# Efficient Quantum Feature Extraction for CNN-based Learning

Tong Dou1, Guofeng Zhang2, and Wei Cui1<EMAIL_ADDRESS>1School of
Automation Science and Engineering, South China University of Technology,
Guangzhou 510640, China  2Department of Applied Mathematics, The Hong Kong
Polytechnic University, Hongkong, China

###### Abstract

Recent work has begun to explore the potential of parametrized quantum
circuits (PQCs) as general function approximators. In this work, we propose a
quantum-classical deep network structure to enhance classical CNN model
discriminability. The convolutional layer uses linear filters to scan the
input data. Moreover, we build PQC, which is a more potent function
approximator, with more complex structures to capture the features within the
receptive field. The feature maps are obtained by sliding the PQCs over the
input in a similar way as CNN. We also give a training algorithm for the
proposed model. The hybrid models used in our design are validated by
numerical simulation. We demonstrate the reasonable classification
performances on MNIST and we compare the performances with models in different
settings. The results disclose that the model with ansatz in high
expressibility achieves lower cost and higher accuracy.

††preprint: APS/123-QED

##  I Introduction

Quantum Computers use the principles of quantum mechanics for computing, which
are considered more powerful than classical computers in some computing
problems. In view of the rapid progresses in quantum computing hardware
arute2019quantum , we are entering into so called NISQ era preskill2018quantum
. Noisy intermediate scale quantum (NISQ) devices will be the only quantum
devices that can be reached in near-term, where we can only use a limited
number of qubits with little error correction. Thus, developing a useful
computational algorithm using the NISQ devices is the urgent task at present.

Quantum machine learning (QML) biamonte2017quantum  is gaining attention in
the hope of speeding up machine learning tasks with quantum computers. Combing
quantum computing and machine learning, QML attempts to utilize the power of
quantum computers to achieve computational speedups or better performance for
some machine learning tasks. Many quantum machine learning algorithms, such as
qSVM rebentrost2014quantum , qPCA lloyd2014quantum  and quantum Boltzmann
machine amin2018quantum , have been developed. Recently, several NISQ
algorithms, such as quantum transfer learning mari2020transfer  and quantum
kernel methods havlivcek2019supervised ; schuld2019quantum , have been
proposed. On the other hand, parameterized quantum circuits
benedetti2019parameterized  provide an another path for QML toward quantum
advantages in NISQ era. Compared to conventional quantum algorithms, such as
Shor’s algorithm shor1999polynomial , Grover’s algorithm grover1997quantum ,
HHL algorithm harrow2009quantum , PQC based quantum-classical hybrid
algorithms are inherently robust to noise and could benefit from quantum
advantages, by taking advantage of both the high dimensional Hilbert space of
a quantum system and the classical optimization scheme. Several popular
related algorithms have been proposed, including variational quantum
eigensolvers (VQE) peruzzo2014variational ; wecker2015progress ;
mcclean2016theory  and quantum approximate optimization algorithm (QAOA)
farhi2014quantum ; hadfield2019quantum .

In addition, quantum neural networks (QNNs) schuld2014quest ;
farhi2018classification ; lloyd2020quantum  based on parameterized quantum
circuits have been proposed. Some of QNN models utilized the thoughts of
Convolutional Neural Network (CNN), which is a popular machine learning model
on the classical computer. Cong et al. cong2019quantum  designed a quantum
circuit model with a similar hierarchy to classical CNNs, which dealt with
quantum data and could be used to recognize phases of quantum states and to
devise a quantum error correction scheme. The convolutional and pooling layers
were approximated by quantum gates with adjustable parameters. Ref.
henderson2020quanvolutional  proposed a new quanvolutional (short for quantum
convolutional) filter, in which a random quantum circuit was deployed to
transform input data locally. Quanvolutional filters were embedded into
classical CNNs, forming a quantum-classical hybrid structure. In addition, a
more complex QCNN design was presented in recent works Kerenidis2020Quantum ;
li2020quantum ; zheng2021speeding , where delicate quantum circuits were
employed to accomplish quantum inner product computations and approximate
nonlinear mappings of activation functions.

In this work, we propose a type of hybrid quantum-classical neural network
based on PQCs for in analogy with the CNN. As a popular scheme in machine
learning field, CNN replaces the fully connected layer with a convolutional
layer. The convolutional layer only connects each neuron of the output to a
small region of the input which is referred to as a feature map, thus greatly
reducing the number of parameters. However, the filter in classical CNN model
is a generalized linear model (GLM). It is difficult for linear filters to
extract the concepts are generally highly nonlinear function of the data
patch. In Network-in-Network (NiN) lin2013network , the linear filter replaced
with a multilayer perceptron which is a general function approximator. As a
“micro network”, multilayer perceptron can improve the abstraction ability of
the model. In the field of QML, PQCs are considered as the “quantum network”
structures. Recent works schuld2021effect ; goto2021universal ; liu2021hybrid
shown that there exist PQCs which are universal function approximators with a
proper data encoding strategy. Combining these ideas, we replace the linear
filter with a PQC. The resulting structure is called the quantum feature
extraction layer.

The key idea of our hybrid neural network is to implement the feature map in
the convolutional layer with quantum parameterized circuits, and
correspondingly, the output of this feature map is a correlational measurement
on the output quantum state of the parameterized circuits. Different from Ref.
henderson2020quanvolutional , which uses parameters fixed circuits, we can
iteratively update the parameters of circuits to get better performance. We
also give the training algorithm of this hybrid quantum-classical neural
network which are similar to backpropagation (BP) algorithm, meaning that our
model is trained as a whole. Hence it can be trained efficiently with NISQ
devices. Therefore, our scheme could utilize all the features of classical
CNN, and is able to utilize the power of current NISQ processors. Notice that
the methods of data encoding and decoding are different from
henderson2020quanvolutional .

