# Incomplete Multi-view Clustering via Cross-view Relation Transfer

Yiming Wang, Dongxia Chang, Zhiqiang Fu, and Yao Zhao

###### Abstract

In this paper, we consider the problem of multi-view clustering on incomplete
views. Compared with complete multi-view clustering, the view-missing problem
increases the difficulty of learning common representations from different
views. To address the challenge, we propose a novel incomplete multi-view
clustering framework, which incorporates cross-view relation transfer and
multi-view fusion learning. Specifically, based on the consistency existing in
multi-view data, we devise a cross-view relation transfer-based completion
module, which transfers known similar inter-instance relationships to the
missing view and recovers the missing data via graph networks based on the
transferred relationship graph. Then the view-specific encoders are designed
to extract the recovered multi-view data, and an attention-based fusion layer
is introduced to obtain the common representation. Moreover, to reduce the
impact of the error caused by the inconsistency between views and obtain a
better clustering structure, a joint clustering layer is introduced to
optimize recovery and clustering simultaneously. Extensive experiments
conducted on several real datasets demonstrate the effectiveness of the
proposed method.

Figure 1: The framework of the proposed CRTC. In the figure, two-view data is
used as a showcase. As shown, it consists of two main components: cross-view
relation transfer completion and multi-view fusion clustering. Specifically,
the cross-view relation transfer completion module recovers missing data by
transferring similar inter-instances relationships of existing views. And the
multi-view fusion clustering network adaptively fuses the features of each
view to obtain a consistent representation suitable for clustering.

## Introduction

With the advance of data acquisition devices, real data often appears in
multiple modalities or comes from multiple sources (Blum and Mitchell 1998).
Multi-view Clustering (MVC), as one of the most important unsupervised multi-
view learning tasks, has attracted lots of attention. The goal of MVC is to
improve clustering performance by integrating features from multiply views.
Towards this goal, a variety of related methods have been proposed (Yu et al.
2012; Li et al. 2015; Liu et al. 2019). Moreover, MVC is widely applied in
various applications, such as object segmentation, information retrieval, and
saliency detection. A common assumption for most MVC methods is that all of
the views are complete. However, in real-world applications, some views often
suffer from data missing, which makes most multi-view clustering methods
inevitably degenerate. So how to reduce the impact of missing views and learn
a consistent representation are challenging problems of incomplete multi-view
learning.

To solve the above challenging problem, many incomplete multi-view clustering
(IMC) methods have been proposed in the last few decades. These methods can be
divided into three main categories: Non-negative Matrix Factorization (NMF)
based methods, kernel based methods, and Deep Neural Networks (DNNs) based
methods. NMF based methods (Zhao, Ding, and Fu 2017; Hu and Chen 2018) learn a
common latent space for complete instances and private latent representations
for incomplete instances. For example, Li et al. (Li, Jiang, and Zhou 2014a)
first proposed NMF based partial multi-View clustering method to learn a
latent subspace over two views, which achieves notable clustering performance.
Motivated by these methods, a lot of partial multi-view clustering approaches
(Liu et al. 2015) extend NMF to data with more views. Kernel based methods
(Shao, Shi, and Yu 2013; Liu et al. 2019) leverage kernel matrices of the
complete views for completing the kernel matrix of incomplete view. For
example, MVL-IV (Xu, Tao, and Xu 2015) proposes an algorithm to accomplish
multi-view learning with incomplete views by exploiting the connections of
multiple views. These two types of methods show satisfactory performance, but
suffer from two main disadvantages: (a) It is inefficient to employ them for
large-scale datasets. (b) They need high time consumption. The third kind of
methods is based on DNNs. Recently, DNNs have shown amazing power in handling
multi-view data. Many recent studies (Wen et al. 2020a; Zhang et al. 2020)
employ DNNs to solve IMC challenges, showing noticeable improvement on
clustering performance. These DNNs based methods typically employ Autoencoders
(AEs) to learn the common representations for clustering and employ Generative
Adversarial Networks (GANs) to generate missing data. However, GAN-based
methods have difficulty avoiding the problem of training instability.
Furthermore, few methods consider cross-view consistency.

Different from existing deep incomplete multi-view clustering approaches, we
focus on recovering missing data based on the similar relationships between
instances in the existing views. Considering consistency of multiple views,
similarity relationships between instances in the existing views have
availability in the missing view. On this basis, we design a novel completion
model based on cross-view relation transfer and proposed a joint incomplete
multi-view clustering framework named Cross-view Relation Transfer Clustering
(CRTC). The main contributions of our method are summarized as follows:

  * •

We devised a cross-view relation transfer completion module based on the
consistency existing in multi-view data. To the best of our knowledge, our
method is the first approach to recover missing data by transferring
similarity relationships between instances in the existing views.

  * •

Inspired by transformer technology, we introduce an attention-based fusion
layer to obtain the common representation, which can efficiently reduce the
negative influence of missing views.

  * •

Missing data completion and multi-view clustering are incorporated to obtain
the final clustering assignments. Moreover, extensive experiments show that
CRTC makes a considerable improvement over the state-of-the-art methods.

## Related Work

Deep Incomplete Multi-view Clustering. With the development of deep learning,
a number of recent studies (Zhang et al. 2020; Lin et al. 2021) employ DNNs to
solve incomplete multi-view clustering challenges, showing noticeable
improvement on clustering performance. Based on producing missing data or not,
most existing deep incomplete multi-view clustering methods could be roughly
classified into two categories, _i.e._ , learning a uniform representation
without generating missing data (Wen et al. 2020b, a; Wang et al. 2021b) and
generating missing instances using adversarial learning (Wang et al. 2021a,
2018; Zhang et al. 2020; Xu et al. 2021). The first class of methods reduces
the negative influence of missing views by modifying the multi-view fusion
representation learning model. For example, CDIMC-net (Wen et al. 2020a)
introduces a self-paced strategy to select the most confident samples for
model training to reduce the negative influence of outliers. Moreover, DIMC-
net (Wen et al. 2020b) further develops a weighted fusion layer to obtain the
common representation shared by all views. Instead, iCmSC (Wang et al. 2021b)
employs subspace learning to solve IMC, and maximize the intrinsic
correlations among different modalities by deep canonical correlation analysis
(DCCA) (Andrew et al. 2013). Adversarial learning-based methods for missing
completions typically use discriminators to optimize the generation process of
missing instances. For example, CPM-Nets (Zhang et al. 2020) uses encoding
networks to generate missing data and introduces an adversarial strategy to
enforce the generated data for missing ones in each view to obey distribution
of the observed data. GP-MVC (Wang et al. 2021a) introduces view-specific GAN
with multi-view cycle consistency to generate the missing data of one view
conditioning on the shared representation given by other views. Unlike these
two types of methods, COMPLETER (Lin et al. 2021) generates the
representations of the missing views by minimizing the conditional entropy of
different views by dual prediction. Different from the current methods, we
focus on generating missing data based on the similarity relationships of
corresponding instances in the existing views.

Graph Neural Networks. To mine non-Euclidean graph information, researchers
have devised graph neural networks (GNNs) (Kipf and Welling 2017; Velickovic
et al. 2018) capable of encoding both graph structure and node characteristics
for node latent representation. GNNs have successfully expanded deep learning
techniques to non-Euclidean graph data with remarkable achievement made in
multiple graph tasks, such as graph classification (Li et al. 2020) and visual
question answering (Li et al. 2019). Additionally, thanks to the properties of
graph convolution, many GNN-based graph clustering methods (Wang et al. 2019;
Fan et al. 2020) have been proposed. These methods typically construct
adjacency matrices based on the K-nearest-neighbor (KNN) graph (Bo et al.
2020) or attributed graph (Cheng et al. 2020), and learn graph embedding using
graph autoencoder (GAE) (Kipf and Welling 2016). In our model, GNN is not
directly used to learn common representations but to generate missing
instances through inter-instances relationships.

## The proposed CRTC

### Motivation and Notations

As aforementioned analysis, existing incomplete multi-view clustering methods
can be roughly grouped into three categories, _i.e._ , NMF based methods,
kernel based methods, and DNN based methods. Among them, methods of inferring
missing data by NMF and kernel may suffer problems such as high time
consumption. Moreover, most DNN based methods recover missing data using GAN,
which may face training problems such as difficult convergence and gradient
explosion. Meanwhile, similar inter-instance relationships in the existing
view are hardly utilized in current multi-view clustering methods. From this
point, we propose Cross-view Relation Transfer Clustering (CRTC), which
performs multi-view recovery and clustering simultaneously. As illustrated in
Fig. 1, CRTC consists of two main components: cross-view relation transfer
completion and multi-view fusion clustering, which are jointly trained to
recover missing data and obtain clustering assignments.

In the incomplete multi-view clustering, instances are characterized by
multiple views and some views may be destroyed. Formally, given a multi-view
incomplete dataset $\mathcal{X}=\\{X^{(1)},...,X^{(V)}\\}$ consisting of $N$
instances of $V$ views. And
$X^{(v)}=\\{x^{(v)}_{1},...,x^{(v)}_{N}\\}\in\mathbb{R}^{{D_{v}}\times N}$
denotes the feature matrix in the $v$-th view, where $D_{v}$ is the feature
dimension of $v$-th view. For convenience, matrix $M\in\mathbb{R}^{N\times V}$
is used to record the view available and missing information, where
$M_{i,v}=0$ means the $i$-th instance is available in the $v$-th view,
otherwise $M_{i,v}=1$. And the goal of CRTC is to group all the $N$ instances
into $C$ clustering.

### Cross-view Relation Transfer Completion

In the task of incomplete multi-view clustering, recovering missing data has
been a challenge and has a large impact on the clustering performance.
Existing DNN based methods typically obtain the missing data using GAN, which
fail to consider inter-instance relationship in the existing views and may
suffer from training difficulties. To this end, we devise a cross-view
relation transfer completion module, which transfers the known similar inter-
instance relationships in the existing views to recover missing views.

