# Sparse LiDAR and Stereo Fusion (SLS-Fusion) for Depth Estimation and 3D
Object Detection

Author Affiliation Country Contact Details

###### Abstract

The ability to accurately detect and localize objects is recognized as being
the most important for the perception of self-driving cars. From 2D to 3D
object detection, the most difficult is to determine the distance from the
ego-vehicle to objects. Expensive technology like LiDAR can provide a precise
and accurate depth information, so most studies have tended to focus on this
sensor showing a performance gap between LiDAR-based methods and camera-based
methods. Although many authors have investigated how to fuse LiDAR with RGB
cameras, as far as we know there are no studies to fuse LiDAR and stereo in a
deep neural network for the 3D object detection task. This paper presents SLS-
Fusion, a new approach to fuse data from 4-beam LiDAR and a stereo camera via
a neural network for depth estimation to achieve better dense depth maps and
thereby improves 3D object detection performance. Since 4-beam LiDAR is
cheaper than the well-known 64-beam LiDAR, this approach is also classified as
a low-cost sensors-based method. Through evaluation on the KITTI benchmark, it
is shown that the proposed method significantly improves depth estimation
performance compared to a baseline method. Also when applying it to 3D object
detection, a new state of the art on low-cost sensor based method is achieved.

#### Keywords:

Autonomous vehicle, 3D object detection, Depth completion, LiDAR stereo
fusion, Pseudo LiDAR

##  1 Introduction

Accurately detecting obstacles in their surroundings is key for safe driving
of an autonomous vehicle. The current top-performance methods in 3D object
detection [1, 2, 3] are mainly based on LiDAR technology. Alternatively,
systems based only on camera sensors also receive much attention because of
their low cost and wide range of use [4, 5, 6, 7, 8, 9, 10] . Even though much
research has focused on camera-based methods, the gap performance between
monocular or stereo-based methods and LiDAR-based methods is still
significant, meriting more research.

The work reported here aims at designing a 3D object detection architecture
from low-cost sensors because we are interested in the practical applications
of using them in self-driving vehicles like an autonomous bus. There are many
3D object detection architectures that will be presented in the next section.
In recent years, a new interesting and promising branch of research proposed
by You et al. [5], namely Pseudo-LiDAR, generates pseudo point clouds based on
the estimated depth map from images using the pinhole camera model. Then, they
can be treated as a LiDAR signal input for any LiDAR-based 3D object detector.
This method has shown a significant improvement in performance compared to
previous image-based object detection methods with this simple conversion.

Real LiDAR sensors can provide depth information with high accuracy in the
form of 3D points. However, these are very sparse point clouds depending on
the properties of the objects in the scene. Hence, point density is normally
uneven and it causes difficulties for recognizing obstacles. Furthermore, in
denser point clouds obtained by the Pseudo-LiDAR pipeline [6], distant or
small objects such as pedestrians and cyclists, for which LiDARs usually
generate less points, can get more points and so be more easily detected and
localized if the predicted depth map from monocular or stereo-based method is
good enough. Finally, another advantage of Pseudo-LiDAR is that it can combine
the state of the art of depth estimation and of 3D object detection.
Therefore, the performance for 3D object detection of this kind of methods
based on this idea strongly depends on the estimated depth map. Instead of
using expensive 64 laser beams which costs about $75,000, You et al. [5]
simulate the 125 times cheaper 4-beam LiDAR Scala by sparsifying the original
64-beam signals and use these points to correct their estimated depth map (by
Stereo Depth Net (SDN)) by applying a graph-based depth correction (GDC).

Starting from this work, a neural network is proposed here that takes 4-beam
LiDAR and stereo images as input. The new method estimates a dense depth map
and then uses it for the 3D object detection task. Unlike [5], taking stereo
images to estimate the depth map and then integrating the 4-beam LiDAR as a
post-processing step to correct the estimated depth map, this approach fuses
the 4-beam LiDAR with stereo images in deep neural network to obtain a more
accurate depth map. Fig. 1 illustrates that when using this method, the
resulting pseudo point cloud is closer to the ground truth than the baseline
Pseudo-LiDAR++ (SDN) method [5].

Figure 1: An example of pseudo point cloud generated by the proposed depth
network and the baseline Pseudo-LiDAR++ (SDN) [5]. The figure shows the 3D
point cloud (green) from bird’s-eye view (BEV) representation corresponding to
a car at 33m in the top picture in the KITTI dataset. They are respectively
point cloud from the proposed method (denoted by Ours$*$) (1), ground truth
(2) and the Pseudo-LiDAR++ (SDN) (denoted by Baseline$*$) (3). The 3D bounding
box (red) denotes the ground truth location of the car.

To summarize, the main contributions in this paper are as follows:

  * •

A Sparse LiDAR and Stereo Fusion network (SLS-Fusion) is proposed, which is
the first method fusing LiDAR and stereo together in a deep neural network to
focus on 3D object detection.

  * •

By integrating the GDC proposed in [5] with the proposed SLS-Fusion,
experimental results on the validation KITTI object detection dataset
demonstrate that the proposed approach can produce very accurate depth maps
thanks to LiDAR-Stereo fusion. This outperforms all previous low-cost based
methods for 3D object detection task.

##  2 Related work

Figure 2: The pipeline of SLS-Fusion. The left and right input stereo images,
$I_{l}$ and $I_{r}$, are fed to two weight-sharing pipelines consisting of
ResNet blocks to encode features shown on the top flow (the same way for the
left and right projected 4-beam LiDAR images, $S_{l}$ and $S_{r}$, on the
bottom flow). In the decoding phase, each LiDAR and image pair are fused,
($S_{l}$, $I_{l}$) and ($S_{r}$, $I_{r}$) and corresponding feature tensors
are obtained, left feature and right feature, for each LiDAR-image input pair.
Then left and right feature tensors go through Depth Cost Volume (DeCV)
proposed in [5] to directly learn the depth.

Depth Completion. Depth completion aims to predict a denser depth map from
image and sparse depth map. Qiu et al. [11] proposed the first study extending
the surface normal as guidance to estimate depth map for outdoor environments.
They combined a single RGB image, LiDAR point cloud and the surface normal to
produce a dense depth map, but their network is very large (around 144 million
parameters), and therefore time consuming needing a powerful architecture for
training.

3D Object Detection. Most 3D object detection methods [1, 2, 3] achieving good
performance are strongly based on LiDAR exploiting its precision. Meanwhile,
methods based only on camera sensors [4, 5, 6, 7, 8, 10, 12] face difficulties
to get precise depth. Chatbot et al. [4] presented one of the first studies in
monocular-based 3D object detection. It is mainly based on detecting 2D
keypoints and finding the best position through 2D and 3D matching. However,
it is time consuming and requires a 3D model dataset including several types
of vehicles. A recent study by Wang et al. [6] showed surprising results by
detecting objects on pseudo point clouds. Some other methods [13, 14] try to
fuse LiDAR point cloud and image. Qi et al. [13] proposed to detect 2D
bounding boxes from a 2D image and further extract corresponding 3D bounding
frustums. Finally, they predict 3D bounding boxes on interior points. However,
its performance heavily relies on the 2D object detector which may fail due to
some drawbacks of the camera such as illumination or ill-posed problem. Sensor
fusion is a hot topic but how to effectively fuse these sensors is difficult
and still needs more research.

From 2D to 3D object detection. Detecting and localizing objects in 3D from 2D
is challenging. Existing approaches usually generate several proposal boxes
from an image or a bird’s eye view projection and then predict the 3D bounding
box. Recently, a new research direction has been proposed in [6] which detects
objects on pseudo LiDAR generated from the estimated depth map by using a
simple conversion. It achieved good results and inspired many other studies
[5, 10, 12, 15]. You et al. [5] proposed SDN network with Depth Cost Volume to
improve depth map accuracy and integrated a depth correction step to
regularize the pseudo point cloud. Wang et al. [15] studied the effect of
foreground and background pixels and they proposed a foreground-background
separation for depth estimation. [10, 12] improved the accuracy of the
predicted depth map by integrating semantics from RGB images.

LiDAR-Stereo Fusion. There have been very few attempts to fuse LiDAR and
stereo in autonomous driving. Previous methods [16, 17, 18] were designed for
the depth completion and, as far as we know, there is no fusion involved for
3D object detection benchmark dataset. Cheng et al. [16] reported existing
noise in LiDAR signals such as misalignment between LiDAR and stereo cameras.
They then proposed an unsupervised learning method to predict the dense depth
map by fusing LiDAR and stereo which guides noise filtering in LiDAR point
cloud. Wang et al. [18] extracted features from LiDAR and stereo images and
proposed the Conditional Cost Volume Normalization. The main aim is to
adaptively regularize cost volume optimization based on LiDAR signals.

The approach proposed here is specifically intended for the depth completion
task and tested on a 3D object detection task. Both Pseudo-LiDAR++ (SDN) [5]
and the proposed SLS-Fusion method use Depth Cost Volume (DeCV) [5] to
directly minimize the depth error instead of the disparity error. Thereby, we
range differences can be avoided between depth $D$ and disparity $d$ value for
each pixel $(u,v)$ expressed through $D(u,v)=\frac{f_{U}\times b}{d(u,v)}$,
where $f_{U}$ and b are respectively the horizontal focal length and the
baseline in the stereo system. Unlike [5], a new backbone is designed that
enables to take left, right RGB images and the image projection of LiDAR. This
paper presents a new approach which fuses stereo images and 4-beam LiDAR and
predict a dense depth map.