This paper is organized as follows. Section II is the preliminary, in which we
first review the classical convolution neural network and the framework of
PQCs. Then, the proposed quantum-classical hybrid model is described in detail
in Section III. In Section IV, we present numerical simulation where we apply
the hybrid model to image recognition and analyze the result. Conclusion and
discussion are given in Section V.

##  II Preliminaries

###  II.1 Supervised learning: construct a classifier

In a supervised classification task, we are given a training set $T$ and a
test set $S$ on a label set $C$. Assuming there exists a map $m:T\cup
S\rightarrow C$ unknown to the algorithm. Our task is to construct a model to
infer an approximate map on the test set $\tilde{m}:S\rightarrow C$ only
receiving the labels of the training set such that the difference between
$m(s)$ and $\tilde{m}(s)$ is small on the test set $s\in S$. For such a
learning task to be meaningful it is assumed that there is a correlation
between the labels given for training set and the true map. A popular approach
to this problem is to use neural networks(NNs), where the training data goes
through the linear layer and the activation function repeatedly to map the
true label. After constructing a NNs model that is accurate enough, we can
perform the prediction of test set.

###  II.2 Convolution neural network(CNN)

CNN dates back decades lecun1989backpropagation ; lecun1998gradient  which is
a specialized kind of neural network for processing data that has a known
grid-like topology. Deep CNNs have recently shown an explosive popularity
partially due to its success in image classification krizhevsky2012imagenet .
It also has been successfully applied to other fields, such as objection
detection redmon2016you , semantic segmentation long2015fully  and pedestrian
detection zhang2016faster , even for natural language processing(NLP)
vaswani2017attention .

Though there are several architectures of CNN models such as LeNet
lecun1998gradient , AlexNet krizhevsky2012imagenet , GoogLeNet
szegedy2015going , VGG simonyan2014very , ResNet he2016deep  and DenseNet
huang2017densely , they use three main types of layers to build CNN models:
convolutional layer, pooling layer and fully-connected layer. Broadly, a CNN
model can be divided into two parts: feature extraction module and classifier
module. The former includes convolutional layer and pooling layer while the
classifier module consists of fully-connected layers.

Formally, a convolutional layer is expressed as an operation $f_{1}$:

$\displaystyle f_{1}(\textbf{X})=g(\textbf{W}_{1}*\textbf{X}+\textbf{B}_{1}),$
(1)

where $\textbf{W}_{1}$ and $\textbf{B}_{1}$ represent the filters weights and
biases respectively, and ‘$*$’ denotes the convolution operation. Concretely,
$\textbf{W}_{1}$ corresponds to $n_{c}$ filters of support $f\times f\times
c$, where $c$ is the number of channels in the input X, $f$ is the spatial
size of a filter. Intuitively, $\textbf{W}_{1}$ applies $n_{c}$ convolutions
on the input, and each convolution has a kernel size $f\times f\times c$. The
output is composed of $n_{c}$ feature maps. $\textbf{B}_{1}$ is a
$n_{c}$-dimensional vector, whose each element is associated with a filter.
$g(\cdot)$ is an activation function which element-wisely apply a nonlinear
transformation ,such as ReLU or Tanh, on the filter responses.

A pooling layer is generally added after the convolutional layer. Its function
is to progressively reduce the spatial size of the representation to reduce
the amount of parameters and computation in the network. Two commonly used
pooling operations are max pooling and average pooling. The former calculates
the maximum value for each patch of the feature map while the latter
calculates the average value for each patch on the feature map. And the units
in a fully-connected layer have full connections to all activations in the
previous layer, as seen in regular neural networks.The activations are
computed with a matrix multiplication followed by a bias offset, as shown in
the operation $f_{2}$:

$\displaystyle f_{2}(\textbf{x})=h(\textbf{W}_{2}\textbf{x}+\textbf{B}_{2}),$
(2)

where x, $\textbf{W}_{2}$, $\textbf{B}_{2}$ are a input vector, a weight
matrix and a bias vector, respectively. $h(\cdot)$ is an activation function
like $g(\cdot)$ mentioned.

###  II.3 General framework of PQCs

Parametrized quantum circuits, also known as “variational circuits”, are a
kind of quantum circuits that have trainable parameters subject to iterative
optimizations. Previous results show that such circuits are robust against
quantum noise inherently and therefore suitable for the NISQ devices.
Recently, there have been many developments in PQCs-based algorithms which
combine both quantum and classical computers to accomplish a particular task.
In addition to VQE and QAOA, the PQCs framework has been extended for
applications in generative modeling dallaire2018quantum , quantum data
compression romero2017quantum , quantum circuit compiling sharma2020noise  and
so on.

Algorithms involving PQCs usually work in the following way. First, prepare a
initial state $|\varphi_{0}\rangle$ by encoding input into the quantum device.
Second, we need to choose an appropriate ansatz, that is, designing the
circuit structure of a PQC $U(\bm{\theta})$, and apply $U(\bm{\theta})$ to
$|\varphi_{0}\rangle$, where $\bm{\theta}$ is parameters of the circuit. Then
measure the circuit repeatedly on a specific observable $\hat{O}$ to estimate
an expectation value $\langle\hat{O}(\bm{\theta})\rangle$. Based on the
$\langle\hat{O}(\bm{\theta})\rangle$ which is fed into a classical optimizer,
we compute a cost function $L(\langle\hat{O}(\bm{\theta})\rangle)$ to be
minimized by updating $\bm{\theta}$.

Several factors are of central importance in the PQC algorithms. First, it is
necessary to find a suitable $\hat{O}$ for the given problem. That is, we
require the minimum of $L(\langle\hat{O}(\bm{\theta})\rangle)$ to correspond
to the solution of the task. How easily $\hat{O}$ can be measured on a quantum
computer and the locality of $\hat{O}$(i.e. the number of qubits it acts non-
trivially on) are also relevant.

