# QDCNN: Quantum Dilated Convolutional Neural Network

Yixiong Chen  IBM Corporation
Email<EMAIL_ADDRESS>
###### Abstract

In recent years, with rapid progress in the development of quantum
technologies, quantum machine learning has attracted a lot of interest. In
particular, a family of hybrid quantum-classical neural networks, consisting
of classical and quantum elements, has been massively explored for the purpose
of improving the performance of classical neural networks. In this paper, we
propose a novel hybrid quantum-classical algorithm called quantum dilated
convolutional neural networks (QDCNNs). Our method extends the concept of
dilated convolution, which has been widely applied in modern deep learning
algorithms, to the context of hybrid neural networks. The proposed QDCNNs are
able to capture larger context during the quantum convolution process while
reducing the computational cost. We perform empirical experiments on MNIST and
Fashion-MNIST datasets for the task of image recognition and demonstrate that
QDCNN models generally enjoy better performances in terms of both accuracy and
computation efficiency compared to existing quantum convolutional neural
networks (QCNNs).

##  1 Introduction

Convolutional neural networks (CNNs), proposed by Yann LeCun et al [1] in
1989, are one of the most powerful algorithms in the context of deep learning.
The main advantage of CNNs is that they use multiple feature extraction stages
to automatically and accurately learn important features from the data without
any human supervision. Due to this advantage, CNNs have been tremendously
successful in a broad array of high-level computer vision problems, including
image recognition [2, 3, 4, 5] , object detection [6, 7, 8], and image
segmentation [9, 10, 11]. In recent years, with further development in deep
learning , CNNs have also been demonstrated to show promising performances in
other machine learning areas such as time series forecasting [12, 13], speech
recognition [14] and recommendation system [15].

Parallelly, with recent achievements in quantum technologies (e.g. noisy
intermediate-scale quantum (NISQ) processors are currently availlabe), the
domain of quantum machine learning has attracted growing concerns and
triggered an enormous amount of work. Quantum machine learning is a research
area with the purpose of utilizing quantum mechanical effects such as
superposition and entanglement to improve the performance of machine learning
algorithms. Even though quantum machine learning is a new discipline, it has
witnessed a number of successful quantum extensions to classical machine
learning problems, including support vector machine [16], clustering [17, 18],
and principal component analysis [19].

Among quantum machine learning algoritms, quantum convolutional neural
networks (QCNNs), also known as hybrid quantum-classical convolutional neural
networks, are a family of variational quantum algorithms and they have
recently become a very active research field. The central idea of QCNNs is to
construct a quantum convolutional layer within neural networks based on
parameterized quantum circuits to estimate complex kernel functions in high
dimensional Hilbert space. Inspired by CNNs, Liu et al. [20] proposed the
first QCNN model and implement it for image recognition. Afterwards, the QCNN
model was investigated further in various work [21, 22, 23, 24, 25, 26].
Recently, it has been demonstrated in [27] that QCNN models can also achieve
promising results in speech recognition.

Despite these successes, QCNNs suffer from computational bottlenecks which
make it time consuming to train QCNNs. Firstly, quantum operations applied on
n-qubit quantum circuits require unitary matrices of size $2^{n}\times 2^{n}$
which will scale exponentially as the size of quantum circuit. Moreover, the
calculation of gradients, due to the parameter-shift rule [28, 29], result in
more quantum circuit executions, when QCNNs are trained on a real quantum
device. As an example, a quantum filter with $p$ trainable parameters will add
$2p$ more quantum circuit executions for each training sample to compute the
required gradients. Even though this problem can be mitigated when QCNNs are
implemented on quantum simulators that support more efficient gradient
computation methods such as back-propagation [30, 31] and adjoint
differentiation method [32], it is inevitable for QCNNs to face another
challenge. In CNNs, a convolutional layer, due to local connectivity, performs
a large amount of element-wise matrix multiplication operations. For example,
an output $100\times 120$ feature map of a convolutional layer is obtained
from $100\times 120=12,000$ multiplication operations. The computational cost
will increase significantly with the feature map size. Fortunately, this
computational issue in CNNs can be handled by using vectorization techniques
[33, 34]. QCNNs, as the counterpart of CNNs, have the same problem. However,
unlike CNNs, most of current quantum devices, including quantum hardware and
quantum simulator, do not support vectorization. Despite the availability of
more mature quantum devices in the NISQ era, executing a large number of
quantum circuits would be impractical in general.

A few works have been done to investigated how to reduce the runtime
complexity of QCNNs. In the first family of works, a small number of qubits
required for the quantum circuit is achieved by using classical data pre-
processing techniques to reduce the dimension of the input features fed into
the quantum (convolutional) layer. For instance, Pramanik et al. [35] employ
principal component analysis (PCA) to reduce the VGG-16 features for the
quantum variational classifier (VQC), while Hur et al. [36] adopt autoencoding
(AutoEnc) for the dimensionality reduction. Nevertheless, the performance of
the model trained in this way is likely to be compromised by the limited
expressive power of the reduced features, as shown in [35]. The second family
of works focus on how to efficiently encode classical data into quantum
states. Schuld and Killoran [37] propose and implement the _amplitude
encoding_ for variational quantum circuits, which is explored further in [38]
for Flexible Representation of Quantum Images (FRQI). This type of encoding
method is efficient in terms of required qubits for data encoding but it
relies on too deep quantum circuits which are unpractical on NISQ devices. In
a different direction, some recent researchers [39, 40, 26] propose angel
encoding (also referred to as qubit encoding) and its variants (e.g. dense
angel encoding) which use a constant quantum circuit depth for state
preparation. This encoding scheme requires one qubit to encode one or a
limited number of components of the input feature vector and thus is not
efficient for high-dimensional input features from a resource prospective. To
trades off these two encoding methods mentioned above, Hur et al. [36] further
develop a hybrid encoding approach which requires fewer number of qubits than
the angel encoding and use shallower quantum circuit depth than the amplitude
encoding. Moreover, Henderson et al. [22] employ a threshold based encoding
technique to reduce the input-state space and made it possible to obtain the
output feature map through a look-up table during the quantum convolution
process without needing to execute the same quantum circuit repeatedly on
image segments. This method is easy to implement, but it is infeasible on real
quantum devices, as mentioned in [22].