##  3 Method

The goal is to detect and localize 3D bounding box of objects by using stereo
RGB images and 4-beam LiDAR. Given left and right sparse depth maps $S_{l}$,
$S_{r}$ generated by projecting the sparse LiDAR point cloud on the left and
right image planes using the calibration parameters and the left and right RGB
image $I_{l}$, $I_{r}$, the Sparse Lidar and Stereo Fusion network estimates
the dense depth map $D$ of the whole image. The overall pipeline of the depth
network is shown in Fig. 2.

###  3.1 Motivation

Wang et al. [6] presented Pseudo LiDAR which improves the performance of the
previous 3D object detection methods based on RGB/D image by changing the
nature of the input. You et al. [5] created their own depth estimation network
to improve the accuracy of the depth map and thereby to increase the 3D object
detection performance. We think that adding LiDAR, which allows to have strong
and accurate depth information into the network to have more features, can
improve the performance of the model. The inexpensive 4-beam LiDAR system, is
integrated instead of using the well-known but expensive 64-beam LiDAR. This
hypothesis has led to the design of the SLS-Fusion network.

###  3.2 Input

To enrich the representation for a normal stereo (RGBs) matching network, a
decision has been taken to join the geometry information from the LiDAR point
cloud. However, instead of directly using a 3D point cloud from LiDAR, like in
[18], the 4-beam LiDAR point cloud is reprojected to both left and right image
coordinates using the calibration parameters to obtain the two sparse 4-beam
LiDAR depth maps corresponding to stereo images. Unlike [18] that used a
simple early fusion paradigm by concating stereo images with their
corresponding sparse LiDAR depth maps, the proposed method uses a late fusion
approach, presented in the following section.

###  3.3 SLS-Fusion Network

The model can be divided into two parts: feature extraction and Depth Cost
Volume [5].

Feature extraction. Qiu et al. [11] proposed a Deep Completion Unit which is
an encoder-decoder network taking a late fusion strategy, where extracted
features from RGB images and LiDAR point cloud are combined in the decoding
phase. Inspired by this work, a network architecture is proposed here that
combines LiDAR and image in a late fusion strategy. To further leverage the
stereo input, as in [5, 19], a weight-sharing pipeline is used for both LiDAR
and image, ($I_{l}$, $S_{l}$) and ($I_{r}$ , $S_{r}$), instead of left and
right images. The encoder consists of a series of ResNet blocks followed by a
convolution layer with a stride of 2 to downsize the feature maps by a ratio
of 1/16 of input to obtain more detailed features for small objects in the
image. To upsize the feature maps and to integrate features from both encoders
for LiDAR and image, an up-projection layer is used [20] for the decoder.
However, unlike [11], only 3 up-projection layers are used to obtain a tensor
for a balance between feature resolution (1/4 of input) and the number of
feature channels to later put into the Depth Cost Volume, which is
computationally expensive.

Depth Cost Volume. The left and right features obtained from the decoding
stage are passed to the Depth Cost Volume (DeCV) proposed by You et al. [5] to
learn the depth. In fact, the two features are fed to Disparity Cost Volume
(DiCV) [19] to form a 4D cost tensor and then it is converted to DeCV to
directly optimize the distance loss instead of the disparity loss. The smooth
L1 loss function from [5] is used.

##  4 Experiments

###  4.1 Experimental Setup

Datasets. Datasets are divided into 2 parts: A) for depth estimation and B)
for 3D object detection.

A) For depth estimation, we take advantage of the diversity of a large scale
dataset, Scene Flow dataset [21], which contains 35,454 training and 4,370
testing images of size $960\times 540$ from an array of synthetic scenes. This
is a large scale synthetic dataset which provides dense and elaborate
disparity maps as ground truth to train the proposed network.

[6, 22] use the training data from KITTI depth completion dataset which
overlaps with KITTI object detection validation part and induces overfitting.
Meanwhile [5, 6] use the LiDAR 64 beams as depth map ground truth for training
their depth network. Instead, we use the KOD dataset provided by Wang et al.
[15], which provides the relation between the KITTI object detection dataset
and the KITTI depth completion dataset. So, for the ground truth, instead of
using LiDAR 64 beams depth maps, we have denser depth maps from KITTI depth
completion dataset which combines 11 frames of LiDAR.

B) For 3D object detection, the proposed approach is evaluated on the KITTI 3D
object detection benchmark [23], which is the most well known dataset for
self-driving cars. It contains 7,481 training samples and 7,518 testing
samples. LiDAR point clouds, RGB images and also camera calibration matrices
are provided as input. Following Chen et al. [24], the training dataset is
split into a training part (3,712 samples) and a validation part (3,769
samples). The benchmark has annotations for 3 classes, which are cars,
pedestrians and cyclists.

Evaluation metrics. For the evaluation of object detection, we report average
precision (AP) with IoU (Intersection over Union) thresholds at 0.5 and 0.7,
as per normal practice. We denote AP for the 3D and bird’s eye view (BEV)
tasks as $AP_{3D}$ and $AP_{BEV}$, respectively. Object detection is divided
into three levels of difficulty: easy, moderate and hard, depending on the 2D
bounding box sizes, occlusion, and truncation extent following the KITTI
definitions. Like [10], experiments are compared to six recent stereo-based
detectors: TLNet [8], Stereo-RCNN [7], DSGN [9], CG-Stereo: PointRCNN [10],
Pseudo-LiDAR: PointRCNN [6], the baseline method Pseudo-LiDAR++: PointRCNN [5]
and also the original LiDAR-based detector PointRCNN [3]. But unlike [10], we
don’t compare with the result of OC-Stereo [12] which used the AVOD [25]
detector while the proposed method uses the PointRCNN [3] detector.

Results are also reported to evaluate the depth network. Following the KITTI
depth completion benchmark, the root mean squared error [mm] (RMSE) is used as
the main metrics. Root mean squared error of the inverse depth [1/km] (iRMSE),
mean absolute error [mm] (MAE) and mean absolute error of the inverse depth
[1/km] (iMAE) are also reported. A comparison is made with the Stereo Depth
Network (SDN), which is the depth network of Pseudo-LiDAR++ (SDN) (You et al.
[5]), to test the advantage of LiDAR feature in the deep network.

Method | Input | 0.5 IoU | 0.7 IoU
---|---|---|---
Easy | Moderate | Hard | Easy | Moderate | Hard
TLNet [8] | S | 62.46 / 59.51 | 45.99 / 43.71 | 41.92 / 37.99 | 29.22 / 18.15 | 21.88 / 14.26 | 18.83 / 13.72
Stereo-RCNN [7] | S | 87.13 / 85.84 | 74.11 / 66.28 | 58.93 / 57.24 | 68.50 / 54.11 | 48.30 / 36.69 | 41.47 / 31.07
Pseudo LiDAR [6] | S | 88.4 / 88.0 | 76.6 / 73.7 | 69.0 / 67.8 | 73.4 / 62.3 | 56.0 / 44.9 | 52.7 / 41.6
DSGN [9] | S | - | - | - | 83.24 / 73.2 | 63.91 / 54.27 | 57.83 / 47.71
CG-Stereo [10] | S |  97.04 / 90.58 | 88.58/ 87.01 | 80.34/ 79.76 | 87.31 / 76.17 | 68.69 / 57.82 | 65.80 / 54.63
Baseline$*$ [5] | S |  89.8 / 89.7 |  83.8 / 78.6 |  77.5 / 75.1 |  82.0 / 67.9 |  64.0 / 50.1 |  57.3 / 45.3
Ours$*$ | L4+S |  90.13 / 89.9 |  83.89 / 78.81 |  78.03 / 76.14 |  82.38 / 68.08 |  65.42 / 50.81 |  57.81 / 46.07
Baseline$**$ [5] | L4+S | 90.3 / 90.3 | 87.7 / 86.9 |  84.6 / 84.2 |  88.2 / 75.1 |  76.9 / 63.8 | 73.4 / 57.4
Ours$**$ | L4+S |  93.16 / 93.02 |  88.81 / 86.19 | 83.35 / 84.02 | 87.51 / 76.67 | 76.88 / 63.90 |  73.55 / 56.78
PointRCNN [3] | L64 | 97.3 / 97.3 | 89.9 / 89.8 | 89.4 / 89.3 | 90.2 / 89.2 | 87.9 / 78.9 | 85.5 / 77.9
Table 1: $AP_{BEV}$/ $AP_{3D}$ results on KITTI validation set for ”Car”
category with IoU at 0.5 and 0.7 and on three levels of difficulty: Easy,
Moderate and Hard. The results are evaluated using the original KITTI metric
with 11 recall positions. All methods are ranked based on the moderately
difficult $AP_{3D}$ results (red). S, L4 and L64 respectively denote stereo,
simulated 4-beam LiDAR and 64-beam LiDAR inputs. Baseline$*$, Baseline$**$,
Ours$*$ and Ours$**$ respectively are Pseudo-LiDAR++ (stereo), Pseudo-LiDAR++
(stereo + LiDAR 4 beam), the proposed method SLS-Fusion (stereo) and SLS-
Fusion (stereo + LiDAR 4 beam).

Implementation Details. The procedure used is quite similar to the common
pipeline of Pseudo-LiDAR.