Another aspect that determines the success of a PQC algorithm is the choice in
ansatz for $U(\bm{\theta})$. While discrete parameterizations are possible,
$\bm{\theta}$ usually are continuous parameters, such as gate rotation angles,
in a PQC. In general, a PQC is of form

$\displaystyle U(\bm{\theta})=\prod_{j=1}^{N}U_{j}(\theta_{j})W_{j},$ (3)

where $\bm{\theta}=(\theta_{1},\dots,\theta_{N})$ are parameters, and
$\\{W_{j}\\}_{j=1}^{N}$ is a set of unitaries without parameters while
$U_{j}=e^{-i\theta_{j}V_{j}}$ is a rotation gate of angle $\theta_{j}$
generated by a Hermitian operator $V_{j}$ such that $V_{j}^{2}=I$. The
rotation angles $\\{\theta_{j}\\}$ are typically considered to be independent.

|  |  |
---|---|---|---
(a) convolutional layer | (b) Mlpconv layer | (c) Quanvolutional layer | (d) QFE layer
Figure 1: Comparison of convolutional layer, mlpconv layer, quanvolutional
layer and QFE layer. The convolutional layer includes a linear filter. The
mlpconv layer includes a multilayer perceptron. The quanvolutional layer
includes a random quantum circuit without trainable parameters. The QFE layer
includes a “micro quantum network”(we choose PQC in this paper).

##  III Convolutional Network with Quantum Feature Extraction Layers

In this section, we will first describe the quantum-classical hybrid model
with quantum feature extraction layers and give a training algorithm. Then the
possible experimental setup for its realization is discussed.

###  III.1 Quantum Feature Extraction Layers

A quantum feature extraction(QFE) layer is a transformational layer using
learnable parameteried quantum circuits. It is similar to a convolutional
layer in classical CNNs. Combining pooling layers, fully-connected layer even
with convolutional layer, we can construct a hybrid quantum-classical neural
network.

Suppose a QFE layer learns a mapping $F$, which conceptually consists of three
parts, namely encoding, variational evolution, decoding.

  1. 1.

_Encoding._ This operation encodes patches from the input data $\bm{x}$(or the
output of previous layer) into a quantum circuit by applying $U_{en}(\bm{x})$
to the initialized states $|0\rangle^{n}$. Hereafter, we abbreviate
$|0\rangle^{n}$ as $|0\rangle$ if there is no confusion. Note that the output
state $|\bm{x}\rangle$ are characterized by input $x$.

  2. 2.

_Variational evolution._ In this part, we apply a parameterized unitary
circuit $U_{var}(\bm{\theta})$ to the quantum state $|x\rangle$.
$U_{var}(\bm{\theta})$ maps the quantum state $|\bm{x}\rangle$ onto another
quantum state $|\bm{x};\bm{\theta}\rangle$ using a serial of quantum gates.
This operation can introduce quantum properties such as superposition,
entanglement and quantum parallelism.

  3. 3.

_Decoding._ This operation include two parts: measurement and classical post-
processing. Firstly, at the end of the circuit, measurements are performed on
an observable $\hat{H}$ to get an expectation value. Then after the classical
post-processing, we get a output of one step in QFE layer. Analogously to a
classical convolution layer, each output is mapped to a different channel of a
single pixel value.

The structure of a QFE layer is compared with convolutional , quanvolutional
and mlpconv layer in Fig. 1. Next we detail our definition of each operation.

####  III.1.1 Encoding

There are two popular strategy to encode classical data into quantum circuit,
including the basis encoding and the amplitude encoding. The basis encoding
method treats two basic states of a qubit, $|0\rangle$ and $|1\rangle$, as
binary values of a classical bit, 0 and 1. While the amplitude encoding method
uses the probability amplitudes of a quantum state to store classical data. As
described below, to update the parameters effectively, we need to calculate
the partial derivative of input. However, the two methods mentioned above are
difficult to achieve it. To satisfy the condition, we treat a single input
value as the angle of a parametrized rotation gate. Combining with parameter-
shift rulemitarai2018quantum , we can calculate the analytical gradients of
input on quantum circuits. Formally, the encoding part is expressed as an
operation $F_{1}$:

$\displaystyle F_{1}(\bm{x})=U_{en}(\bm{x})|0\rangle=|\bm{x}\rangle.$ (4)

Here $\bm{x}=(x_{1},x_{2},\dots,x_{n})^{\bm{T}}$ is the input patch of QFE
layer where $n$ is the dimension of $\bm{x}$(usually in a $3\times 3$ or
$5\times 5$ size) and represents the size of the quantum circuit.
$U_{en}(\bm{x})$ is a unitary to embed $\bm{x}$ into a quantum circuit. For
example, as we used in the experiments, $U_{en}(\bm{x})$ can expressed as:

$\displaystyle U_{en}(\bm{x})=\bigotimes_{i=1}^{n}R_{y}(x_{i}).$ (5)

####  III.1.2 Variational evolution

The first part encodes the each of input patch $\bm{x}$ as a quantum state
$|\bm{x}\rangle$. In the second part, applying a unitary
$U_{var}(\bm{\theta})$, we map the state $|\bm{x}\rangle$ into another state
$|\bm{x};\bm{\theta}\rangle$. Utilizing the power of quantum mechanics, such
as superposition and entanglement, we hope that it can improve the usefulness
of features. The operation of the second part is:

$\displaystyle
F_{2}(\bm{x})=U_{var}(\bm{\theta})|\bm{x}\rangle=|\bm{x};\bm{\theta}\rangle,$
(6)

where $U(\bm{\theta})$ is an ansatz characterized by the parameters $\theta$
and $\theta$ is a vector whose dimension depends on the type of ansatz. It is
important to find an expressive ansatz that can achieve a big range of quantum
states. More generally, it is possible to repeat $U_{var}(\bm{\theta})$ to
reach more quantum states. But this can increase the complexity of the model,
and thus demands more training time.