Having reviewing all these challenges and developments, in this work, we
propose a novel hybrid quantum-classical architecture which we will call
_quantum dilated convolutional neural network_ (QDCNN). Our approach,
motivated by the dilated convolution in deep learning, is an extension of the
architectures presented in [20] and [22], and helps reduce the computational
cost of QCNNs in a different way compared to the aforementioned approaches.
Dilated convolution, also known as atrous convolution, was originally
developed for efficiently computating the undecimated discrete wavelet
transform [41]. In recent years, dilated convolution has attracted more and
more attention, and is widely used in semantic segmentation [42, 43, 44, 45,
11, 46]. Following these successes, dilated convolution has also been adopted
for a broader set of tasks, such as object localization [47], time series
forecasting [12, 13] and sound classification [48]. The advantage of dilated
convolution is that it allows for effectively expanding the field of view of
filters to capture larger context without increasing the number of parameters
or the computational complexity. By virtue of dilated convolution, the
proposed QDCNNs can generally improve the computational efficiency of existing
QCNNs while achieving the better task performance.

In summary, the contributions of our work are

  * •

We propose a novel architecture of quantum convolutional neural network based
on quantum dilated convolution operation. To the best of our knowledge, our
work is the first attempt to combine the concept of dilated convolution with
variational quantum circuits.

  * •

We conduct experiments using MNIST and Fashin-MNIST datasets and demonstrate
the superior performance of QDCNN models over QCNN models.

##  2 Method

###  2.1 Preliminaries

####  2.1.1 Convolution Operation

The convolutional layer, which performs an operation called a “convolution“,
plays a central role in CNNs. In the context of convolutional networks, a
convolution is a linear operation that involves the multiplication of a set of
weights with the input. For a convolution operation, a _kernel_ or _filter_ is
defined as a feature extractor which is a two-dimensional (2-D) array of
learnable weights. A filter is applied to a filter-size patch of the input
image called _receptive field_ and a dot product is performed between the
pixels within the receptive field and the weight values in the filter.
Afterwards, the filter shifts to the next patch according to a step size
called _stride_ , and repeats the above process until it has swept across the
entire image. The final output from the series of dot products between the
filter weights and the values underneath the filter, is called a _feature
map_. Let us denote the output feature map by $y$ and the input image by $x$.
In the 2-D convolution process, the feature map $y$ is obtained by applying a
filter $k$ to the input image $x$:

$\displaystyle y[i,j]=\sum_{q}\sum_{l}k[q,l]\cdot x[i+q,j+l]$ (1)

where $i$ and $j$ are location indices of $y$. The output feature map, due to
the convolution operation, usually has smaller spatial resolution than the
input image. This reduction in dimensions can be avoided by employing _zero
padding_ technique, namely adding a border of pixels with value zero around
the edges of the input image before the application of a filter. A
hyperparameter called _padding_ can be defined to determine how many zero
values to add to the border of the image. Generally, the spatial resolution
$o_{w}$ and $o_{h}$ of the resulting feature map, extracted from an
$i_{w}\times i_{h}$ input image by a $m\times n$ kernel, can be calculated as

$\displaystyle o_{w}$ $\displaystyle=$
$\displaystyle\left(\frac{i_{w}-m+2p}{s}\right)+1$ (2) $\displaystyle o_{h}$
$\displaystyle=$ $\displaystyle\left(\frac{i_{h}-n+2p}{s}\right)+1$ (3)

where $p$ and $s$ represent padding and stride respectively.

####  2.1.2 Dilated Convolution

Dilated convolution is a type of convolution that expands the kernel by
inserting holes (i.e. points with weight of zero) between the consecutive
kernel elements. In simple terms, dilated convolution is just a convolution
applied to the input with defined gaps. Compared to standard convolution,
dilated convolution introduces an extra hyperparameter called _dilation rate_
that determines the stride with which the input pixels are sampled. According
to the definition of dilated convolution, $r-1$ zero values are inserted
between two consecutive filter values, if the dilation rate is denoted by $r$.
In this spirit, Eq. (1) needs to be reformulated as

$\displaystyle y[i,j]=\sum_{q}\sum_{l}k[q,l]\cdot x[i+q\cdot r,j+l\cdot r]$
(4)

in the context of dilated convolution. It can be seen from Eq. (4) that
dilated convolution is able to capture a larger receptive field without
introducing more learnable parameters compared to standard convolution with
the same kernel size. Moreover, for dilated convolution, we also need to
rewrite Eq. (2) and Eq. (3) as

$\displaystyle o_{w}$ $\displaystyle=$
$\displaystyle\left(\frac{i_{w}-m-(m-1)(r-1)+2p}{s}\right)+1$ (5)
$\displaystyle o_{h}$ $\displaystyle=$
$\displaystyle\left(\frac{i_{h}-n-(n-1)(r-1)+2p}{s}\right)+1$ (6)

which indicate that dilated convolution generally results in a feature map
with smaller size compared to standard convolution for the same set of
hyperparameters. It is worth noting that the standard convolution can be
regarded as a special case of dilated convolution with dilation rate $r=1$.

####  2.1.3 Quantum Convolution

In contrast to classical convolution, quantum convolution is a new type of
convolution based on quantum circuits and it generally consists of three
modules:

  * •

_ENCODING MODULE_. In this module, classical data is encoded into a quantum
state which will be further processed in the quantum convolutional circuit.
There exist various encoding methods such as angle encoding, amplitude
encoding and basis encoding. A summary of them can be found in the literature
[49]. Among these methods, angle encoding is the most commonly used encoding
approach. In this encoding scheme, the classical input is treated as the
rotation angle of a single-qubit rotation gate (e.g. $RY$ rotation gate). For
example, a classical variable or feature $a$ can be encoded by $RY(a)$ which
is applied on some initial state (e.g. vacuum state $|0\rangle$). In this
sense, we can say that the classical information $a$ is encoded into the
initial state of a qubit. This type of angle encoding is called _one
variable/qubit encoding_. This approach requires $n$ qubits to encode $n$
input variables. To reduce the required qubits, we can also encode multiple
variables by sequential rotations applied on a single qubit. For example,
input variables $a_{1}$, $a_{2}$ and $a_{3}$ can be encoded using $RX(a_{1})$,
$RY(a_{2})$ and $RZ(a_{3})$ rotation gates applied successively on a single
qubit. This angle encoding is called _multiple variables/qubit encoding_ or
_dense angel encoding_. In this paper, we focus on one variable/qubit encoding
method. Let us denote by $E(x)$ the encoding operator where $x$ is the input
vector. Then the encoded quantum state is obtained by

$\displaystyle|x\rangle=E(x)|0\rangle.$ (7)

It is worth noting that $E(x)$ usually contain the Hadmard gate which
transforms the initial state into a superpostion state.

