figurec

# Reflection Equivariant Quantum Neural Networks
for Enhanced Image Classification

Maxwell West<EMAIL_ADDRESS>School of Physics, The University
of Melbourne, Parkville, 3010, VIC, Australia  Martin Sevior  School of
Physics, The University of Melbourne, Parkville, 3010, VIC, Australia
Muhammad Usman<EMAIL_ADDRESS>School of Physics, The University of
Melbourne, Parkville, 3010, VIC, Australia  Data61, CSIRO, Clayton, 3168, VIC,
Australia

###### Abstract

Machine learning is among the most widely anticipated use cases for near-term
quantum computers, however there remain significant theoretical and
implementation challenges impeding its scale up. In particular, there is an
emerging body of work which suggests that generic, data agnostic quantum
machine learning (QML) architectures may suffer from severe trainability
issues, with the gradient of typical variational parameters vanishing
exponentially in the number of qubits. Additionally, the high expressibility
of QML models can lead to overfitting on training data and poor generalisation
performance. A promising strategy to combat both of these difficulties is to
construct models which explicitly respect the symmetries inherent in their
data, so-called geometric quantum machine learning (GQML). In this work, we
utilise the techniques of GQML for the task of image classification, building
new QML models which are equivariant with respect to reflections of the
images. We find that these networks are capable of consistently and
significantly outperforming generic ansatze on complicated real-world image
datasets, bringing high-resolution image classification via quantum computers
closer to reality. Our work highlights a potential pathway for the future
development and implementation of powerful QML models which directly exploit
the symmetries of data.

1\. Introduction

The significant interest in the possibility of realising superior machine
learning algorithms on quantum computers has led over the last few years to
intensive study of the capabilities and limitations of quantum machine
learning (QML) models [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Although remarkable
improvements in the capability of QML methods over their classical
counterparts have been reported for some specific use cases [11, 12, 13], the
extent to which they can be expected to perform well on general datasets
remains an open question. In fact, recent work has suggested that generic,
commonly employed QML architectures such as variational quantum circuits will
face significant limitations due to their high expressibility and potentially
low trainability resulting from barren plateaus in their training landscapes
[14, 15, 16, 17, 18, 19]. Efforts to address this, while still maintaining
sufficiently complicated circuits to allow for the possibility of quantum
advantage, are ongoing [18, 19, 20, 21, 22], but the ultimate resolution of
the problem of barren plateaus remains unclear, and a key research direction
of QML. Separately, in the case of image classification, the application of
QML is still limited to relatively simple and low resolution datasets [23] by
the current hardware limitations, with the development of high performance QML
models for complex image datasets another important open problem. In this work
we utilise techniques developed to tackle poor generalisation and barren
plateaus to design new QML models which explicitly exploit the reflection
symmetry inherent in many image datasets (see Figure 1). We establish that
these reflection equivariant models can provide superior performance on
complex images at the forefront of the current capabilities of QML, both
obtaining higher accuracies and possessing parameter gradients which vanish
more slowly than those of a generic counterpart.

Symmetry has long played an important role in physics, facilitating both the
discovery and understanding of the laws of nature [24]. More recently, the
symmetries of data have been recognised to play an important role in machine
learning, informing the design of some of the most effective classical
classification algorithms [25]. Convolutional neural networks (CNNs), for
example, which have been famously successful in the classification of image
data, begin by applying a set of so-called convolutional filters, which act on
all regions of the image identically – they have translation equivariance [26,
27]. The loss of expressibility that this necessarily entails has been found
to be more than compensated for by their increased trainability, with CNNs
having now dominated image classification for years [26, 28]. More generally,
in recent years the new field of geometric deep learning (GDL) [29, 30, 31,
25] has begun to explore the role of symmetry respecting neural network
architectures beyond such Euclidean translational equivariance, studying for
example symmetries of data which lives on graphs [32] or Riemannian manifolds
[33]. The observed success of these strategies naturally raises the question
of whether such techniques could also help to build QML models which benefit
from utilising the symmetries of their data.