Figure 2: Illustration of cross-view relation transfer completion module

Here we use the two-view data shown in Fig. 2 to show the process of the
cross-view relation transfer completion. For the missing data $x^{(1)}_{a}$ in
the first view, the corresponding existing instance in the second view can be
denoted as $x^{(2)}_{a}$. Since IMC is an unsupervised task, to obtain the
existing instances related to $x^{(2)}_{a}$ in the second view, KNN graph is
constructed based on the distance metric. And the $K$ neighbors of
$x^{(2)}_{a}$ can be denoted by
$K^{(2)}_{a}=\\{x^{(2)}_{a^{1}},...,x^{(2)}_{a^{K}}\\}$. Considering the
consistency of multiple views, similarity relationships between instances in
existing views are valid for the missing views. Therefore, we can transfer the
KNN graph to the first view to find similar instances to the missing data
$x^{(1)}_{a}$. Since some of instances in $K^{(2)}_{a}$ are missing in the
first view, we remove the missing instances and obtain the transferred KNN
graph $K^{(1)}_{a}=\\{x^{(1)}_{a^{1}},...,x^{(1)}_{a^{K^{\prime}}}\\}$, where
$K^{\prime}\leq K$. In this way, we can obtain the transferred KNN graph of
all missing data. For datasets with more than two views, there may be more
known instances for missing data $x^{(1)}_{a}$, which leads to more
transferred KNN graphs. We combine these graphs into a single graph to infer
the missing data.

To recover the missing data, we apply a graph convolutional network (GCN) to
infer the missing data based on the transferred relations. For convenience,
the recovery of $x^{(1)}_{a}$ is presented as an example to introduce the
completion process, and the input of GCN is the KNN graph $K^{(1)}_{a}$
consisting of features and edges. Then the recovered feature can be obtained
by:

$\hat{x}_{a}^{(1)}=\sigma(b+\sum_{j\in
K^{(1)}_{a}}\frac{1}{|K^{(1)}_{a}|}wx_{a_{j}}^{(1)})$ (1)

where $\sigma(\cdot)$ is an activation function, $b$ is the bias, $w$ is the
parameter matrix, and $x_{a_{j}}^{(1)}$ denotes the features of instances in
$K^{(1)}_{a}$. Since the feature of $x^{(1)}_{a}$ is ‘NaN’, we remove the
self-loop, which is a typical setting in GNN.

Then, in order to recover all missing data according to corresponding
transferred relations and existing instances, we minimize the reconstruction
loss as

$L_{cr}=\sum_{i=1}^{N}\sum_{v=1}^{V}\sum_{j\in
K^{(i)}_{a}}||\hat{x}_{i}^{(v)}-x_{i_{j}}^{(v)}||^{2}M_{i,v}$ (2)

### Multi-view Fusion Clustering

To obtain the final clustering assignments, we develop a multi-view fusion
clustering network, which mainly contains three parts: view-specific encoding
and decoding, an attention fusion layer, and a clustering layer. In fact, in
addition to the consistency existing in multi-view data, each view is also
complementary and contains distinctive features that other views do not have.
Therefore, to capture the view-specific features of different views, we use
view-specific encoders to embed original heterogeneous data to the same size
features. And the $i$-th embedding features of $v$-th view can be obtained as

$h_{i}^{(v)}=g^{(v)}(x_{i}^{(v)})$ (3)

where $g^{(v)}$ denotes the encoder for the $v$-th view. In order to flexibly
integrate the view-specific features and mitigate the effects of missing data,
we devise an attention fusion layer to obtain a common representation based on
the learned view-specific features. Specifically, for a view-specific features
$h^{(v)}_{i}$, we calculate the attention distribution $\alpha^{(v)}_{i}$ as

$\displaystyle\alpha_{i}^{(v)}=$ $\displaystyle{\rm
softmax}(s(w_{i}^{(v)}h_{i}^{(v)},h_{i}^{+}))$ (4) $\displaystyle=$
$\displaystyle\frac{exp(s(w_{i}^{(v)}h_{i}^{(v)},h_{i}^{+}))}{\sum_{j=1}^{V}exp(s(w_{i}^{(j)}h_{i}^{(j)},h_{i}^{+}))}$

where $w_{i}^{(v)}$ is the parameter matrix of $v$-th view for the $i$-th
instance, $s(\cdot)$ is a similarity function, and $h_{i}^{+}$ is given by

$h_{i}^{+}=\frac{1}{V}\sum_{v=1}^{V}h_{i}^{(v)}$ (5)

Then, common representation $\tilde{h}_{i}$ is obtained by the weighted
average of features with attention distribution $\alpha_{i}^{(v)}$:

$\tilde{h}_{i}=\sum_{v=1}^{V}\alpha_{i}^{(v)}h_{i}^{(v)}$ (6)

After obtaining the common representation $\\{\tilde{h}_{i}\\}_{i=1}^{N}$,
view-specific decoders $d^{(v)}$ is employed to reconstruct the original
multi-view data. And the traditional element-wise mean squared error shown in
Eq.(7) is utilized to evaluate reconstruction in the pre-training.

$L_{mr}=\sum_{v=1}^{V}\sum_{i=1}^{N}||x_{i}^{(v)}-d^{(v)}(\tilde{h}_{i})||^{2}$
(7)

Algorithm 1 The CRTC algorithm

Input: A multi-view incomplete dataset
$\mathcal{X}=\\{X^{(1)},...,X^{(V)}\\}$, number of clusters: $C$, maximum
iterations: $MaxIter$, stopping threshold: $\delta$, target distribution
update interval: $T$.

Output: Clustering assignments $R$.

Initialization: Initialize parameters of the whole model, construct KNN graphs
$K_{a}$ for missing data.

Pre-train cross-view relation transfer completion module using Eq.(2).

Pre-train multi-view fusion clustering network using Eq.(7).

Obtain cluster centers $\\{\mu_{j}\\}_{j=1}^{C}$ using K-means.

Obtain target distribution $P$ according to Eq.(10).

for $Iter=0$ to $MaxIter$ do

if $Iter\%T=0$ then

Recover missing data according to Eq.(1).

Compute the common representation $\tilde{H}$.

Compute $P$ and $Q$ using Eq.(9) and Eq.(10).

Save last clustering assignments: $r_{old}=r$.

Compute clustering assignments $r$ via
$r_{i}=\mathop{\arg\min}\limits_{j}q_{ij}$.

if $sum(r_{old}\neq r)/N\leq\delta$ then

Stop training.

end if

end if

Update cross-view relation transfer completion module by Eq.(11).

Update multi-view fusion clustering network by Eq.(8).

end for

return Clustering assignments $R$.

Table 1: Clustering results on five datasets with different missing-view rates
or paired-view rates (Average$\pm$Standard deviations,$\%$). Note: the results
of PVC-GAN and GP-MVC shown in the table are from (Wang et al. 2021a).