For the depth estimation, the depth network is implemented in Pytorch and its
backbone is inspired from that of DeepLiDAR. For faster convergence, full
depth maps from Scene Flow dataset are firstly used for training. Here the
LiDAR input is set as zeros because this synthetic dataset is designed
exclusively for RGB stereo and no LiDAR scans are provided. Then fine-tuning
is done on the 3,712 training samples of the KOD dataset for 100 epochs, with
batch size 4 and the learning rate set to 0.001. The Adam optimizer was used
to optimize the network on 2 NVIDIA GeForce GTX 1080 Ti GPUs with 11G memory.
You et al. [5] is followed to generate the simulated 4-beam LiDAR (as similar
as possible to 4-beam LiDAR ScaLa sensor) from 64-beam LiDAR and then project
it on the left and right image planes to feed them into the depth network.

Once the predicted depth map $D$ is obtained, with $D(u,v)$ being the depth
corresponding to each image pixel $(u,v)$, a pseudo point cloud can be
generated by using the pinhole camera model. Given the depth estimate and
camera intrinsic matrix, deriving the 3D location ($X_{c}$, $Y_{c}$, $Z_{c}$)
in the camera coordinate system for each pixel $(u,v)$ is simply as:

$\displaystyle\textrm{(depth) }Z_{c}$ $\displaystyle=D(u,v)$ (1a)
$\displaystyle\textrm{(width) }X_{c}$ $\displaystyle=\frac{(u-c_{U})\times
Z_{c}}{f_{U}}$ (1b) $\displaystyle\textrm{(height) }Y_{c}$
$\displaystyle=\frac{(v-c_{V})\times Z_{c}}{f_{V}}$ (1c)

where $c_{U}$ and $c_{V}$ is the pixel location corresponding to the camera
center; $f_{U}$ and $f_{V}$ is the horizontal and the vertical focal length.
Then this point is transformed into ($X_{l}$, $Y_{l}$, $Z_{l}$) in the LiDAR
coordinate system (the real world coordinate system). Given the camera
extrinsic matrix $C=\begin{bmatrix}R&t\\\ 0&1\end{bmatrix}$, where $R$ and $t$
are respectively the rotation matrix and translation vector. The pseudo point
clouds can be obtained by computing:

$\displaystyle\begin{bmatrix}X_{l}\\\ Y_{l}\\\ Z_{l}\\\
1\end{bmatrix}=C^{-1}\begin{bmatrix}X_{c}\\\ Y_{c}\\\ Z_{c}\\\ 1\end{bmatrix}$
(2)

To make pseudo point cloud more similar to real LiDAR signals, following [6,
26], reflectance is set to 1 for each point and points higher than 1m are
removed. Afterwards, LiDAR-based 3D object detector can be applied on these
points.

For 3D object detector, PointRCNN [3] is applied, which is a LiDAR-based
method with a high performance and used by many other methods as a baseline.
As it was designed to take into account the sparse point cloud, the dense
point cloud is sub-sampled to 64 lines LiDAR. Then, the released
implementations of PointRCNN is used directly and their guidelines followed to
train it on the training set of KITTI object detection dataset only for the
“Car” class because car is one of the main objects and occupies the largest
percentage in KITTI dataset which causes the unbalance between “Car” and other
classes.

Method | RMSE | MAE | iRMSE | iMAE
---|---|---|---|---
(mm) | (mm) | (1/km) | (1/km)
Baseline$*$ [5] | 1506.92 | 443.81 | 5.59 | 1.90
Ours$*$ | 845.21 | 307.36 | 2.72 | 1.28
Table 2: Study on depth estimation. The results show the mean error (shown in
different metrics) for a pixel from all test images of KOD dataset [15]. RMSE
(red) is the main metric to rank methods.

###  4.2 Experimental Results

This section firstly presents an evaluation of the depth estimation accuracy
of the SLS-Fusion network. The results are shown in Tab. 2. The results show
that the proposed network outperforms the baseline Pseudo-LiDAR++ (SDN) [5] by
a large margin over all metrics evaluated on pixels with a valid depth ground
truth in the range [1m, 80m] on the KOD dataset [15]. They support the idea
that adding LiDAR in stereo network can boost the accuracy of the predicted
depth map.

Following the promising results from the predicted map, results for 3D object
detection task are reported in Tab. 1, showing two results for the proposed
approach. Ours$*$ is the base proposed model, while Ours$**$ includes the GDC
proposed in [5]. Pseudo-LiDAR++ is taken as a baseline and therefore in the
same way, Baseline$*$ and Baseline$**$ denote respectively Pseudo-LiDAR++
(SDN) and Pseudo-LiDAR++ (SDN+GDC). The proposed approach is also compared
with six other outstanding methods: Pseudo LiDAR [6], TLNet [8], Stereo-RCNN
[7], DSGN [9], CG-Stereo [10] which are based on low-cost sensors and the
original 3D object detector PointRCNN [3] which uses 64-beam LiDAR. Firstly,
comparing Ours$*$ with the Baseline$*$, the proposed method outperforms the
baseline over all indicators, thereby showing the effect of 4-beam LiDAR in
the proposed stereo matching model. Secondly, Ours$**$ outperforms the
Baseline$**$ and other stereo-based methods on some indicators. The various
results can be explained by the cumbersome of the model or the training way.
The proposed method gets the best result (63.90%) on 3D AP with IoU=0.7 which
is the main indicator to rank all methods on KITTI. Lastly, compared to the
original detector PointRCNN [3] which used 64-beam LiDAR as input, Ours$*$ and
Ours$**$ have inferior performance, which is expected, but it helps to get
closer to the method trained on real LiDAR.

The good results achieved by CG-Stereo [10] on 0.5 IoU (easy and moderate)
could be explained by the method used. In fact, this latter splits the scene
into foreground/ background and applies on each part independent depth
estimation. This leads to easier detection of ”easy” objects in the
foreground. However, we notice that our method is more stable because it
achieves better results on hard objects.

##  5 Conclusion

This paper has introduced SLS-Fusion network, a novel depth prediction
framework taking 4-beam LiDAR and stereo images as input which are low-cost
sensors in the autonomous driving domain, for 3D object detection. To the best
of our knowledge, there are no studies to fuse LiDAR and stereo together in
deep neural network which focus on the 3D object detection task in the
literature. In contrast to existing fusion-based method in other tasks, often
found in indoor scenes, a network is proposed that can take advantage of
features from both LiDAR and images by using a encoder-decoder network and
then optimize the depth loss via a Depth Cost Volume [5] for the road scenes.
Experimental results on the validation KITTI dataset demonstrate that this
model can improve both depth estimation and 3D object detection accuracy.

Perspectives and Future work. We plan to train and submit our results on
testing part (without ground truth) of KITTI object detection dataset to have
a more objective assessment about this approach. We also evaluate results on
other classes such as pedestrian or cyclist and to study the density of pseudo
point cloud instead of sub-sampling pseudo point cloud like in PointRCNN
detector [3].

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]These authors contributed equally to this work.

]These authors contributed equally to this work.

]These authors contributed equally to this work.

# Observation of Emergent $\mathbb{Z}_{2}$ Gauge Invariance in a
Superconducting Circuit

Zhan Wang  [  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100190, China  Zi-Yong Ge  [  Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  School of Physical Sciences, University of Chinese
Academy of Sciences, Beijing 100190, China  Zhongcheng Xiang  [  Beijing
National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  Xiaohui Song  Beijing
National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  Rui-Zhen Huang  Kavli
Institute for Theoretical Sciences, University of Chinese Academy of Sciences,
Beijing 100190, China  Pengtao Song  Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100190, China  Xue-Yi Guo  Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  Luhong Su  Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100190, China  Kai Xu  Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  Songshan Lake Materials Laboratory, Dongguan 523808,
Guangdong, China  CAS Center for Excellence in Topological Quantum
Computation, UCAS, Beijing 100190, China  Dongning Zheng<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190,
China  Songshan Lake Materials Laboratory, Dongguan 523808, Guangdong, China
CAS Center for Excellence in Topological Quantum Computation, UCAS, Beijing
100190, China  Heng Fan<EMAIL_ADDRESS>Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  Songshan Lake Materials Laboratory, Dongguan 523808,
Guangdong, China  CAS Center for Excellence in Topological Quantum
Computation, UCAS, Beijing 100190, China  Beijing Academy of Quantum
Information Sciences, Beijing 100193, China

###### Abstract

Lattice gauge theories (LGTs) are one of the most fundamental subjects in
many-body physics, and has recently attracted considerable research interests
in quantum simulations. Here we experimentally investigate the emergent
$\mathbb{Z}_{2}$ gauge invariance in a 1D superconducting circuit with 10
transmon qubits. By precisely adjusting staggered longitudinal and transverse
fields to each qubit, we construct an effective Hamiltonian containing an LGT
and gauge-broken terms. The corresponding matter sector can exhibit a
localization, and there also exists a 3-qubit operator, of which the
expectation value can retain nonzero for a long time in low-energy regimes.
The above localization can be regarded as the confinement of matter fields,
and the 3-body operator is the $\mathbb{Z}_{2}$ gauge generator. These
experimental results demonstrate that, despite the absence of gauge structure
in the effective Hamiltonian, $\mathbb{Z}_{2}$ gauge invariance can still
emerge in low-energy regimes. Our work provides a method for both
theoretically and experimentally studying the rich physics in quantum many-
body systems with emergent gauge invariance.

Introduction.—Gauge invariance is one of the most fundamental principles of
quantum field theories, and lattice gauge theories (LGTs) [1, 2, 3, 4] play a
significant role in a wide range of modern physics, e.g., helping us
understand the confinement of quarks [1]. Recently, as the rapid development
of quantum simulations [5, 6], studying LGTs in synthetic quantum many-body
systems becomes possible and has drawn many interests from both theoretical
and experimental physicists [7, 8, 9, 10, 11, 12, 13]. On the one hand, many
unresolved problems associated with LGTs are potentially solvable via large-
scale quantum simulations. On the other hand, quantum simulations provide a
new viewpoint to study LGTs, i.e., nonequilibrium dynamics [14, 15, 16, 17].
The corresponding experimental studies have been demonstrated in ultracold
atoms [5, 20, 19].