  * •

_ENTANGLEMENT MODULE_. In this module, a cluster of single- and multi-qubit
gates are applied to the encoded quantum state obtained from the previous
module. Mutli-qubit gates are usually $CNOT$ gate and parametric controlled
rotation gate (e.g. $CRZ(\theta)$ where $\theta$ is a trainable parameter),
and they are used to generate correlated quantum states, namely entangled
states. Single-qubit gates are mainly parametric rotation gates. This
combination of single- and multi-qubit gates is referred to as parameterized
layer in a QCNN and is designed to extract task-specific features. This
parameterized layer is usually repeated multiple times to extend the feature
space. If we denote all unitary operations in the entanglement module by
$U(\theta)$ for simplicity, the output quantum state will be

$|x,\theta\rangle=U(\theta)|x\rangle.$ (8)

  * •

_DECODING MODULE_. At this stage, certain local observable $A^{\otimes m}$
(e.g. Pauli Z operator $\sigma_{z}^{\otimes m}$) is measured in the final
quantum state $|x,\theta\rangle$ from the entanglement module, where $m$ is
equal or smaller than the total number of qubits $n$ in the quantum system.
The expectation value of the chosen observable $A^{\otimes m}$ can be obtained
by repeated measurements:

$\displaystyle f(x,\theta)=\langle x,\theta|A^{\otimes m}|x,\theta\rangle$ (9)

So the purpose of this layer is to extract a classical output vector
$f(x,\theta)$ by using the mapping from the quantum state to a classical
vector:

$\displaystyle\mathcal{M}:\quad|x,\theta\rangle\rightarrow f(x,\theta).$ (10)

This classical vector $f(x,\theta)$ can be used as the input features for the
subsequent layer in the QCNN.

Figure 1: An example of a quantum dilated convolutional layer (QDC) with
kernel size $=2\times 2$ and dilation rate $=2$. In contrast to standard
quantum convolution, dilated convolution is applied to the input with defined
gaps (one gap in this examle) to enlarge the receptive field. Encoding,
entanglement, decoding modules for the QDC layer are highlighted with green,
purple and yellow colors respectively.

###  2.2 QDCNN

The proposed QDCNN is designed in the same fashion as QCNNs described in
literatures [20, 22]. Our model integrates quantum layers with classical
layers and the quantum circuit ansatz can be placed anywhere in the model
(e.g. at the beginning of the network, at intermediate layers in the network).

The key difference between our method and existing QCNNs is that the dilated
convolution is employed for the quantum convolutional layer. So the quantum
layer in QDCNNs is called _quantum dilated convolutional (QDC) layer_. An
example of a QDC layer is illustrated in Fig. 1. Due to the mechanism of
dilated convolution , the quantum kernel in our model generally covers larger
image patches (i.e. receptive fields ). For example, a $2\times 2$ quantum
dilated convolution with dilation rate of 3 has a receptive field of $4\times
4$ while the standard quantum convolution with the same kernel size has only a
receptive field of $2\times 2$ . It is noteworthy that even though the quantum
dilated convolution is able to expand the receptive field the number of data
points that are fed into the quantum convolution circuit is the same as the
one for the standard quantum convolution. This means that the quantum dilated
convolution does not requires more qubits than the standard quantum
convolution with the same kernel size.

Our QDCNN model has mainly two advantages. Firstly, the QDC layer in our
model, thanks to the enlarged receptive field, requires less number of times
that the quantum kernel slides across the image (if there is no padding and
the stride is the same), compared to the existing QCNN models. This can be
understood by comparing Eqs. (2) and (3) with Eqs. (5) and (6) respectively.
Therefore, using the QDC layer helps reduce the number of quantum circuit
executions during the quantum convolution process. In the NISQ era, long
training time is one of the biggest challenges facing the QCNN models. This
difficulty mainly stems from the large number of quantum circuit executions
from the quantum layers. In the quantum feature mapping process, due to the
probabilistic characteristics, quantum measurement is usually performed
multiple times (e.g. 1024) to get expectation values of some observables which
can be considered as the extracted quantum feature maps. So how to reduce the
number of quantum circuit executions plays a crucial role in mitigating the
long-running-time problem of QCNNs. Our proposed quantum dilation is a
powerful tool to explicitly control the amount of quantum circuit executions
in the quantum layer.

The second advantage of our model is that it can improve the performance (e.g.
classification accuracy) of existing QCNN models. Due to the expanded
receptive field, the QDC layer in our model generally reduce the spatial
resolution of the resulting feature maps. However, these feature maps are
extracted from larger receptive fields of the image and hence contain long-
range context which plays an essential role in many machine learning tasks
such as image recognition and image segmentation.

##  3 Experiments

In this section, we conduct two experiments to evaluate the performance of our
proposed QDCNN model and compare it with the existing QCNN model. In
_Experiment A_ and _Experiment B_ , we construct quantum convolutional models
with non-trainable and trainable quantum filters, respectively.

###  3.1 Experiment settings

####  3.1.1 Dataset

We choose the image benchmark MNIST and Fashion-MNIST datasets [50, 51] for
our experiments. The MNIST dataset contains 10 different classes of
handwritten digits from ‘0’ to ‘9’ , while the Fashion-MNIST dataset is a
collection of 10 different shapes of t-shirts, dresses, shoes, etc. Both of
these datasets have 60,000 training samples, and 10,000 test samples of
28-by-28 gray scale pixel images. Due to the expensive training and
validation, we pick two subsets of the entire MNIST and Fashion-MNIST
datasets, respectively, both of which consist of 1,000 balanced training
samples and 200 balanced testing samples.

####  3.1.2 Tested Models

In this research, we consider two types of models:

  * •

_QDCNN Model._ We employ the architecture of the most basic convolution-
inspired hybrid quantum-classical neural network. Our QDCNN model consists of
one QDC layer with one filter and one fully-connected layer with 10 neurons.
The kernel size and stride for the QDC layer is selected as $2\times 2$ and 2
respectively without specification. The quantum circuit ansatz of the QDC
layer is designed as below. The 1 variable/qubit encoding scheme is adopted to
encode the input image. Specifically, $2\times 2$ pixels are encoded into a
4-qubit state using RY rotation gates. Note that, these $2\times 2$ pixels are
not adjacent to each other in the input image, due to the quantum dilated
convolution. The resulting 4-qubit state is further transformed by a following
random parameterized quantum circuit which might creates the entanglement. The
decoding method follows the same spirit of [52], in which each expectation
value is mapped to a different channel of a single output pixel. Consequently,
even though there is only one filter, the quantum layer can transform the
input 2-D image into four feature maps. This type of quantum layer might
benefit the model performance as it allows for correlation among channels of
the output feature maps. In both cases of MNIST and Fashion-MNIST datasets,
the QDC layer extracts from the $28\times 28$ input image a feature tensor of
size $13\times 13\times 4$, which is then transformed to 10 output
probabilities by the fully-connected layer with softmax activation. To
evaluate how the dilation rate impacts the model performance, we consider two
QDCNN models with dilation rate $r=2$ and $r=3$. We refer to these two models
as _QDCNN_r2_ and _QDCNN_r3_ respectively for the rest of the paper.