| ACC (%) | NMI (%)
---|---|---
Datasets | Method\$p$ | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7
BDGP | BSV | 36.73$\pm$1.96 | 42.21$\pm$2.04 | 44.72$\pm$4.33 | 50.15$\pm$2.47 | 25.27$\pm$0.72 | 30.67$\pm$1.11 | 35.00$\pm$2.52 | 38.74$\pm$1.05
Concat | 40.66$\pm$1.60 | 45.74$\pm$1.53 | 48.46$\pm$2.23 | 55.41$\pm$2.61 | 28.57$\pm$1.18 | 33.58$\pm$1.58 | 36.50$\pm$1.97 | 43.28$\pm$2.48
PVC | 42.54$\pm$0.00 | 57.84$\pm$0.00 | 57.10$\pm$0.00 | 60.47$\pm$0.00 | 23.59$\pm$0.00 | 34.79$\pm$0.00 | 33.86$\pm$0.00 | 38.33$\pm$0.00
MIC | 34.22$\pm$1.66 | 41.23$\pm$2.62 | 51.55$\pm$1.90 | 58.78$\pm$4.57 | 13.33$\pm$2.00 | 20.31$\pm$2.56 | 30.76$\pm$0.50 | 35.82$\pm$3.48
IMG | 48.33$\pm$0.89 | 50.78$\pm$0.17 | 53.61$\pm$0.16 | 55.75$\pm$0.84 | 31.81$\pm$1.20 | 34.62$\pm$0.12 | 37.95$\pm$0.75 | 38.73$\pm$1.15
DAIMC | 50.34$\pm$6.38 | 57.22$\pm$5.22 | 62.92$\pm$5.20 | 67.49$\pm$1.14 | 25.05$\pm$4.86 | 34.30$\pm$3.67 | 44.43$\pm$2.16 | 50.21$\pm$0.89
PVC-GAN | 52.10$\pm$0.90 | 67.11$\pm$1.07 | 86.31$\pm$0.43 | 91.54$\pm$1.07 | - | - | - | -
GP-MVC | 58.74$\pm$2.49 | 78.68$\pm$2.34 | 88.79$\pm$1.28 | 93.19$\pm$0.82 | - | - | - | -
iCmSC | 53.04$\pm$2.18 | 67.90$\pm$2.09 | 80.27$\pm$0.74 | 89.63$\pm$0.98 | 39.27$\pm$1.38 | 44.63$\pm$2.64 | 61.28$\pm$1.32 | 64.92$\pm$0.95
CDIMC-net | 57.43$\pm$3.05 | 74.72$\pm$2.05 | 77.65$\pm$1.50 | 83.69$\pm$2.98 | 36.25$\pm$4.19 | 53.26$\pm$3.10 | 62.02$\pm$2.33 | 67.63$\pm$3.59
CRTC | 70.28$\pm$5.89 | 85.62$\pm$3.88 | 90.34$\pm$2.09 | 93.88$\pm$1.29 | 45.45$\pm$5.78 | 67.55$\pm$5.33 | 75.77$\pm$3.33 | 81.75$\pm$1.91
MNIST | BSV | 34.03$\pm$0.97 | 39.79$\pm$1.96 | 43.24$\pm$1.37 | 49.15$\pm$1.65 | 30.00$\pm$0.86 | 33.76$\pm$1.25 | 37.00$\pm$0.95 | 41.32$\pm$1.00
Concat | 40.00$\pm$0.78 | 41.07$\pm$1.06 | 43.99$\pm$2.27 | 48.17$\pm$1.89 | 34.40$\pm$0.53 | 35.63$\pm$0.81 | 37.03$\pm$0.91 | 41.49$\pm$1.08
PVC | 39.88$\pm$0.00 | 44.20$\pm$0.00 | 46.97$\pm$0.00 | 44.43$\pm$0.00 | 35.03$\pm$0.00 | 37.34$\pm$0.00 | 39.59$\pm$0.00 | 39.75$\pm$0.00
MIC | 36.31$\pm$4.39 | 30.83$\pm$2.84 | 36.78$\pm$1.57 | 38.24$\pm$2.50 | 26.08$\pm$3.91 | 25.75$\pm$2.43 | 31.00$\pm$1.50 | 33.40$\pm$1.90
IMG | 43.15$\pm$1.03 | 43.38$\pm$0.24 | 46.20$\pm$0.98 | 48.08$\pm$1.46 | 33.83$\pm$0.38 | 36.03$\pm$0.11 | 36.58$\pm$0.43 | 38.68$\pm$0.19
DAIMC | 41.39$\pm$3.87 | 48.75$\pm$2.69 | 50.75$\pm$1.32 | 51.14$\pm$1.50 | 33.53$\pm$2.92 | 40.99$\pm$2.20 | 43.42$\pm$0.95 | 44.14$\pm$1.21
PVC-GAN | 45.17$\pm$0.86 | 48.36$\pm$0.71 | 52.80$\pm$0.78 | 52.02$\pm$0.70 | - | - | - | -
GP-MVC | 56.46$\pm$2.47 | 55.42$\pm$3.30 | 57.76$\pm$1.69 | 59.63$\pm$0.88 | - | - | - | -
iCmSC | 46.97$\pm$0.26 | 51.11$\pm$0.58 | 56.05$\pm$1.12 | 58.65$\pm$0.80 | 48.66$\pm$1.35 | 53.27$\pm$0.40 | 54.21$\pm$1.20 | 55.48$\pm$1.79
CDIMC-net | 51.65$\pm$0.14 | 57.64$\pm$1.44 | 58.28$\pm$0.68 | 59.15$\pm$0.21 | 48.25$\pm$0.47 | 50.54$\pm$1.26 | 51.70$\pm$0.67 | 52.87$\pm$0.48
CRTC | 58.27$\pm$0.36 | 58.64$\pm$0.91 | 59.88$\pm$1.48 | 61.02$\pm$0.59 | 52.89$\pm$1.10 | 55.75$\pm$0.82 | 56.04$\pm$1.94 | 56.80$\pm$0.99
Caltech20 | BSV | 32.80$\pm$0.65 | 33.62$\pm$0.94 | 33.03$\pm$1.95 | 36.59$\pm$1.55 | 36.27$\pm$0.46 | 42.00$\pm$0.60 | 45.05$\pm$1.11 | 49.67$\pm$0.78
Concat | 33.65$\pm$1.97 | 31.74$\pm$1.50 | 31.96$\pm$2.27 | 36.33$\pm$2.12 | 38.34$\pm$0.61 | 42.13$\pm$1.11 | 46.60$\pm$1.09 | 51.23$\pm$1.48
PVC | 36.43$\pm$0.00 | 36.34$\pm$0.00 | 41.75$\pm$0.00 | 44.25$\pm$0.00 | 50.12$\pm$0.00 | 50.65$\pm$0.00 | 53.43$\pm$0.00 | 57.10$\pm$0.00
MIC | 30.91$\pm$2.32 | 31.86$\pm$2.15 | 32.14$\pm$2.01 | 37.66$\pm$2.31 | 40.70$\pm$1.35 | 43.93$\pm$1.64 | 46.27$\pm$1.15 | 51.90$\pm$0.60
IMG | 39.37$\pm$0.83 | 42.65$\pm$0.24 | 43.93$\pm$0.57 | 44.91$\pm$0.26 | 49.56$\pm$1.28 | 53.27$\pm$0.27 | 54.16$\pm$0.26 | 56.08$\pm$0.22
DAIMC | 31.97$\pm$3.18 | 41.14$\pm$2.91 | 45.08$\pm$2.18 | 46.53$\pm$2.34 | 46.04$\pm$1.59 | 50.00$\pm$2.23 | 52.68$\pm$1.26 | 54.68$\pm$1.31
iCmSC | 46.81$\pm$0.59 | 45.88$\pm$1.17 | 51.90$\pm$1.68 | 53.06$\pm$1.84 | 60.05$\pm$0.91 | 59.49$\pm$0.47 | 61.80$\pm$1.59 | 64.15$\pm$1.34
CDIMC-net | 46.38$\pm$5.29 | 49.63$\pm$3.77 | 50.46$\pm$1.73 | 51.23$\pm$2.56 | 59.65$\pm$0.40 | 62.21$\pm$0.28 | 64.73$\pm$0.82 | 65.15$\pm$0.86
CRTC | 61.96$\pm$2.75 | 63.83$\pm$2.04 | 65.33$\pm$3.15 | 66.71$\pm$4.09 | 62.11$\pm$3.65 | 63.90$\pm$2.26 | 64.87$\pm$3.01 | 66.10$\pm$3.26
Animal | BSV | 35.64$\pm$0.90 | 43.39$\pm$1.28 | 49.11$\pm$1.04 | 55.62$\pm$1.13 | 49.03$\pm$0.20 | 55.69$\pm$0.51 | 61.10$\pm$0.47 | 67.16$\pm$0.45
Concat | 38.69$\pm$1.14 | 43.71$\pm$1.24 | 49.05$\pm$1.67 | 54.65$\pm$1.65 | 52.61$\pm$0.52 | 56.50$\pm$0.42 | 60.49$\pm$0.58 | 64.85$\pm$0.50
MIC | 34.26$\pm$0.93 | 39.02$\pm$1.79 | 41.29$\pm$0.89 | 48.45$\pm$1.87 | 48.36$\pm$1.14 | 51.22$\pm$1.37 | 55.11$\pm$0.89 | 59.88$\pm$0.59
IMG | 48.91$\pm$0.11 | 54.52$\pm$0.23 | 56.81$\pm$0.19 | 59.67$\pm$0.08 | 54.16$\pm$0.08 | 58.58$\pm$0.08 | 61.40$\pm$0.10 | 65.27$\pm$0.13
DAIMC | 41.45$\pm$2.22 | 51.51$\pm$1.00 | 55.65$\pm$1.13 | 57.36$\pm$0.84 | 49.45$\pm$1.18 | 56.34$\pm$0.76 | 60.72$\pm$0.42 | 64.34$\pm$0.35
iCmSC | 37.40$\pm$1.58 | 40.27$\pm$1.01 | 44.31$\pm$4.00 | 47.71$\pm$1.20 | 52.82$\pm$1.01 | 54.74$\pm$1.51 | 55.73$\pm$0.70 | 59.45$\pm$0.93
CDIMC-net | 36.29$\pm$0.67 | 45.43$\pm$1.37 | 49.61$\pm$1.44 | 54.80$\pm$1.56 | 51.55$\pm$0.25 | 56.52$\pm$0.27 | 60.03$\pm$0.66 | 63.46$\pm$0.68
CRTC | 44.49$\pm$1.13 | 55.02$\pm$1.69 | 62.41$\pm$1.03 | 66.63$\pm$1.76 | 54.56$\pm$0.15 | 60.01$\pm$0.36 | 68.25$\pm$0.52 | 72.51$\pm$0.69
HW | BSV | 69.05$\pm$4.10 | 53.00$\pm$2.36 | 37.67$\pm$1.04 | 28.04$\pm$0.57 | 63.69$\pm$1.55 | 50.12$\pm$0.41 | 38.17$\pm$1.15 | 28.00$\pm$0.78
Concat | 57.10$\pm$2.67 | 46.49$\pm$2.35 | 35.58$\pm$2.05 | 26.52$\pm$0.10 | 54.36$\pm$1.29 | 43.98$\pm$1.61 | 33.83$\pm$1.16 | 24.75$\pm$0.15
PVC | 62.15$\pm$0.00 | 59.50$\pm$0.00 | 42.93$\pm$0.00 | 33.86$\pm$0.00 | 55.73$\pm$0.00 | 57.28$\pm$0.00 | 36.79$\pm$0.00 | 31.06$\pm$0.00
MIC | 70.05$\pm$1.05 | 68.43$\pm$1.96 | 59.87$\pm$0.59 | 38.45$\pm$3.12 | 68.89$\pm$1.82 | 65.31$\pm$2.53 | 50.12$\pm$1.31 | 30.69$\pm$2.56
IMG | 73.07$\pm$6.45 | 66.44$\pm$5.02 | 57.91$\pm$2.72 | 42.63$\pm$0.09 | 67.04$\pm$3.95 | 58.58$\pm$2.96 | 50.30$\pm$0.74 | 39.78$\pm$0.29
DAIMC | 83.29$\pm$4.82 | 82.22$\pm$3.72 | 75.86$\pm$5.72 | 49.51$\pm$6.14 | 76.19$\pm$3.21 | 73.58$\pm$2.80 | 66.01$\pm$3.29 | 40.35$\pm$5.81
CDIMC-net | 95.13$\pm$0.39 | 94.67$\pm$0.77 | 91.68$\pm$1.24 | 87.98$\pm$0.39 | 91.33$\pm$0.75 | 89.85$\pm$1.09 | 84.84$\pm$1.76 | 82.13$\pm$0.42
CRTC | 97.31$\pm$0.71 | 96.80$\pm$0.69 | 95.18$\pm$1.15 | 90.24$\pm$1.87 | 93.20$\pm$0.99 | 92.63$\pm$0.71 | 89.47$\pm$0.93 | 81.89$\pm$2.03

### Joint Training and Clustering

By optimizing the above Eq.(2) and Eq.(7), we can obtain the recovery data and
common representation $\tilde{H}=\\{\tilde{h}_{1},...,\tilde{h}_{N}\\}$.
However, there may be several potential mistakes in the transferred relations
as the distinctness between views. Meanwhile, whether the recovery data and
the common representation are suitable for clustering is unidentified. For
this reason, we develop a dual self-supervised training module as a solution
to confront this challenge in both modules. It can optimize recovery and
clustering simultaneously to obtain the clustering assignments.