In order to implement the ansatz in a real quantum device, we select
$\\{H,R_{X},R_{Y},R_{Z},CNOT\\}$ to compose the ansatz In this case, motivated
by the universality of QAOA lloyd2018quantum ; morales2020universality , we
use a QAOA-heuristic circuit as an ansatz. The circuit is as shown in Fig. 2.

Figure 2:  The QAOA-heuristic circuit.

####  III.1.3 Decoding

To access the information of the quantum states, we need to “decode” the
quantum state $|\bm{x};\bm{\theta}\rangle$ into classical data, which consists
of measurement and classical post-processing. At the measurement part, we
measure a expectation value of a observable $\hat{H}$. Then, in classical
post-processing part, we transform this expectation value to get the output by
adding a bias $b$. The output is expressed as:

$\displaystyle
F_{3}(\bm{x})=\sigma(Tr(|\bm{x};\bm{\theta}\rangle\langle\bm{x};\bm{\theta}|\hat{H})+b),$
(7)

where $|\bm{x};\bm{\theta}\rangle$ is the output state from the variational
evolution part and $b$ is the bias. $\hat{H}$ is an observable composed of
$\\{\hat{H_{k}}\\}_{k=1}^{l}$ where $H_{k}$ is a fixed Hamiltonian.
$Tr(\cdot)$ is the trace operator. $\sigma(\cdot)$ is the activation function.
Its function is to manipulate the outputs into a range of $[0,2\pi]$

###  III.2 Backward pass of QFE layer

In classical NNs, gradient descent based optimization algorithms are usually
applied to update the parameters. A powerful routine in training is the
backpropagation algorithm, which combines the chain rule for derivatives and
stored values from intermediate layers to efficiently compute gradients.
However, this strategy can not generalized straightforwardly to the QFE layer,
because intermediate values from a quantum circuit can only be accessed via
measurements, which will cause the collapse of quantum states. Using the
parameter-shift rule and the chain rule, we proposed a training algorithm to
compute the gradients of QFE Layer which can be embedded in the standard
backpropagation. Hence we can train the hybrid model as a whole.

Suppose a QFE layer with parameters $\bm{\theta}$, $b$, whose input is $Z$ and
output is $A$. The size of $Z$ is $m\times m$. The size of kernel is $f\times
f$ and the stride is $s$, therefore, the size of $A$ is
$(\lfloor\frac{m-f}{s}\rfloor+1)\times(\lfloor\frac{m-f}{s}\rfloor+1)$. Let
$n=\lfloor\frac{m-f}{s}\rfloor+1$ to make the procedure clearer. Then, $Z$ and
$A$ can be expressed as:

$\displaystyle Z=\begin{pmatrix}z_{11}&z_{12}&\cdots&z_{1m}\\\
z_{21}&z_{22}&\cdots&z_{2m}\\\ \vdots&\vdots&\ddots&\vdots\\\
z_{m1}&z_{m2}&\cdots&z_{mm}\end{pmatrix},$ (8) $\displaystyle
A=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\\ a_{21}&a_{22}&\cdots&a_{2n}\\\
\vdots&\vdots&\ddots&\vdots\\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{pmatrix}.$ (9)

---
(a) The model that includes the stacking of two QFE layers and a classical
“classifier”.
(b) The model that includes the stacking of three QFE layers and a global
average layer.
Figure 3: The overall structures of evaluated models.

$L$ denotes the loss function. Firstly, consider the condition $s=1$ which is
be commonly used. Assuming that the partial derivative $\frac{\partial
L}{\partial A}$ was known. $\frac{\partial a_{ij}}{\partial z_{uv}}$ and
$\frac{\partial a_{ij}}{\partial\theta_{k}}$ where $i,j=1,\cdots,n$,
$u,v=1,\cdots,m$ and $k=1,\cdots,l$ can be calculated by parameter-shift rule.
To implement the backpropagation of the QFE layer, we need to get
$\frac{\partial L}{\partial\bm{\theta}}$, $\frac{\partial L}{\partial b}$ and
$\frac{\partial L}{\partial Z}$. For convenience, we flatten $\frac{\partial
L}{\partial Z}$ and $\frac{\partial L}{\partial A}$ row by row as follow:

$\displaystyle\frac{\partial L}{\partial A}=\begin{pmatrix}\frac{\partial
L}{\partial a_{11}}&\cdots&\frac{\partial L}{\partial a_{1n}}\\\
\frac{\partial L}{\partial a_{21}}&\cdots&\frac{\partial L}{\partial
a_{2n}}\\\ \vdots&\ddots&\vdots\\\ \frac{\partial L}{\partial
a_{n1}}&\cdots&\frac{\partial L}{\partial a_{nn}}\\\
\end{pmatrix}\rightarrow\frac{\partial
L}{\partial\overline{A}}=\begin{pmatrix}\frac{\partial
L}{\partial\overline{a}_{1}}\\\ \frac{\partial L}{\partial\overline{a}_{2}}\\\
\vdots\\\ \frac{\partial L}{\partial\overline{a}_{n\times n}}\end{pmatrix}.$
(10)

For $\frac{\partial L}{\partial b}$, by the chain rule, we have

$\displaystyle\frac{\partial L}{\partial b}=\sum_{i,j}\frac{\partial
a_{i,j}}{\partial b}\frac{\partial L}{\partial
a_{i,j}}=\sum_{i,j}\frac{\partial L}{\partial a_{i,j}}=\sum\frac{\partial
L}{\partial A}.$ (11)