  * •

_QCNN Model._ We choose the standard QCNN model as our benchmark model. The
QCNN model follows the same structure of our QDCNN model with the only
difference that it uses standard quantum kernel rather than dilated quantum
kernel.

The random quantum circuit in each of these models consists of two 4-qubit
random layers, each of which has four non-trainable or trainable parameters.
For fair comparison, all of these random circuits share the same architecture
generated by the same random seed.

####  3.1.3 Training Setup

In Experiment A, after applying the non-trainable quantum filter to transform
the original image data into feature maps, we use a mini-batch of 32 and Adam
optimizer with a learning rate of 0.01 to train each model for 30 epochs. In
Experiment B, due to the computational cost of training parametric quantum
circuits involved in the trainable quantum filter, we reduce the batch size to
four and train all models for 20 epochs with other hyperparameters remaining
unchanged.

####  3.1.4 Experimental Environment

Experiments are conducted on the local computer with a 6-core CPU (2.2 GHz) by
using PennyLane [53], Qulacs [54] and PyTorch [55]. PennyLane is an open
source python-based framework that enables the automatic differentiation for
hybrid quantum-classical computations. It is compatible with mainstream
machine learning frameworks such as TensorFlow [56] and PyTorch, and it has a
large plugin ecosystem which offers access to numerous quantum devices (i.e.
simulators and hardware) from different vendors including IBM, Google,
Microsoft, Rigetti and QunaSys. In Experiment A, we perform the quantum
processing of the original image data by using the _Qulacs_ simulator [57]
which is a high-performance C++ quantum simulator and made available through
the community contributed PennyLane-Qulacs plugin [58]. In Experiment B,
considering the large amount of quantum circuit executions required in the
scheme of parameter-shift rule, we train all hybrid models by using instead
the built-in Pennylane simulator _default.qubit_ which supports back-
propagation method for PyTorch interface.

###  3.2 Results

As demonstrated in Table I and Table II, our proposed QDCNN models exhibit
significant performance benefits over the QCNN model in terms of run-time
efficiency. More specifically, QDCNN models in Experiment A and B speed up the
model training process by up to 15.24% and 18.42% respectively, compared with
the QCNN model (QDCNN_r2 and QDCNN_r3 have similar training time because their
QDC layers output feature maps of the same size in both experiments). These
speed-ups result from the reduced number of quantum circuit executions, as
discussed in the subsection 2.2. Take Experiment A as an example. In this
experiment, $13\times 13$ quantum circuits need to be executed for both the
QDC layers with dilation rate $=2$ and dilation rate $=3$ while $14\times 14$
quantum circuits for standard quantum convolutional layer. This means that
QDCNN_r2 and QDCNN_r3 require 27 fewer quantum circuit executions per image
than QCNN models. Compared with Experiment A, it generally takes much longer
time to train hybrid models in Experiment B, even though the quantum filter in
each of them has only eight trainable parameters coming from the random
circuit. This is mainly due to the fact that PennyLane does not support
vectorization for quantum circuit executions. Nevertheless, quantum dilated
convolution can still help reduce the training time significantly in this
case.

Furthermore, it can also be seen from Table I and Table II that our QDCNN
models generally enjoy higher recognition accuracy than the QCNN model. In
particular, QDCNN_r3 achieves the best performance with regard to both
validation loss and accuracy across all tasks. In light of the QDC layer with
dilation rate of 3, QDCNN_r3 provides up to 31.74% lower validation loss and
up to 3% higher validation accuracy, compared with the QCNN model. This
observed model performance boost mainly stems from the contextual information
in larger scales captured by the QDC layer.

Dataset | Method | Test acc | Test loss | Running time
---|---|---|---|---
MNIST | QCNN | 88.00% | 0.4389 | 624.648s
QDCNN_r2 | 88.50% | 0.3858 |  529.472s
QDCNN_r3 |  91.00% | 0.3466 | 530.270s
Fashion-MNIST | QCNN | 78.50% | 0.6319 | 610.978s
QDCNN_r2 | 80.00% | 0.6354 | 526.060s
QDCNN_r3 |  81.00% | 0.6031 |  525.182s

TABLE I: Results of Experiment A on MNIST and Fashion-MNIST datasets, reported
for three hybrid-classical neural network models with non-trainable quantum
filters. QDCNN_r2 and QDCNN_r3 are hybrid models with one QDC layer with
dilation rate $r=2$ and $r=3$, respectively. QCNN represents the hybrid model
with one standard quantum convolutional layer or equivalently QDC layer with
dilation rate $r=1$.

Dataset | Method | Test acc | Test loss | Running time
---|---|---|---|---
MNIST | QCNN | 86.50% | 0.5341 | 7.420h
QDCNN_r2 | 86.50% | 0.5315 | 6.436h
QDCNN_r3 |  89.50% | 0.3646 |  6.053h
Fashion-MNIST | QCNN | 79.50% | 0.8923 | 7.443h
QDCNN_r2 | 77.00% | 1.1038 |  6.222h
QDCNN_r3 |  80.50% | 0.8837 | 6.372h

TABLE II: Results of Experiment B on MNIST and Fashion-MNIST datasets,
reported for three hybrid-classical neural network models with trainable
quantum filters.

##  4 Conclusion

In this work, we propose the QDCNN model, which adopts the idea of dilated
convolution in deep learning to the quantum neural network. We show through
empirical evidence that the QDCNN model outperforms the recent QCNN method in
terms of computation time and recognition accuracy. In particular, we find
that the quantum dilated convolution with a larger dilation rate generally
contribute to a better model performance. Dilated convolution has been
extensively studied in the area of deep learning, but little work has been
done to explore it in the context of quantum machine learning. Our work
constitutes a first step in this direction. With the promising results on both
MNIST and Fashion-MNIST datasets, our QDCNN approach deserves further
investigation in the future.