In condensed matter physics, gauge invariance is generally not a necessary
element and thus absent in the Hamiltonian. However, due to strong quantum
fluctuations of quantum many-body systems, gauge invariance may emerge in low-
energy regimes leading to various novel quantum phases, such as quantum spin
liquid [21, 22, 23] and deconfined quantum critical points [24]. In addition,
it is challenging to construct an exact gauge-invariant Hamiltonian in an
artificial quantum system [9, 5]. Thus one natural question is whether we can
construct a quantum many-body system that the gauge structure is absent in the
Hamiltonian but emergent in the low-energy physics. Moreover, this emergent
gauge invariance may lead to novel dynamics, for instance, the confinement
induced localization of matter degrees of freedom [17, 5], which can be
detected by state-of-the-art quantum simulators.

Figure 1: Set-up and protocol. (a) Optical micrograph of the superconducting
circuit. There are 10 transmon qubits arranged into a chain, where the odd and
even qubits are used for realizing $s$\- and $\tau$-spins in the following
experiment, respectively. Each qubit couples to a microwave line for the $XY$
driving ($\text{XY}_{j}$), a flux bias line for the $Z$ pulse
($\text{Z}_{j}$), and a readout resonator for measurement ($\text{R}_{j}$).
The NN qubits are directly coupled via a capacitor, and the capacitance of
each qubit itself contributes to the NNN coupling. The parameters of the
device are presented in the Supplemental Material (SM) in detail [44]. (b)
Lattice skeleton of the effective Hamiltonian (S2). There are two types of
spins, i.e., $s$ (red sites) and $\tau$ (blue sites), corresponding to matter
and gauge fields, respectively. The purple arrows represent the three-body
couplings, the orange arrows are direct NNN two-body couplings, and the blue
arrows are longitudinal/transverse fields. (c) Pulse sequences of the
experiment. The solid and dashed lines represent XY drivings and Z pulses,
respectively. First, all qubits are at their idle frequencies for preparing
the initial state $\ket{\psi_{0}}$ via XY driving. Then, each qubit is biased
to the working point with $Z$ pulses, and meanwhile, the XY driving is applied
to each even qubit. After the system evolves for time $t$, all qubits are
biased back to idle points for readout.

In this Letter, we experimentally demonstrate the emergent $\mathbb{Z}_{2}$
gauge invariance in a superconducting processor with 10 qubits arranged into a
chain. Following the scheme in Ref. [6], we apply stagger longitudinal and
transverse fields to the system to construct an effective Hamiltonian, which
is a mixture of a $\mathbb{Z}_{2}$ LGT and some gauge-broken terms. To detect
the emergent $\mathbb{Z}_{2}$ gauge invariance, we first study the charge
spreading of the matter sector. The experimental result shows that it can
exhibit a localization under proper parameters, which is a strong dynamical
signature of the confinement induced by emergent gauge invariance. Then,
benefiting from the joint readout of three qubits under the arbitrary basis,
we can study the time evolution of 3-qubit operators. We find that there is a
3-qubit operator, whose expectation value can retain nonzero for a long time.
We demonstrate that this operator can be regarded as the $\mathbb{Z}_{2}$
gauge generator. Therefore this experimental result is a direct evidence of
the emergent $\mathbb{Z}_{2}$ gauge invariance. Furthermore, according to
relations between charge spreading and initial states, we find that the gauge
structure mainly emerges in low-energy regimes.

Model and set-up.—Due to the scalability, long coherent time, and high-
precision full control, superconducting circuits [26, 27] become one of the
most competitive candidates for achieving the universal quantum computation
[28, 29], and they are also a more suitable platform for performing quantum
simulations [35, 34, 30, 31, 33, 36, 32, 38, 37, 39, 40, 41, 42, 2]. Several
theoretical schemes have been proposed to simulate LGTs via specific
superconducting circuits [11, 12]. However, the corresponding experimental
realization is still absent, which may be due to the technological challenges
for preparing such superconducting circuits. Here inspired by the scheme in
Ref. [6], we find that the realization of LGTs is feasible by using
conventional chain like transmon qubits.

This experiment is performed in a superconducting circuit with 10 transmon
qubits ($Q_{1}$–$Q_{10}$) arranged into a chain, see Fig. 1(a). Due to the
large and staggered anharmonicity [44], the system can be described by an
isotropic 1D $XY$ model with tunable transverse and longitudinal fields [5,
20, 19]. In addition, the next-nearest neighbor (NNN) coupling cannot be
neglected in the following experiment. Therefore the Hamiltonian can be
written as

$\displaystyle\hat{H}=$
$\displaystyle\sum_{j}(g_{j}\hat{\sigma}^{+}_{j}\hat{\sigma}^{-}_{j+1}+\lambda_{j}\hat{\sigma}^{+}_{j}\hat{\sigma}^{-}_{j+2}+\text{H.c.})$
$\displaystyle+\sum_{j=1}(\frac{V_{j}}{2}\hat{\sigma}^{z}_{j}+h_{x,j}\hat{\sigma}_{j}^{x}),$
(1)

where $\hat{\sigma}^{\pm}=(\hat{\sigma}^{x}\pm i\hat{\sigma}^{y})/2$,
$\hat{\sigma}^{x,y,z}$ are Pauli matrices, $g_{j}/2\pi\approx 12$ MHz and
$\lambda_{j}/2\pi\approx 0.7-1.1$ MHz are the nearest-neighbor (NN) and NNN
coupling strengths [44], respectively, $V_{j}$ is a longitudinal field tuned
by Z pulses, and $h_{x,j}$ is a transverse field controlled by XY drivings.

Here we let the longitudinal field at each odd qubit $V_{2\ell-1}/2\pi=-80$
MHz and the detuning between NN qubits is much larger than the coupling
strength, i.e., $V_{2\ell}-V_{2\ell-1}\gg g_{j}$. In addition, the transverse
field is only applied at the even qubits with equal strength $h_{x}$. Thus
according to Ref. [6], we can obtain an effective Hamiltonian
$\hat{H}_{\text{eff}}$, which reads

$\displaystyle\hat{H}_{\text{eff}}=\hat{H}_{1}+\hat{H}_{2}+\hat{H}_{3},$ (2)
$\displaystyle\hat{H}_{1}=\sum_{\ell=1}^{4}\tilde{g}_{s,\ell}(\hat{s}^{+}_{\ell}\hat{\tau}^{z}_{\ell+\frac{1}{2}}\hat{s}^{-}_{\ell+1}+\text{H.c.})+\sum_{\ell=1}^{5}h_{x}\hat{\tau}_{\ell+\frac{1}{2}}^{x},$
$\displaystyle\hat{H}_{2}=-\sum_{\ell=1}^{4}\tilde{g}_{\tau,\ell}(\hat{\tau}^{+}_{\ell-\frac{1}{2}}\hat{s}^{z}_{\ell}\hat{\tau}^{-}_{\ell+\frac{1}{2}}+\text{H.c.}),$
$\displaystyle\hat{H}_{3}\\!=\\!\sum_{\ell=1}^{4}(\lambda_{2\ell-1}\hat{s}^{+}_{\ell}\hat{s}^{-}_{\ell+1}\\!+\\!\lambda_{2\ell}\hat{\tau}^{+}_{\ell+\frac{1}{2}}\hat{\tau}^{-}_{\ell+\frac{3}{2}}+\text{H.c.})\\!+\\!\sum_{\ell=1}^{5}h_{z}\hat{\tau}_{\ell+\frac{1}{2}}^{z},$

where $\hat{s}_{\ell}=\hat{\sigma}_{2\ell-1}$ and
$\hat{\tau}_{\ell+\frac{1}{2}}=\hat{\sigma}_{2\ell}$ are also Pauli matrices
labeling the odd and even qubits, respectively, and
$\tilde{g}_{s/\tau,\ell}\approx g_{j}^{2}/\Delta\approx-2\pi\times 1.8$ MHz is
the effective three-body coupling strength, see Fig. 1(b). Here the effective
longitudinal field $h_{z}$ is the summation of the original longitudinal field
of even qubits ($V_{2\ell}$) and the Lamb shift [44]. In the experiment,
$V_{2\ell}$ is adjustable by controlling the detuning between qubit
frequencies and XY driving frequencies.

Here $\hat{H}_{1}$ is nothing but a $\mathbb{Z}_{2}$ lattice gauge field
($\tau$-spins) coupled to a matter field ($s$-spins), where the corresponding
$\mathbb{Z}_{2}$ gauge generator reads
$\hat{G}^{0}_{\ell}=\hat{\tau}^{x}_{\ell-\frac{1}{2}}\hat{s}^{z}_{\ell}\hat{\tau}^{x}_{\ell+\frac{1}{2}}$,
i.e., $[\hat{G}_{\ell}^{0},\hat{H}_{1}]=0$ [4]. However, we can find that
additional terms $\hat{H}_{2}$ and $\hat{H}_{3}$ both violate this gauge
invariance, so the whole Hamiltonian is not a rigorous LGT. However, by
numerical simulation via density matrix renormalization group (DMRG) method
[7, 8] (the details are shown in the SM [44]), we find that a new
$\mathbb{Z}_{2}$ gauge invariance can still emerge in the ground state for a
large transverse field $h_{x}$ and proper longitudinal field $h_{z}$. The
corresponding new gauge generator is

$\displaystyle\hat{G}_{\ell}={\tilde{\tau}}_{\ell-\frac{1}{2}}^{x}\hat{s}_{\ell}^{z}{\tilde{\tau}}_{\ell+\frac{1}{2}}^{x},$
(3)

where $\tilde{\tau}^{x}=\sin\beta\hat{\tau}^{z}+\cos\beta\hat{\tau}^{x}$ and
$\beta=\arctan(h_{z}/h_{x})$.