For $\frac{\partial L}{\partial\bm{\theta}}$, by the chain rule, we have

$\displaystyle\frac{\partial L}{\partial\theta_{k}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\sum_{j=1}^{m}\frac{\partial
a_{ij}}{\partial\theta_{k}}\frac{\partial L}{\partial
a_{ij}}=\sum_{i=1}^{m\times
m}\frac{\partial\overline{a}_{i}}{\partial\theta_{k}}\frac{\partial
L}{\partial\overline{a}_{i}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{\partial\overline{a}_{1}}{\partial\theta_{k}}&\frac{\partial\overline{a}_{2}}{\partial\theta_{k}}&\cdots&\frac{\partial\overline{a}_{m\times
m}}{\partial\theta_{k}}\end{pmatrix}\begin{pmatrix}\frac{\partial
L}{\partial\overline{a}_{1}}\\\ \frac{\partial L}{\partial\overline{a}_{2}}\\\
\vdots\\\ \frac{\partial L}{\partial\overline{a}_{m\times
m}}\end{pmatrix}=\frac{\partial\overline{A}}{\partial\theta_{k}}^{T}\frac{\partial
L}{\partial\overline{A}}.$

Thus,

$\displaystyle\frac{\partial
L}{\partial\bm{\theta}}=\begin{pmatrix}\frac{\partial
L}{\partial\theta_{1}}\\\ \frac{\partial L}{\partial\theta_{2}}\\\ \vdots\\\
\frac{\partial
L}{\partial\theta_{l}}\end{pmatrix}=\frac{\partial\overline{A}}{\partial\bm{\theta}}^{T}\frac{\partial
L}{\partial\overline{A}},$ (13)

where
$\frac{\partial\overline{A}}{\partial\bm{\theta}}^{T}=\begin{pmatrix}\frac{\partial\overline{a}_{1}}{\partial\theta_{1}}&\frac{\partial\overline{a}_{2}}{\partial\theta_{1}}&\cdots&\frac{\partial\overline{a}_{m\times
m}}{\partial\theta_{1}}\\\
\frac{\partial\overline{a}_{1}}{\partial\theta_{2}}&\frac{\partial\overline{a}_{2}}{\partial\theta_{1}}&\cdots&\frac{\partial\overline{a}_{m\times
m}}{\partial\theta_{2}}\\\ \vdots&\vdots&\ddots&\vdots\\\
\frac{\partial\overline{a}_{1}}{\partial\theta_{l}}&\frac{\partial\overline{a}_{l}}{\partial\theta_{1}}&\cdots&\frac{\partial\overline{a}_{m\times
m}}{\partial\theta_{l}}\\\ \end{pmatrix}$.

To compute the $\frac{\partial L}{\partial Z}$, we define $\frac{\partial
A}{\partial Z}$ as

$\displaystyle\frac{\partial A}{\partial
Z}=\left(\begin{array}[]{cccc}\frac{\partial A}{\partial
z_{11}}&\frac{\partial A}{\partial z_{12}}&\cdots&\frac{\partial A}{\partial
z_{1m}}\\\ \frac{\partial A}{\partial z_{21}}&\frac{\partial A}{\partial
z_{22}}&\cdots&\frac{\partial A}{\partial z_{2m}}\\\
\vdots&\vdots&\ddots&\vdots\\\ \frac{\partial A}{\partial
z_{m1}}&\frac{\partial A}{\partial z_{m2}}&\cdots&\frac{\partial A}{\partial
z_{mm}}\\\ \end{array}\right),$ (18)

where $\frac{\partial A}{\partial z_{ij}}=\begin{pmatrix}\frac{\partial
a_{11}}{\partial z_{ij}}&\frac{\partial a_{12}}{\partial
z_{ij}}&\cdots&\frac{\partial a_{1n}}{\partial z_{ij}}\\\ \frac{\partial
a_{21}}{\partial z_{ij}}&\frac{\partial a_{22}}{\partial
z_{ij}}&\cdots&\frac{\partial a_{2n}}{\partial z_{ij}}\\\
\vdots&\vdots&\ddots&\vdots\\\ \frac{\partial a_{n1}}{\partial
z_{ij}}&\frac{\partial a_{n2}}{\partial z_{ij}}&\cdots&\frac{\partial
a_{nn}}{\partial z_{ij}}\\\ \end{pmatrix}$.

Then, the $\frac{\partial L}{\partial Z}$ can be expressed as

$\displaystyle\frac{\partial L}{\partial Z}=\frac{\partial A}{\partial
Z}*\frac{\partial L}{\partial A},$ (19)

where ‘$*$’ denotes the convolution operation with the stride $n$.

##  IV Experiments

In this section, we measure the performance of our hybrid models on MNIST
lecun1998gradient  dataset by numerical simulations. All of the experiments
are performed with Yao luo2020yao  package in Julia language.

###  IV.1 Dataset

The MNIST dataset is composed of 10 classes of hand written digits 0-9. There
are 60,000 training images and 10,000 testing images. Each image is a gray
image of size $28\times 28$. To save the training time, we cut the images into
$22\times 22$ without information loss, as the original images are $20\times
20$. For training and testing procedure, we use 6000 and 600 samples of all
categories evenly for training set and test set, respectively. The labels use
one-hot encoding.

###  IV.2 Effectiveness of proposed method

Architecture selection: Here, to verify the feasibility of the scheme, we
evaluated models of following architectures. The models all consist of stacked
QFE layers instead of linear convolutional layers.

  1. 1.

_QFE layers with fully-connected layers._ A model, which is similar to the
classical CNN, replaces the linear convolutional layers with QFE layers. The
structure is as followed: QFE1 - POOL1 - QFE2 - POOL2 - FC1 - FC2 - FC3. As
shown in the Fig. 3(a) In this model, the fully-connected layers work as a
“classifier”. We regard this model as model 1.

  2. 2.

_QFE layers with global average pooling._ A model uses global average pooling
(GAP) instead of fully-connected layers at the top of the network, with
following structure: QFE1 - POOL1- QFE2 - QFE3 - GAP. As shown in Fig. 3(b).
In this case, GAP introduced as a regularizer in Ref. lin2013network  is used
to remove the “classical factors” of the network. We regard this model as
model 2.