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#  The virtual-state character of the 9Be $1/2^{+}$ state in the
9Be($\gamma$,$n$)8Be reaction

Myagmarjav Odsuren  Nuclear Research Center, National University of Mongolia,
Ulaanbaatar 210646, Mongolia<EMAIL_ADDRESS>Yuma Kikuchi  RIKEN Nishina
Center, Wako 351-0198, Japan  Takayuki Myo  General Education, Faculty of
Engineering, Osaka Institute of Technology, Osaka 535-8585, Japan  Research
Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan
Masayuki Aikawa  Nuclear Reaction Data Centre, Faculty of Science, Hokkaido
University, Sapporo 060-0810, Japan  Kiyoshi Katō  Nuclear Reaction Data
Centre, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan

###### Abstract

We study the character of the first excited $1/2^{+}$ state of 9Be, which is
observed as a low-lying sharp peak in the cross section of
9Be($\gamma$,$n$)$2\alpha$ just above the 8Be + $n$ threshold. Using the
$\alpha$ \+ $\alpha$ \+ $n$ three-body model, we describe the ground and
excited unbound states of 9Be above the $\alpha$ \+ $\alpha$ \+ $n$ threshold.
Applying the complex scaling method to the three-body model, we find no
$1/2^{+}$ resonant solutions with the scaling angle of $\theta\leq 15$
degrees, while the low-lying peak in the photodisintegration cross section is
reproduced in the present calculation. It is found that the low-lying peak is
dominantly explained by the 8Be + $n$ component. Furthermore, using the
analytical continuation of the coupling constant of the three-body interaction
for the $\alpha$ \+ $\alpha$ \+ $n$ system, we discuss the virtual-state
character of the $1/2^{+}$ state.

###### pacs:

21.45.-v, 21.60.Gx, 26.30.-k, 27.20.+n

Introduction. The neutron capture $(n,\gamma)$ and its inverse $(\gamma,n)$
reactions are interesting topics in nuclear physics since these reactions play
important roles in nuclear astrophysics. In neutron capture reactions, the
$\alpha(\alpha n,\gamma)^{9}$Be reaction is one of the most interesting ones.
It has been suggested that the reaction rate of this reaction is crucial to
understand the productions of heavy elements in supernova explosions
Terasawa01 ; Sasaqui05 .

To determine the reaction rate of the $\alpha(\alpha n,\gamma)^{9}$Be, the
9Be$(\gamma,n)^{8}$Be reaction has been performed by several experiments
Gibbons59 ; John62 ; Fujishiro82 ; Utsunomiya00 ; Burda10 ; Arnold12 . In the
$\alpha(\alpha n,\gamma)^{9}$Be reaction, a sequential process,
$\alpha(\alpha,\gamma)^{8}$Be$(n,\gamma)^{9}$Be, has been considered as a
dominant one. However, owing to the short life-time of the 8Be ground state
($\sim 10^{-16}$s), a direct measurement of the 8Be$(n,\gamma)^{9}$Be reaction
is impossible. For an alternative way, the cross section of its inverse
reaction, 9Be$(\gamma,n)^{8}$Be, has been measured to deduce the cross section
of 8Be$(n,\gamma)^{9}$Be.

In the experiments, there is an inconsistency among the observed cross
sections of 8Be$(n,\gamma)^{9}$Be in particular the peak just above the 8Be +
$n$ threshold Gibbons59 ; John62 ; Fujishiro82 ; Utsunomiya00 ; Burda10 ;
Arnold12 . The main difference among the cross sections comes from the
contribution from the first excited $1/2^{+}$ state of 9Be via the $E1$
transition. This low-lying $1/2^{+}$ state can have an impact on the reaction
rate of 8Be$(n,\gamma)^{9}$Be in stellar environments and supernova explosions
Terasawa01 ; Sasaqui05 . For theoretical side, it is interesting problem to
answer how the low-lying $1/2^{+}$ state of 9Be contributes to the
8Be$(n,\gamma)^{9}$Be reaction.

Theoretically, the structure and the reaction mechanism of 9Be have been
studied by using the $\alpha$ \+ $\alpha$ \+ $n$ three-body models Efros99 ;
Arai03 ; Garrido10 ; A_Rodriguez10 ; Garrido11 ; Casal14 . In previous works,
the structure of the first excited $1/2^{+}$ state has been discussed. Efros
et al. Efros99  discussed the structure of $1/2^{+}$ state from the scattering
length for the 8Be + $n$ scattering. They show that the scattering length is
obtained as a large negative value, which characterizes the first excited
$1/2^{+}$ state as a virtual state. Also, Arai et al. Arai03  show that the
virtual-state character of the $1/2^{+}$ state from the $R$-matrix analysis.
On the other hand, in Refs. Garrido10 ; A_Rodriguez10 , the first excited
$1/2^{+}$ state is discussed as a three-body resonance of $\alpha$ \+ $\alpha$
\+ $n$. In these calculations, they show the importance of a strong mixture of
the 5He + $\alpha$ configuration at the internal region of the $1/2^{+}$
state. There exists the contradictions among the theories and the structure of
the first $1/2^{+}$ is still unclear. Additionally, in Ref. Garrido11 ;
Casal14 , the importance of the direct three-body capture for 9Be below the
8Be + $n$ threshold is suggested recently. It is important to investigate the
relation between the structure of the first excited $1/2^{+}$ state and the
mechanism of the capture reaction. Comprehensive understanding for the
structure of $1/2^{+}$ of 9Be is now required.

The purpose of this work is to understand the structure of the low-lying
$1/2^{+}$ state of 9Be and its contribution to the 9Be$(\gamma,n)^{8}$Be
reaction. We describe 9Be by using the $\alpha$ \+ $\alpha$ \+ $n$ three-body
model. To treat the photodisintegration into unbound excited states of 9Be, we
apply the complex scaling method (CSM) Aguilar71 ; Balslev71 ; Ho83 ;
Moiseyev98 ; Aoyama06 ; Myo14 ; Odsuren14  to the $\alpha$ \+ $\alpha$ \+ $n$
three-body model. The CSM is a powerful tool to investigate many-body
resonances and has been extensively used to discuss their structures Myo11b ;
Myo12 ; Myo14 . It has been shown that the CSM enables us to describe the
resonances from two- to five-body system successfully Aoyama06 ; Myo11b ;
Myo12 ; Myo14 ; Odsuren14 . Furthermore, the CSM has been developed to
describe the breakups of unstable nuclei. The CSM has been applied to the
Coulomb breakup reactions of two-neutron halo nuclei into core + $n$ \+ $n$
scattering states and well reproduces the observed cross sections Myo01 ;
Myo08 ; Kikuchi13a . From the above, the CSM is promising to discuss the
excited states of 9Be and the photodisintegration into unbound excited states
of 9Be on the same footing.

