#  Observation of reentrant metal-insulator transition in a random-dimer
disordered SSH lattice

Ze-Sheng Xu<EMAIL_ADDRESS>Department of Applied Physics, KTH Royal Institute
of Technology, Albanova University Centre, Roslagstullsbacken 21, 106 91
Stockholm, Sweden  Jun Gao<EMAIL_ADDRESS>Department of Applied Physics, KTH
Royal Institute of Technology, Albanova University Centre, Roslagstullsbacken
21, 106 91 Stockholm, Sweden  Adrian Iovan  Department of Applied Physics, KTH
Royal Institute of Technology, Albanova University Centre, Roslagstullsbacken
21, 106 91 Stockholm, Sweden  Ivan M. Khaymovich  Nordita, Stockholm
University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12,
SE-106 91 Stockholm, Sweden  Institute for Physics of Microstructures, Russian
Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia  Val Zwiller
Department of Applied Physics, KTH Royal Institute of Technology, Albanova
University Centre, Roslagstullsbacken 21, 106 91 Stockholm, Sweden  Ali W.
Elshaari<EMAIL_ADDRESS>Department of Applied Physics, KTH Royal Institute
of Technology, Albanova University Centre, Roslagstullsbacken 21, 106 91
Stockholm, Sweden

###### Abstract

The interrelationship between localization, quantum transport, and disorder
has remained a fascinating focus in scientific research. Traditionally, it has
been widely accepted in the physics community that in one-dimensional systems,
as disorder increases, localization intensifies, triggering a metal-insulator
transition. However, a recent theoretical investigation [Phys. Rev. Lett. 126,
106803] has revealed that the interplay between dimerization and disorder
leads to a reentrant localization transition, constituting a remarkable
theoretical advancement in the field. Here, we present the experimental
observation of reentrant localization using an experimentally friendly model,
a photonic SSH lattice with random-dimer disorder, achieved by incrementally
adjusting synthetic potentials. In the presence of correlated on-site
potentials, certain eigenstates exhibit extended behavior following the
localization transition as the disorder continues to increase. We directly
probe the wave function in disordered lattices by exciting specific lattice
sites and recording the light distribution. This reentrant phenomenon is
further verified by observing an anomalous peak in the normalized
participation ratio. Our study enriches the understanding of transport in
disordered mediums and accentuates the substantial potential of integrated
photonics for the simulation of intricate condensed matter physics phenomena.

Since the pioneering work of Rolf Landauer and Philip. W. AndersonLa ; And ,
the phenomenon of wave function localization in disordered media has been
extensively investigated, exploring the quantum phase transition and the
transport physics in disordered systems. Anderson localization predicts the
absence of diffusive behavior in the single-particle wave function when
subjected to a disorder potential, arising from wave interference effects
between multiple propagation paths. As the disorder strength increases,
transport becomes suppressed resulting in the metal-insulator transition.
Extensive studies have been conducted on this transition in diverse systems,
including electron gases, doped semiconductors, random media, photonic
lattices, ultra-cold optical lattices, ultrasound, and atomic system Cut ; PAL
; Diederik ; Martin ; Tal ; Topol ; Hefei ; AA ; Donald . The prior work on
low-dimensional systems ($D\leq 2$) establishes that uncorrelated disorder can
shift all eigenstates to a localized phase, regardless of the disorder degree
EAbr . In contrast, a quasiperiodic potential reveals that a non-zero critical
disorder strength can differentiate the metallic and insulator phases,
generating an intermediate critical region where both extended and localized
states coexist SD ; HP ; SA ; YW ; MG ; SG ; beta-footnote . This critical
region is defined by the single-particle mobility edge PAL ; NMO ; Biddle2009
; XLI ; SD2 ; SD3 ; MGR ; JBI , symbolizing a critical energy threshold
distinguishing the boundary between the extended and localized states in the
system.

Historically, the prevailing belief held that the system would undergo a
unidirectional transition from the extended state to the localized state as
the disorder increases. Contrary to this understanding, recent theoretical
studies illustrate that the reentrant localization can be achieved in a Su-
Schrieffer-Heeger (SSH) chain with staggered quasiperiodic disorder SRO1 ;
SRO2  or random-dimer disorder ZWZ . This implies that with a further increase
in disorder, two transitions ensue, leading to an extended phase with the
mobility edge and further to the second localized phase, respectively. The
genesis of this phenomenon resides in the increase of the disorder strength,
which results in the re-extension of some previously localized states. At even
higher disorder strength these states converge to the localized phase, showing
a strong correlation with multiple critical regions that accommodate the
mobility edge within the spectrum.

In this work, we implement a correlated disorder by imposing a specific
potential on the random dimer of an SSH chain. To achieve this, we employ a
Si3N4 waveguide array fabricated using electron beam lithography and reactive
ion etching JG1 ; ZSX ; JG2 , facilitating effective control over the nearest-
neighbor hopping amplitude. In addition, we tailor the width of array
waveguides to introduce a synthetic on-site potential JG3 ; JG4 . To examine
the system, we selectively excite a specific single site within the waveguide
array. Subsequently, we adopt a top imaging strategy to probe the intensity
distribution at the end of the array by analyzing the scattered light. These
images are recorded for different disorder strengths, and the normalized
participation ratios (NPR), characterizing the localization-delocalization
phase diagram, are extracted. In addition to observing light diffusion or
localized behavior corresponding to different quantum transport phases, we
also discern a consistent trend in the NPR of single-site excitation that
aligns with our theoretical model. The concordance between experimental
observations and theoretical predictions provides compelling evidence of the
non-monotonic metal-insulator transition in this one-dimensional photonic
system. This reentrant localization transition, observed experimentally, not
only opens a promising pathway for studying condensed matter physics within an
on-chip nano-photonic system but also broadens the frontiers of quantum
encoding and transport in low-dimensional systems, with potential applications
in quantum information processing and communication FUL ; DAN ; ANDR ; JIA .

Here, we consider a one-dimensional SSH model with random-dimer disorder,
which is described by the following Hamiltonian

$\displaystyle
H=t\sum_{n=1}^{N-1}\left[1-(-1)^{n}\Delta](c_{n}^{\dagger}c_{n+1}+\text{ H.c.
}\right)$ (1)
$\displaystyle+\sum_{n=1}^{N/2}\epsilon_{n}(c_{2n-1}^{\dagger}c_{2n-1}+c_{2n}^{\dagger}c_{2n}),$

$\epsilon_{n}=\begin{cases}\epsilon,&\text{ probability}=p\\\ 0,&\text{
probability }=1-p\end{cases}$ (2)

This model comprises $N/2$ unit cells, each containing two sites, where
$c_{n}^{\dagger}$ and $c_{n}$ represent the creation and annihilation
operators corresponding to the site $n$. The nearest-neighbor hopping
$t(1\pm\Delta)$, with the dimerization parameter $\Delta$, represent the
intra-cell and inter-cell coupling strength between sites, respectively.
$\epsilon_{n}$ designates the bimodally distributed on-site potential applied
to the random dimer, being $\epsilon$ with the probability $p$ and zero
otherwise. By tuning the value of $\epsilon$, we can probe the quantum phase
transitions occurring at different disorder strengths. This model presents the
reentrant localization at the specific critical values of $\epsilon$, which
can be examined by analyzing the $\mathrm{NPR}$ in conjunction with the
inverse participation ratio ($\mathrm{IPR}$ ) XLI ; XLI2 . The definitions of
these metrics are provided as follows:

$\mathrm{NPR}_{n}=\left(N\sum_{i=1}^{N}\left|\phi_{n}^{i}\right|^{4}\right)^{-1},\quad\mathrm{IPR}_{n}=\sum_{i=1}^{N}\left|\phi_{n}^{i}\right|^{4}.$
(3)