Specifically, for the common representation $\tilde{H}$, if its cluster center
is denoted by $\mu_{j}$, the clustering loss function for the multi-view
fusion clustering network can be written as

$L_{mc}=KL(P||Q)=\sum_{i=1}^{N}\sum_{j=1}^{C}p_{ij}\rm{log}\frac{\it
p_{ij}}{\it q_{ij}}$ (8)

In Eq.(8), $q_{ij}$ can be calculated using the Student’s
t-distribution(Maaten and Hinton 2008) as

$q_{ij}=\frac{(1+||\tilde{h}_{i}-\mu_{j}||^{2})^{-1}}{\sum_{k=1}^{C}(1+||\tilde{h}_{i}-\mu_{k}||^{2})^{-1}}$
(9)

where $\mu_{j}$ is initialized by K-means (Arthur and Vassilvitskii 2007) on
the pre-trained common representation. $q_{ij}$ can be seen as the probability
of assigning node $\tilde{h}_{i}$ to cluster $j$. $Q=[q_{ij}]$ is the
distribution of the assignments of all nodes. $P$ is the target distribution
of $Q$ and can be calculated as

$p_{ij}=\frac{q_{ij}^{2}/\sum_{i=1}^{N}q_{ij}}{\sum_{k=1}^{C}(q_{ik}^{2}/\sum_{i=1}^{N}q_{ik})}$
(10)

By minimizing the KL divergence loss between distributions $P$ and $Q$, a more
cluster-friendly representation can be produced according to the high
confidence predictions.

To obtain more cluster-friendly recovery data and reduce the impact of
possible errors in the transferred relations, KL divergence between the
recovery data and relational instances is used to train the cross-view
relation transfer completion module. And the loss function can be written as

$L_{cc}=KL(Q_{a}||Q_{K})=\sum_{i=1}^{N_{a}}\sum_{j\in
K_{a}}\sum_{k=1}^{C}q_{ik}\rm{log}\frac{\it q_{ik}}{\it q_{ijk}}$ (11)

where $N_{a}$ is the number of missing instances. By optimizing Eq.(11), the
clustering information can be introduced into the missing data recovery
process to obtain more clustering-friendly completion features. Our CRTC
method is summarized in Algorithm 1.

(a) BSV

(b) Concat

(c) IMG

(d) MIC

(e) DAIMC

(f) CDIMC-net

(g) CRTC-$h$

(h) CRTC-$q$

Figure 3: $t$-SNE visualizations of representations learned by various methods
on BDGP with a missing ratio of 30%. Figure 4: Data recovery on MNIST dataset.

## Experiments

In this section, we evaluate the proposed CRTC method on five widely-used
multi-view datasets with several state-of-the-art clustering methods.

### Experimental Settings

Datasets: Five widely-used datasets are used to assess the proposed CRTC
method. 1) BDGP (Cai et al. 2012) contains 2500 images of five categories, and
each image is described by a 1750-D visual vector and a 79-D textual feature
vector. 2) MNIST is a widely-used dataset composed of 70000 digital images.
Since some baselines cannot handle such a large dataset, we follow (Wen et al.
2020a; Wang et al. 2018) to use a subset of MNIST, which contains 4000 samples
from 10 categories. Pixel feature and edge feature are extracted as two views.
Please refer to the supplementary material for the results of the full MNIST
dataset. 3) Caltech20 (Li et al. 2015) consists of 2386 images of 20 subjects,
and we follow (Lin et al. 2021) to use two views, _i.e._ , HOG and GIST
features. 4) Animal consists of 10,158 images from 50 classes, and two types
of deep features extracted with DECAF (Krizhevsky, Sutskever, and Hinton 2012)
and VGG19 (Simonyan and Zisserman 2015) are used as two views. 5) HW contains
2000 instances from ten numerals. We use five views from Fourier coefficients,
profile correlations, KL coefficient, Zernike moments, and pixel average
extractors.

Compared methods and evaluate metrics: We compare CRTC with several baselines
include: best single view K-means++ (BSV) (Zhao, Liu, and Fu 2016), concat
feature K-means++ (Concat) (Zhao, Liu, and Fu 2016), PVC (Li, Jiang, and Zhou
2014b), MIC (Shao, He, and Yu 2015), IMG (Qian et al. 2016), DAIMC (Hu and
Chen 2018), PVC-GAN (Wang et al. 2018), CDIMC-net (Wen et al. 2020a), GP-MVC
(Wang et al. 2021a), and iCmSC (Wang et al. 2021b). And we evaluate the
clustering performance with three statistical metrics: Accuracy (ACC),
Normalized Mutual Information (NMI), and Adjusted Rand Index (ARI). For all
the metrics, higher value indicates a better performance. Please refer to the
supplementary material for the results of ARI.

Implementation details: For datasets with more than two views, we randomly
remove $p(p\in\\{0.1,0.3,0.5,0.7\\})$ instances under the condition that all
samples at least have one view. For datasets with two views, we randomly
select $p(p\in\\{0.1,0.3,0.5,0.7\\})$ instances as paired samples whose views
are complete, and the remaining samples are treated as single view samples. To
better illustrate the effectiveness of the algorithm, we perform ten times and
report the average clustering results and the corresponding standard deviation
of all the methods. Note that our model is not sensitive to the $k$ of initial
KNN graphs in a large range. We set $k=10$ for datasets with two views and set
$k=5$ for the HW dataset. Please refer to the supplementary material for the
details of network architectures and parameter settings.

### Performance with Different Missing Rates

Table 1 shows the clustering performance of CRTC and baseline methods. The
best results are highlighted in bold. Here, we summarize some observations as
follows: (1) In terms of both ACC and NMI, CRTC achieves relatively promising
performance compared with all baselines, which validates the effectiveness of
the proposed method. (2) We can observe that incomplete multi-view methods
achieve better results than BSV and Concat in most cases, which illustrates
that exploring features from multiple sources can improve clustering
performance. (3) CRTC achieves much better performance than GAN-based methods
GP-MVC and PVC-GAN, validating that recovering missing data based on relation
transfer can improve the performance of incomplete multi-view clustering.

We visualize the results of different methods on the BDGP dataset with a
missing ratio of 30% with $t$-SNE(Maaten and Hinton 2008). The results are
shown in Fig. 3, where CRTC-$h$ and CRTC-$q$ denote the common representation
$\tilde{H}$ and the probability $Q$, respectively. As can be seen from Fig. 3,
our method can recover a better cluster structure of data since it has smaller
intra-cluster scatter and larger inter-cluster scatter, which demonstrates the
effectiveness of the proposed CRTC method. Meanwhile, we can find that
CRTC-$q$ has fewer intra-cluster errors and smaller intra-cluster scatter than
CRTC-$h$, which illustrates that the clustering layer is valid for learning a
better clustering structure.

### Model Analysis

To visually present the recovered results of our method, we show the recovered
images by our method on the MNIST dataset with a missing ratio of 30$\%$ on
Fig. 4, in which lines 1 and 4 are complete views, lines 2 and 5 are missing
views, and lines 3 and 6 are the recovered results from the complete view. We
can see our method can recover the missing data for both the original and edge
images, which illustrates that recovering missing data based on the similar
relationships of existing instances is effective.

Figure 5: Performance comparisons of different approaches on BDGP dataset.

We further conduct a series of experiments with different approaches to show
the effectiveness of the cross-view relation transfer completion. Fig. 5 shows
the clustering accuracy on the BDGP dataset for six approaches: (a) BSV, (b)
Concat, (c) BSV based on recovered data using the cross-view relation transfer
(Rec-BSV), (d) Concat based on recovered data using the cross-view relation
transfer (Rec-Concat), (e) Filling the missing data using the average value of
the existing data and clustering by multi-view fusion clustering (Ave-MFC),
and (f) the complete CRTC. According to Fig. 5, we have the following
observations: (1) Compared with BSV and Concat, recovering data using the
cross-view relation transfer can substantially improve the performance of
k-means++ (Arthur and Vassilvitskii 2007) for about $14\%$. The improvement is
more significant with the decrease of paired-view rates. (2) The complete CRTC
outperforms Ave-MFC in all paired-view rates, especially with less complete
data, which demonstrates the validity of recovering data using the cross-view
relation transfer.

(a) ACC

(b) NMI

Figure 6: ACC and NMI of the proposed CRTC and two degraded approaches on
Caltech20.

In order to show the effect of the attention fusion layer and joint training,
two degraded approaches of CRTC, _i.e._ , CRTC without joint training (CRTC-
wjd) and CRTC without attention fusion layer (CRTC-waf), are compared with the
complete CRTC. The ACC and NMI results on the Caltech20 dataset are shown in
Fig. 6. It can be seen that CRTC vastly outperforms these two degraded
approaches in all paired-view rates, which illustrates that the joint training
and the attention fusion layer can effectively improve the clustering
performance.

(a) $p=0.3$

(b) $p=0.5$

Figure 7: The clustering performance influence of parameter $k$ on BDGP
dataset with a missing ratio of 30$\%$.

The number of the nearest neighbors $k$ is an important parameter in
constructing the $k$-NN graph and has a great impact on the performance of
most graph-based algorithms. To examine the effect of $k$, we design a
$k$-sensitivity experiment on the BDGP datasets with $k\in\\{5,15\\}$. It can
be seen from Fig. 7 that our model is insensitive with $k$ in a larger range.
It proves that our method is not sensitive to $k$ and can obtain a stable
clustering even when there are fewer relational instances or some spurious
relations in the transferred relation graph.

## Conclusion

In this paper, we propose a novel algorithm for incomplete multi-view
clustering named CRTC, which can simultaneously learn a clustering structure
and recover the incomplete views. Based on the consensus that consistency
exists between views, we construct similarity relations in the known views and
transfer them to the missing views to recover incomplete data. Then, to fully
explore the information in the transferred relations, GCN is employed to
recover missing data based on the transferred relation graph and the existing
instances. Compared with existing GAN-based approaches, the recovered views
can fully exploit the relationships in existing views and are consistent with
the clustering structure. Then, a multi-view fusion clustering network
incorporating an attention-based fusion layer is introduced to obtain the
clustering assignments. Moreover, a clustering layer is designed to optimize
recovery and clustering simultaneously using the cluster-based KL-divergence
loss. Comprehensive experiments validate the performance improvement of the
proposed CRTC compared with state-of-the-art methods.