Each QFE layer uses “quantum filters” of size $3\times 3$. Max pooling is used
in the pooling layers which downsamples the size of input by a factor of 2.

Training strategy: For fair comparison, we adopt the same dataset for
training. To save training time, we also explore a step-by-step training
procedure. First, we train the network from scratch with a relatively small
dataset. Then when the training is saturated, we add more training data for
fine-tuning and repeat this step. With this strategy, the training converges
earlier and faster than training with the larger dataset from the beginning.
For initialization, the weights of QFE layers are initialized by drawing
randomly from a uniform distribution with a range of $[-\pi,\pi]$, while the
weights of fully-connected layers are initialized by a Gaussian distribution
with zero mean and standard deviation 0.001. The models are trained using Adam
optimization algorithm with mini-batches of size 50. The initial learning rate
is 0.01 and is adjusted manually during training.

###  IV.3 Investigation of different settings

To investigate the property of the model, we design a set of controlling
experiments with different settings of two variables \- the type of the ansatz
and the number of layers of ansatz, based on the structure of model 2 (i.e.
QFE1 - POOL1- QFE2 - QFE3 - GAP).

####  IV.3.1 Type of ansatz

Recent work by Sim etc.sim2019expressibility  studies expressibility the
entangling capability of quantum circuits, showing these descriptors are
related to the performance of PQCs-based algorithms. Here, we compare the
model performance with different ansaetze. Specifically, we choose circuit 1,
2, 9, 14 and 15 in sim2019expressibility  besides the QAOA-heuristic circuit.

####  IV.3.2 Number of layers

In general, the performance would improve if we increase the number of layers
of ansatz at the cost of running time. Here, we examine the model sensitivity
to different number of layers. In previous experiments, we set the number of
layers of ansatz $L=3$ for each ”quantum” filter. To be consistent with
previous experiments, we keep the kinds of ansaetze but change the number of
layers from 1 to 5. All the other settings remain the same with model 2.

Totally, we examine 30 models for the same training data with 2560 samples for
9 epochs. The experimental setup is shown in the Table I.

Table 1: Experimental setup to investigate different settings Epoch | Learning rate | batch size
---|---|---
1 | 0.01 | 32
2 - 3 | 0.005 | 32
4 - 6 | 0.001 | 32
7 - 9 | 0.0005 | 16

###  IV.4 Results

Figure 4:  Training cost and accuracy for the model that includes QFE layers
with fully-connected layers. Figure 5:  Training cost and accuracy for the
model that includes QFE layers with GAP.

We first report results on classification to demonstrate that our method works
with fully-connected layers or not. Then we compare the performance between
different settings.

Classification. training performances between model 1 and model 2 are shown in
Fig. 4 and Fig. 5. For test set error rates, model1 and model2 achieve 3.1%
and 6.8% respectively. It is reasonable for the toy models. However, removing
classical fully-connected layers, there is a small performance drop of 2 - 3%.

Compare using different settings. To further compare our approach with
different settings mentioned above, we train the models with the same dataset
and the results are shown in Fig.6. Note that these results are obtained using
different ansaetze with different layers. We can observe that the model
converges lower cost and achieves higher accuracy with deeper layers in
ansatz. However, we note that “saturation” phenomenon exists, which means
adding addition layers to PQC, the performance of the model does not always
continue to improve. This phenomenon appears more and less in all of the
ansaetze. Take circuit 9 as an example, the model with 2-layer PQC converges
on a lower value and a higher accuracy than that with 1-layer PQC, while the
curves of cost and accuracy are almost the same respectively, when the models
equip with circuit 9 in 4 layers and 5 layers. The “saturation” phenomenon may
relate with expressibility saturationsim2019expressibility  or barren
plateausmcclean2018barren . To understand how expressibility correlates with
the model performance further, future research is needed.

This section aims to demonstrate a proof of concept for the proposed method on
a real-world dataset. Because the training is time-consuming, we construct our
architectures based on the small model which has only two or three layers.
Note that model 2 which has no classical fully-connected layers achieves 6.8%
test set error. This shows the the effectiveness of QFC layers and their
training algorithm.

|  |
---|---|---
(a) costs of circuit 1 | (c) costs of circuit 2 | (e) costs of circuit 9
|  |
(b) accuracies of circuit 1 | (d) accuracies of circuit 2 | (f) accuracies of circuit 9
|  |
(g) costs of circuit 14 | (i) costs of circuit 15 | (k) costs of QAOA-like circuit
|  |
(h) accuracies of circuit 14 | (j) accuracies of circuit 15 | (l) accuracies of QAOA-like circuit
Figure 6: The performances (cost and accuracy) of models with different
settings

##  V Conclusions

This paper proposed a hybrid quantum-classical network with deep
architectures. The new structure consists of QFE layers which use PQCs to
“convolve” the input. QFE layers generalize convolution to the high
dimensional hilbert space and model the local patches better. It has been
shown that our hybrid models can achieve competitive performances based on
simple architectures. Besides, we compare the performances of models with
different ansaetze in different depths, showing that the model with ansatz in
high expressibility performs better. In practice, the network architecture and
the initialization method have a significant impact on performance. However,
there is a no effective method to avoid the barren plateaus yet. On the other
hand, since the modern classical network are so deep and our method provides
many possibilities via PQCs, we cannot perform exhaustive tests to find the
best sequence of QFE layers and the best combination of circuit ansaetze. Due
to the large number of choices of PQCs, we cannot perform a brute force search
for all the possibilities, while this opens a new space for the construction
of hybrid quantum-classical networks. We expect the introduction of QFE layers
in more architectures and extensive hyperparameter searches can improve the
performance.

Besides, it is noted that quantum neural tangent kernel (QNTK)
theorynakaji2021quantum ; shirai2021quantum ; liu2021representation  is
developed recently, which can be applied to QNNs. It will be interesting to
analyze the model with QFE layers based on the QNTK theory. This open question
is left for future research.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
under Grant 61873317.