Where $\phi_{n}^{i}$ denotes the $n$-th eigenstate coefficient at the site
$1\leq i\leq N$, while $N$ signifies the number of sites in the lattice. It is
theoretically proposed that in the thermodynamic limit, as $N$ tends towards
infinity, the $\mathrm{NPR}$ stays finite when the $\mathrm{IPR}$ vanishes for
an ergodically extended state. Conversely, the $\mathrm{NPR}$ approaches zero
when the $\mathrm{IPR}$ stays finite for a localized state. Further, we will
use the mean values across the spectrum of the above two measures, defined as
follows

$\langle\mathrm{NPR}\rangle=\frac{1}{N}\sum_{i=1}^{N}\mathrm{NPR}_{n},\quad\langle\mathrm{IPR}\rangle=\frac{1}{N}\sum_{i=1}^{N}\mathrm{IPR}_{n}.$
(4)

Figure 1: SSH lattice with random-dimer disorder and dimerization amplitude
$\Delta$ = 0.25 featuring an on-site potential. (a) Schematic representation
of the random-dimer disordered SSH model. The two green nodes within the
dotted box depict a situation devoid of on-site potential, whereas the two red
nodes symbolize a dimer with a chosen finite on-site potential. The on-site
potential is randomly allocated with a probability $0.5$, analogous to the
likelihood of getting heads in a fair coin toss. In a broad sense, our model’s
on-site potential can be regarded as a bimodal telegraph noise. (b)
Eigenenergy spectrum of the random-dimer disordered SSH model with varying on-
site potential $\varepsilon$. The spectrum is clearly separated into four
bands: energies of two of them increase linearly with $\epsilon$, while those
of the other two are insensitive to it. The corresponding wave functions of
linear (constant) in $\epsilon$ live mostly at disordered
$\epsilon_{n}=\epsilon$ (clean $\epsilon_{n}=0$) dimers. In each pair of
bands, these wave functions are also symmetric for the lower and antisymmetric
for the higher one of two bands, see Supplemental Information. As the disorder
strength approaches two critical values, we observe the gap closure and
reopening, indicative of the disappearing and reappearing in-gap edge states,
associated with the non-monotonic insulator-metal-insulator transition. The
blue curve represents the Energy gap in this system.

Figure 2: Light transport in the lattice and average NPR under varying random-
dimer on-site potential. (a) The four simulated trajectories of light in
$1000~{}\mu$m long SSH lattices comprising $100$ sites with varying degrees of
random dimer disorder are displayed. I, II, III, and IV correspond to random-
dimer on-site potentials, $\varepsilon=0$, $0.1$, $0.185$, and $0.3$
respectively, with random-dimer assignment probability of $p=0.5$. In I, a
ballistic light propagation takes place without any disorder present; in II
and IV, light becomes localized within the lattice at the designed random-
dimer on-site potentials, despite significant propagation distance. III
demonstrates the phenomenon of reentrant localization, where both extended and
localized states coexist. (b) The numerically computed mean NPR as a function
of random-dimer potential $\epsilon$ over a range of lattice sites from $30$
to $1000$. As the number of lattice sites increases, the mean NPR exhibits a
declining trend, yet the counterintuitive reentrant peaks consistently emerge.
The four light transport simulations coincide with the different regimes in
the mean NPR plot, as denoted by the color coding and labeling I to IV.

.

When both the average values $\langle\mathrm{NPR}\rangle$ and
$\langle\mathrm{IPR}\rangle$ are simultaneously finite, it suggests the
coexistence of localized and extended states. This gives rise to intermediate
states, triggering the phenomenon of reentrant localization and inducing the
non-monotonic metal-insulator transition within the system. However, relying
solely on the $\mathrm{NPR}$ or $\mathrm{IPR}$ values in our investigation
poses difficulties in accurately identifying the coexistence of eigenstates
within the extended and localized phases. One feasible approach is the
simultaneous calculation of the $\mathrm{NPR}$ and $\mathrm{IPR}$ along the
random-dimer onsite potential, followed by the analysis of the non-zero region
for both these parameters. An alternative strategy involves computing a
defined parameter $\eta$ as
$\eta=\log_{10}[\langle\mathrm{IPR}\rangle\times\langle\mathrm{NPR}\rangle]$
XLI2 . By assessing the ratio of extended to localized states within all N
eigenstates, we establish the criteria for the reentrant regime. The detailed
derivation can be found in the Supplementary information:

$\mathrm{log_{10}}\left(\frac{2}{N}\right)\leq\eta\leq\mathrm{log_{10}}\left(\frac{1}{4}+\frac{1}{2N}\right)$
(5)

In our experiment, we select a lattice with $N=100$ sites, as depicted in Fig.
1(a), and generate a random-dimer number array with a probability $p=0.5$
according to Eq. (2). We choose $t=0.08$ and $\Delta=0.25$, so the
intra/inter-cell coupling strength is $0.1$ and $0.06$, which sets the lattice
at the topologically trivial phase ZWZ . Fig. 1(b) presents the results of
sweeping the on-site potential from $0$ to $0.4$ across the random-dimer
number array. Subsequently, we conduct numerical calculations to determine the
lattice’s spectrum, unveiling the system’s energy levels and their
corresponding IPR. The spectrum of the system is given by four well
distinguished bands, centered around $E_{\nu,\pm}=\nu\epsilon\pm t(1+\Delta)$,
with wave functions, living mostly on the dimers with the on-site potential
$\epsilon_{n}=\nu\epsilon$, $\nu=0,1$, and being symmetric (antisymmetric) at
each of such dimers with the corresponding energy shift $\mp t(1+\Delta)$,
respectively, see Supplemental Information. The width of each band is given by
the smaller hopping $2t(1-\Delta)$. The blue curve uncovers the closing and
reopening of the energy gap in the system, signifying the topological phase
transition of the eigenstates. It is this gap closure which is responsible for
the reentrant delocalization. Indeed, close to the intersection of $2$ bands,
the effective disorder is given by the difference of energy centers
$E_{\nu,\pm}$. As soon as $2$ bands cross, this difference goes to zero and a
disorder-free sector of the Hamiltonian with delocalized states emerges at
this crossing-band pair. The states in the other pair of bands stay localized.
This explains the reentrant character of delocalization-localization phase
diagram.

Fig. 2(a) displays the simulated pattern of the $41$st single-site excitation
obtained using the coupled mode theory SEM ; JRP ; HAH  to showcase the
evolution of light within lattices having a length of $1000~{}\mu$m. The on-
site potentials $\epsilon$ of these lattices are $0$, $0.1$, $0.185$ and
$0.3$, from left to right. Comparing Fig. 2(a) III with II and I, we observe
that the light undergoes a distinct behavior in III. In this intermediate
regime III, a part of the light intensity stays near the initial location as
in II, but another part ballistically spreads as in I1. This gives evidence of
a coexistence of localized and extended light, observed simultaneously.
Unexpectedly this coexistence appears even after the light has transitioned
into the fully localized regime II upon reaching the critical value of the on-
site potential, $\epsilon$. Furthermore, as we continuously increase
$\epsilon$, the light evolution pattern in regime IV reverts back to
exhibiting localized behavior, clearly demonstrating the occurrence of
reentrant localization in this system. Fig. 2(b) shows the numerically
calculated Mean NPR for the random-dimer on-site potential system with pre-
defined parameters above while varying lattice sites numbers. In the limit of
a large number of lattice sites, the $\langle\mathrm{NPR}\rangle$ approaches
its lower bound, which scales as $\sim\frac{1}{\mathrm{N}}$. Despite the
decreasing trend of the lower limits as the number of sites increases, the
curves consistently exhibit an anomalous peak followed by a subsequent
decrease. This observed behavior of the curves provides convincing evidence
for the existence of reentrant localization in this system and justifies that
the reentrant localization in our system with $100$ sites does exist and can
be experimentally observed.