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# Exploring variational quantum eigensolver ansatzes for the long-range XY
model

Jia-Bin You  0000-0001-8815-1855<EMAIL_ADDRESS>Institute of
High Performance Computing, A*STAR (Agency for Science, Technology and
Research), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632  Dax Enshan
Koh  0000-0002-8968-591X<EMAIL_ADDRESS>Institute of High
Performance Computing, A*STAR (Agency for Science, Technology and Research), 1
Fusionopolis Way, #16-16 Connexis, Singapore 138632  Jian Feng Kong
0000-0001-5980-4140 Institute of High Performance Computing, A*STAR (Agency
for Science, Technology and Research), 1 Fusionopolis Way, #16-16 Connexis,
Singapore 138632  Wen-Jun Ding  Institute of High Performance Computing,
A*STAR (Agency for Science, Technology and Research), 1 Fusionopolis Way,
#16-16 Connexis, Singapore 138632  Ching Eng Png  Institute of High
Performance Computing, A*STAR (Agency for Science, Technology and Research), 1
Fusionopolis Way, #16-16 Connexis, Singapore 138632  Lin Wu  Institute of High
Performance Computing, A*STAR (Agency for Science, Technology and Research), 1
Fusionopolis Way, #16-16 Connexis, Singapore 138632  Science, Mathematics and
Technology (SMT), Singapore University of Technology and Design (SUTD), 8
Somapah Road, Singapore 487372

###### Abstract

Finding the ground state energy and wavefunction of a quantum many-body system
is a key problem in quantum physics and chemistry. We study this problem for
the long-range XY model by using the variational quantum eigensolver (VQE)
algorithm. We consider VQE ansatzes with full and linear entanglement
structures consisting of different building gates: the CNOT gate, the
controlled-rotation (CRX) gate, and the two-qubit rotation (TQR) gate. We find
that the full-entanglement CRX and TQR ansatzes can sufficiently describe the
ground state energy of the long-range XY model. In contrast, only the full-
entanglement TQR ansatz can represent the ground state wavefunction with a
fidelity close to one. In addition, we find that instead of using full-
entanglement ansatzes, restricted-entanglement ansatzes where entangling gates
are applied only between qubits that are a fixed distance from each other
already suffice to give acceptable solutions. Using the entanglement entropy
to characterize the expressive powers of the VQE ansatzes, we show that the
full-entanglement TQR ansatz has the highest expressive power among them.

##  I Introduction

Simulating quantum many-body systems composed of many particles and learning
their properties are, in general, hard problems for classical computers. In
contrast, many of these problems may be solved more efficiently by
appropriately designed quantum algorithms [1, 2, 3, 4, 5, 6] running on large-
scale fault-tolerant quantum computers, which promise to efficiently simulate
these systems and provide important insights into their properties [7, 8, 9].
However, such fault-tolerant quantum computers currently still do not exist.
The most powerful quantum computers we have today are noisy intermediate-scale
quantum (NISQ) [10, 11] devices that are too small and noisy to run many
quantum algorithms, including quantum phase estimation [2, 12], which require
fully coherent evolution. Nevertheless, these devices have already succeeded
at solving certain sampling problems believed to be intractable for classical
computers [13, 14, 15]. To harness the full power of these devices to solve a
larger number and range of problems, it is essential to ensure efficient
management of a limited number of qubits with finite coherence times.

Hybrid classical-quantum algorithms—such as variational quantum algorithms
(VQAs) [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]—leverage classical resources to
reduce the required number of quantum gates, and hold the promise of
outperforming current purely classical techniques. The structure of the VQA
ansatzes generally depend on the tasks at hand as well as the quantun
resources available [36]. For example, the hardware-efficient ansatz [44] aims
at reducing the circuit depth; the unitary coupled ansatz [16] is used for
problems in quantum chemistry; and the quantum approximate optimization
algorithm (QAOA) ansatz is used to obtain approximate solutions to
combinatorial optimization problems [45, 46]. The central idea of VQAs is to
design a parameterized quantum circuit that depends on a set of gate
parameters and whose architecture is dictated by the structure of the NISQ
devices [32]. These variational parameters can be optimized using quantum-
classical optimization loops by extremizing a cost function which is designed
to encode the solution of the optimization problem into its extremum. In
theory, the approximation improves with increasing ansatz depth, though a
larger number of noisy gates and variational parameters undermine performance
in practice [47]. Noise grows rapidly with circuit depth and affects the
fidelity of the prepared quantum state, and so shallow circuits could
potentially achieve a better performance on current NISQ devices compared to
their deeper counterparts.

Here we focus on the variational quantum eigensolver (VQE) [16, 17, 18, 19,
20, 21, 22, 23, 24], a VQA that is designed to approximate the ground state of
quantum many-body systems. The VQE—one of the most promising near-term
applications using NISQ devices—works by preparing a quantum many-body state
on a parameterized quantum circuit and executing the circuit without quantum
error correction, which works if the circuit is short enough and the fidelity
of gates high enough [17]. The parameterized quantum circuit provides a
variational ansatz for the ground state, and a classical optimizer tunes the
circuit parameters to give a quantum circuit that rotates the product state
$\otimes_{n}|0_{n}\rangle$ into (an approximation of) the ground state of a
given Hamiltonian of an $n$-qubit quantum many-body system [18]. The
efficiency of optimization depends on the number of iterations required for
convergence and the number of gates involved in each preparation and
measurement cycle of the quantum subroutine [48]. To date, several small-scale
experimental demonstrations of the VQE for molecules and quantum magnets have
been performed [49, 25, 50, 51, 44] on various quantum platforms, including
photonic chips [16], ion traps [51] and superconducting circuits [49, 44].

In this work, we apply the VQE algorithm to search for the ground state energy
and wavefunction of the long-range XY model with length $N$. The Hamiltonian
of the system can be written as

$\displaystyle H=-J\sum_{i<j}(X_{i}X_{j}+Y_{i}Y_{j})-h\sum_{i}Z_{i},$ (1)

where $J$ is the coupling strength; $h$ is the Zeeman field; and $X_{i}$,
$Y_{i}$, $Z_{i}$ denote the Pauli matrices at site $i$. Note that this model
can be used to describe Bardeen-Cooper-Schrieffer (BCS) superconductivity [52,
53], which is an important prototype for modern condensed matter physics.

In general, the ground state of a spin chain can be expressed as a collection
of Bloch vectors defined by
$(\langle{X_{i}}\rangle,\langle{Y_{i}}\rangle,\langle{Z_{i}}\rangle)$ on the
Bloch sphere for each site, as well as the correlations between sites, such as
$\langle{X_{i}X_{j}}\rangle$, $\langle{Y_{i}Z_{j}}\rangle$ and so on. The
Bloch vector at site $i$ can be rotated from initial state $|0_{i}\rangle$ by
single-qubit rotation gates and the correlations in the ground state can be
generated by the entangler which consists of two-qubit gates. To fully
describe the ground state of the spin chain, a mean field layer and an
entangler should be included in an ansatz.

We first analytically obtain a mean-field solution for the ground state energy
based on the mean-field ansatz. The mean-field solution, which neglects
entanglement, will serve as a benchmark for other VQE ansatzes that take
entanglement into account. We then introduce quantum entanglement into the VQE
ansatz by constructing entanglers with different building gates and different
entanglement structures. More specifically, we consider the following three
building gates: (i) CNOT gate, (ii) parametric controlled-rotation (CRX)
gates, and (iii) parametric two-qubit rotation (TQR) gates. Note that we
gradually increase the number of variational parameters for the three
ansatzes, allowing for finer-grained control over the minimization of the
Hamiltonian. We consider the following two entanglement structures for each of
these building gates: (a) linear entanglement and (b) full entanglement.

Amongst the above ansatzes, we find that the only ansatzes that well describe
the ground state energy and wavefunction of the long-range XY model are the
full-entanglement CRX and TQR ansatzes; for these two ansatzes, linear
entanglement is insufficient. However, we proceed to show that full
entanglement is, in fact, not necessary to describe well the ground state
energy and wavefunction of the Hamiltonian: it suffices to use CRX and TQR
ansatzes in which the entangling gates are restricted to act only between
qubits that are a fixed distance away from each other. Using these restricted-
entanglement ansatzes in lieu of their full-entanglement counterparts reduces
the overall running time of the optimization step in the VQE algorithm. In
addition, we find that the TQR ansatz can reduce the circuit depth by
increasing the number of variational parameters introduced in each layer.
Increasing the number of parameters allows for finer-grained control over the
minimization of the Hamiltonian. While introducing more parameters within one
layer, a corresponding reduction in circuit depth preserves the overall
performance of the minimization problems. Finally, to elucidate the
differences in performance of the different ansatzes, we use the entanglement
entropy to characterize their expressive powers. We find that the full-
entanglement TQR ansatz, which gives a good VQE performance, also has the
largest expressive power amongst all the ansatzes considered.

##  II VQE algorithm

Figure 1: Schematic of the VQE algorithm. The long-range XY model is recast
into a summation of Pauli strings. An initial guess for the variational
parameters is input into the quantum circuit. The estimation of variational
energy is obtained by a series of Pauli measurements from the circuits. The
classical optimization routine estimates a new value for the circuit
parameters. The process is repeated until achieving convergence on the total
energy. The red blocks summarize the operations performed by the classical
computer, whereas the blue blocks summarize the operations performed by the
quantum computer.

A schematic of the VQE algorithm is shown in Fig. 1. A general spin model
Hamiltonian $H$ can be represented as a sum of Pauli strings, viz.
$H=\sum_{i}p_{i}P_{i}$, where the $i$-th Pauli string in the sum is
$P_{i}=\otimes_{j}\sigma_{j}^{(i)}$, where $\sigma_{j}$ denotes a Pauli matrix
(i.e., either $\mathbbm{1}$, $X$, $Y$, or $Z$) acting on site $j$. The ground
state of $H$ can be approximated by a quantum circuit ansatz realized by
applying a parameterized unitary operation $U(\vec{\theta})$—built from a
series of parameterized single- and two-qubit gates—to the computational basis
state $\otimes_{j}|0_{j}\rangle$, where $\vec{\theta}$ are the variational
parameters of the circuit. We denote the ansatz from the above construction by
$|\Psi(\vec{\theta})\rangle=U(\vec{\theta})\otimes_{j}\\!|0_{j}\rangle$. A
guess for the variational parameters $\vec{\theta}^{(0)}$ is inputted into the
circuit for initialization. The variational energy $E(\vec{\theta})$ of $H$
can be estimated by performing a series of Pauli measurements to obtain the
expectation values
$\langle\Psi(\vec{\theta})|P_{i}|\Psi(\vec{\theta})\rangle$, which are then
plugged into the formula
$E(\vec{\theta})=\sum_{i}p_{i}\langle\Psi(\vec{\theta})|P_{i}|\Psi(\vec{\theta})\rangle/\langle\Psi(\vec{\theta})|\Psi(\vec{\theta})\rangle$.
An optimization subroutine is performed to minimize the variational energy
(i.e. we solve the optimization problems
$\min_{\vec{\theta}}[E(\vec{\theta})]$ and
$\mathrm{argmin}_{\vec{\theta}}[E(\vec{\theta})]$) to find the ground state
energy and wavefunction. This subroutine is performed by a classical
optimizer, which updates new values for the circuit parameters in each step,
via $\vec{\theta}^{(n+1)}\leftarrow\vec{\theta}^{(n)}+d\vec{\theta}$. This
process is repeated until convergence of the ground state energy is achieved.
The ground state energy and wavefunction can be calculated as
$E(\vec{\theta}^{\text{final}})$ and
$|\Psi(\vec{\theta}^{\text{final}})\rangle=U(\vec{\theta}^{\text{final}})\otimes_{j}|0_{j}\rangle$.
It should be noted that the classical optimizer may get stuck in a local
minimum and fail to find the global minimum for the VQE ansatz. To address
this issue, we run the VQE algorithm repeatedly starting from different random
parameter seeds. While this approach does not guarantee that the global
minimum is found, from a practical point of view it increases the odds of
finding an acceptable solution, at the expense of potentially having to run
the algorithm an increased number of times. In Fig. 1, the red blocks indicate
the operations performed by the classical computer, whereas the blue blocks
indicate the operations performed by the quantum computer.