Generally, the resonances are obtained as the pole of $S$-matrix with complex
energy eigenvalues in the fourth quadrant of the complex energy plane. On the
other hand, the virtual state is obtained on the negative energy axis of the
second Riemann sheet. In the CSM, we can obtain the solution of the resonances
but not of the virtual state. Such a pole of the virtual state has been
discussed by using an analytical continuation method within the two-body
system Arai03 . The properties of the virtual state of a three-body system
such as 9Be is still unclear. In this work, we investigate not only the
eigenvalue distribution but also the photodisintegration cross section in
order to clarify the structure of the $1/2^{+}$ state whether the resonance or
the virtual state.

Formalism. We solve the Schrödingier equation for the $\alpha$ \+ $\alpha$ \+
$n$ system using the complex-scaled orthogonality condition model Saito77 .
The complex-scaled Schrd̈ingier equation is given as

$\hat{H}^{\theta}\Psi^{\nu}_{J}(\theta)=E^{\theta}_{\nu}\Psi^{\nu}_{J}(\theta),$
(1)

where $J$ is the total spin of the $\alpha$ \+ $\alpha$ \+ $n$ system and
$\nu$ is the state index. The complex-scaled Hamiltonian and wave function are
given as

$\hat{H}^{\theta}=U(\theta)\hat{H}U^{-1}(\theta)\ \ \ \text{and}\ \ \
\Psi^{\nu}_{J}(\theta)=U(\theta)\Psi^{\nu}_{J},$ (2)

respectively. The complex scaling operator $U(\theta)$ transforms the relative
coordinate $\boldsymbol{\xi}$ as

$U(\theta):\boldsymbol{\xi}\rightarrow\boldsymbol{\xi}e^{i\theta},$ (3)

where $\theta$ is the scaling angle being a positive real number.

The Hamiltonian for the relative motion of the $\alpha$ \+ $\alpha$ \+ $n$
three-body system for 9Be is given as

$\hat{H}=\sum^{3}_{i=1}t_{i}-T_{\text{c.m.}}+\sum^{2}_{i=1}V_{\alpha
n}(\boldsymbol{\xi}_{i})+V_{\alpha\alpha}+V_{\text{PF}}+V_{3},$ (4)

where $t_{i}$ and $T_{\text{c.m.}}$ are kinetic operators for each particle
and the center-of-mass of the system, respectively. The interactions between
the neutron and the $i$-th $\alpha$ particle is given as $V_{\alpha
n}(\boldsymbol{\xi}_{i})$, where $\boldsymbol{\xi}_{i}$ is the relative
coordinate between them. We here employ the KKNN potential Kanada79  for
$V_{\alpha n}$. For the $\alpha$-$\alpha$ interaction $V_{\alpha\alpha}$ we
employ a folding potential of the effective $NN$ interaction Schmid61  and the
Coulomb interaction:

$V_{\alpha\alpha}(r)=v_{0}\exp{\left(-ar^{2}\right)}+\frac{4e^{2}}{r}\text{erf}\left(\beta
r\right),$ (5)

where $v_{0}=-106.09$ MeV, $a=0.2009$ fm-2, and $\beta=0.5972$ fm-1. The
pseudo potential $V_{\text{PF}}=\lambda|\Phi_{PF}\rangle\langle\Phi_{PF}|$ is
the projection operator to remove the Pauli forbidden states from the relative
motions of $\alpha$-$\alpha$ and $\alpha$-$n$ Kukulin86 . The Pauli forbidden
state is defined as the harmonic oscillator wave functions by assuming the
$(0s)^{4}$ configuration whose oscillator length is fixed to reproduce the
observed charge radius of the $\alpha$ particle. In the present calculation,
$\lambda$ is taken as $10^{6}$ MeV.

To discuss the photodisintegration of 9Be, it is important to reproduce the
breakup threshold into the $\alpha$ \+ $\alpha$ \+ $n$. In the present
calculation, we introduces the $\alpha$ \+ $\alpha$ \+ $n$ three-body
potential $V_{3}$ to reproduce the binding energy of the 9Be ground state,
$E_{\text{gs}}$, measured from the $\alpha$ \+ $\alpha$ \+ $n$ threshold. The
explicit form of $V_{3}$ is given as

$V_{3}=v_{3}\exp{(-\mu\rho^{2})},$ (6)

where $\rho$ is the hyper-radius of the $\alpha$ \+ $\alpha$ \+ $n$ system.

We solve the eigenvalue problem given in Eq. (1) with Gaussian expansion
method hiyama03 , and obtain the energy eigenvalues and eigenstates (their
biorthogonal states) as $\\{E^{\theta}_{\nu}\\}$ and
$\\{\Psi^{\nu}_{J}(\theta)\\}$ ($\\{\tilde{\Psi}^{\nu}_{J}(\theta)\\}$),
respectively. Using them, we define the complex-scaled Green’s function
$\mathcal{G}^{\theta}(E;\boldsymbol{\xi},\boldsymbol{\xi}^{\prime})$ as

$\mathcal{G}^{\theta}(E;\boldsymbol{\xi},\boldsymbol{\xi}^{\prime})=\left\langle\boldsymbol{\xi}\left|\frac{1}{E-H^{\theta}}\right|\boldsymbol{\xi}^{\prime}\right\rangle=\sum_{\nu}\int\frac{\Psi^{\nu}(\theta)\tilde{\Psi}^{\nu}(\theta)}{E-E^{\theta}_{\nu}}.$
(7)

In the derivation of the right-hand side of Eq. (7), we use the extended
completeness relation, whose detailed explanation is given in Ref. Myo97 . It
is noticed that we take into account outgoing boundary conditions for all open
channels of a three-body system in the form of complex energy eigenvalues
$E^{\theta}_{\nu}$. The complex-scaled Green’s function in Eq. (7) enables us
to describe the scattering observables for many-body systems, such as the
photodisintegration cross section.