Figure 3: Details of the experimental setup and collected data. (a)
Illustration of the experimental setup. A $786$ nm continuous wave laser is
utilized for the excitation of the random-dimer disordered SSH photonic
lattice through a lensed fiber. A Y-splitter is incorporated to control
polarization and optimize the coupling to the photonic chip. The Transverse
Electric (TE) mode is excited by accurately adjusting the paddle of the
polarization controller and monitoring the detector counts transmitted through
the chip. A polarizing beam splitter enables exciting the TE modes by
minimizing the detector count. The intensity of the facet light distribution
is imaged using a microscope objective and a CCD camera. (b) Extraction of the
output light distribution. Two images, both with identical focus, are captured
to ascertain the distribution of light intensity. II represents the output
pattern of the lattice, while I serves as a reference pattern of the
illuminated lattice that aids in pinpointing exact coordinates in pixel space.
The section within the dotted line in both images indicates the position of
the lattice, obtained from the reference image, to extract the light intensity
at each lattice site. (c) Comparison between experimentally measured and
simulated light intensity distributions. I, II, III, and IV correspond to the
output intensity distribution at a distance of $1000~{}\mu$m, following the
$41$st single-site excitation with random-dimer on-site potentials
$\varepsilon=0$, $0.1$, $0.185$, and $0.3$, respectively. Figure 4: Phase
diagram and retrieved NPR for single-site excitation. (a) The phase diagram of
$\eta$ over the dimerization parameter $\Delta$ and on-site potential
$\epsilon$ plane for a $100$-site dimer-disordered SSH lattice. The blue
region signifies the concurrent non-zero values of both
$\langle\mathrm{IPR}\rangle$ and $\langle\mathrm{NPR}\rangle$, according to
the definition
$\eta=\log_{10}[\langle\mathrm{IPR}\rangle\times\langle\mathrm{NPR}\rangle]$.
Four small yellow circles in the diagram represent the experimental sampling
points we selected, corresponding to the four regions presented in Fig. 2(b).
The reentrant localization is validated by the coexistence of localized and
extended states, a consequence of the system hosting a mobility edge. (b)
Phase diagram of $\eta$ in the on-site potential $\epsilon$ and on-site
probability $\mathrm{p}$ plane. The near-vertical border of the reentrant
localization region accentuates that the critical values remain stable as the
on-site probability varies widely ($0.15<q<0.8$), confirming that our choice
of on-site probability $p=0.5$ does not add any specificity. (c) The trend of
the average NPR results from the ensemble of statistical behaviors of all NPRs
from single-site excitations. Specifically, the NPRs from single-site
excitations which are linked with extended eigenstates contribute to the peak
of the average NPR in the reentrant localization and trigger a non-monotonic
metal-insulator transition in the one-dimensional random-dimer disordered SSH
system.

To investigate our system’s properties, we employ a full-vectorial mode solver
FDTD  for numerical simulations of the waveguide’s propagation constants.
These propagation constants serve as a representation of the on-site potential
within our model. Detailed analysis is presented in the Supplementary
information. In our experiment, we utilized a CMOS-compatible Si3N4 photonic
platform to successfully integrate nano-sized SSH devices with random-dimer
disorder. Each device consists of $N=100$ waveguides, with dimensions of $250$
nm high and $450-518$ nm wide according to the different on-site potentials,
with SiO2 bottom and air top cladding. These waveguides support only the
fundamental modes at a $786$ nm wavelength. The width of and the gap between
these waveguides are well-engineered to satisfy the designed hopping amplitude
and the on-site potentials. The experimental setup is shown in Fig. 3(a). We
excite the lattice with a continuous-wave coherent laser centered at $786$ nm.
Using a lensed fiber fixed on a six-axis nano-positioning stage, the laser is
finely coupled into the nanophotonic chip. A polarizing beam splitter with a
three-paddle polarization controller is installed to selectively excite only
the transverse electric (TE) mode. The on-chip Y-splitter divides the light
into two paths, the first leading to the random-disordered SSH lattice, with
the second acting as the monitor waveguide for tuning the coupling efficiency
and controlling the polarization. The output pattern of the light intensity at
the end of each lattice is collected from the scattered signal imaged with a
$40$X objective and captured by a charge-coupled device (CCD) camera. Fig.
3(b) presents captured images obtained under both non-illuminated and
illuminated conditions, with the focus remaining consistent. By analyzing the
region marked by dashed lines in Fig. 3(b) I , we extract the pixel
coordinates corresponding to the photonic lattice structure. By combining this
information with Fig. 3(b) II which depicts the output pattern of the lattice,
we establish a relationship between the light intensity distribution and the
lattice site numbers ranging from $1$ to $100$. This enables us to retrieve
the $\mathrm{NPR}$ for the single-site excitation scenario, providing valuable
insights into the system’s light behavior. Fig. 3(c) shows the experimentally
measured light distribution along the sites compared with the theoretically
predicted light distribution. We can clearly see the distinct signature of
extended and localized wave packets transport in different regimes from I to
IV. I indicates extended behavior with $\epsilon=0$, while II and IV
demonstrate localized behavior. III represents the intermediate case.
Specifically, in region II and IV the wave packet is highly confined within
few lattice sites, while in region III we see a clear signature that indicates
the coexistence of extended and localized behaviour. The peak and overall
behavior of the measured light intensity distribution align well with the
theoretical prediction. Notably, the laser light transport shows consistency
with the eigenstates’ behavior, implying the non-monotonic metal-insulator
transition and the reentrant localization.

In order to explore the reentrant localization regime, we plot the phase
diagram according to Eq. 5. Fig. 4(a) reveals the $\eta$ over different
dimerization parameters $\Delta$ and on-site potentials $\mathrm{\epsilon}$,
while Fig. 4(b) shows the $\eta$ for different on-site potentials
$\mathrm{\epsilon}$ and on-site probabilities $\mathrm{p}$. The white edges in
these phase diagrams separate the localized phases from the delocalized one
with the mobility edge. Non-monotonic behavior with the on-site potential
shows the reentrant localization which we are interested in. The four yellow
circles mark the experimental sampling points for collecting the light
distribution and calculating the single-site excitation $\mathrm{NPR}$. Fig.
4(c) presents the $\mathrm{NPR}$ of single-site excitation compared with the
simulated $\langle\mathrm{NPR}\rangle$. Four highlighted regions I, II, III,
IV correspond to the four specific regimes depicted in Fig. 2, respectively.
The measured $\mathrm{NPR}$ from the $41$st single-site excitation contributes
to the statistical behavior observed in the averaged
$\langle\mathrm{NPR}\rangle$ and exhibits a consistent trend, as depicted in
the plot. The choice of the $41$st site to be excited is not specific and for
most of the sites the behavior is expected to be similar. The $\mathrm{NPR}$
function shows a non-monotonic trend, it decreases in value as the disorder
strength increases, subsequently revealing an anomalous peak in the
intermediate regime, and ultimately undergoing a secondary decrease for large
disorder strength. Based on the experimental results, we unambiguously observe
the occurrence of reentrant localization behavior in the random-dimer
disordered SSH photonic lattices, both through intuitive visual inspection and
rigorous multifractal analysis.