Figure 1: Reflection Equivariant Quantum Neural Networks. (a) We consider
image data whose labels are left invariant by the action of a group
$\mathcal{G}$. In the case shown (from the CIFAR-10 dataset [34]), the label
“horse” applies to the image both before and after a reflection about the
central vertical axis. The action of the symmetry group on the encoded quantum
states of the images is determined by a unitary representation $R$ of
$\mathcal{G}$ satisfying
$\ket{\psi(g(\boldsymbol{x}))}=R_{g}\ket{\psi(\boldsymbol{x})}\ \forall
g\in\mathcal{G},\boldsymbol{x}\in\mathcal{X}$, i.e. the diagram commutes. (b)
A schematic depiction of the generic quantum variational classifier (QVC) that
we employ for image classification. The pale green subcircuit $\mathcal{E}$
implements amplitude encoding. The variational component of the circuit
consists of the unit in the dotted box repeated 100 times (with different
parameters in each layer). Finally, $\sigma_{z}$ measurements are made on each
qubit to determine the label prediction. (c) A modified circuit which
possesses reflection equivariance. The encoding subcircuit
$\tilde{\mathcal{E}}$ now includes an additional change of basis following the
amplitude encoding (see the text and Figure 3). In the variational section,
which again consists of the unit in the dotted box repeated 100 times, gates
which commute wth the symmetry operations are chosen. Finally, $\sigma_{z}$
measurements are again made to determine the label prediction, although this
time products of the measurement outcomes on neighbouring qubits are used in
order to retain equivariance (see the text). While (b) and (c) depict 3-qubit
cartoons of the two models, the actual circuits employ either 10 or 12 qubits
depending on the dataset due to the resolution of the images:
1$\times$28$\times$28 for MNIST, and 3$\times$32$\times$32 for CIFAR-10 and
CelebA (see Figure 2).

Indeed, promising recent work has incorporated ideas from GDL into QML,
resulting in the emerging subject of geometric quantum machine learning (GQML)
[35, 36, 37, 38, 39, 40, 41, 42]. For example, this approach has been used to
construct QML models which, classifying data that enjoys permutation symmetry,
provably avoid barren plateaus [38]. In this work we turn the techniques of
GQML to the problem of image classification, an important example of a problem
for which deep learning frameworks drastically outperform all other methods.
The translational equivariance of CNNs has already been imported to the
quantum setting, with quantum convolutional neural networks (QCNNs) offering a
pathway to efficient image classification on quantum computers [8]. Here we
consider the alternative symmetry of reflections, noting that the labels
assigned to an image will often be independent of reflections of that image
about various axes (see Figure 1(a)), as for example in most common object
identification problems. We construct reflection equivariant quantum neural
networks (REQNNs, see Figure 1(c)) and show that they consistently outperform
a generic model (see Figure 1(b)) when benchmarked across three standard image
datasets (CIFAR-10 [34], MNIST [26], and Celeb-A [43] – see Figure 2 for
examples of images from each dataset), despite having access to fewer
trainable parameters. These results provide concrete evidence supporting the
emerging philosophy that sacrificing generality and expressibility in favour
of targeting a smaller but more meaningful fraction of the model space is a
promising approach forward for QML. Moreover, by demonstrating enhanced
performance for image classification through the use of reflection equivariant
networks, separate to the previous use of a translation equivariant
convolutional structure in QCNNs [8], our results encourage the future
development of QML models which exploit as much of the available symmetry
information of their data as possible, including extensions to additional
symmetries and the simultaneous consideration of multiple symmetries. The
practical realisation of tailored QML models such as these will bring the
possibility of successfully applying quantum computing to the many domains of
ML, both image-based and otherwise, closer to reality.