Pauli measurements are performed on the ansatz $|\Psi(\vec{\theta})\rangle$ to
read out the variational energy during the energy estimation subroutine. The
expectation of a Pauli string $P_{i}$,
$\langle{P_{i}}\rangle=\langle\Psi(\vec{\theta})|P_{i}|\Psi(\vec{\theta})\rangle/\langle\Psi(\vec{\theta})|\Psi(\vec{\theta})\rangle$
can be reconstructed from the countings of computational basis states
$\\{|0_{j}\rangle,|1_{j}\rangle\\}$ of $|\Psi(\vec{\theta})\rangle$ by Pauli
measurements
$\\{\langle{X_{j}}\rangle,\langle{Y_{j}}\rangle,\langle{Z_{j}}\rangle\\}$ in
the $x$, $y$ and $z$ directions. Our simulations were implemented using
Qiskit, IBM’s open-source framework for quantum computing [54]. In particular,
we used the QASM Simulator backend in Aer for all our simulations, except for
Fig. 10 and the fidelity calculations, where we used the Statevector Simulator
backend in Aer. The Pauli-Z measurement can simply be obtained by counting the
number of states $|0_{j}\rangle$ and $|1_{j}\rangle$ at qubit $j$ and
calculating the difference,
$\langle{Z_{j}}\rangle=(N_{|0_{j}\rangle}-N_{|1_{j}\rangle})/N_{s}$, where
$N_{s}$ is the number of shots in simulations. The Pauli-X and Pauli-Y
measurements can be achieved by first rotating $x$ and $y$ axes to $z$ axis,
equivalent to applying $R_{y}(-\pi/2)$ and $R_{x}(\pi/2)$ gates to the
circuits respectively, then subsequently performing the Pauli-Z measurement.
Here the single-qubit rotation gates are defined as
$R_{y}(\theta_{j})=e^{-i\theta_{j}Y_{j}/2}$ and
$R_{x}(\theta_{j})=e^{-i\theta_{j}X_{j}/2}$ respectively. The measurement
circuits are shown in Fig. 1. Similarly, we can obtain the expectations of the
products of two Pauli operators, such as
$\\{X_{i}X_{j},Y_{i}Y_{j},Z_{i}Z_{j},X_{i}Y_{j},Y_{i}Z_{j},Z_{i}X_{j}\\}$, by
rotating the $x$ and $y$ axes to $z$ axis, resulting in the calculation of the
expectation of $\langle{Z_{i}Z_{j}}\rangle$ which can be achieved by
$\langle{Z_{i}Z_{j}}\rangle=(N_{|0_{i}0_{j}\rangle}-N_{|0_{i}1_{j}\rangle}-N_{|1_{i}0_{j}\rangle}+N_{|1_{i}1_{j}\rangle})/N_{s}$.

##  III Analytical solution for the long-range XY model

Before discussing the VQE algorithm, we will first give an analytical solution
for the isotropic long-range XY model. By using the collective Pauli
operators, $S_{x}=\sum_{i}X_{i}$, $S_{y}=\sum_{i}Y_{i}$, and
$S_{z}=\sum_{i}Z_{i}$, the Hamiltonian can be recast as

$\begin{split}H=-2JS^{+}S^{-}+(J-h)S_{z}+JN,\end{split}$ (2)

where $S^{\pm}=\frac{1}{2}(S_{x}\pm iS_{y})$. Note that
$[S^{+}S^{-},S_{z}]=0$, and so the Hamiltonian and the particle number
operator $S^{+}S^{-}$ have the same eigenstates. We can thus represent the
wavefunctions in different excitation subspaces. It is found that in the
$n$-excitation subspace, the symmetric state,
$\psi_{\text{sym}}\sim\sum_{\\{i_{1},i_{2},\ldots,i_{n}\\}}\sigma_{i_{1}}^{+}\sigma_{i_{2}}^{+}\cdots\sigma_{i_{n}}^{+}|\mathbf{0}\rangle$,
has the lowest energy per site,

$\begin{split}E_{\text{sym}}^{(n)}/N=(1-2n/N)h-2Jn(1-n/N).\end{split}$ (3)

Here $\sigma_{i_{k}}^{+}$ is the raising operator at site $i_{k}$ and
$\\{i_{1},i_{2},\ldots,i_{n}\\}$ are all the possible combinatorics from
$\\{0,1,\ldots,N-1\\},(0\leq{n}\leq{N})$. Thus the ground state energy per
site of the long-range XY model is
$E_{\text{GS}}/N=\min\\{E_{\text{sym}}^{(n)}/N,(n=0,1,\ldots,N)\\}$.

The quantum phase transitions occur at the critical points defined by
$E_{\text{sym}}^{(n+1)}/N=E_{\text{sym}}^{(n)}/N$. After some calculations, we
find that the critical point for transition from $n$-excitation phase to
$n+1$-excitation phase is $h/J=2n+1-N$, where $n$ ranges from $0$ to $N-1$.
The various transitions are essentially of the same nature, i.e., transition
from $n$ excitations to $n\pm 1$ excitations, equivalently,
$\langle{S_{z}}\rangle=s$ to $s\pm 2$ sectors. $\langle{S_{z}}\rangle$ changes
by 2 every time across each transition, and thus the slope $dE_{\text{GS}}/dh$
changes by 2 each time. When $N$ is odd, the ground state magnetization cannot
be zero and is always in the paramagnetic phases with different polarizations.
Interestingly, when $N$ is even, there exists a ferromagnetic phase when
$|{h/J}|\leq 1$ and the net magnetization is zero. The system undergoes a
ferromagnetic to paramagnetic phase transition across the critical point
$|h/J|=1$. We will focus on this phase transition behavior in the rest of the
paper.

##  IV VQE algorithm for mean field ansatz

Figure 2: VQE algorithm with mean-field ansatz. a: quantum circuits for the
mean-field ansatz. b: ground state energy per site $E_{\text{GS}}/N$ and
fidelity between VQE algorithm and exact solution as a function of the Zeeman
field $h$. The parameters are $J=1$, $N=4$. The number of shots is $2^{14}$ in
all simulations.

We first consider the mean-field ansatz where the ground state is approximated
as a product state of single qubits,

$\begin{split}|\Psi_{\text{MF}}\rangle=\bigotimes_{p}\left[\begin{array}[]{*{20}ccccccccc}{\cos{\tfrac{\theta_{p}}{2}}}\\\
{\sin{\tfrac{\theta_{p}}{2}}e^{i\varphi_{p}}}\\\
\end{array}\right].\end{split}$ (4)

Notice that there are no correlations within the mean-field ansatz. We can
thus decouple the hopping terms and write the mean-field variational energy as
$E_{\text{MF}}=-J\sum_{p<q}(\langle{X_{p}}\rangle\langle{X_{q}}\rangle+\langle{Y_{p}}\rangle\langle{Y_{q}}\rangle)-h\sum_{p}\langle{Z_{p}}\rangle$.
In the mean-field ansatz, the expectations of Pauli matrices are
straightforwardly calculated as
$\langle{X_{p}}\rangle=\sin{\theta_{p}}\cos{\varphi_{p}}$,
$\langle{Y_{p}}\rangle=\sin{\theta_{p}}\sin{\varphi_{p}}$, and
$\langle{Z_{p}}\rangle=\cos{\theta_{p}}$, which can be represented as a point
on the Bloch sphere. Therefore, we can obtain an analytical solution for the
mean-field variational energy,

$\begin{split}E_{\text{MF}}&=-J\sum_{p<q}\sin{\theta_{p}}\sin{\theta_{q}}\cos{(\varphi_{p}-\varphi_{q})}-h\sum_{p}\cos{\theta_{p}}.\\\
\end{split}$ (5)

Minimizing the energy $E_{\text{MF}}$ with respect to the variational
parameters $\\{\theta_{p},\varphi_{p}\\}$, we can estimate the ground state
energy within the mean-field variational class.

On the other hand, the mean-field ansatz in Eq. (4) can readily be implemented
by the quantum circuit shown in Fig. 2a. This state can be realized by
sequentially applying two single-qubit rotation gates $R_{y}(\theta_{p})$ and
$R_{z}(\varphi_{p})$ on each qubit $p$ initially prepared in the state
$|0_{p}\rangle$. We can then use VQE to estimate the ground state energy. We
first perform the Pauli measurements to evaluate the variational energy
$E_{\text{MF}}$, and then update the variational parameters
$\\{\theta_{p},\varphi_{p}\\}$ to minimize the variational energy
$E_{\text{MF}}$ by a classical optimizer and finally to obtain the ground
state energy. Notice that for the $N$-qubit mean-field ansatz of the long-
range XY model, we have $N^{2}$ Pauli strings (corresponding to the number of
terms in the Hamiltonian) and $3N$ Pauli measurements
$\\{\langle{X_{p}}\rangle,\langle{Y_{p}}\rangle,\langle{Z_{p}}\rangle\\}$
$(p=0,1,\ldots,N-1)$ to evaluate the expectation of Hamiltonian.