In conclusion, we have demonstrated the observation of the reentrant
localization in a one-dimensional disordered system. This system is based on a
nano-fabricated Si3N4 platform realized in photonic waveguide devices. The
numerically calculated $\langle\mathrm{NPR}\rangle$ and the phase diagram for
the defined parameter $\eta$ guided our choice of sampling points with
different on-site potentials $\epsilon$ to illustrate the transitions. Our
technique of fine-tuning the width of waveguides allows us to control the on-
site potential for each dimer. By using a single-shot approach, which involves
probing the light intensity at each lattice site using an edge-scattered
structure, we can investigate and observe the transport of light within the
system. This approach enables us to retrieve the normalized participation
ratio ($\mathrm{NPR}$) for the single-site excitation scenario. The counter-
intuitive drop and increase of $\mathrm{NPR}$ indicate the occurrence of non-
monotonic localization transitions, thus pointing to the presence of reentrant
localization. Our work contributes to a deeper comprehension of the underlying
mechanisms governing the behavior of quantum systems subjected to disorder.
Such insights are invaluable in elucidating the complex nature of localization
transitions and their implications in various physical systems. From a
technical perspective, we incorporate a nanometer-level precision electron
beam lithography process, resulting in a precise implementation of complex on-
site potentials and hopping parameters. The field of micro-nano photonics,
particularly with the inclusion of the reentrant localization, holds a
significant potential in various areas such as information processing, optical
computing, hybrid photonic circuits, topological photonics, and photonic
quantum technologies AWE1 ; BAE ; TOZ ; LLU ; CPO .

###### Acknowledgements.

A.W.E acknowledges support Knut and Alice Wallenberg (KAW) Foundation through
the Wallenberg Centre for Quantum Technology (WACQT), Swedish Research Council
(VR) Starting Grant (Ref: 2016-03905), and Vinnova quantum kick-start project
2021. V.Z. acknowledges support from the KAW and VR. Work at Nordita was
supported by European Research Council under the European Union Seventh
Framework ERS-2018-SYG HERO, KAW 2019.0068 and the University of Connecticut.

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# A Quantum Convolutional Neural Network on NISQ Devices

ShiJie Wei<EMAIL_ADDRESS>Beijing Academy of Quantum Information Sciences,
Beijing 100193, China  State Key Laboratory of Low-Dimensional Quantum Physics
and Department of Physics, Tsinghua University, Beijing 100084, China  YanHu
Chen  Institute of Information Photonics and Optical Communications, Beijing
University of Posts and Telecommunications, Beijing 100876, China  ZengRong
Zhou  Beijing Academy of Quantum Information Sciences, Beijing 100193, China
State Key Laboratory of Low-Dimensional Quantum Physics and Department of
Physics, Tsinghua University, Beijing 100084, China  GuiLu Long
<EMAIL_ADDRESS>State Key Laboratory of Low-Dimensional Quantum Physics
and Department of Physics, Tsinghua University, Beijing 100084, China  Beijing
Academy of Quantum Information Sciences, Beijing 100193, China  Beijing
National Research Center for Information Science and Technology and School of
Information Tsinghua University, Beijing 100084, China  Frontier Science
Center for Quantum Information, Beijing 100084, China

###### Abstract

Quantum machine learning is one of the most promising applications of quantum
computing in the Noisy Intermediate-Scale Quantum(NISQ) era. Here we propose a
quantum convolutional neural network(QCNN) inspired by convolutional neural
networks(CNN), which greatly reduces the computing complexity compared with
its classical counterparts, with $O((log_{2}M)^{6})$ basic gates and
$O(m^{2}+e)$ variational parameters, where $M$ is the input data size, $m$ is
the filter mask size and $e$ is the number of parameters in a Hamiltonian. Our
model is robust to certain noise for image recognition tasks and the
parameters are independent on the input sizes, making it friendly to near-term
quantum devices. We demonstrate QCNN with two explicit examples . First, QCNN
is applied to image processing and numerical simulation of three types of
spatial filtering, image smoothing, sharpening, and edge detection are
performed. Secondly, we demonstrate QCNN in recognizing image, namely, the
recognition of handwritten numbers. Compared with previous work, this machine
learning model can provide implementable quantum circuits that accurately
corresponds to a specific classical convolutional kernel. It provides an
efficient avenue to transform CNN to QCNN directly and opens up the prospect
of exploiting quantum power to process information in the era of big data.

##  I Introduction

Machine learning has fundamentally transformed the way people think and
behave. Convolutional Neural Network(CNN) is an important machine learning
model which has the advantage of utilizing the correlation information of
data, with many interesting applications ranging from image recognition to
precision medicine.

Quantum information processing (QIP)1, 2, which exploits quantum-mechanical
phenomena such as quantum superpositions and quantum entanglement, allows one
to overcome the limitations of classical computation and reaches higher
computational speed for certain problems3, 4, 5. Quantum machine learning, as
an interdisciplinary study between machine learning and quantum information,
has undergone a flurry of developments in recent years6, 7, 8, 9, 10, 11, 12,
13, 14, 15. Machine learning algorithm consists of three components:
representation, evaluation and optimization, and the quantum version16, 17,
18, 19, 20 usually concentrates on realizing the evaluation part, the
fundamental construct in deep learning21.

A CNN generally consists of three layers, convolution layers, pooling layers
and fully connected layers. The convolution layer calculates new pixel values
$x_{ij}^{(\ell)}$ from a linear combination of the neighborhood pixels in the
preceding map with the specific weights,
$x_{i,j}^{(\ell)}=\sum_{a,b=1}^{m}w_{a,b}x_{i+a-2,j+b-2}^{(\ell-1)}$, where
the weights $w_{a,b}$ form a $m\times m$ matrix named as a convolution kernel
or a filter mask. Pooling layer reduces feature map size, e.g. by taking the
average value from four contiguous pixels, and are often followed by
application of a nonlinear (activation) function. The fully connected layer
computes the final output by a linear combination of all remaining pixels with
specific weights determined by parameters in a fully connected layer. The
weights in the filter mask and fully connected layer are optimized by training
on large datasets.

In this article, we demonstrate the basic framework of a quantum convolutional
neural network(QCNN) by sequentially realizing convolution layers, pooling
layers and fully connected layers. Firstly, we implement convolution layers
based on linear combination of unitary operators(LCU)22, 23, 24. Secondly, we
abandon some qubits in the quantum circuit to simulate the effect of the
classical pooling layer. Finally, the fully connected layer is realized by
measuring the expectation value of a parametrized Hamiltonian and then a
nonlinear (activation) function to post-process the expectation value. We
perform numerical demonstrations with two examples to show the validity of our
algorithm. Finally, the computing complexity of our algorithm is discussed
followed by a summary.

##  II Framework of Quantum Neural Networks

###  II.1 Quantum Convolution Layer

Figure 1: Comparison of classical convolution processing and quantum
convolution processing. $F$ and $G$ are the input and output image data,
respectively. On the classical computer, a $M\times M$ image can be
represented as a matrix and encoded with at least $2^{n}$ bits
[$n=\lceil\log_{2}(M^{2})\rceil$]. The classical image transformation through
the convolution layer is performed by matrix computation $F*W$, which leads to
$x_{i,j}^{(\ell)}=\sum_{a,b=1}^{m}w_{a,b}x_{i+a-2,j+b-2}^{(\ell-1)}$. The same
image can be represented as a quantum state and encoded in at least $n$ qubits
on a quantum computer. The quantum image transformation is realized by the
unitary evolution $U$ on a specific quantum state.