2\. Geometric Quantum Machine Learning

We begin by briefly summarising the aspects of GQML relevant to our
construction of REQNNs, introducing the notation and formalism required to
discuss equivariance and symmetry operations on data encoded into quantum
states. Interested readers can find further details of GQML in Refs. [36, 37].
We consider the classification of (image) data $\boldsymbol{x}\in\mathcal{X}$,
with associated labels $y(\boldsymbol{x})\in\mathcal{Y}$. Our QML models will
follow a standard three-step procedure: a data encoding circuit which maps the
classical image data $\boldsymbol{x}$ to a quantum state,
$\boldsymbol{x}\mapsto\ket{\psi(\boldsymbol{x})}$, followed by a variational
circuit $U_{\theta}$, followed by measurements of a set of operators
$\\{M_{j}\\}_{j=1}^{n_{\mathrm{classes}}}$ to determine the class label. For a
given set of parameters $\theta\in\Theta$ the prediction
$\hat{y}_{\theta}(\boldsymbol{x})$ of the model on an input $\boldsymbol{x}$
is given by

$\hat{y}_{\theta}(\boldsymbol{x})=\operatorname*{argmax}_{j}\bra{\psi(\boldsymbol{x})}\mathcal{U}_{\theta}^{\dagger}M_{j}\mathcal{U}_{\theta}\ket{\psi(\boldsymbol{x})}$
(1)

We also introduce a symmetry group $\mathcal{G}$ which we will identify with
its action on $\mathcal{X}$, i.e. writing
$\mathcal{G}:\mathcal{X}\to\mathcal{X}$, with $g(\boldsymbol{x})$ for
$g\in\mathcal{G}$ the image obtained by performing the symmetry transformation
$g$ on $\boldsymbol{x}$. We emphasise that the symmetries are expected to be
respected only at the level of the labels of the data, i.e.
$y(g(\boldsymbol{x}))=y(\boldsymbol{x})\ \forall\boldsymbol{x}\in\mathcal{X},\
g\in\mathcal{G}$, but perhaps $g(\boldsymbol{x})\neq\boldsymbol{x}$. In this
work we will focus on the group which enacts reflections about the central
vertical axis of the images, so that $\mathcal{G}\cong\mathbb{Z}_{2}$. The
group will act on the Hilbert space $\mathcal{H}$ of the quantum computer via
a unitary representation $R:\mathcal{G}\to\mathrm{Aut}(\mathcal{H})$ which
satisfies $R(g)\ket{\psi(\boldsymbol{x})}=\ket{\psi(g(\boldsymbol{x}))}$.
Henceforth we write $R_{g}\equiv R(g)$. We wish for the predictions of our QNN
to be $\mathcal{G}$-invariant, i.e.

$\hat{y}_{\theta}(\boldsymbol{x})=\hat{y}_{\theta}(g(\boldsymbol{x}))\qquad\forall
g\in\mathcal{G},\boldsymbol{x}\in\mathcal{X},\theta\in\Theta$ (2)

From Equation 1 we have that

$\displaystyle\hat{y}_{\theta}(g(\boldsymbol{x}))$
$\displaystyle=\operatorname*{argmax}_{j}\bra{\psi(g(\boldsymbol{x}))}\mathcal{U}_{\theta}^{\dagger}M_{j}\mathcal{U}_{\theta}\ket{\psi(g(\boldsymbol{x}))}$
(3)
$\displaystyle=\operatorname*{argmax}_{j}\bra{\psi(\boldsymbol{x})}R_{g}^{\dagger}\mathcal{U}_{\theta}^{\dagger}M_{j}\mathcal{U}_{\theta}R_{g}\ket{\psi(\boldsymbol{x})}$
(4)

and therefore the condition of Equation 2 will be satisfied if

$\left[R_{g},\
\mathcal{U}_{\theta}^{\dagger}M_{j}\mathcal{U}_{\theta}\right]=0\qquad\forall
g\in\mathcal{G},\boldsymbol{x}\in\mathcal{X},\theta\in\Theta$ (5)

We refer to QNNs satisfying this condition as reflection equivariant.