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##  Appendix A The Dirac Equation

In this section, we will investigate how fermionic quantum fields evolve in a
$(2+1)$-dimensional spacetime equipped with the following line element:

$\displaystyle
ds^{2}=-\bigg{(}1+\frac{|x|}{\lambda}\bigg{)}^{2}c^{2}dt^{2}+dx^{2}+dy^{2},$
(101)

which is written in some coordinates $(t,x,y)$ whose interpretation depends on
the choice of parameter $\lambda$. The limit in which it diverges, i.e.
$\lambda\rightarrow\infty$, we recover the flat Minkowski metric expressed in
inertial coordinates $(t,x,y)$, which we could also re-label with $(T,X,Y)$ as
was done in (18):

$\displaystyle\lim\limits_{\lambda\rightarrow\infty}ds^{2}=-c^{2}dt^{2}+dx^{2}+dy^{2},$
(102)

whereas in the opposite limit where this parameter is small, i.e.
$\lambda\rightarrow 0$, we recover the flat Minkowski metric written in terms
of the Rindler coordinates $(t,x,y)$:

$\displaystyle\lim\limits_{\lambda\rightarrow
0}ds^{2}=-\frac{x^{2}}{\lambda^{2}}c^{2}dt^{2}+dx^{2}+dy^{2},$ (103)

and thus here $\lambda$ plays the role of $x_{\text{min}}=\frac{c^{2}}{a}$
which is the closest distance of approach from the origin at $x=0$, of a
Rindler observer accelerating with $a$. To derive the Dirac equation in these
two limits, we will write it using the most general metric (101). The Dirac
equation describing the evolution for massless or Weyl fermions in arbitrary
spacetime is as follows:

$i\gamma^{a}e^{\mu}_{a}\nabla_{\mu}\hat{\psi}(x)=0$ (104)

where $\hat{\psi}(x)$ is the massless Dirac spinor (or Weyl spinor) and can be
written as a two component spinor, which due to zero rest mass are decoupled
from each other
$\hat{\psi}^{\text{T}}(x)=\begin{bmatrix}\hat{\psi}_{\text{R}}(x),\leavevmode\nobreak\
\hat{\psi}_{\text{L}}(x)\end{bmatrix}$. Also, the covariant derivative is
defined as
$\nabla_{\mu}=\partial_{\mu}-\frac{i}{4}\omega^{ab}_{\mu}\sigma_{ab}$, where
$\sigma_{ab}=\frac{i}{2}\big{[}\gamma_{a},\gamma_{b}\big{]}$, where the Dirac
matrices satisfy the Clifford algebra
$\big{\\{}\gamma^{a},\gamma^{b}\big{\\}}=2\eta^{ab}$. From the line element in
(101), we can write down the metric components as follows:

$\displaystyle
g_{\mu\nu}=\text{diag}\bigg{[}-c^{2}\bigg{(}1+\frac{|x|}{\lambda}\bigg{)}^{2},1,1\bigg{]},$
(105)

which is diagonal and hence simplifies our derivation. Tetrads are objects
that take us from an arbitrary metric to the local flat metric of the tangent
space at a point. They are as defined as:

$g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\eta_{ab},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \eta_{ab}=\begin{bmatrix}-1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\ \end{bmatrix}.$ (106)

where $\eta_{ab}$ is the Minkowski tensor. In our notation we make use of
Greek $(\mu,\nu,..)$ indices to denote curved spacetime labels such as
$(t,x,y)$, and Roman $(a,b,..)$ indices signify that we are in the tangent
space at a particular point in spacetime and therefore can take values
$(0,1,2)$. By comparing (105) and (106) we get the following tetrads:

$\displaystyle
e^{a}_{\mu}=\text{diag}\bigg{[}c\bigg{(}1+\frac{|x|}{\lambda}\bigg{)},1,1\bigg{]},$
(107)

These tetrads can now be used to derive the spin connections
$\omega_{\mu}^{ab}$ which take into account the spin-precession of fermions
due to the curvature of spacetime. They are defined as follows:

$\displaystyle\omega_{\mu}^{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{\nu
a}\big{(}\partial_{\mu}e^{b}_{\nu}-\partial_{\nu}e^{b}_{\mu}\big{)}-\frac{1}{2}e^{\nu
b}\big{(}\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu}\big{)}$ (108)
$\displaystyle-$ $\displaystyle\frac{1}{2}e^{\rho a}e^{\sigma
b}\big{(}\partial_{\rho}e_{\sigma c}-\partial_{\sigma}e_{\rho
c}\big{)}e^{c}_{\mu}.$

which is manifestly anti-symmetric $\omega_{\mu}^{ab}=-\omega_{\mu}^{ba}$. The
only surviving components of the spin connection in the metric of (101) are as
follows:

$\displaystyle\omega_{t}^{01}=-\omega_{t}^{10}$ $\displaystyle=$
$\displaystyle\frac{c}{\lambda}\text{sgn}(x),$ (109)

where $\text{sgn}(x)$ is the signum function, i.e. it returns $+1$ for
positive values and $-1$ for negative entries. Finally, we will be needing the
Dirac matrices in the Weyl or Chiral representation:

$\gamma^{0}=\begin{bmatrix}0&-I_{2}\\\ -I_{2}&0\\\
\end{bmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\gamma^{i}=\begin{bmatrix}0&\sigma^{i}\\\ -\sigma^{i}&0\\\
\end{bmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\gamma^{5}=\begin{bmatrix}I_{2}&0\\\ 0&I_{2}\\\ \end{bmatrix},$ (110)

where $\sigma^{i}$ are the Pauli matrices:

$\sigma^{1}=\begin{bmatrix}0&1\\\ 1&0\\\ \end{bmatrix},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \gamma^{2}=\begin{bmatrix}0&-i\\\
i&0\\\ \end{bmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \sigma^{3}=\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix}.$
(111)