Next, we will experimentally investigate quench dynamics of this system to
probe the emergent gauge invariance. Here the gauge invariance is almost
independent of the filling of $s$-spins [44]. Thus without loss of generality,
we consider the system containing only one $s$-spin, i.e.,
$\sum_{\ell=1}^{5}\hat{s}_{\ell}^{+}\hat{s}_{\ell}^{-}=1$. The initial state
is chosen as $|\psi_{0}\rangle=|s\rangle\otimes|\tau\rangle$, where
$|s\rangle$ and $|\tau\rangle$ label the states of $s$\- and $\tau$-spins,
respectively. We let $|s\rangle=\ket{00100}$ and
$|\tau\rangle=\ket{\Phi_{\theta}\Phi_{\theta}\Phi_{\theta}\Phi_{\theta}\Phi_{\theta}}$,
where$\ket{\Phi_{\theta}}=\cos\frac{\theta}{2}\ket{1}+\sin\frac{\theta}{2}\ket{0}$.
Here according to the effective Hamiltonian $\hat{H}_{\text{eff}}$, we know
that different $\theta$ means different energy during the quench dynamics. The
experimental procedure can be summarized as follows: First, all qubits are at
their idle frequencies, and we use single-qubit rotational gates to prepare
the initial state $\ket{\psi_{0}}$. Then, we bias the frequency of each qubit
to the corresponding working point with Z pulses, i.e., let the local
potential of each qubit be $V_{j}$. Meanwhile, we apply ac drivings with
amplitude $h_{x}$ to even qubits through XY lines to realize transverse
fields. Finally, after the system evolves with time $t$, we bias back all
qubits to their idle points, and read out the corresponding observable. The
pulse sequence of each qubit is shown in Fig. 1(c).

Figure 2: Spreading of $s$ spins up to $1$ $\mu$s for (a) $h_{x}/2\pi=2$ MHz,
$V_{2\ell}/2\pi=0$ MHz, $\theta=-\pi/3$; (b) $h_{x}/2\pi=6$ MHz,
$V_{2\ell}/2\pi=15$ MHz, $\theta=-\pi/3$; (c) $h_{x}/2\pi=6$ MHz,
$V_{2\ell}/2\pi=15$ MHz, $\theta=-\pi/2$; and (d) $h_{x}/2\pi=6$ MHz,
$V_{2\ell}/2\pi=15$ MHz, $\theta=\pi$. Here the $s$-spin shows localization in
the cases of (b–c) indicating the presence of confinement, while it transports
freely in (a) and (d) indicating the absence of confinement. Each point is the
average of $8,000$ single-shot readouts.

Confinement dynamics.—In 1D $\mathbb{Z}_{2}$ LGTs, the system is generally in
a confined phase. In this case, similar to the confinement of quarks in
particle physics, the presence of a gauge field will induce a long-range
potential of matter fields, e.g., linear potential [48]. This long-range
potential can suppress the transport of matter field and results in a
localization [4, 5]. Thus we can study the dynamics of $s$-spins to
demonstrate whether they can localize, which is a signature for identifying
the existence of emergent gauge invariance. Here we measure the density
distributions of the photon at odd qubits, i.e., the spin-density distribution
of the $s$ sector, defined as

$\displaystyle
P_{j}(t):=\bra{\psi(t)}\hat{\sigma}^{+}_{j}\hat{\sigma}^{-}_{j}\ket{\psi(t)},$
(4)

where $\ket{\psi(t)}=\mathrm{\exp}(-i\hat{H}t)|\psi_{0}\rangle$ is the wave
function of the system at time $t$.

In the case of $h_{x}/2\pi=2$ MHz, $V_{2\ell}/2\pi=0$ MHz, and
$\theta=-\pi/3$, we can find that $s$-spins delocalize and spread to the whole
system very quickly, see Fig. 2(a). However, the situation is different when
$h_{x}/2\pi=6$ MHz, $V_{2\ell}/2\pi=15$ MHz. Figs. 2(b–c) show that $s$ spin
can indeed localize when $\theta=-\pi/3$ or $-\pi/2$, which is an evidence of
existing confinement. When we continue to change the initial state, e.g.,
$\theta=\pi$, the localization of $s$ spins disappears, see Fig. 2(d). These
experimental results indicate that the confinement of $s$ spins can indeed
emerge for the proper transverse and longitudinal fields, and also depend on
the system energy.

Figure 3: Extended imbalance $\mathcal{I}$ versus initial states. (a) Time
evolution of $\mathcal{I}$ with $h_{x}/2\pi=6$ MHz, $V_{2\ell}/2\pi=15$ MHz
and different $\theta$, i.e., cases in Figs. 2(b–c). The dots are experimental
data, while the solid curves are the corresponding numerical results. (b) The
relation between $\mathcal{I}_{\infty}$ and $\theta$. Here,
$\mathcal{I}_{\infty}$ is calculated as the average of $\mathcal{I}$ from 0.2
to 1 $\mu$s, and the standard deviation is the corresponding variance. Using
Gaussian fitting, we obtain the peak of experimental result is at
$\theta=\theta_{m}\approx-0.35$. The numerical results are obtained via
Hamiltonian in Eq. (Observation of Emergent $\mathbb{Z}_{2}$ Gauge Invariance
in a Superconducting Circuit), where decoherence and dephasing are both
neglected.

Next, we further explore the relation between confinement and the system
energy, which can be characterized by the localization strength of $s$ spins.
Here we define the extended imbalance of $s$ spins [42]

$\displaystyle\mathcal{I}:=\sum_{j=\text{odd}}\eta_{j}P_{j},$ (5)

where $\eta_{j}=1/N_{1}\ (-1/N_{0})$ if the initial state of $Q_{j}$ is
$\ket{1}\ (\ket{0})$, and $N_{1}\ (N_{0})$ is the number of $\ket{1}\
(\ket{0})$ for the initial state of the $s$-sector. In Fig. 3(a), we show the
dynamics of extended imbalance for different $\theta$ under the condition of
$h_{x}/2\pi=6$ MHz and $V_{2\ell}/2\pi=15$ MHz. It shows that $\mathcal{I}$
can stabilize at different values indicating the different localization
strength. Now we use the steady value of extended imbalance
$\mathcal{I}_{\infty}$ to quantify the localization strength, which can be
calculated as the average of $\mathcal{I}$ during the last 0.8 $\mu$s. Here
the larger $\mathcal{I}_{\infty}$ means the stronger localization strength. In
Fig. 3(b), the relation between $\mathcal{I}_{\infty}$ and $\theta$ is
presented. We can find that, when $\theta=\theta_{m}\approx-0.35$, the
corresponding $\mathcal{I}_{\infty}$ is the largest indicating the strongest
localization strength in this case. In the following discussion, we will
verify that this is because the emergent gauge invariance can only exist in a
low-energy regime, and the initial state is close to the ground state when
$\theta=\theta_{m}$.

Figure 4: Gauge invariance. (a) The relation between
$\braket{\hat{G}_{3}}_{\infty}$ and $\alpha$ with the parameters being
identical to the cases in Figs. 2(b). Here, $\braket{\hat{G}_{3}}_{\infty}$,
calculated as the average of $\mathcal{I}$ from 0.2 to 1 $\mu$s with the
standard deviation being the error bar, represents the steady expectation
value of 3-qubit operator $\hat{G}_{3}(\alpha)$, defined in Eq. (S10). The
curve is almost a sine-shape curve with the valley at $\alpha\approx-1.19$.
(b) Time evolution of 3-qubit operator $\braket{\hat{G}_{3}(\alpha=-1.19)}$ up
to 1 $\mu$s. It shows that $\braket{\hat{G}_{3}}$ can retain a nonzero value
for long time, where the oscillation is from the high-order term [44].

Gauge invariance.—Now we start to directly study the gauge invariance of the
system. Here when deriving the effective Hamiltonian $\hat{H}_{\text{eff}}$,
the correction of longitudinal field $h_{z}$ from high-order terms can hardly
be directly confirmed accurately. Thus the $\beta$ or $\mathbb{Z}_{2}$ gauge
generator $\hat{G}_{\ell}$ is in fact unknown. However, we can define an
ansatz of $\mathbb{Z}_{2}$ gauge generator as

$\displaystyle\hat{G}_{\ell}(\alpha):=\hat{T}_{\ell-\frac{1}{2}}(\alpha)\hat{s}_{\ell}^{z}\hat{T}_{\ell+\frac{1}{2}}(\alpha),$
(6)

where
$\hat{T}_{\ell-\frac{1}{2}}(\alpha)=\cos(\alpha)\hat{\tau}^{x}_{\ell-\frac{1}{2}}+\sin(\alpha)\hat{\tau}^{z}_{\ell-\frac{1}{2}}$,
so $\hat{T}_{\ell+\frac{1}{2}}(\beta)=\tilde{\tau}^{x}_{\ell-\frac{1}{2}}$ and
$\hat{G}_{\ell}(\beta)=\hat{G}_{\ell}$ is the emergent $\mathbb{Z}_{2}$ gauge
generator defined in Eq. (S5).