3\. Results

Our first task in building REQNNs is to establish the unitary representation
of the symmetry group $\mathcal{G}$. As in our case
$\mathcal{G}\cong\mathbb{Z}_{2}$, there is only one nontrivial symmetry
operator $R_{g}$, namely the one which maps a state to the state encoding the
reflection of the image encoded by the original state. As we have
$R_{g}\ket{\psi(\boldsymbol{x})}=\ket{\psi(g(\boldsymbol{x}))}$ for all images
$\boldsymbol{x}$, $R_{g}$ will be determined by the form of the data encoding
map, to which we now turn. Due to our desire to classify high dimensional
image data using only the relatively small number of qubits available to
classical simulators we need to employ a method of encoding which is highly
efficient in the number of qubits required. For this reason we choose
amplitude encoding, i.e., if $\boldsymbol{x}$ is a vector containing the pixel
values of an image then we construct the state

$\boldsymbol{x}\mapsto\ket{\psi(\boldsymbol{x})}=\sum_{i}x_{i}\ket{i}$

As there are $2^{n}$ amplitudes available for an $n$ qubit state, only
$\lceil\log_{2}(C\texttimes L\texttimes W)\rceil$ qubits are needed, where $C$
is the number of channels of the image, $L$ is the length, and $W$ the width.
For MNIST, we have $(C,L,W)=(1,28,28)$ and therefore require 10 qubits, and
for CIFAR-10 and CelebA $(C,L,W)=(3,32,32)$, requiring 12. In both cases we
append and prepend zeros equally in order to obtain an input vector of length
a power of two in order to proceed with the amplitude encoding.

Figure 2: Image datasets. We consider three standard image datasets from the
ML literature: MNIST [26], CIFAR-10 [34] and CelebA [43]. The MNIST dataset
consists of handwritten digits, CIFAR-10 of various objects and animals, and
CelebA of human faces. In the case of MNIST we restrict our attention to the
digits “0”, “1” and “8”. Having done this, all of the data we consider has
labels which are unchanged under a horizontal reflection and is therefore
suitable for classification by our REQNNs.  Figure 3: Amplitude Encoding.
Encoding for an example $4\times 4$ image. The pixels are numbered by the
index of the basis state into the amplitude of which they are encoded. (a) The
standard case. For example, the pixel value of the pixel in the top right
corner is encoded into the amplitude of $\ket{3}\equiv\ket{0011}$. (b) Here we
permute the order of the encoding. For example, the pixel value of the pixel
in the top right corner is now encoded into the amplitude of
$\ket{15}\equiv\ket{1111}$. In this case a horizontal reflection is
represented by $X^{\otimes n}$.  Figure 4: Quantum neural network
performances.  We compare the performance of the generic and reflection
equivariant QNNs across a range of image datasets. As the different datasets
have varying resolutions, the number of qubits needed to implement the models
also changes (see the Results section), providing another axis along which to
contrast the performance of the two classes of models. We find that the
reflection equivariant QNNs learn more quickly than their generic counterparts
consistently across different datasets, number of classes, and number of
qubits used to implement the QNNs. The plotted accuracies refer to 500 test
samples, calculated at various points throughout the training process. (a, b)
Two and four class classification using the CIFAR-10 dataset, respectively.
(c) Three class classification with the MNIST dataset. Although this dataset
is quite simple, and QNNs have previously been used to achieve high accuracy
on all ten classes [13], we restrict here to the digits “0”, “1” and “8” as
they are the only ones which (approximately) respect the reflection symmetry.
Both models achieve high test accuracy ($>$96%), but the reflection
equivariant model learns more quickly. (d) The CelebA dataset consists of
images of human faces. We consider the classification task of determining the
gender of the imaged person, again finding that the reflection equivariant
model significantly outperforms its generic counterpart.