The results for the mean field ansatz are shown in Fig. 2b. In the upper plane
of Fig. 2b, we study the ground state energy per site $E_{\text{GS}}/N$ as a
function of Zeeman field $h$. The blue-solid, red-dashed, and green-dotted
lines result from the VQE algorithm, the mean-field solution and the exact
solution respectively. It is found that the VQE solution based on the mean-
field ansatz is consistent with the mean-field solution. However, due to the
ignorance of intrinsic correlations in the long-range XY model, there is a
deviation between the VQE solution and the exact solution, which is also
evidenced by the low fidelity
$F=|\langle\psi_{\text{VQE}}|\psi_{\text{exact}}\rangle|^{2}<0.5$ between the
VQE algorithm and the exact solution shown in the lower plane of Fig. 2b. The
fidelity becomes even worse near the critical point of phase transition $h=1$.
The asymmetry of the energy and the fidelity results from the sampling error
of the qasm simulator in IBM qiskit package. This sampling error will induce
the error in estimating the gradient descent in the classical optimization
routine which is liable to become trapped in local minima.

##  V VQE algorithm with CNOT ansatz

Figure 3: VQE algorithm with CNOT ansatz. (a) quantum circuits for the linear
and the full CNOT ansatzes. (b) and (c) are ground state energy per site
$E_{\text{GS}}/N$ and fidelity between VQE algorithm and exact solution as a
function of Zeeman field $h$ for the full and the linear ansatzes
respectively. The parameters are $J=1$, $N=4$.

We find that the mean field VQE ansatz is consistent with the analytical mean-
field solution, but deviates significantly from the exact solution. This is
because there is some entanglement in the ground state of the model which is
not captured by the mean field ansatz since the product state indicates no
entanglement in the ansatz. We thus consider adding some degree of
entanglement into the system by adding entanglers into the circuits as shown
in Fig. 3a. Here we consider adding a full CNOT entangler or a linear CNOT
entangler as an entangler after the mean-field block
$U_{\text{MF}}(\vec{\theta})$ has been applied. The full CNOT entangler
includes a series of CNOT gates linking all the pairs of qubits while the
linear one only links the nearest neighbor pairs of qubits. Note that there
are no variational parameters in the CNOT entanglers; the variational
parameters in the mean-field block are optimized for the CNOT ansatz. Due to
the entangler, the CNOT ansatz can no longer be expressed as a product state.
We should instead directly evaluate the expectation of the energy as
$\langle{H}\rangle=-J\sum_{i<j}(\langle{X_{i}X_{j}}\rangle+\langle{Y_{i}Y_{j}}\rangle)-h\sum_{i}\langle{Z_{i}}\rangle$,
where the correlators of Pauli matrices are included.

The results for the full and linear CNOT ansatzes are shown in Fig. 3b and 3c
respectively. For the full CNOT ansatz, it is found that the VQE energy is
still significantly different from the exact solution, implying the poor
expressive power of the full CNOT ansatz for the long-range XY model. However,
since the VQE energy for the ferromagnetic phase ($|h|<1$) is between the
mean-field solution and exact solution, the ansatz can capture part of the
entanglement in the ferromagnetic phase; while for the paramagnetic phase
($|h|>1$), the VQE energy is even worse than the mean-field solution. The
fidelity is lower than 0.9 in both phases. The result for linear CNOT ansatz
is shown in Fig. 3c. We find that in the paramagnetic phase, the VQE energy
for linear CNOT ansatz is higher than that for the full CNOT ansatz, implying
that the full entangler has better expressive power than its linear
counterpart. This can also be reflected in the lower fidelity ($<0.4$) for the
linear ansatz. In the ferromagnetic phase, the full and linear ansatzes have
similar capacity to minimize the VQE energy.

##  VI VQE algorithm with CRX ansatz

Figure 4: VQE algorithm with CRX ansatz. a: quantum circuits for the linear
and the full CRX ansatzes. b and c are ground state energy per site
$E_{\text{GS}}/N$ and fidelity between VQE algorithm and exact solution as a
function of Zeeman field $h$ for the full and the linear ansatzes
respectively. The parameters are $J=1$, $N=4$.

We find that the CNOT ansatzes cannot approximate the ground state energy of
the long-range XY model well. It is because the CNOT entanglers, having no
variational parameters, have limitations in bringing entanglement into the
ansatzes. To improve the ansatzes, we can introduce variational parameters
into the CNOT entanglers to gradually introduce the entanglement into the
ansatzes. A natural way is to generalize the CNOT gates to the controlled-
rotation gates, for example, the controlled-$R_{x}$ gate, to tune the degree
of entanglement more flexibly. We thus consider the full and the linear CRX
entanglers as shown in Fig. 4a.

In Fig. 4b, we consider the full CRX ansatz and find that this ansatz
significantly improves the accuracy. The VQE energy almost approaches the
exact solution with some fluctuation resulting from the sampling error and the
intrinsic expressive power of the ansatz. This can also be observed in the
fidelity. It is found that the more the VQE energy deviates from the exact
solution, the lower the fidelity. We also find that the fidelity is low near
the critical point $h=1$, implying poor expressive power near the phase
transition. As a comparison, we consider the linear CRX ansatz shown in Fig.
4c. We find that the VQE energy of the linear ansatz has lower expressive
power than the full one to describe the ground state energy of the long-range
XY model. It only approaches to the mean-field solution and deviates from the
exact solution. This can also be reflected in the low fidelity shown in the
figure.

##  VII VQE algorithm with TQR ansatz

Figure 5: VQE algorithm with the TQR ansatz. a: quantum circuits for the
linear- and the full-entanglement TQR ansatzes. The box labeled
“$R_{xx},R_{yy}$” in the circuit diagram is meant to represent the operation
of performing the $R_{xx}$ and $R_{yy}$ gates in sequence. Also, note that
unlike previous diagrams, the entangler is performed before the mean-field
ansatz. It turns out that this performs better than the case where the order
is flipped. b and c are ground state energy per site $E_{\text{GS}}/N$ and
fidelity between VQE algorithm and exact solution as a function of Zeeman
field $h$ for the full and the linear ansatzes respectively. The parameters
are $J=1$, $N=4$.

To further improve the performance of the VQE ansatz, we propose a problem-
inspired ansatz in Fig. 5a based on the two-qubit rotation gates
$R_{xx}(\alpha_{ij})$ and $R_{yy}(\beta_{ij})$, which are defined as
$R_{xx}(\alpha_{ij})=e^{-i\alpha_{ij}{X_{i}{\otimes}X_{j}}/2}$ and
$R_{yy}(\beta_{ij})=e^{-i\beta_{ij}{Y_{i}{\otimes}Y_{j}}/2}$ respectively. We
consider the full and the linear TQR ansatzes for the long-range XY model.

We demonstrate the results for the full TQR ansatz in Fig. 5b. It is found
that this ansatz can fully capture the properties of the long-range XY model
both in the ferromagnetic and the paramagnetic phases. The VQE energy for the
ground state is almost the same as the exact solution; furthermore, compared
with the full CRX ansatz, the fidelity is close to one except for the critical
points $|h|=1$. In Fig. 5c, we show the result for the linear TQR ansatz. It
is found that the linear TQR ansatz is a good candidate for the paramagnetic
phase of the long-range XY model which can be demonstrated by the VQE energy
nearly identical to the exact solution and the fidelity close to one. However,
the linear TQR ansatz is not expressive enough to represent a state in the
ferromagnetic phase. The VQE energy deviates from the plateau in the exact
solution in the ferromagnetic phase. We observe that a new plateau is
developed with lower fidelity in the ferromagnetic phase. Strikingly, apart
from the plateau in the VQE energy, the fidelity even drops to zero. However,
the VQE energy is in between the mean-field solution and the exact solution,
implying that some degree of entanglement is captured by the ansatz. Comparing
with the linear CRX ansatz shown in Fig. 4d, the linear TQR ansatz performs
better than the linear CRX ansatz.

##  VIII Gate order dependence of different VQE ansatzes

Note that the two-qubit gates in the entanglers of the three VQE ansatzes
(CNOT, CRX and TQR) do not commute with each other, and so a natural question
is to study whether the gate order in the entangler affects the VQE results.
For the four-qubit VQE ansatzes with full entanglers, there are $6!\times
2^{6}=46080$ ways to arrange the gates in the circuit. We do not exhaust all
the possibilities here but randomly choose 4 (10) realizations for full CNOT
(CRX, TQR) ansatzes to investigate the dependence on the gate order.

Figure 6: Gate order dependence of the full CNOT ansatz. Subfigures a–d are
ground state energy per site $E_{\text{GS}}/N$ and fidelity between VQE
algorithm and exact solution as a function of Zeeman field $h$ for the four
random realizations of the full CNOT ansatz. The parameters are $J=1$, $N=4$.

The gate order dependence for the full CNOT ansatz is shown in Fig. 6. For
each realization, we obtain the VQE ground state energy and fidelity. The gate
order in the realization is shown in the title of each figure. For example,
$(1,3)(1,0)(3,2)(1,2)(3,0)(0,2)$ in Fig. 6a means that we consecutively append
CNOT gates to the circuit in the order specified. The notation $(i,j)$ means
that we apply the CNOT gate to the qubit pair $(i,j)$, with $i$ being the
control qubit and $j$ being the target qubit. It is shown that for the full
CNOT ansatz, the VQE energy and fidelity are indeed dependent on the gate
order and are always far from the exact energy. This is due to the
insufficient entanglement inside the full CNOT ansatz, which we will
illustrate in detail in Sec. X.

Figure 7: Gate order dependence of the full CRX ansatz. a-j are ground state
energy per site $E_{\text{GS}}/N$ and fidelity between VQE algorithm and exact
solution as a function of Zeeman field $h$ for the ten random realizations of
the full CRX ansatz. The parameters are $J=1$, $N=4$.

The gate order dependence for the full CRX ansatz is shown in Fig. 7. Since
the full CRX ansatz contains more variational parameters than the full CNOT
ansatz, we sample ten realizations for the full CRX ansatz. Compared to the
full CNOT ansatz, it is found that while some of the realizations can well
approximate the exact ground energy, from the view of the fidelity, no
realizations can well describe the ground state wavefunction of the model.