If the specific eigenstate has emergent $\mathbb{Z}_{2}$ gauge invariance and
the initial state has a large overlap with this state, then the expectation
value of $\hat{G}_{\ell}$ during the quench dynamics will be nearly time-
independent. Furthermore, the steady expectation value of
$\hat{G}_{\ell}(\alpha)$ during the dynamics approaches the minimum/maximum
when $\alpha=\beta$ [44]. Therefore, based on these considerations, we can
determine $\beta$ and thus fix $\hat{G}_{\ell}$ in the experiment. To measure
the expectation value of $\hat{G}_{\ell}(\alpha)$, we need the joint readout
of adjacent three qubits under the specific basis, which is accessible in
superconducting circuits. We should measure the expectation values of
$\hat{\tau}^{z}_{\ell-\frac{1}{2}}\hat{s}_{\ell}^{z}\hat{\tau}^{z}_{\ell+\frac{1}{2}}$,
$\hat{\tau}^{z}_{j-\frac{1}{2}}\hat{s}_{\ell}^{z}\hat{\tau}^{x}_{\ell+\frac{1}{2}}$,
$\hat{\tau}^{x}_{\ell-\frac{1}{2}}\hat{s}_{\ell}^{z}\hat{\tau}^{z}_{\ell+\frac{1}{2}}$,
and
$\hat{\tau}^{x}_{\ell-\frac{1}{2}}\hat{s}_{\ell}^{z}\hat{\tau}^{x}_{\ell+\frac{1}{2}}$,
respectively. Then, we combine these four values linearly according to Eq.
(S10).

In Fig. 4(a), we show the relation between $\braket{\hat{G}_{3}}_{\infty}$ and
$\alpha$ when $h_{x}/2\pi=6$ MHz, $V_{2\ell}/2\pi=15$ MHz, and
$\theta=-\pi/3$. Here, $\braket{\hat{G}_{3}}_{\infty}$ is the average of
$\braket{\hat{G}_{3}}$ during the last 0.8 $\mu$s representing the steady
value of $\braket{\hat{G}_{3}}$. We find that $\braket{\hat{G}_{3}}_{\infty}$
is $\alpha$-dependent, and it can approach the minimum when
$\alpha=\beta\approx-1.19$. According to Fig. 4(b), we can find that
$\braket{\hat{G}_{3}(-1.19)}$ is stabilized at a nonzero value (The
oscillation is from the high-order term and absent for effective Hamiltonian
$\hat{H}_{\text{eff}}$ [44].). This result provides evidence that some
specific eigenstates of $\hat{H}_{\text{eff}}$ are almost the eigenstates of
$\hat{G}_{\ell}$. That is, these eigenstates of the effective Hamiltonian can
emerge a $\mathbb{Z}_{2}$ gauge invariance.

According to $\hat{H}_{\text{eff}}$, when the external field
$\sqrt{h_{x}^{2}+h_{z}^{2}}$ is much larger than the coupling strength, the
$\tau$ sector will become the leading contribution of the energy, and the
initial state is much close to the ground state when $\theta=-\pi/2-\beta$.
From Figs. 3(b) and 4(a), we can find that $\theta_{m}$ almost equals
$-\pi/2-\beta$, and the localization of the $s$-sector mainly exists at the
vicinity of $\theta_{m}$, i.e., in low-energy regimes. Therefore, we
conjecture that the $\mathbb{Z}_{2}$ gauge invariance mainly emerges in the
low-energy states of $\hat{H}_{\text{eff}}$. In the SM [44], by obtaining wave
functions of all eigenstates of $\hat{H}_{\text{eff}}$ by exact
diagonalization method, we can find that the gauge invariance indeed mainly
emerges in low-energy regimes.

Summary.—In conclusion, we have experimentally investigated the emergent
$\mathbb{Z}_{2}$ gauge invariance in a 10-qubit superconducting processor. Our
experimental results demonstrate that $\mathbb{Z}_{2}$ gauge invariance can
indeed emerge in a low-energy regime, even though the $\mathbb{Z}_{2}$ gauge
structure is absent in the effective Hamiltonian. Moreover, this emergent
gauge invariance can lead to exotic dynamical behaviors, for instance,
confinement-induced localization, which has been observed in this experiment.
Our results can scale up to larger quantum systems and enable the further
study of emergent LGTs in superconducting circuits. For instance, the dynamics
of string breaking [16], the thermalization of the effective Hamiltonian
$\hat{H}_{\text{eff}}$ and whether existing disorder-free many-body
localization [49, 50] are interesting issues. In addition, how to realize a
truly gauge-invariant Hamiltonian on superconducting circuits is an another
relevant question.

###### Acknowledgements.

This work was supported by the State Key Development Program for Basic
Research of China (Grant No. 2017YFA0304300), the Key-Area Research and
Development Program of Guangdong Province, China (Grant No. 2020B0303030001)
the National Natural Science Foundation of China (Grant Nos. T2121001, and
11934018), the Strategic Priority Research Program of Chinese Academy of
Sciences (Grant No. XDB28000000), Scientific Instrument Developing Project of
Chinese Academy of Sciences (Grant No. YJKYYQ20200041), Beijing Natural
Science Foundation (Grant No. Z200009),.

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Supplemental Material
Observation of Emergent $\mathbb{Z}_{2}$ Gauge Invariance in a Superconducting
Circuit

In this Supplemental Material, we mainly present more details about
experiments, including information of the device, corrections of pulses, and
other extended experimental and numerical data.

##  I  Experimental Device

###  I.1 Parameters of the chip

The chip of this experiment is a chain-like superconducting circuit consisting
of 10 Xmon qubits (from left to right of the chain is $Q_{1}$ to $Q_{10}$, see
Fig. 1(a) in the main text), which are fabricated on a 10 mm$\times$10
mm$\times$0.43 mm sapphire substrate with two steps of aluminum deposition
[2].

The detailed parameters of the device are listed in Tab. S1. The frequency of
readout resonator $f_{r}$ increases by about 20 MHz from $Q_{1}$ to $Q_{10}$
distributed from 6.51 GHz to 6.69 GHz. Each Xmon qubit can reach its maximum
frequency $f_{m}$ at the sweetpoint, at which the qubit is insensitive to flux
noise and exhibits long dephasing time. In this experiment, all qubits are
initialized to the ground state ($\ket{0}$) at their idle frequencies $f_{i}$,
which are in the range from 4.934 GHz to 5.692 GHz. The single-qubit gate used
in the experiment is also performed at their idle frequencies. Additionally,
in this experiment, the working points of odd and even qubits are about 5.14
GHz and 5.22 GH, respectively. Due to the different capacitors between odd and
even qubits, anharmonicity $\eta$ of the qubits is staggered. Here, the energy
relaxation time (decoherence time) $T_{1}$ and the dephasing time $T_{2}^{*}$
are both measured at the idle point. We optimize the quadrature correction
term with DRAG coefficient $\alpha$ to minimize leakage to higher levels [3].
Finally, we use the randomized benchmark (RB) method to characterize the error
of $X/2$ and $Y/2$ gate. The nearest neighbor (NN) coupling strength is about
12 MHz, and the next nearest neighbor (NNN) coupling strength between the odd
qubits and between the even qubits are different due to the different
effective capacitance of even and odd qubits.

| $Q_{1}$ | $Q_{2}$ | $Q_{3}$ | $Q_{4}$ | $Q_{5}$ | $Q_{6}$ | $Q_{7}$ | $Q_{8}$ | $Q_{9}$ | $Q_{10}$
---|---|---|---|---|---|---|---|---|---|---
$f_{r}$(GHz) | 6.514 | 6.536 | 6.554 | 6.575 | 6.596 | 6.614 | 6.634 | 6.651 | 6.671 | 6.689
$f_{m}$(GHz) | 5.223 | 5.553 | 5.207 | 5.593 | 5.156 | 5.695 | 5.181 | 5.660 | 5.142 | 5.656
$f_{i}$(GHz) | 5.037 | 5.534 | 4.934 | 5.543 | 5.064 | 5.692 | 5.00 | 5.468 | 5.113 | 5.570
$\eta$(MHz) | -204 | -253 | -206 | -253 | -204 | -249 | -207 | -251 | -205 | -247
$T_{1}(us)$ | 31.8 | 32.8 | 34.8 | 32.2 | 38.9 | 25.9 | 37.0 | 18.8 | 25.5 | 35.5
$T_{2}^{*}(us)$ | 1.688 | 3.326 | 0.685 | 1.83 | 4.539 | 2.034 | 1.242 | 2.197 | 5.657 | 3.809
$F_{ej}$ | 0.897 | 0.874 | 0.865 | 0.916 | 0.880 | 0.932 | 0.908 | 0.845 | 0.888 | 0.893
$F_{gj}$ | 0.970 | 0.958 | 0.966 | 0.989 | 0.959 | 0.984 | 0.976 | 0.968 | 0.956 | 0.979
X/2 Error(%) | 0.15 | 0.71 | 0.25 | 0.62 | 0.15 | 0.30 | 0.20 | 0.20 | 0.05 | 0.30
Y/2 Error(%) | 0.20 | 0.51 | 0.25 | 0.56 | 0.40 | 0.30 | 0.35 | 0.35 | 0.10 | 0.30
| $Q_{1-2}$ | $Q_{2-3}$ | $Q_{3-4}$ | $Q_{4-5}$ | $Q_{5-6}$ | $Q_{6-7}$ | $Q_{7-8}$ | $Q_{8-9}$ | $Q_{9-10}$ | -
$g_{j,j+1}$(MHz) | 12.05 | 12.2 | 11.90 | 11.90 | 11.90 | 11.76 | 11.90 | 12.05 | 12.35 | -
| $Q_{1-3}$ | $Q_{2-4}$ | $Q_{3-5}$ | $Q_{4-6}$ | $Q_{5-7}$ | $Q_{6-8}$ | $Q_{7-9}$ | $Q_{8-10}$ | - | -
$g_{j,j+2}$(MHz) | 1.10 | 0.69 | 1.10 | 0.69 | 1.10 | 0.61 | 1.10 | 0.71 | - | -