The first step for performing quantum convolution layer is to encode the image
data into a quantum system. In this work, we encode the pixel positions in the
computational basis states and the pixel values in the probability amplitudes,
forming a pure quantum state. Given a 2D image $F=(F_{i,j})_{M\times L}$,
where $F_{i,j}$ represents the pixel value at position $(i,j)$ with
$i=1,\dots,M$ and $j=1,\dots,L$. $F$ is transformed as a vector $\vec{f}$ with
$ML$ elements by puting the first column of $F$ into the first $M$ elements of
$\vec{f}$, the second column the next $M$ elements , etc. That is,

$\displaystyle\vec{f}=(F_{1,1},F_{2,1},\dots,F_{M,1},F_{1,2},\dots,F_{i,j},\dots,F_{M,L})^{T}.$
(1)

Accordingly, the image data $\vec{f}$ can be mapped onto a pure quantum state
$|f\rangle=\sum_{k=0}^{2^{n}-1}c_{k}|k\rangle$ with
$n=\lceil\log_{2}(ML)\rceil$ qubits, where the computational basis $|k\rangle$
encodes the position $(i,j)$ of each pixel, and the coefficient $c_{k}$
encodes the pixel value, i.e., $c_{k}=F_{i,j}/(\sum{F_{i,j}^{2}})^{1/2}$ for
$k<ML$ and $c_{k}=0$ for $k\geq ML$. Here $(\sum{F_{i,j}^{2}})^{1/2}$ is a
constant factor to normalizing the quantum state.

Without loss of generality, we focus on the input image with $M=L=2^{n}$
pixels. The convolution layer transforms an input image $F=(F_{i,j})_{M\times
M}$ into an output image $G=(G_{i,j})_{M\times M}$ by a specific filter mask
$W$. In the quantum context, this linear transformation, corresponding to a
specific spatial filter operation, can be represented as
$|g\rangle=U|f\rangle$ with the input image state $|f\rangle$ and the output
image state $|g\rangle$. For simplicity, we take a $3\times 3$ filter mask as
an example

$W=\left[\begin{matrix}w_{11}&w_{12}&w_{13}\\\ w_{21}&w_{22}&w_{23}\\\
w_{31}&w_{32}&w_{33}\\\ \end{matrix}\right].$ (2)

The generalization to arbitrary $m\times m$ filter mask is straightforward.
Convolution operation will transform the input image $F=(F_{i,j})_{M\times M}$
into the output image as $G=(G_{i,j})_{M\times M}$ with the pixel
$G_{i,j}=\sum_{u,v=1}^{3}w_{uv}F_{i+u-2,j+v-2}$ ($2\leq i,j\leq M-1$). The
corresponding quantum evolution $U|f\rangle$ can be performed as follows. We
represent input image $F=(F_{i,j})_{M\times M}$ as an initial state

$\displaystyle|f\rangle=\sum_{k=0}^{M^{2}-1}c_{k}|k\rangle,$ (3)

where $c_{k}=F_{i,j}/(\sum{F_{i,j}^{2}})^{1/2}$. The $M^{2}\times M^{2}$
linear filtering operator $U$ can be defined as25:

$U=\left[\begin{matrix}E&&&&&\\\ V_{1}&V_{2}&V_{3}&&&\\\
&\ddots&\ddots&\ddots&&\\\ &&\ddots&\ddots&\ddots&\\\ &&&V_{1}&V_{2}&V_{3}\\\
&&&&&E\\\ \end{matrix}\right],$ (4)

where $E$ is an $M$ dimensional identity matrix, and $V_{1}$, $V_{2}$, $V_{3}$
are $M\times M$ matrices defined by

$V_{1}=\left(\begin{array}[]{ccccc}0&&&&\\\ w_{11}&w_{21}&w_{31}&&\\\
&\ddots&\ddots&\ddots&\\\ &&w_{11}&w_{21}&w_{31}\\\ &&&&0\\\
\end{array}\right)_{M\times M}V_{2}=\left(\begin{array}[]{ccccc}1&&&&\\\
w_{12}&w_{22}&w_{32}&&\\\ &\ddots&\ddots&\ddots&\\\ &&w_{12}&w_{22}&w_{32}\\\
&&&&1\\\ \end{array}\right)_{M\times
M}V_{3}=\left(\begin{array}[]{ccccc}0&&&&\\\ w_{13}&w_{23}&w_{33}&&\\\
&\ddots&\ddots&\ddots&\\\ &&w_{13}&w_{23}&w_{33}\\\ &&&&0\\\
\end{array}\right)_{M\times M}.$ (5)

Generally speaking, the linear filtering operator $U$ is non-unitary that can
not be performed directly. Actually, we can embed $U$ in a bigger system with
an ancillary system and decompose it into a linear combination of four unitary
operators 26. $U=U_{1}+U_{2}+U_{3}+U_{4}$, where
$U_{1}=(U+U^{\dagger})/2+i\sqrt{I-(U+U^{\dagger})^{2}/4}$,
$U_{2}=(U+U^{\dagger})/2-i\sqrt{I-(U+U^{\dagger})^{2}/4}$,
$U_{3}=(U-U^{\dagger})/2i+i\sqrt{I+(U-U^{\dagger})^{2}/4}$ and
$U_{4}=(U-U^{\dagger})/2i-i\sqrt{I+(U-U^{\dagger})^{2}/4}$. However, the basic
gates consumed to perform $U_{i}$ scale exponentially in the dimensions of
quantum systems, making the quantum advantage diminishing. Therefore, we
present a new approach to construct the filter operator to reduce the gate
complexity. For convenience, we change the elements of the first row, the last
row, the first column and the last column in the matrix $V_{1},V_{2},$ and
$V_{3}$, which is allowable in imagining processing, to the following form

$V^{\prime}_{1}=\left(\begin{array}[]{ccccc}w_{21}&w_{31}&&&w_{11}\\\
w_{11}&w_{21}&w_{31}&&\\\ &\ddots&\ddots&\ddots&\\\ &&w_{11}&w_{21}&w_{31}\\\
w_{31}&&&w_{11}&w_{21}\\\ \end{array}\right)_{M\times
M}V^{\prime}_{2}=\left(\begin{array}[]{ccccc}w_{22}&w_{32}&&&w_{12}\\\
w_{12}&w_{22}&w_{32}&&\\\ &\ddots&\ddots&\ddots&\\\ &&w_{12}&w_{22}&w_{32}\\\
w_{32}&&&w_{12}&w_{22}\\\ \end{array}\right)_{M\times
M}V^{\prime}_{3}=\left(\begin{array}[]{ccccc}w_{23}&w_{33}&&&w_{13}\\\
w_{13}&w_{23}&w_{33}&&\\\ &\ddots&\ddots&\ddots&\\\ &&w_{13}&w_{23}&w_{33}\\\
w_{33}&&&w_{13}&w_{23}\\\ \end{array}\right)_{M\times M}.$ (6)

Defining the adjusted linear filtering operator $U^{\prime}$ as

$U^{\prime}=\left[\begin{matrix}V^{\prime}_{2}&V^{\prime}_{3}&&&&V^{\prime}_{1}\\\
V^{\prime}_{1}&V^{\prime}_{2}&V^{\prime}_{3}&&&\\\ &\ddots&\ddots&\ddots&&\\\
&&\ddots&\ddots&\ddots&\\\ &&&V^{\prime}_{1}&V^{\prime}_{2}&V^{\prime}_{3}\\\
V^{\prime}_{3}&&&&V^{\prime}_{1}&V^{\prime}_{2}\\\ \end{matrix}\right],$ (7)

Next, we decompose $V^{\prime}_{\mu}(\mu=1,2,3)$ into three unitary matrices
without normalization,
$V^{\prime}_{\mu}=V^{\prime}_{1\mu}+V^{\prime}_{2\mu}+V^{\prime}_{3\mu}$,
where