Figure 8: Gate order dependence of the full TQR ansatz. a-j are ground state
energy per site $E_{\text{GS}}/N$ and fidelity between VQE algorithm and exact
solution as a function of Zeeman field $h$ for the ten random realizations of
the full TQR ansatz. The parameters are $J=1$, $N=4$.

The gate order dependence for the full TQR ansatz is shown in Fig. 8. We also
give ten realizations for the full TQR ansatz. We find that among the CNOT,
CRX and TQR ansatzes, only TQR ansatz can satisfactorily describe both the
ground state energy and the wavefunction of the model. Furthermore, it is
shown that except for the points near the phase boundary $|h|=1$, the gate
order has no effect on the TQR ansatz, unlike the case for the CNOT and CRX
ansatzes.

##  IX Multi-layer setup for different VQE ansatzes

Figure 9: Multi-layer setup for the three full entangler ansatzes. a,b: layer
dependence ($L=2,3$) for full CNOT ansatz. c,d: layer dependence ($L=4,5$) for
full CRX ansatz. e,f: layer dependence ($L=2,3$) for full TQR ansatz. The
parameters are $J=1$, $N=4$.

In this section, we will study the multi-layer setup for all the three
ansatzes. To this end, we repeat the entangler in each of the three full
ansatzes (CNOT, CRX and TQR) $L$ times with independent parameters for each
layer and study the VQE energy and fidelity with an increasing number of
layers (entanglers).

In Fig. 9, we present the layer dependence for the three full entangler
ansatzes. In Figs. 9a and 9b, we find that the full CNOT ansatz cannot
approach the exact ground state energy and wavefunction in spite of the
increase in the number of layers. This will be further explained from the
point of view of the entanglement entropy in Sec. X.

In Figs. 9c and 9d, we find that the full CRX ansatz can approximate the exact
ground state energy and wavefunction when the number of layers $L\geq 4$.
However, there are some fluctuations for the VQE wavefunction near the phase
boundary $|h|=1$, which can be mitigated by increasing the number of layers.
Again, we find that the best multi-layer ansatz for the long-range XY model is
the full TQR ansatz. Even for the single-layer setup in Fig. 5, the TQR ansatz
can well describe the ground state energy and wavefunction. Furthermore, from
Fig. 9e and 9f, the stability improves with the increase in the number of
layers.

##  X Expressive power of different VQE ansatzes

In the above sections, we find that there are significant differences among
the VQE ansatzes in representing the ground state of the long-range XY model.
Here we will characterize the expressive power of different ansatzes from the
point of view of the entanglement entropy [55, 56]. Consider an $N$-qubit
chain with sites labeled $0,1,2,\ldots,N-1$. We assume open boundary
conditions and label the bonds of the chain by $x$ ($0\leq{x}\leq{N}$). Here
$x=0$ and $x=N$ are two virtual bonds to the left of qubit $0$ and to the
right of qubit $N-1$ respectively. Since the quantum circuits consist of only
unitary gates, the full density matrix $\rho=|\Psi\rangle\\!\langle\Psi|$ is a
pure state. We now consider the entanglement entropy across a single cut of
the bond at position $x$. We first define the reduced density matrix
$\rho_{x}$ by chopping the chain into two parts at $x$ and tracing out the
right-hand side. The entanglement entropy is defined as the von Neumann
entropy for a cut at $x$, which is given by
$S(x)=-\text{Tr}\rho_{x}\log_{2}\rho_{x}$. Note that there are boundary
conditions on the entropy: $S(0)=S(N)=0$.

Figure 10: (a) Half chain entanglement entropy as a function of Zeeman field
$h$ for the exact solution. The entropy has centain values to the left and
right of the critical points $|h|=1$. (b) Half chain entanglement entropy as a
function of number of qubits $N$ (even) for the exact solution and different
VQE ansatzes. The black solid ($\square$) and red solid ($\bigcirc$) lines are
ferromagnetic and paramagnetic entropies for the exact solution of the model.
The blue dotted line is the minimum entropy for all VQE ansatzes. The pink
dashed ($\Diamond$) line is the maximum entropy for the linear and full CNOT
entanglers, the linear CRX and TQR entanglers. The olive triangle
($\bigtriangleup$) line is the maximum entropy for the full CRX and TQR
entanglers. (c) Maximum half chain entanglement entropy as a function of
neighborhood size $r$ for different number of qubits $N$. Note that for $N=2$,
the entropy is equal to 1 only when $r=1$. (d) The ground state energy per
site $E_{\text{GS}}/N$ and fidelity between VQE algorithm and exact solution
as a function of Zeeman field $h$ for different neighborhood $r$. The
parameters are $J=1$, $N=4$.

Figure 11: The growth of entanglement entropy for a chain with length $N=4$.
(a, b) Half chain entanglement entropy as a function of iteration $n$ for the
full CRX and TQR entanglers. (c, d) Entanglement entropy as a function of cut
$x$, for several successive iterations ($n=380,550,910$ for CRX entangler and
$n=20,40,80$ for TQR entangler). This shows that the state evolves from a
product state to a maximally-entangled one.

For even $N$, we first study the half chain entropy $S(N/2)$ as a function of
the Zeeman field $h$ for the long-range XY model as shown in Fig. 10a. It is
found that the entropy remains a constant within the same quantum phase while
has a jump at the phase boundary. The entropy also increases with the number
of qubits $N$ increases. We summarize the paramagnetic (low value) and
ferromagnetic (high value) entropies for different $N$ in Fig. 10a and shown
in the black solid ($\square$) and red solid ($\bigcirc$) lines in Fig. 10b.

We then study the half chain entropy for the VQE ansatzes as shown in Fig.
10b. It is easy to check that the minimum entropy is zero for all ansatzes
since the variational parameters in the ansatzes can be tuned to make the
resulting state become separable. The maximum entropy for the linear and full
CNOT entanglers, and the linear CRX and TQR entanglers is $1$ while $N/2$ for
the full CRX and TQR entanglers which is the largest possible entropy for an
$N$-qubit chain. Therefore, from the entanglement entropy point of view, it is
clear that only the full CRX and TQR entanglers have enough expressive power
to represent the ground state of the long-range XY model. This has been shown
in the previous sections.

We find that the convergence accuracy of the ground state energy can be
improved by gradually introducing the two-qubit rotation gates
$R_{xx}(\alpha_{ij})$ and $R_{yy}(\beta_{ij})$ into the circuits. We first
define the distance of a pair of qubits by $r_{ij}=|i-j|$, where $i,j$ are the
index of qubits. Then we create the entanglers by including the two-qubit
rotation gates $R_{xx}(\alpha_{ij})$ and $R_{yy}(\beta_{ij})$ with the
distance $r_{ij}\leq{r}$. In Fig. 10c, it is found that for a circuit with
number of qubits $N$, the half-chain entropy can reach to the maximal entropy
$N/2$ before the full entangler is applied. Therefore, it is possible to
obtain an acceptable solution without taking into account the full entangler
and this has the advantage of faster simulation time. In Fig. 10d, we plot the
ground state energy and fidelity for $N=4$ and for different neighborhood $r$,
and find that when $r$ goes from $1$ (linear) to $3$ (full), the accuracies of
ground state energy and fidelity are gradually improved. An acceptable
solution for the long-range XY model can be achieved when $r=2$.

We will further interpret the expressive powers of different VQE ansatzes,
especially the full CRX and TQR ansatzes, from a dynamical point of view. In
Fig. 11a and 11b, we study the entanglement entropy growth (half chain
entropy) as a function of the number of iteration for the full CRX and TQR
ansatzes. The optimization object is to maximize the half chain entropy
$S(N/2)$ and is done by the Powell method in the minimize routine of the SciPy
package. It is found that the full TQR ansatz converges about one magnitude
faster than the full CRX ansatz. In Fig. 11c and 11d, we give the growth of
the entanglement entropy as a function of the cut $x$, for several successive
iterations. It is found that the maximal entropy profile,
$S_{\text{max}}(x)=\big{|}\left|x-N/2\right|-N/2\big{|}$ can be reached for
both the full CRX and full TQR ansatzes; however, the full TQR ansatz
approaches faster to the maximally-entangled state than the full CRX ansatz.
In addition, we find that the transient entropy profiles are asymmetric. This
is because the transient step is for a particular realization of the quantum
circuit with circuit parameters randomly distributed in the parameter space.
However, the profile will eventually go to the maximally-entangled state with
a symmetric profile.

##  XI Conclusion

In this paper, we studied the ground state of the long-range XY model by
utilizing the VQE algorithm. Among different VQE ansatzes, i.e., mean-field,
linear and full CNOT, CRX, and TQR ansatzes, we found that the full CRX and
TQR ansatzes can represent the ground state energy of the long-range XY model
well. In contrast, only the full TQR ansatz can represent the ground state
wavefunction with a fidelity close to one. The TQR ansatz can reduce the
circuit depth by increasing the number of variational parameters introduced in
each layer. This is practical for NISQ devices where the noise grows rapidly
with circuit depth and affects the fidelity of the prepared quantum state. We
also found that an acceptable solution can be achieved using restricted-
entanglement ansatzes where entangling gates are applied only between qubits
that are a fixed distance from each other. Using these restricted-entanglement
ansatzes instead of full-entanglement ansatzes allows for faster simulation.
Finally, we discuss the performance of VQE ansatzes from the point of view of
the entanglement entropy and find that the full TQR ansatz is more powerful in
representing the ground state energy and wavefunction of the long-range XY
model than the full CRX ansatz.

## Acknowledgments

We would like to thank Mirko Consiglio and Tony J. G. Apollaro for useful
discussions. The IHPC A*STAR Team acknowledges support from the A*STAR Career
Development Award (C210112010), A*STAR (#21709), National Research Foundation
Singapore (NRF2021-QEP2-02-P01, QEP-SF1, NRF2021-QEP2-02-P03) and A*STAR
Computational Resource Centre through the use of its high performance
computing facilities. Wu gratefully acknowledges the Start-Up Research Grant
from Singapore University of Technology and Design via Grant No. SRG SMT 2021
169, and National Research Foundation Singapore via Grant No.
NRF2021-QEP2-03-P09 and NRF-CRP26-2021-0004. We acknowledge the use of IBM
Quantum services for this work. The views expressed are those of the authors,
and do not reflect the official policy or position of IBM or the IBM Quantum
team.

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