Table S1: Basic device parameters. $f_{r}$ is the readout resonator frequency,
$f_{m}$ is the qubit maximum frequency, and $f_{i}$ is the qubit idle
frequency. $\eta$ is the qubit anharmonicity. $T_{1}$ and $T_{2}^{*}$ are the
energy relaxation time and dephasing time of the qubit at idle point. $F_{gj}$
and $F_{ej}$ are the readout fidelities for the ground and first-excited
states, respectively. The errors of $X/2$ and $Y/2$ gates are also presented.
In addition, $g_{j,j+1}$ and $g_{j,j+2}$ are the coupling strengths of
nearest-neighbor (NN) and next-nearest-neighbor (NNN) qubits, respectively.

###  I.2  Experimental setup

In Fig. S1, we present the diagram of the experimental setup. This system is
consisted of some main function boards and some auxiliary boards. The main
function boards, displayed at the left top, includes control board, bias
board, and readout board. The control board is composed of 6 DACs controlled
by FPGA, and is used to realize the full XY control and partial Z control.
DACs together with the nearby microwave source output microwaves for the XY
control of each qubits. The dc-bias wires compose the bias board, which is
used to complete Z control together with the Z wire in control board and bias
tee. The readout board provides the measurement function of qubits. The DAC in
the readout board and the nearby microwave source output a ten-tone microwave
pulse targeting all readout resonators. The readout signal is amplified
sequentially by the Josephson parametric amplifier (JPA) (JPA is unused in
this experiment), high electron mobility transistor (HEMT), and room
temperature amplifiers before demodulated by the ADC. All control lines go
through various stages of attenuation and filter to prevent unwanted noises
from disturbing the operation of the device.

##  II  CALIBRATION

###  II.1 Readout calibration

The qubit readout pulse consists of a 1.6 $\mu$s microwave pulse, which
contains information of all readout resonance. After demodulation by FPGA, we
can obtain IQ data, see Fig. S2. Due to the unwanted noise, readout has
errors. Here, we can use the calibration matrix to calibrate this error, which
reads

$\left(\begin{array}[]{cc}P_{g}^{\text{c}}\\\ P_{e}^{\text{c}}\\\
\end{array}\right)=\left(\begin{array}[]{cc}F_{gj}&1-F_{ej}\\\
1-F_{gj}&F_{ej}\\\ \end{array}\right)\left(\begin{array}[]{cc}P_{g}\\\
P_{e}\\\ \end{array}\right),$ (S1)

where $F_{gj}$ and $F_{ej}$ are readout fidelities of state $\ket{0}$ and
$\ket{1}$ of $Q_{j}$, respectively, see Tab. S1, and $P_{g}^{\text{c}}$
($P_{e}^{\text{c}}$) and $P_{g}$ ($P_{e}$) are calibrated and original
probabilities (calculated directly from IQ data) of $\ket{0}$ ($\ket{1}$),
respectively.

###  II.2 Crosstalk correction of Z pulse

The crosstalk of Z lines will decrease the accuracy of the experiment, and
thus should be corrected. Firstly, we need determine the Z crosstalk matrix
$M_{z}$, which can be calculated by measuring the offset and the compensation
offset of the qubit through different Z lines. The measured Z crosstalk matrix
in this system is shown in Tab. S2. Here, if we applied the Z pulse to each
qubit with strength $Z_{\text{app}}$, then the qubits can actually feel the
strength $Z_{\text{act}}=M_{z}\cdot Z_{\text{app}}$. Thus, we can get the
compensation value through the above formula to calibrate the error caused
from Z crosstalk.

| $Q_{1}$ | $Q_{2}$ | $Q_{3}$ | $Q_{4}$ | $Q_{5}$ | $Q_{6}$ | $Q_{7}$ | $Q_{8}$ | $Q_{9}$ | $Q_{10}$
---|---|---|---|---|---|---|---|---|---|---
$Q_{1}$ | 1.0000 | -0.0032 | 0.0112 | 0.0031 | -0.0012 | 0.0064 | 0.0033 | 0.0037 | 0.0027 | 0.0034
$Q_{2}$ | -0.0023 | 1.0000 | -0.0041 | -0.0033 | 0.0000 | -0.0036 | -0.0020 | -0.0026 | -0.0021 | -0.0024
$Q_{3}$ | -0.0058 | -0.0377 | 1.0000 | -0.0077 | 0.0005 | -0.0075 | -0.0049 | -0.0065 | -0.0052 | -0.0057
$Q_{4}$ | -0.0028 | -0.0068 | -0.0085 | 1.0000 | -0.0029 | -0.0026 | 0.0031 | -0.0027 | -0.0028 | -0.0028
$Q_{5}$ | 0.0051 | 0.0102 | 0.0081 | 0.0297 | 1.0000 | 0.0039 | 0.0050 | 0.0069 | 0.0055 | 0.0049
$Q_{6}$ | 0.0033 | 0.0061 | 0.0040 | 0.0090 | -0.0102 | 1.0000 | 0.0082 | 0.0064 | 0.0045 | 0.0037
$Q_{7}$ | -0.0049 | -0.0085 | -0.0050 | -0.0098 | 0.0060 | 0.0188 | 1.0000 | -0.0220 | -0.0103 | -0.0061
$Q_{8}$ | -0.0030 | -0.0049 | -0.0028 | -0.0053 | 0.0028 | 0.0052 | 0.0010 | 1.0000 | -0.0113 | -0.0042
$Q_{10}$ | 0.0031 | 0.0045 | 0.0024 | 0.0048 | -0.0026 | -0.0053 | -0.0025 | -0.0075 | -0.0072 | 1.0000

Table S2: Z crosstalk matrix $M_{z}$.

###  II.3 Distortion calibration of Z pulse

In the experiment, we need adjust the qubit level by applying a square wave
pulse to the Z line. However, due to the presence of parasitic inductance and
capacitance, an ideal square pulse is usually distorted when reaching to the
chip showing overshoot or undershoot near the rising edge and tailed falling
edge. In Fig. S3, we show the results for distortion calibration of Z pulse.
We can find that, after third order calibration, Z pulse can nearly become a
perfect square wave.

##  III Low-energy physics of the effective Hamiltonian

In this section, we use matrix product state (MPS) based methods to study the
ground state and quench dynamics properties of the effective Hamiltonian,
i.e., Eq. (2) in the main text. Without loss generality, we consider a
homogeneous system, namely, we fix $\tilde{g}_{s/\tau,\ell}=g=1.8$,
$\lambda_{2\ell-1}=\lambda_{s}=1.1$, and $\lambda_{2\ell}=\lambda_{\tau}=0.7$.
Thus, the Hamiltonian reads

$\displaystyle\hat{H}_{\text{eff}}=\hat{H}_{1}+\hat{H}_{2}+\hat{H}_{3},$ (S2)
$\displaystyle\hat{H}_{1}=-\sum_{\ell=1}g(\hat{s}^{+}_{\ell}\hat{\tau}^{z}_{\ell+\frac{1}{2}}\hat{s}^{-}_{\ell+1}+\text{H.c.})+\sum_{\ell=1}h_{x}\hat{\tau}_{\ell+\frac{1}{2}}^{x},$
$\displaystyle\hat{H}_{2}=\sum_{\ell=1}g(\hat{\tau}^{+}_{\ell-\frac{1}{2}}\hat{s}^{z}_{\ell}\hat{\tau}^{-}_{\ell+\frac{1}{2}}+\text{H.c.}),$
$\displaystyle\hat{H}_{3}\\!=\\!\sum_{\ell=1}(\lambda_{s}\hat{s}^{+}_{\ell}\hat{s}^{-}_{\ell+1}\\!+\\!\lambda_{\tau}\hat{\tau}^{+}_{\ell+\frac{1}{2}}\hat{\tau}^{-}_{\ell+\frac{3}{2}}+\text{H.c.})\\!+\\!\sum_{\ell=1}h_{z}\hat{\tau}_{\ell+\frac{1}{2}}^{z}.$

Thus, there are only two driving parameters, i.e., $h_{x}$ and $h_{z}$. Here,
we note that $h_{z}$ has two parts: one is original longitudinal fields of
$\tau$-spins (even qubits), i.e., $V_{2\ell}$ in Eq.(1) of main text, and the
other is the high-order correction from Schrieffer-Wolf transformation, i.e.,
Lamb shift. In Ref. [6], $h_{z}$ is fixed to $0$ by choosing a proper
$V_{2\ell}$, but we let $h_{z}$ as a driving parameter in this experiment. The
Lamb shift can hardly be confirmed accurately, since we cannot calculate all
orders of Schrieffer-Wolf transformation. However, $V_{2\ell}$ is adjustable
in the experiment, which is the detuning between the corresponding qubit
frequencies and XY driving frequencies.

