
possible graph states with $4$ vertices, we choose a fraction for training and
the other part for testing. We use two different instances of this
partitioning- for _case 1_ we have 45 data-points for training and 18 for
testing, while for case 2 we have 52 and 11 data-points respectively for
training and testing. In both cases, the training data is well balanced
between the two classes, while the test data has a large fraction of connected
graphs. We augment the training and test set with multiple copies of the
distinct data-points. As the convolutional ansatze, we use the
$S_{n}$-equivariant ansatze shown in Fig. 7(c) in the first layer, and its
two-qubit analog in the second layer. The choice of the gates is again
arbitrary. In addition to EQCNN, we also test the performance of
$S_{n}$-equivariant QNN without pooling layer, for which we replace the second
convolutional layer by another layer of the four-qubit ansatze in Fig. 7(c).
In this case we measure Pauli-Z operator on all the qubits to obtain the loss
function. Note that the number of trainable parameters is same for the EQCNN
and EQNN. For non-equivariant QCNN, we use the architecture of QCNN in Ref.
[51] with Fig. 7(a) and 7(b) as convolutional and pooling ansatze. All models
are trained using Adam optimizer with a learning rate 0.01, and we use mean-
squared error as loss function. We obtain the mean training loss, training
accuracy and test accuracy from 10 randomly initialized training schemes with
a batch size $10$. The results are presented in Fig.
LABEL:compare_accuracy_sn.

On average, we find that the $S_{n}$-EQNN has a good generalization behaviour,
and the corresponding test accuracy is better than the other two models for
case 1 when the train-test ratio is $5:3$ approximately. The performance of
$S_{n}$-EQNN does not differ much for case 1 and case 2. On the other hand,
the $S_{n}$-EQCNN overfits the training data in case 1. However, when the
train-test ratio is approximately $5:1$ in case 2 and there are more data-
points to learn, the performance of the EQCNN improves and it shows higher
average test accuracy than $S_{n}$-equivariant QNN. In this case, out of the
10 instances of initial parameters, there are more cases for which the maximum
test accuracy $1$ is obtained for EQCNN compared to EQNN. In Fig. 9 we show
the training and test accuracy for one such instance. The non-equivariant QCNN
has a much lower test accuracy than both the equivariant architectures in both
cases.

##  IV Discussions and Conclusion

In this work, we proposed a pixel-to-qubit embedding order that facilitates
applying the quantum convolutional ansatze in an equivariant manner for angle-
embedded classical images with reflection and rotational symmetry of image
labels. The resulting EQCNNs show better classification accuracy than non-
equivariant QCNNs. Though angle embedding requires as many qubits as the
number of pixels compared to the logarithmically less number of qubits in
amplitude embedding, the latter needs an exponentially deep circuit to encode
an arbitrary quantum superposition of $n$ qubits, thus subsuming the advantage
due to logarithmically shallow circuit of QCNN. Compared to that, angle
embedding requires a circuit of constant depth, since each qubit can be
manipulated individually in parallel. With this advantage, and the fast
growing dimension of NISQ devices, angle embedding remains a more potent
choice for quantum embedding of classical data. Moreover, one can also pre-
process a large classical image to reduce its dimension using principal
component analysis (PCA) or a classical autoencoder, before angle-embedding it
into a smaller number of available qubits. In this case however, the pre-
processing must preserve the label symmetry, in order to use an EQCNN for
classification. It is to be noted that, though our embedding trick works for
small subgroups of $S_{n}$, it will become increasingly difficult to design a
translationally symmetric equivariant convolutional ansatze for bigger
permutation groups, e.g. $p4m$ symmetry group, which again arises frequently
in highly symmetric classical images. In such cases, one may choose to take
into account fewer symmetries. This is supported by our numerical data showing
a better performance compared to equivariant circuit which takes into account
all existing symmetries. One can also use a more flexible ansatze that weakly
breaks the symmetries [45].