Unfortunately, standard amplitude encoding will render $R_{g}$ a complicated,
non-local operator which will be difficult to work with in practice,
especially on real hardware with constrained qubit connectivity. In order to
rectify this we consider, for our equivariant models, a slight modification of
standard amplitude encoding which entails rearranging the order in which the
pixel values are encoded so as to produce an encoding with respect to which we
have $R_{g}=X^{\otimes n}$ (see Figure 3). The standard order in which the
pixel values are read is shown in Figure 3(a) for an example of a 4 × 4
greyscale (i.e. only one channel) image, and our alternate encoding in Figure
3(b). The encoding strategy of Figure 3(a) is employed in the generic model
(Figure 1(b)), and the strategy of Figure 3(b) in the equivariant model
(Figure 1(c) of the main text). In the second case, a reflection of the image
is represented at the Hilbert space level by the operator $X^{\otimes n}$,
which exchanges the basis states $\ket{i}$ and $\ket{2n-1-i}$. The case of
three channel RGB images is similar, with the order that the data is encoded
chosen so as to enforce the requirement that, for every pixel and every
channel, the amplitude of the states $\ket{i}$ and $\ket{2n-1-i}$ are the
values (in a given channel) of a pair of pixels related by the vertical
reflection. With this choice of encoding we have that
$R_{g}\ket{\psi(\boldsymbol{x})}=\ket{\psi(g(\boldsymbol{x}))}$ for $g=e,r$
the identity and horizontal reflection operations on the images respectively,
with $R_{e}=I$, $R_{r}=X^{\otimes n}$. Armed with the representation of our
symmetry group (consisting simply of the operators $\\{I,X^{\otimes n}\\}$) we
are ready to begin constructing REQNNs.

Given a symmetry group $\mathcal{G}$ and a unitary representation $R$, various
ways of constructing equivariant QNNs have been proposed. Ref. [35], for
example, takes a standard set of gates and “symmetrises” them, building new
gates which are guaranteed to commute with the symmetry representation. Here,
benefiting from the simplicity of our representation, we adopt a different
approach, simply manually selecting gates and measurements which commute with
$X^{\otimes n}$ and therefore satisfy Equation 5. Schematic depictions of the
two models considered in this work are shown in Figure 1(b,c). The model of
Figure 1(b) is a “generic ansatz” consisting of amplitude encoding followed by
a standard variational circuit followed by $\sigma_{z}$ measurements on the
first $n_{\mathrm{classes}}$ qubits. The prediction of the model is taken to
be the class corresponding to the qubit which reports the largest such
measurement (i.e. $M_{j}=\sigma^{(j)}_{z}$ in the notation of Equation 1, with
$\sigma^{(j)}_{z}$ the Pauli $z$ operator acting on the $j$th qubit). The
reflection equivariant model of Figure 1(c) differs in several ways. First,
the encoding stage is slightly modified as previously discussed (see also
Figure 3). Second, the variational component is built from $R_{x}$ and
$R_{yy}$ gates, both of which commute with $X^{\otimes n}$. Finally, the class
labels are determined by measurements
$M_{j}=\sigma^{(j)}_{z}\otimes\sigma^{(j+1\ \mathrm{mod}\ n)}_{z}$, which also
commute with $X^{\otimes n}$. This QNN therefore satisfies the equivariance
condition of Equation 5.

We implement the networks within the Pennylane framework [44], and train them
using the Nesterov momentum optimiser [45]. The results are shown in Figure 4.
Our results show that the reflection equivariant model consistently
outperforms the generic model, despite its lower expressibility and the
generic model containing 50% more trainable parameters. The increased
performance is particularly notable in the cases of the more complicated
datasets, CIFAR-10 and Celeb-A, which consist of colour (RGB) images. In the
case of MNIST, and although the difference in final test accuracies is
negligible, we nonetheless see the reflection equivariant model learning more
quickly in the initial stage of training, with its focus narrowed to the more
meaningful subset of reflection insensitive decision functions.