$V^{\prime}_{1\mu}=\left(\begin{array}[]{ccccc}&&&&w_{1\mu}\\\ w_{1\mu}&&&&\\\
\ddots&\ddots&\ddots&&\\\ &&w_{1\mu}&&\\\ &&&w_{1\mu}&\\\
\end{array}\right)_{M\times M}\\\
V^{\prime}_{2\mu}=\left(\begin{array}[]{ccccc}w_{2\mu}&&&&\\\ &w_{2\mu}&&&\\\
&\ddots&\ddots&\ddots&\\\ &&&w_{2\mu}&\\\ &&&&w_{2\mu}\\\
\end{array}\right)_{M\times M}\\\
V^{\prime}_{3\mu}=\left(\begin{array}[]{ccccc}&w_{3\mu}&&&\\\ &&w_{3\mu}&&\\\
&&\ddots\ddots&\ddots&\\\ &&&&w_{3\mu}\\\ w_{3\mu}&&&&\\\
\end{array}\right)_{M\times M}.$ (8)

Thus, the linear filtering operator $U^{\prime}$ can be expressed as

$\displaystyle
U^{\prime}=\sum_{\mu=1}^{3}\sum_{v=1}^{3}(V^{\prime}_{\mu\mu}/w_{\mu\mu})\otimes
V^{\prime}_{v\mu}.$ (9)

which can be simplified to

$\displaystyle U^{\prime}=\sum_{k=1}^{9}\beta_{k}Q_{k},$ (10)

where $Q_{k}=(V^{\prime}_{\mu\mu}/w_{\mu\mu})\otimes
V^{\prime}_{v\mu}/w_{v\mu}$ is unitary, and $\beta_{k}$ is a relabelling of
the indices.

Now, we can perform $U^{\prime}$ through the linear combination of unitary
operators $Q_{k}$. The number of unitary operators is equal to the size of
filter mask. The quantum circuit to realize $U^{\prime}$ is shown in Fig.2.
The work register $|f\rangle$ and four ancillary qubits $|0000\rangle_{a}$ are
entangled together to form a bigger system.

Firstly, we prepare the initial state $|f\rangle$ using amplitude encoding
method or quantum random access memory(qRAM). Then, performing unitary matrix
$S$ on the ancillary registers to transform $|0000\rangle_{a}$ into a specific
superposition state $|\psi\rangle_{a}$

$\displaystyle
S|0000\rangle_{a}=|\psi\rangle_{a}=\sum_{v=1}^{9}\beta_{k}/N|k\rangle$ (11)

where $N_{c}=\sqrt{\sum_{k=1}^{9}\beta_{k}^{2}}$ and $S$ satisfies

$S_{k,1}=\begin{cases}\beta_{k}/N_{c}&\textrm{if $k\leq 9$}\\\ 0&\textrm{if
$k>9$.}\end{cases}$ (12)

$S$ is a parameter matrix corresponding to a specific filter mask that
realizes a specific task.

Then, we implement a series of ancillary system controlled operations
$Q_{k}\otimes|k\rangle\langle k|$ on the work system $|f\rangle$ to realize
LCU. Nextly, Hadamard gates $H^{T}=H^{\otimes 4}$ are acted to uncompute the
ancillary registers $|\psi\rangle_{a}$. The state is transformed to

$\displaystyle|g^{\prime}\rangle=\sum_{i=1}^{16}\frac{1}{N_{c}}|i\rangle\sum_{k=1}^{16}H^{T}_{(i:)}S_{(:1)}Q_{k}|f\rangle,$
(13)

where $H^{T}_{(i:)}$ is the i-th row in matrix $H^{T}$ and $S_{(:1)}$ is the
first column in matrix $S$. The first term equals to

$\displaystyle\frac{1}{N_{c}}|0\rangle\sum_{k=1}^{9}\beta_{k}Q_{k}|f\rangle,$
(14)

which corresponds to the filter mask $W$. The $i$-th term equals to filter
mask $W^{i}(i=2,3,\dots,16)$, where

$W^{i}=\left[\begin{matrix}H^{T}_{i1}w_{11}&H^{T}_{i4}w_{12}&H^{T}_{i7}w_{13}\\\
H^{T}_{i2}w_{21}&H^{T}_{i5}w_{22}&H^{T}_{i8}w_{23}\\\
H^{T}_{i3}w_{31}&H^{T}_{i6}w_{32}&H^{T}_{i9}w_{33}\\\ \end{matrix}\right].$
(15)

Totally, $16$ filter masks are realized, corresponding to ancilla qubits in
$16$ different state $|i\rangle(i=1,2,\dots,16)$. Therefore, the whole effect
of evolution on state $|f\rangle$ without considering the ancilla qubits, is
the linear combination of the effects of 16 filter masks.

If we only need one filter mask $W$, measuring the ancillary register and
conditioned on seeing $|0000\rangle$. We have the state
$\frac{1}{N_{c}}|0000\rangle U^{\prime}|f\rangle$, which is proportional to
our expected result state $|g\rangle$. The probability of detecting the
ancillary state $|0000\rangle$ is

$\displaystyle
P_{s}=\parallel\sum_{k=1}^{9}\beta_{k}Q_{k}|f\rangle\parallel^{2}/{N_{c}^{2}}$
(16)

Figure 2: Quantum circuit for realizing the QCNN. $|f\rangle$ denotes the
initial state of work system after encoded the image data, and the ancillary
system is a four qubits system in the $|0000\rangle_{a}$ state. The squares
represent unitary operations and the circles represent the state of the
controlling system. Unitary operations $Q_{1}$,$Q_{2}$,$\cdots$, $Q_{9}$, are
activated only when the auxiliary system is in state
$|0000\rangle$,$|0001\rangle$$\cdots$,$|1000\rangle$ respectively.

After obtaining the final result $\frac{1}{N_{c}}U^{\prime}|f\rangle$, we can
multiply the constant factor $N_{c}$ to compute
$|g^{\prime}\rangle=U^{\prime}|f\rangle$. In conclusion, the filter operator
$U^{\prime}$ can be decomposed into a linear combination of nine unitary
operators in the case that the general filter mask is $W$. Only four qubits or
a nine energy level ancillary system is consumed to realize the general filter
operator $U^{\prime}$, which is independent on the dimensions of image size.

The final stage of our method is to extract useful information from the
processed results $|g^{\prime}\rangle$. Clearly, the image state $|g\rangle$
is different from $|g^{\prime}\rangle$. However, not all elements in
$|f\rangle$ are evaluated, the elements corresponding to the four edges of
original image remain unchanged. One is only interested in the pixel values
which are evaluated by $W$ in $|f\rangle$. These pixel values in
$|g^{\prime}\rangle$ are as same as that in $|g\rangle$(see proof in
supplementary materials). So, we can obtain the informations of
$G=(G_{i,j})_{M\times M}$ ($2\leq i,j\leq M-1$) by evaluating the $|f\rangle$
under operator $U^{\prime}$ instead of $U$.