Our construction of $S_{n}$-equivariant QCNN proposed also a probabilistic
picture of QNN, which has not been explored in detail in literature yet. We
use graph states as dataset to analyze the performance of the
$S_{n}$-equivariant EQCNN, which shows a better performance than non-
equivariant QCNN. When sufficient training data are used, this EQCNN performs
better than $S_{n}$-equivariant QNN on average. This probabilistic application
of $S_{n}$-equivariant QCNN can be viewed as a special type of dropout
mechanism in QNN [56, 10, 54, 55], in analogy to dropout in CNN to prevent
overfitting [60]. Particularly in our case, the important difference with the
conventional dropout is that, here no trainable parameters are dropped, rather
some of the qubits on which the corresponding gates apply. One can also apply
with equal probability the equivariant ansatze for every symmetry group when
there exist more than one symmetry group, as also discussed in the last
paragraph.

It should be mentioned that there exist alternative approaches for designing a
quantum convolutional neural network in quantum systems [13, 39, 61].
Particularly in references [39, 61], the authors conceptualize the qubit
permutation symmetry as the quantum analog of translational symmetry in
classical images. They propose as ‘convolutional quantum ansatze’ a
permutation-equivariant ansatze which is in turn constructed by employing
global $SU(d)$-equivariant Hamiltonians, since by Schur’s lemma the
representation theory of $S_{n}$ can be described using the representation
theory of $SU(d)$, and vice versa. As a result, their architecture can be used
to find the ground states of many-body Hamiltonians that exhibit $SU(d)$
symmetry. This is in contrast to our work which proposes a permutation-
equivariant QCNN that manifests the translation symmetry in a different way
[11].

Data classification is an ubiquitous task in a plethora of everyday
applications. Classical state-of-the-art classifiers are supremely successful,
yet the potency of quantum classifiers are worth the investigation. The
quantum convolutional neural networks are proved to be efficient classifiers.
Thus, our construction of equivariant QCNNs for spatial symmetry of images is
a significant addition to the growing literature. The $S_{n}$-equivariant QCNN
proposed in this work is extremely relevant due to the frequent presence of
permutation symmetry in nature. We also believe that the stochastic nature of
the variational quantum ansatz is an interesting direction to be pursued for
future research.

###### 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.

##  Appendix A $S_{n}$-equivariant channel and measurements

Denoting the input state as $\rho$, the state after the first convolutional
layer is

$\rho_{1}=C_{1}\rho C_{1}^{\dagger},$ (14)

where $C_{1}$ is the unitary corresponding to the first convolutional layer.

The initial state in the auxiliary register $A_{1}$ is

$\displaystyle|\Psi_{A_{1}}\rangle=\frac{1}{\sqrt{12}}\big{(}|0000\rangle+|0001\rangle+|0010\rangle+|0011\rangle+|0100\rangle$
$\displaystyle+|0101\rangle+|0110\rangle+|0111\rangle+|1000\rangle+|1001\rangle$
$\displaystyle+|1010\rangle+|1011\rangle\big{)}.$ (15)

After the controlled application of second convolutional layer on different
pairs of qubits, the input state becomes,

$\displaystyle\rho_{2}=\frac{1}{6}\big{(}C_{2}^{12}\rho_{1}C_{2}^{12^{\dagger}}+C_{2}^{13}\rho_{1}C_{2}^{13^{\dagger}}+C_{2}^{14}\rho_{1}C_{2}^{14^{\dagger}}$
$\displaystyle+C_{2}^{23}\rho_{1}C_{2}^{23^{\dagger}}+C_{2}^{24}\rho_{1}C_{2}^{24^{\dagger}}+C_{2}^{34}\rho_{1}C_{2}^{34^{\dagger}}\big{)},$
(16)

where $C_{2}^{ij}$ is the second convolutional layer acting on the qubit-pair
$(i,j)$.