We calculate the cross section of the photodisintegration of
${}^{9}\textrm{Be}(3/2^{-})~{}+~{}\gamma\rightarrow\alpha~{}+~{}\alpha~{}+~{}n$
in terms of the multipole response. The cross section is expressed as the
following form;

$\sigma_{E\lambda}^{\gamma}(E_{\gamma})=\frac{(2\pi)^{2}(\lambda+1)}{\lambda[(2\lambda+1)!!]^{2}}\left(\frac{E_{\gamma}}{\hbar
c}\right)^{2\lambda-1}\frac{dB(E\lambda,E_{\gamma})}{dE_{\gamma}},$ (8)

where $E_{\gamma}$ is the incident photon energy and $B(E\lambda)$ is electric
transition strength with the rank $\lambda$. We here calculate the
photodisintegration cross section from the ground $3/2^{-}$ state to $1/2^{+}$
states in 9Be and consider only the $E1$ transition here. Using the CSM and
the complex-scaled Green’s function in Eq. (7), the $E1$ transition strength
is given as

$\begin{split}\frac{dB(E1,E_{\gamma})}{dE_{\gamma}}=&\frac{1}{2J_{\text{gs}}+1}\sum_{\nu}\int\left\langle\tilde{\Psi}_{\text{gs}}||(\hat{O}^{\theta})^{\dagger}(E1)||\Psi^{\nu}_{1/2^{+}}(\theta)\right\rangle\\\
&\times\frac{1}{E-E^{\theta}_{\nu}}\left\langle\tilde{\Psi}^{\nu}_{1/2^{+}}(\theta)||\hat{O}^{\theta}(E1)||\Psi_{\text{gs}}\right\rangle,\end{split}$
(9)

where $J_{\text{gs}}$ and $\Psi_{\text{gs}}$ represent the total spin and the
wave function of the ground state, respectively. The energy $E$ is related to
$E_{\gamma}$ as $E=E_{\gamma}-E_{\text{gs}}$. From the Eqs. (8) and (9), we
finally obtain the photodisintegration cross section as

$\sigma_{E1}^{\gamma}(E_{\gamma})=\frac{16\pi^{3}}{9\hbar
c}E_{\gamma}\frac{dB(E1,E_{\gamma})}{dE_{\gamma}}.$ (10)

Results. We first show the calculated ground-state properties of 9Be. In TABLE
1, the calculated binding energy and charge and matter radii are listed.
Without the three-body potential in Eq. (6), the binding energy of 9Be ground
state is overbound and the charge radius is slightly small compared to
experiments. To reproduce these quantities, we need the repulsive three-body
potential whose parameters are given as $v_{3}=1.10$ MeV and $\mu=0.02$ fm-2.
As results, we reproduce the binding energy and charge radius of 9Be ground
state simultaneously, while the matter radius is slightly larger than the
observed one. We also confirm that no resonance of the $1/2^{+}$ state is
found with the three-body potential in TABLE 1 for the $\theta=15$ degrees
case.

Table 1:  Ground-state properties in comparison with experiments. The calculated binding energies ($E_{\text{gs}}$, unit in MeV), charge radii ($R_{\text{ch}}$, unit in fm), and matter ones ($R_{\text{m}}$, unit in fm) with and without the three-body potential $V_{3}$ are listed. | $E_{\text{gs}}$ | $R_{\text{ch}}$ | $R_{\text{m}}$ | $v_{3}$ | $\mu$
---|---|---|---|---|---
with $V_{3}$ | 1.57 | 2.53 | 2.42 | 1.10 | 0.02
without $V_{3}$ | 2.14 | 2.50 | 2.39 | - | -
Exp. | 1.5736111Reference Tilley04  | 2.519$\pm$0.012222Reference Nortershauser09  | 2.38$\pm$0.01333Reference Tanihata88  |  |

Next we discuss the photodisintegration cross section to $1/2^{+}$ states. In
the present calculation, we fix the ground-state wave function obtained with
the three-body potential in TABLE 1. In FIG. 1, we show the calculated cross
sections using Eq. (10) in comparison with the two sets of the observed data
which commonly have peaks just above the 8Be + $n$ threshold. The red (dashed)
and blue (dotted) lines show the cross sections with and without the three-
body potential for excited $1/2^{+}$ states, respectively, whose parameters
are same as those in TABLE 1. In both results, the calculated cross sections
underestimate the low-lying peak above the 8Be + $n$ threshold.

Figure 1:  Calculated photodisintegration cross sections in comparison with
the experimental data. The red (dashed) and blue (dotted) lines are results
with and without the three-body potential, where $v_{3}=1.10$ MeV and
$\mu=0.02$ fm. The black (solid) line represents the cross section calculated
by using an attractive three-body potential with $v_{3}=-1.02$ MeV. The
experimental data below $E_{\gamma}=2.2$ MeV are taken from Refs. Utsunomiya00
and Arnold12 . The arrow indicates the threshold energy of the 8Be(0+) + $n$
channel.

To discuss the observed sharp peak just above the 8Be + $n$ threshold in the
photodisintegration cross section, we change the strength, $v_{3}$, for the
$1/2^{+}$ state to fit the observed data but its range is fixed as the same as
used in the ground state. We here take the strength as $v_{3}=-1.02$ MeV for
the $1/2^{+}$ state and obtain the cross section as shown as the black (solid)
line in FIG. 1. Our result reproduces the observed peak by using the
attractive three-body potential. We confirm that the calculated cross section
rapidly increases just above the 8Be + $n$ threshold and there is negligibly
small strength below this threshold. We also find that the calculated cross
sections show the strong dependence on the strengths of the three-body
potentials as shown in FIG. 1. This result is interesting and suggests the
existence of the three-body unbound state of 9Be($1/2^{+}$), such as a
resonance or virtual state. In relation to the cross section, we discuss the
character of the $1/2^{+}$ state.

Figure 2:  Distribution of energy eigenvalues of $1/2^{+}$ states measured
from the $\alpha$ \+ $\alpha$ \+ $n$ threshold with scaling angle $\theta=15$
degrees. We here employ the three-body potential with $v_{3}=-1.02$ MeV and
$\mu=0.02$ fm-2. The red (solid), blue (dashed), and green (dotted) lines
represent the branch cuts for $\alpha$ \+ $\alpha$ \+ $n$, 8Be(0+) + $n$, and
5He(3/2-) + $\alpha$ continua, respectively.