Plugging in the tetrads (107) and the spin-connection (109) pertaining to the
metric (101) into the massless Dirac equation (104), we get two decoupled Weyl
equations for the left and right handed fermions:

$\displaystyle\partial_{t}\hat{\psi}_{\text{L}}$ $\displaystyle=$
$\displaystyle
c\bigg{(}1+\frac{|x|}{\lambda}\bigg{)}\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}\hat{\psi}_{\text{L}}+\frac{c\leavevmode\nobreak\
\text{sgn}(x)}{2\lambda}\sigma^{x}\hat{\psi}_{\text{L}},$
$\displaystyle\partial_{t}\hat{\psi}_{\text{R}}$ $\displaystyle=$
$\displaystyle-c\bigg{(}1+\frac{|x|}{\lambda}\bigg{)}\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}\hat{\psi}_{\text{R}}-\frac{c\leavevmode\nobreak\
\text{sgn}(x)}{2\lambda}\sigma^{x}\hat{\psi}_{\text{R}}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (112)

In the limit of $\lambda\rightarrow\infty$, the above set reduces to the Weyl
equations for fermions in inertial frames (102):

$\displaystyle\partial_{t}\hat{\psi}_{\text{L}}$ $\displaystyle=$
$\displaystyle
c\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}\hat{\psi}_{\text{L}},$
$\displaystyle\partial_{t}\hat{\psi}_{\text{R}}$ $\displaystyle=$
$\displaystyle-c\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}\hat{\psi}_{\text{R}},$
(113)

which is the same as (17) describing massless fermions in a flat graphene
sheet. In the opposite limit $\lambda\rightarrow 0$, we recover the Weyl
equations for massless fermions in uniformly accelerating frames:

$\displaystyle\partial_{t}\hat{\psi}_{\text{L}}$ $\displaystyle=$
$\displaystyle\frac{c|x|}{\lambda}\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}\hat{\psi}_{\text{L}}+\frac{c\leavevmode\nobreak\
\text{sgn}(x)}{2\lambda}\sigma^{x}\hat{\psi}_{\text{L}},$
$\displaystyle\partial_{t}\hat{\psi}_{\text{R}}$ $\displaystyle=$
$\displaystyle-\frac{c|x|}{\lambda}\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}\hat{\psi}_{\text{R}}-\frac{c\leavevmode\nobreak\
\text{sgn}(x)}{2\lambda}\sigma^{x}\hat{\psi}_{\text{R}},$ (114)

which is the same as Eq. (24) that describes how electrons and holes evolve in
a Rindler strained graphene sheet.

##  Appendix B Rindler Horizon in Strained Graphene

In this section, we will discuss how an effective Rindler horizon forms at
$x=0$ in a graphene sheet that is strained according to Eq. (II). Our aim is
to show that, for long-wavelength excitations ($k_{y}\lambda\ll 1$, with
$\lambda$ the parameter characterizing the strain), excitations in graphene
obey an effective Schrödinger equation with a potential that diverges for
$x\to 0$.

To illustrate this, we start with the Hamiltonian in Eq. (13) that results in
the following Dirac equation for right-handed massless Dirac fermions:

$i\hbar\partial_{t}\hat{\psi}_{\text{R}}(\boldsymbol{r})=\sqrt{v(x)}\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\sqrt{v(x)}\leavevmode\nobreak\
\hat{\psi}_{\text{R}}(\boldsymbol{r}),$ (115)

where $v(x)=v_{0}\big{(}1+\frac{|x|}{\lambda}\big{)}$ is the Fermi velocity.
Next, we assume the following ansatz:

$\hat{\psi}_{\text{R}}=\begin{pmatrix}f(x)\\\
g(x)\end{pmatrix}e^{i(k_{y}y-\Omega t)},$ (116)

which is a product of a spinor with component functions $f(x)$ and $g(x)$, and
a plane wave in $y$ and $t$. Substituting this ansatz into Eq. (115), we get a
system of coupled differential equations:

$\displaystyle\bigg{[}\big{(}|x|+\lambda\big{)}\frac{d}{dx}+k_{y}\big{(}|x|+\lambda\big{)}+\frac{\text{sgn}(x)}{2}\bigg{]}g$
$\displaystyle=$ $\displaystyle i\bar{\Omega}f,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (117a)
$\displaystyle\bigg{[}\big{(}|x|+\lambda\big{)}\frac{d}{dx}-k_{y}\big{(}|x|+\lambda\big{)}+\frac{\text{sgn}(x)}{2}\bigg{]}f$
$\displaystyle=$ $\displaystyle i\bar{\Omega}g,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (117b)

where we have defined $\bar{\Omega}=\Omega/\omega_{c}$. Assuming $x>0$, we
eliminate $g(x)$ in favor of $f(x)$. Defining a new function
$F(x)=f(x)/(x+\lambda)$, we arrive at the following second order differential
equation:

$\frac{d^{2}F(x)}{dx^{2}}-\big{[}k_{y}^{2}-V(x)\big{]}F(x)=0,$ (118)

taking the form of an effective Schrödinger equation with the single-particle
potential:

$V(x)=-\frac{k_{y}}{(x+\lambda)}+\frac{1/4+\bar{\Omega}^{2}}{(x+\lambda)^{2}}.$
(119)

We now re-scale the coordinates $x\to x/k_{y}$ and take the long-wavelength
limit, i.e., $k_{y}\lambda\ll 1$, which makes Eq. (118) take the following
form:

$\frac{d^{2}F(x)}{dx^{2}}-\bigg{[}1+\frac{1}{x}-\frac{1/4+\bar{\Omega}^{2}}{x^{2}}\bigg{]}F(x)=0,$
(120)

exhibiting a divergence at $x\to 0$. This shows that in the presence of an
appropriate strain field, electronic quasiparticles in graphene experience an
effective infinite potential barrier reflecting the Rindler horizon.