At this stage, the auxiliary register $A_{2}$ is in the state $|0\rangle$, and
we denote the corresponding qubit with index $0$. After application of
controlled swap gates, the state of the joint system consisting of our input
state register and auxiliary register $A_{2}$ is,

$\begin{split}&\rho_{3}^{\prime}=\frac{1}{12}\big{(}S_{01}(C_{2}^{12}\rho_{1}C_{2}^{12^{\dagger}}\otimes|0\rangle\langle
0|)S_{01}^{\dagger}+S_{02}(C_{2}^{12}\rho_{1}C_{2}^{12^{\dagger}}\otimes|0\rangle\langle
0|)S_{02}^{\dagger}+S_{01}(C_{2}^{13}\rho_{1}C_{2}^{13^{\dagger}}\otimes|0\rangle\langle
0|)S_{01}^{\dagger}\\\
&+S_{03}(C_{2}^{13}\rho_{1}C_{2}^{13^{\dagger}}\otimes|0\rangle\langle
0|)S_{03}^{\dagger}+S_{01}(C_{2}^{14}\rho_{1}C_{2}^{14^{\dagger}}\otimes|0\rangle\langle
0|)S_{01}^{\dagger}+S_{04}(C_{2}^{14}\rho_{1}C_{2}^{14^{\dagger}}\otimes|0\rangle\langle
0|)S_{04}^{\dagger}\\\
&+S_{02}(C_{2}^{23}\rho_{1}C_{2}^{23^{\dagger}}\otimes|0\rangle\langle
0|)S_{02}^{\dagger}+S_{03}(C_{2}^{23}\rho_{1}C_{2}^{23^{\dagger}}\otimes|0\rangle\langle
0|)S_{03}^{\dagger}+S_{02}(C_{2}^{24}\rho_{1}C_{2}^{24^{\dagger}}\otimes|0\rangle\langle
0|)S_{02}^{\dagger}\\\
&+S_{04}(C_{2}^{24}\rho_{1}C_{2}^{24^{\dagger}}\otimes|0\rangle\langle
0|)S_{04}^{\dagger}+S_{03}(C_{2}^{34}\rho_{1}C_{2}^{34^{\dagger}}\otimes|0\rangle\langle
0|)S_{03}^{\dagger}+S_{04}(C_{2}^{34}\rho_{1}C_{2}^{34^{\dagger}}\otimes|0\rangle\langle
0|)S_{04}^{\dagger}\big{)}.\end{split}$ (17)

The expectation value of the measurement on $A_{2}$ is,

$\displaystyle m=\mathrm{Tr}(\rho_{3}^{\prime}M_{0}^{\prime}),$ (18)

where
$M_{0}^{\prime}=\sigma^{z}_{0}\otimes\mathbb{I}_{1}\otimes\mathbb{I}_{2}\otimes\mathbb{I}_{3}\otimes\mathbb{I}_{4}$.

Now, using the cyclic property of trace, we can write the following for an
arbitrary term in Eq. (17),

$\begin{split}m^{ij}_{i}&=\mathrm{Tr}\Big{(}\big{(}S_{0i}(C_{2}^{ij}\rho_{1}C_{2}^{ij^{\dagger}}\otimes|0\rangle\langle
0|)S_{0i}^{\dagger}\big{)}M_{0}^{\prime}\Big{)}\\\
&=\mathrm{Tr}\bigg{(}(S_{0i}^{\dagger}M_{0}^{\prime}S_{0i})(C_{2}^{ij}\rho_{1}C_{2}^{ij^{\dagger}}\otimes|0\rangle\langle
0|)\bigg{)}\\\
&=\mathrm{Tr}\big{(}M_{i}^{\prime}(C_{2}^{ij}\rho_{1}C_{2}^{ij^{\dagger}}\otimes|0\rangle\langle
0|)\big{)}\\\
&=\mathrm{Tr}(M_{i}C_{2}^{ij}\rho_{1}C_{2}^{ij^{\dagger}}).\mathrm{Tr}(\mathbb{I}_{0}|0\rangle\langle
0|)\\\ &=\mathrm{Tr}(M_{i}C_{2}^{ij}\rho_{1}C_{2}^{ij^{\dagger}}),\end{split}$
(19)

where $M_{i}\;(i\neq 0)$ denotes a Pauli-Z measurement on the
$i^{\mathrm{th}}$ qubit. In this way, we design a measurement outcome which is
the weighted sum of measurement outcomes from all equally probable QCNNs.