We investigate the origin of the low-lying peak above the 8Be + $n$ threshold
more in detial. For this purpose, we show the distribution of the energy
eigenvalues of the $1/2^{+}$ states by using the CSM. In the CSM, continuum
states are obtained along the branch cuts which start from the threshold
energies and are rotated down by 2$\theta$. A resonance is obtained as a
solution with complex energy of $E^{\theta}=E_{r}-i\Gamma/2$ isolated from the
continuum ones . On the other hand, the virtual states and broad resonances,
which are located on the second Riemann sheet covered by the rotated first
Riemann sheet, cannot be obtained as the isolated pole in CSM. The
contributions from these states to the cross section are scattered into the
continuum states rotated on the $2\theta$ lines. In FIG. 2, we show the
distribution of the energy eigenvalues for the $1/2^{+}$ states calculated
with $v_{3}=-1.02$ MeV, which reproduces the observed peak as shown in FIG. 1.
In the present calculation, we find no resonances in the energy eigenvalue
distribution. All energy eigenvalues are located on the 2$\theta$-lines,
corresponding to the branch cuts for the $\alpha$ \+ $\alpha$ \+ $n$, 8Be(0+)
+ $n$, and 5He(3/2-) + $\alpha$ continuum states.

We investigate the contributions of two- and three-body continuum states to
the cross section to understand the mechanism of the photodisintegration.
Using the CSM, we decompose the cross section into 8Be + $n$ and $\alpha$ \+
$\alpha$ +$n$ components as shown in FIG. 3, to clarify the dominant
contribution in the cross section Myo01 . The 5He + $\alpha$ contribution is
found to be negligible in the low-lying region, and we do not show it in FIG.
3. From the results, we see that the 8Be + $n$ component is almost identical
to the total cross section. This fact indicates that the 8Be + $n$ decay is
dominant in the photodintegration. This decay mode should be related to the
structure of the $1/2^{+}$ state of 9Be.

Figure 3:  Decomposed photodisintegration cross sections. The red (solid) and
blue (dashed) lines are contributions of the $\alpha$ \+ $\alpha$ \+ $n$ and
8Be + $n$ continuum states. The black thin line is same as that in FIG. 1.
Figure 4:  Pole trajectory of the 9Be $1/2^{+}$ state in complex energy plane
by changing the three-body potential. The closed circles represent the poles
obtained as isolated three-body resonances in CSM. The open circles and the
crosses are speculated pole positions for the virtual states and broad
resonances, respectively.

To investigate the structure of the $1/2^{+}$ state of 9Be, we calculate the
energy eigenvalues of the $\alpha$ \+ $\alpha$ \+ $n$ system by changing the
strength of the three-body potential, $v_{3}$, which is shown in FIG. 4. In
the present calculation, when the strength of the three-body potential
$v_{3}=-1.3$ MeV, the resonance pole suddenly appears just below the 8Be(0+) +
$n$ threshold. This resonance pole with a narrow decay width moves smoothly to
the bound state region as the three-body potential becomes more attractive,
and we finally obtain the 9Be bound state with the region of $v_{3}<-1.8$ MeV.
On the other hand, we consider the pole trajectory in the opposite case of the
three-body potential with $v_{3}>-1.3$ MeV. If the resonance exists, the pole
with a narrow decay width should appear above the 8Be + $n$ threshold as the
analytical continuation from the resonance pole as shown with the crosses in
FIG. 4. However, we found that no resonances appear above the 8Be(0+) + $n$
threshold for $v_{3}>-1.3$ MeV of the three-body potential. These facts in the
pole trajectory show the possibility of the virtual state of the $1/2^{+}$
state consisting of 8Be($0^{+}$) + $n$ when we take $v_{3}=-1.02$ MeV, which
reproduces the experimental cross section. The existence of the virtual state
is consistent with the dominant decay into 8Be + $n$ in the
photodisintegration of 9Be.

The virtual state is often discussed in unstable nuclei, such as 10Li and 11Li
Masui00 ; Myo08 ; Kikuchi13a . For 10Li, Masui et al. Masui00  showed the
explicit pole trajectory of the virtual state of 10Li as 9Li + $n$ picture,
which is similar to the present 8Be($0^{+}$) + $n$ situation. In FIG 4, we
schematically plot the pole trajectory in the region of $v_{3}>-1.3$ MeV,
considering the analogy with the 10Li case as shown in FIG. 3(b) of Masui00 .
This trajectory is located on the second Riemann sheet of the 8Be + $n$ system
which cannot be obtained in the CSM.

We consider the different case when there exists the $1/2^{+}$ resonance to
see how the resonance contributes to the photodisintegaration cross section.
By using the three-body potential with $v_{3}=-1.3$ MeV, the $1/2^{+}$
resonance is obtained with the energy and decay width being 0.091 and 0.002
MeV, respectively, just below the 8Be + $n$ threshold at 0.0918 MeV measured
from the $\alpha$ \+ $\alpha$ \+ $n$ threshold. In FIG. 5, we show the cross
section in this case, which shows quite a sharp peak at the resonance energy.
From the decomposition of the cross section, we find that the peak in the
cross section is dominated by the resonance while the 8Be + $n$ continuum
states have a sizable contribution to the cross section. This trend is much
different from that in FIG. 3. In the result of FIG. 3, the peak in the cross
section is constructed by several low-lying eigenstates located on the 8Be +
$n$ continuum $2\theta$ line in FIG. 2.

Figure 5:  Decomposed photodisintegration cross sections by using the three-
body potential with $v_{3}=-1.3$. The red (solid), blue (dashed) lines are
contributions of the $\alpha$ \+ $\alpha$ \+ $n$ and 8Be + $n$ continuum
states. The green (dotted) line is that of the resonance pole. The black thin
line represents the sum of them. The arrow indicates the threshold energy of
the 8Be(0+) + $n$ channel.

Summary. We investigate the character of the $1/2^{+}$ state of 9Be using the
photodisintegration reaction with the $\alpha$ \+ $\alpha$ \+ $n$ three-body
model and the CSM. The calculated photodisintegration cross sections into the
$1/2^{+}$ states are shown to have a strong dependence on the strength of the
three-body potential for the $1/2^{+}$ state. The experimental cross section
shows the sharp peak just above the 8Be + $n$ threshold, which is nicely
reproduced with the attractive three-body potential. We cannot find any
resonance poles for the $1/2^{+}$ states in explaining the peak in the cross
section. From the decomposition of the calculated cross section, it is shown
that the 8Be + $n$ continuum states dominates the cross section to the
$1/2^{+}$ states. These results indicates the possibility of the virtual-state
nature of the first excited $1/2^{+}$ state. In addition, the pole trajectory
suggests that the pole of the $1/2^{+}$ state is located on the second Riemann
sheet of 8Be + $n$ instead of the broad resonances of $\alpha$ \+ $\alpha$ \+
$n$.

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