###  II.2 Quantum Pooling Layer

The function of pooling layer after the convolutional layer is to reduce the
spatial size of the representation so as to reduce the amount of parameters.
We adopt average pooling which calculates the average value for each patch on
the feature map as pooling layers in our model. Consider a $2*2$ pixels
pooling operation applied with a stride of 2 pixels. It can be directly
realized by ignoring the last qubit and the $m$-th qubit in quantum context.
The input image
$|g^{\prime}\rangle=(g_{1},g_{2},g_{3},g_{4},\dots,\dots,g_{M^{2}})^{T}$ after
this operation can be expressed as the output image

$\displaystyle|p\rangle=\left(\sqrt{g_{1}^{2}+g_{2}^{2}+g_{M+1}^{2}+g_{M+2}^{2}},\sqrt{g_{3}^{2}+g_{4}^{2}+g_{M+3}^{2}+g_{M+4}^{2}},\dots,\sqrt{g_{M^{2}-M-1}^{2}+g_{M^{2}-M}^{2}+g_{M^{2}-1}^{2}+g_{M}^{2}}\right)^{T}.$
(17)

###  II.3 Quantum Fully Connected Layer

Fully connected layers complie the data extracted by previous layers to form
the final output, it usually appear at the end of convolutional neural
networks. We define a parametrized Hamiltonian as the quantum fully connected
layer. This Hamiltonian consists of identity operators $I$ and Pauli operators
$\sigma_{z}$,

$\displaystyle\mathcal{H}=h^{0}I+\sum_{i}h^{i}\sigma_{z}^{i}+\sum_{i,j}h^{ij}\sigma_{z}^{i}\sigma_{z}^{j}+\dots$
(18)

where $h^{0},h^{i},h^{ij},\cdots$ are the parameters, and Roman indices $i,j$
denote the qubit on which the operator acts, i.e., $\sigma^{i}_{z}$ means
Pauli matrix $\sigma_{z}$ acting on a qubit at site $i$. We measure the
expectation value of the parametrized Hamiltonian
$f(p)=\langle{p}|\mathcal{H}|p\rangle$. $f(p)$ is the final output of the
whole quantum neural network. Then, we add an active function to nonlinearly
map $f(p)$ to $R(f(p))$. The parameters in Hamiltonian matrix $\mathcal{H}$
are updated by gradient descent method, i.e., are calculated by
$\frac{\partial f(p)}{\partial h^{i}}=\langle{p}|\sigma_{z}^{i}|p\rangle$ and
$\frac{\partial f(p)}{\partial
h^{ij}}=\langle{p}|\sigma_{z}^{i}\sigma_{z}^{j}|p\rangle$. The parameters in
$S$ matrix can be trained through classical backpropagation.

Now, we construct the framework of quantum neural networks. We demonstrate the
performance of our method in image processing and handwritten number
recognition in the next section.

##  III  Numerical Simulations

Figure 3: Three types of image processing, edge detection, image smoothing and
sharpening, are implemented on an image by classical method and quantum method
respectively.

###  III.1 Image Processing: Edge Detection, Image Smoothing and Sharpening

In addition to constructing QCNN, the quantum convolutional layer can also be
used to spatial filtering which is a technique for image processing27, 28, 29,
25, such as image smoothing, sharpening, edge detection and edge enhancement.
To show the quantum convolutional layer can handle various image processing
tasks, we demonstrate three types of image processing, edge detection, image
smoothing and sharpening with fixed filter mask $W_{de}$, $W_{sm}$ and
$W_{sh}$ respectively

$W_{de}=\left(\begin{array}[]{ccc}-1&-1&-1\\\ -1&8&-1\\\ -1&-1&-1\\\
\end{array}\right),\\\ W_{sm}=\frac{1}{13}\left(\begin{array}[]{ccc}1&1&1\\\
1&5&1\\\ 1&1&1\\\ \end{array}\right),\\\
W_{sh}=\frac{1}{16}\left(\begin{array}[]{ccc}-2&-2&-2\\\ -2&32&-2\\\
-2&-2&-2\\\ \end{array}\right).$ (19)

In a spatial image processing task, we only need one specific filter mask.
Therefore, after performing the above quantum convolutional layer mentioned,
we measure the ancillary register. If we obtain $|0\rangle$, our algorithm
succeeds and the spatial filtering task is completed. The numerical simulation
proves that the output images transformed by a classical and quantum
convolutional layer are exactly the same, as shown in Fig.(3).

###  III.2 Handwritten Number Recognition

Here we demonstrate a type of image recognition task on a real-world dataset,
called MNIST, a handwritten character dataset. In this case, we simulate a
complete quantum convolutional neural network model, including a convolutional
layer, a pooling layer, and a full-connected layer, as shown in Fig.(2). We
consider the two-class image recognition task(recognizing handwritten
characters ${}^{\prime}1^{\prime}$ and ${}^{\prime}8^{\prime}$) and ten-class
image recognition task(recognizing handwritten characters
${}^{\prime}0^{\prime}$-${}^{\prime}9^{\prime}$). Meanwhile, considering the
noise on NISQ quantum system, we respectively simulate two circumstances that
are the quantum gate $Q_{k}$ is a perfect gate or a gate with certain noise.
The noise is simulated by randomly acting a single qubit Pauli gate in
$[I,X,Y,Z]$ with a probability of $0.01$ on the quantum circuit after an
operation implemented. In detail, the handwritten character image of MNIST has
$28\times 28$ pixels. For convenience, we expand $0$ at the edge of the
initial image until $32\times 32$ pixels. Thus, the work register of QCNN
consists of 10 qubits and the ancillary register needs 4 qubits. The
convolutional layer is characterized by 9 learnable parameters in matrix $W$,
that is the same for QCNN and CNN. In QCNN, by abandoning the $4$-th and
$9$-th qubit of the work register, we perform the pooling layer on quantum
circuit. In CNN, we perform average pooling layer directly. Through measuring
the expected values of different Hamiltonians on the remaining work qubits, we
can obtain the measurement values. After putting them in an activation
function, we get the final classification result. In CNN, we perform a two-
layer fully connected neural network and an activation function. In the two-
classification problem, the QCNN’s parametrized Hamiltonian has 37 learnable
parameters and the CNN’s fully-connected layer has 256 learnable parameters.
The classification result that is close to 0 are classified as handwritten
character ${}^{\prime}1^{\prime}$, and that is close to 1 are classified as
handwritten character ${}^{\prime}8^{\prime}$. In the ten-classification
problem, the parametrized Hamiltonian has $10\times 37$ learnable parameters
and the CNN’s fully-connected layer has $10\times 256$ learnable parameters.
The result is a 10-dimension vector. The classification results are classified
as the index of the max element of the vector. Details of parameters, accuracy
and gate complexity are listed in Table.(1).

For the 2 class classification problem, the training set and test set have a
total of 5000 images and 2100 images, respectively. For the 10 class
classification problem, the training set and test set have a total of 60000
images and 10000 images, respectively. Because in a training process, 100
images are randomly chosen in one epoch, and 50 epochs in total, the accuracy
of the training set and the test set will fluctuate. So, we repeatedly execute
noisy QCNN, noise-free, and CNN 100 times, under the same construction. In
this way, we obtain the average accuracy and the field of accuracy, as shown
in Fig.(4). We can conclude that from the numerical simulation result, QCNN
and CNN provide similar performance. QCNN involves fewer parameters and has a
smaller fluctuation range.

Table 1: The important parameters of models Models | Problems |  Data Set | Parameters
---|---|---|---
| Learnable Parameters | Average Accuracy | Gate Complexity
Noisy QCNN | ${}^{\prime}1^{\prime}$ or ${}^{\prime}8^{\prime}$ | training set  | 46 | 0.948 | $O((log_{2}M)^{6})$
test set  | 0.960
${}^{\prime}0^{\prime}\sim^{\prime}9^{\prime}$ | training set  | 379 | 0.742
test set  | 0.740
Noise-free QCNN | ${}^{\prime}1^{\prime}$ or ${}^{\prime}8^{\prime}$ | training set  | 46 | 0.954
test set  | 0.963
${}^{\prime}0^{\prime}\sim^{\prime}9^{\prime}$ | training set  | 379 | 0.756
test set  | 0.743
CNN | ${}^{\prime}1^{\prime}$ or ${}^{\prime}8^{\prime}$ | training set  | 265 | 0.962 | $O(M^{2})$
test set  | 0.972
${}^{\prime}0^{\prime}\sim^{\prime}9^{\prime}$ | training set  | 2569 | 0.802
test set  | 0.804

Figure 4: The performance of QCNN based on MNIST. The blue, red, and green
curves denote the average accuracy of the noisy QCNN, noise-free QCNN, and
CNN, respectively. The shadow areas of the corresponding color denote the
accuracy fluctuation range in the 100 times simulation results. The insets are
the typical images from MNIST set. (a), (b) are the curves representing the
result from the training set and the test set for the 2 class classification
problem respectively. (c), (d) show the result from the training set and the
test set for the 10 class classification problem respectively.