$0$$2{,}000$$4{,}000$$6{,}000$$8{,}000$$10{,}000$$0.4$$0.5$$0.6$$0.7$$0.8$Training
SamplesTest AccuracyCIFAR-10; Trucks and ShipsReflection EquivariantGeneric
AnsatzHybrid Figure 5: Effect of reflection equivariance. In addition to the
considered generic and reflection equivariant models we trial a “hybrid” model
with the same variational circuit structure and final measurement strategy as
the reflection equivariant model, but which uses the standard amplitude
encoding map ${\mathcal{E}}$ (Figure 3 (a)) instead of the modified map
$\tilde{\mathcal{E}}$ (Figure 3 (b)). Therefore, the hybrid model continues to
commute with $X^{\otimes n}$, but this operator no longer represents a
meaningful transformation of the data. The considerable drop in accuracy from
the reflection equivariant model, despite sharing the same variational
architecture, confirms the importance of respecting meaningful symmetries of
the data.
$2$$4$$6$$8$$10$$12$$14$$16$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$n_{\mathrm{qubits}}$Tp$\mathrm{Var}\left[\partial_{\theta_{n}^{(0)}}\langle
M_{1}\rangle\right]$Parameter Gradient VariancepReflection EquivariantGeneric
Ansatz Figure 6: Variance of parameter gradients. We calculate the derivative
of the expectation value of the measured operator $M_{1}$ in both the generic
and reflection equivariant cases with respect to the first variational
parameter in the final layer of the circuits. This is repeated for 3000 random
circuit initialisations, and the variance of the results is plotted as a
function of the number of qubits. At the number of qubits relevant for this
work ($n_{\mathrm{qubits}}=10,12$) we find that the variance of the gradients
in the equivariant case is several orders of magnitude higher than the generic
case. This is consistent with the improved trainability observed for the
equivariant networks, as well as the increased volatility of the test
accuracies seen in Figures 4 and 5.

As the reflection equivariant and generic classifiers considered in this work
are constructed from substantially different quantum circuits, there is a
possibility that the enhanced performance of the reflection equivariant model
is due to some other factor than its respecting of the symmetry. In order to
separate the effect of simply moving to a different circuit architecture from
the role played by the reflection equivariance we consider a “hybrid” model
consisting of the standard encoding $\mathcal{E}$ of Figure 1(b) (see also
Figure 3(a)), and the variational body and measurements of Figure 1(c). This
produces a model with the same restricted expressibility of the reflection
equivariant model, which remains equivariant with respect to the operator
$X^{\otimes n}$, but for which $X^{\otimes n}$ no longer enacts a meaningful
symmetry. The results, shown in Figure 5, show the reflection equivariant
model significantly outperforming this hybrid model, confirming the importance
of building networks which respect genuine symmetries of the data.

Being represented by a group with only two elements, we do not expect the
respecting of reflection symmetry to lead to a provable avoidance of barren
plateaus as has been previously reported in the case of permutation symmetry
[38], but our results here are an encouraging sign that, even in the absence
of such guarantees, considerable gains in accuracy may be realised by such
models in practice. To explore this further we numerically investigate the
variance of the gradient of a measured Pauli observable in both the generic
and reflection equivariant cases (see Figure 6, c.f. Figure 3 of Ref. [14]).
As expected, the (universal) generic model experiences a barren plateau, with
exponentially vanishing gradients. While we also see approximately exponential
decreases in the (non-universal) reflection equivariant model, the rate of
decrease is drastically reduced, indicating that training will remain feasible
for much larger quantum circuits. This is consistent with our expectation
that, while the REQNNs may also asymptotically experience barren plateaus,
they will be able to offer enhanced performance in the highly interesting
region of several tens of qubits explored in this work and accessible on near-
term hardware.

4\. Conclusion

Geometric quantum machine learning is rapidly emerging as a promising research
direction which may ameliorate two of the key challenges facing QML: barren
plateaus and overfitting. We have applied the techniques of GQML to the task
of creating QML models for image classification which are equivariant with
respect to horizontal reflections, finding a consistent improvement over
generic, symmetry agnostic models, despite those models being more expressive.
These encouraging results join the previous GQML literature [35, 36, 37, 38,
39, 40, 41, 42] in demonstrating clear advantages in building tailored QML
models which respect the symmetry of the data they are attempting to classify,
rather than simply employing universal models, and provide further evidence of
the potential of this research direction in the NISQ era and beyond.

The authors acknowledge useful discussions with Jamie Heredge. MW acknowledges
the support of the Australian Government Research Training Program
Scholarship. Computational resources were provided by the National Computing
Infrastructure (NCI) and Pawsey Supercomputing Centre through the National
Computational Merit Allocation Scheme (NCMAS). This work was supported by
Australian Research Council Discovery Project DP210102831.

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