## References

  * 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).
  * 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, Phys. Rev. X 12, 031010 (2022).
  * 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, IEEE Transactions on Quantum Engineering 4, 1 (2023).
  * 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).
  * 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._ [2024] B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Generative quantum machine learning via denoising diffusion probabilistic models, Phys. Rev. Lett. 132, 100602 (2024).
  * Cacioppo  _et al._ [2023] A. Cacioppo, L. Colantonio, S. Bordoni, and S. Giagu, Quantum diffusion models (2023), arXiv:2311.15444 [quant-ph] .
  * 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._ [2022] 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 (2022), 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, npj Quantum Information 9, 47 (2023).
  * 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._ [2024] L. Schatzki, M. Larocca, Q. T. Nguyen, F. Sauvage, and M. Cerezo, Theoretical guarantees for permutation-equivariant quantum neural networks, npj Quantum Information 10, 12 (2024).
  * 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. Le Saux, and S. Vallecorsa, Approximately equivariant quantum neural network for p4m group symmetries in images, in  _2023 IEEE International Conference on Quantum Computing and Engineering (QCE)_, Vol. 01 (2023) pp. 229–235.
  * Das  _et al._ [2024] S. Das, S. Martina, and F. Caruso, The role of data embedding in equivariant quantum convolutional neural networks (2024), arXiv:2312.13250 [quant-ph] .
  * Herrmann  _et al._ [2022] J. Herrmann, S. M. Llima, A. Remm, P. Zapletal, N. A. McMahon, C. Scarato, F. Swiadek, C. K. Andersen, C. Hellings, S. Krinner, N. Lacroix, S. Lazar, M. Kerschbaum, D. C. Zanuz, G. J. Norris, M. J. Hartmann, A. Wallraff, and C. Eichler, Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases, Nature Communications 13, 4144 (2022).
  * 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).
  * Umeano  _et al._ [2023] C. Umeano, A. E. Paine, V. E. Elfving, and O. Kyriienko, What can we learn from quantum convolutional neural networks? (2023), arXiv:2308.16664 [quant-ph] .
  * Monaco  _et al._ [2023] S. Monaco, O. Kiss, A. Mandarino, S. Vallecorsa, and M. Grossi, Quantum phase detection generalization from marginal quantum neural network models, Phys. Rev. B 107, L081105 (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).
  * Vidal [2008] G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101, 110501 (2008).
  * 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] .
  * Kobayashi  _et al._ [2022] M. Kobayashi, K. Nakaji, and N. Yamamoto, Overfitting in quantum machine learning and entangling dropout, Quantum Machine Intelligence 4, 30 (2022).
  * [55] F. Scala, A. Ceschini, M. Panella, and D. Gerace, A general approach to dropout in quantum neural networks, Advanced Quantum Technologies n/a, 2300220, https://onlinelibrary.wiley.com/doi/pdf/10.1002/qute.202300220 .
  * Verdon  _et al._ [2018] G. Verdon, J. Pye, and M. Broughton, A universal training algorithm for quantum deep learning (2018), arXiv:1806.09729 [quant-ph] .
  * 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] .
  * Raussendorf  _et al._ [2003] R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003).
  * Hein  _et al._ [2004] M. Hein, J. Eisert, and H. J. Briegel, Multiparty entanglement in graph states, Phys. Rev. A 69, 062311 (2004).
  * Srivastava  _et al._ [2014] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, Dropout: A simple way to prevent neural networks from overfitting, Journal of Machine Learning Research 15, 1929 (2014).
  * Zheng  _et al._ [2022] H. Zheng, Z. Li, J. Liu, S. Strelchuk, and R. Kondor, On the super-exponential quantum speedup of equivariant quantum machine learning algorithms with su($d$) symmetry (2022), arXiv:2207.07250 [quant-ph] .