##  IV Algorithm complexity

We analyze the computing resources in gate complexity and qubit consumption.
(1) Gate complexity. At the convolutional layer stage, we could prepare an
initial state in $O(poly(log_{2}(M^{2}))$ steps. In the case of preparing a
particular input $|f\rangle$, we employ the amplitude encoding method in
Ref.30, 31, 32. It was shown that if the amplitude $c_{k}$ and
$P_{k}=\sum_{k}|c_{k}|^{2}$ can be efficiently calculated by a classical
algorithm, constructing the $log_{2}(M^{2})$-qubit $X$ state takes
$O(poly(log_{2}(M^{2}))$ steps. Alternatively, we can resort to quantum random
access memory33, 34, 35. Quantum Random Access Memory (qRAM) is an efficient
method to do state preparation, whose complexity is $O(log_{2}(M^{2}))$ after
the quantum memory cell established. Moreover, the controlled operations
$Q_{k}$ can be decomposed into $O((log_{2}M)^{6})$ basic gates (see details in
Appendix A). In summary, our algorithm uses $O((log_{2}M)^{6})$ basic steps to
realize the filter progress in the convolutional layer. For CNN, the
complexity of implementing a classical convolutional layer is $O(M^{2})$.
Thus, this algorithm achieves an exponential speedup over classical algorithms
in gate complexity. The measurement complexity in fully connected layers is
$O(e)$, where $e$ is the number of parameters in the Hamiltonian.
(2) Memory consumption. The ancillary qubits in the whole algorithm are
$O(log_{2}(m^{2}))$, where $m$ is the dimension of the filter mask, and the
work qubits are $O(log_{2}(M^{2}))$. Thus, the total qubits resource needed is
$O(log_{2}(m^{2})+O(log_{2}(M^{2})$.

##  V Conclusion

In summary, we desighed a quantum Neural Network which provides exponential
speed-ups over their classical counterparts in gate complexity. With fewer
parameters, our model achieves similar performance compared with classical
algorithm in handwritten number recognition tasks. Therefore, this algorithm
has significant advantages over the classical algorithms for large data. We
present two interesting and practical applications, image processing and
handwritten number recognition, to demonstrate the validity of our method. We
give the mapping relations between a specific classical convolutional kernel
to a quantum circuit, that provides a bridge between QCNN to CNN. It is a
general algorithm and can be implemented on any programmable quantum computer,
such as superconducting, trapped ions, and photonic quantum computer. In the
big data era, this algorithm has great potential to outperform its classical
counterpart, and works as an efficient solution.

Acknowledgements

###### Acknowledgements.

We thank X. Yao, X. Peng for inspiration and fruitful discussions. This
research was supported by National Basic Research Program of China. S.W.
acknowledge the China Postdoctoral Science Foundation 2020M670172 and the
National Natural Science Foundation of China under Grants No. 12005015. We
gratefully acknowledge support from the National Natural Science Foundation of
China under Grants No. 11974205, and No. 11774197. The National Key Research
and Development Program of China (2017YFA0303700); The Key Research and
Development Program of Guangdong province (2018B030325002); Beijing Advanced
Innovation Center for Future Chip (ICFC).

## Appendix A: Adjusted operator $U^{\prime}$ can provide enough information
to remap the output imagine.

$Proof.$\- The different elements of image matrix after implementing operator
$U^{\prime}$ compared with $U$ are in the edges of image matrix. We prove that
the evolution under operator $U^{\prime}$ can provide enough information to
remap the output image. The different elements between $U^{\prime}$ and $U$
are included in

$U^{\prime}_{k,n}\neq U_{k,n}\begin{cases}(1\leq k\leq M;1\leq n\leq
2M,M^{2}-M\leq n\leq M^{2})\\\ (M^{2}-M\leq k\leq M^{2};1\leq n\leq
M,M^{2}-3M\leq n\leq M^{2}-M)\\\
(k=sM+1;n=1+(s-1)M,2+(s-1)M,sM,sM+1,sM+2,(s+1)M,(s+1)M+1,(s+1)M+2,(s+2)M)\\\
(k=(s+1)M;n=1,sM-1,sM,sM+1,(s+1)M-2,(s+1)M-1,(s+1)M+1,(s+2)M-2,(s+2)M-1)\end{cases}$
(20)

where $1\leq s\leq M-2$.

After performing $U^{\prime}$ and $U$ on quantum state $|f\rangle$
respectively, the difference exits in the elements
$|g^{\prime}_{k}\rangle\neq|g_{k}\rangle(k=1,2,\cdots,M,sM+1,(s+1)M,M^{2}-M+1,\cdots,M^{2})$,
where $1\leq s\leq M-2$. Since $|g^{\prime}\rangle$ can be remapped to
$G^{\prime}$, $U^{\prime}$ will give the output image
$G^{\prime}=(G^{\prime}_{i,j})_{M\times M}$. The elements in $U^{\prime}$
which is different from $U$ only affect the pixel $i,j\not\in{2,\cdots,M-1}$.
Thus, only and if only $i,j\not\in{2,\cdots,M-1}$, the matrix elements satisfy
$G_{i,j}\neq G^{\prime}_{i,j}$. Namely, the output imagine
$G^{\prime}_{i,j}=G_{i,j}$($2\leq i,j\leq M-1$).

## Appendix B: Decomposing operator Q into basic gates

Considering the nine operators $Q_{1}$, $Q_{2}$,$\cdots$, $Q_{9}$ consist of
filter operator $U^{\prime}$. $Q_{k}$ is the tensor product of two of the
following three operators

$E_{1}=\left(\begin{array}[]{cccccc}&&&&1\\\ 1&&&&\\\
\ddots&\ddots&\ddots&&\\\ &&1&&\\\ &&&1&\\\ \end{array}\right)_{M\times M}\\\
E_{2}=\left(\begin{array}[]{cccccc}1&&&&\\\ &1&&&\\\ &\ddots&\ddots&\ddots&\\\
&&&1&\\\ &&&&1\\\ \end{array}\right)_{M\times M}\\\
E_{3}=\left(\begin{array}[]{cccccc}&1&&&\\\ &&1&&\\\ &&\ddots\ddots&\ddots&\\\
&&&&1\\\ 1&&&&\\\ \end{array}\right)_{M\times M}.$ (21)

$E_{2}$ is a $M\times M$ identity matrix not need to be further decomposed.
For convenient, consider a $n$-qubits operator $E_{1}$ with dimension $M\times
M$, where $n=log_{2}(M^{2})$. It can be expressed by the combination of
$O(n^{3})$ CNOT gates and Pauli $X$ gates as shown in Fig.5. Consequently,
$E_{3}$ can be decomposed into the inverse of combinations of basic gate as
shown in Fig.5, because of the fact $E_{3}=E_{1}^{\dagger}$.

Figure 5: Decomposition of operator $E_{1}$ in the form of basic gates.

Thus, $Q_{k}$ can be implemented by no more than $O(n^{6})$ basic gates.
Totally, the controlled $Q_{k}$ operation can be implemented by no more than
$O(n^{6})=O((log_{2}M)^{6})$(ignoring constant number).

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