Here, we can find that $\hat{H}_{1}$ is a typical $\mathbb{Z}_{2}$ LGT coupled
with a matter field [5, 4], where $\tau$ and $s$ are gauge and matter fields,
respectively, and the transverse field $h_{x}$ is the corresponding
$\mathbb{Z}_{2}$ electric field. The $\mathbb{Z}_{2}$ gauge transformation can
be defined as
$\hat{G}^{0}_{\ell}=\hat{\tau}_{j-\frac{1}{2}}^{x}\hat{s}_{j}^{z}\hat{\tau}_{j+\frac{1}{2}}^{x}$
satisfying $[\hat{G}^{0}_{\ell},\hat{H}_{1}]=0$. However, $\hat{H}_{2}$ and
$\hat{H}_{3}$ both lack this $\mathbb{Z}_{2}$ invariance, i.e.,
$[\hat{G}^{0}_{\ell},\hat{H}_{2}]\neq 0$ and
$[\hat{G}^{0}_{\ell},\hat{H}_{3}]\neq 0$ Therefore, the whole effective
Hamiltonian $\hat{H}_{\text{eff}}$ is not $\mathbb{Z}_{2}$ gauge invariant.

To analyze the emergent gauge invariance of $\hat{H}_{\text{eff}}$, we map the
$\tau$-spin (gauge field) to another frame, i.e., performing the following
replacement

$\displaystyle\tilde{\tau}^{x}_{\ell+\frac{1}{2}}=\cos\beta\hat{\tau}^{x}_{\ell+\frac{1}{2}}+\sin\beta\hat{\tau}^{z}_{\ell+\frac{1}{2}},$
(S3)
$\displaystyle\tilde{\tau}^{y}_{\ell+\frac{1}{2}}=\hat{\tau}^{y}_{\ell+\frac{1}{2}},$
$\displaystyle\tilde{\tau}^{z}_{\ell+\frac{1}{2}}=\cos\beta\hat{\tau}^{z}_{\ell+\frac{1}{2}}-\sin\beta\hat{\tau}^{x}_{\ell+\frac{1}{2}},$

where $\beta=\arctan(h_{x}/h_{z})$. Thus, the effective Hamiltonian
$\hat{H}_{\text{eff}}$ can be transformed as

$\displaystyle\hat{H}_{\text{eff}}=\tilde{H}_{1}+\tilde{H}_{2}+\tilde{H}_{3},$
(S4)
$\displaystyle\tilde{H}_{1}=-\sum_{\ell=1}g\cos\beta(\hat{s}^{+}_{\ell}\tilde{\tau}^{z}_{\ell+\frac{1}{2}}\hat{s}^{-}_{\ell+1}+\text{H.c.})+\sum_{\ell=1}\tilde{h}\tilde{\tau}_{\ell+\frac{1}{2}}^{x},$
$\displaystyle\tilde{H}_{2}=\sum_{\ell=1}\hat{s}^{+}_{\ell}(\lambda_{s}-g\sin\beta\tilde{\tau}^{x}_{\ell+\frac{1}{2}})\hat{s}^{-}_{\ell+1}+\text{H.c.},$
$\displaystyle\tilde{H}_{3}=\frac{g}{2}\sum_{\ell=1}[\tilde{\tau}^{y}_{\ell-\frac{1}{2}}(\lambda_{\tau}-\hat{s}^{z}_{\ell})\tilde{\tau}^{y}_{\ell+\frac{1}{2}}\\!+\\!\cos^{2}\\!\beta\tilde{\tau}^{x}_{\ell-\frac{1}{2}}(\lambda_{\tau}-\hat{s}^{z}_{\ell})\tilde{\tau}^{x}_{\ell+\frac{1}{2}}$
$\displaystyle\\!+\\!\sin^{2}\\!\beta\tilde{\tau}^{z}_{\ell-\frac{1}{2}}(\lambda_{\tau}-\hat{s}^{z}_{\ell})\tilde{\tau}^{z}_{\ell+\frac{1}{2}}\\!+\\!\sin\\!\beta\cos\\!\beta\tilde{\tau}^{x}_{\ell-\frac{1}{2}}(\lambda_{\tau}-\hat{s}^{z}_{\ell})\tilde{\tau}^{z}_{\ell+\frac{1}{2}}$
$\displaystyle+\sin\beta\cos\beta\tilde{\tau}^{z}_{\ell-\frac{1}{2}}(\lambda_{\tau}-\hat{s}^{z}_{\ell})\tilde{\tau}^{x}_{\ell+\frac{1}{2}}],$

where $\tilde{h}=\sqrt{h_{x}^{2}+h_{z}^{2}}$. Similarly, we can find that
$\tilde{H}_{1}$ is also a $\mathbb{Z}_{2}$ LGT with gauge generator

$\displaystyle\hat{G}_{\ell}={\tilde{\tau}}_{\ell-\frac{1}{2}}^{x}\hat{s}_{\ell}^{z}{\tilde{\tau}}_{\ell+\frac{1}{2}}^{x}.$
(S5)

However, $\tilde{H}_{2}$ and $\tilde{H}_{3}$ violate such $\mathbb{Z}_{2}$
gauge invariance.

Now we discuss the ground state of $\hat{H}_{\text{eff}}$. When $\tilde{h}\gg
g$, $\tilde{\tau}$ sector is almost polarized in the $\tilde{\tau}^{x}$
channel under the ground state. Thus,
$\braket{\tilde{\tau}_{\ell+\frac{1}{2}}^{x}}\neq 0$, where
$\langle\cdot\rangle$ represents taking expectation value towards the ground
state. Then, using mean-field approximation,

$\displaystyle\tilde{H}_{2}\approx\sum_{\ell=1}\hat{s}^{+}_{\ell}(\lambda_{s}-g\sin\beta\braket{\tilde{\tau}_{\ell+\frac{1}{2}}^{x}})\hat{s}^{-}_{\ell+1}+\text{H.c.}.$
(S6)

Under the specific fields $h_{x}$ and $h_{z}$, we can have
$\lambda_{s}-g\sin\beta\braket{\tilde{\tau}_{\ell+\frac{1}{2}}^{x}}=0$, i.e.,
$\tilde{H}_{2}=0$. For $\tilde{H}_{3}$, we take the first term, i.e.,
$\tilde{h}_{1,\ell}=\tilde{\tau}^{y}_{\ell-\frac{1}{2}}(\lambda_{\tau}-\hat{s}^{z}_{\ell})\tilde{\tau}^{y}_{\ell+\frac{1}{2}}$,
as an example to show how this term to vanish in terms of low-order
perturbations [6]. We can use the following phenomenological description [6]:
As shown in Fig. S4, under the action of $\tilde{H}_{1}$, the system can
firstly have two imaginary processes to make a $s$-spin hop to the NNN
$s$-site by flipping two $\tau$-spins. Then, these two $\tau$-spins can flip
back through the term $\tilde{h}_{1,\ell}$. Here, the energy after the whole
process conserves. Thus, the term $\tilde{h}_{1,\ell}$ can contribute at the
third order perturbation. Following this picture, we can find that the
dominating term of this system is $\tilde{H}_{1}$, which is $\mathbb{Z}_{2}$
gauge invariant.

We also note that, for the small $h_{x}$ or high initial energy, since
$\tau$-spins may not be polarized, the relation
$\lambda_{s}-g\sin\beta\langle{\tilde{\tau}_{\ell+\frac{1}{2}}^{x}}\rangle=0$
is broken. Thus, $\tilde{H}_{2}$ cannot be ignored, so the gauge invariance
will not exist in these cases.

To verify the emergent $\mathbb{Z}_{2}$ gauge invariance of
$\hat{H}_{\text{eff}}$ in the ground state, numerically, we define the
$\mathbb{Z}_{2}$ charge

$\displaystyle\tilde{W}_{s_{z}}(i,j):=\prod_{i\leq k\leq j}\tilde{s}_{k}^{z},$
(S7)

and the flux

$\displaystyle\tilde{C}_{\tau_{x}}(i,j):=\tilde{\tau}_{i-\frac{1}{2}}^{x}\tilde{\tau}_{j+\frac{1}{2}}^{x}.$
(S8)

We know that $\tilde{W}_{s_{z}}(i,j)=\pm\tilde{C}_{\tau_{x}}(i,j)$ must be
satisfied rigorously for $\mathbb{Z}_{2}$ gauge invariant systems (e.g.,
$\tilde{H}_{1}$), which is Gauss law of 1D $\mathbb{Z}_{2}$ LGT. Now, we first
use density matrix renomalized group (DMRG) method to calculate
$\mathbb{Z}_{2}$ charge and flux at the ground sate. Here, open boundary
condition is used for the numerical simulation. According to the above
picture, we can find that the emergent Gauss’s law mainly originates from the
polarization of $\tau$-spins. Thus, the filling of $s$-spins can hardly affect
the effective Hamiltonian, i.e., the effective Hamiltonian $\tilde{H}_{1}$ is
valid for the arbitrary filling factor. Without loss of generality, we set the
$s$-spins to be half-filling, i.e., $\sum_{j=1}^{L}\hat{s}_{j}^{z}=0$. From
Figs. S5(a-c), one can find, in our system,

$\displaystyle\langle{\tilde{C}_{\tau_{x}}(i,j)}\rangle\simeq(-1)^{f(i,j)}\langle{\tilde{W}_{s_{z}}(i,j)}\rangle,$
(S9)

when $h_{x}=6$ and $h_{z}=-4.45$, i.e., $\beta\approx-0.638$. Here,
$f(i,j)=\frac{(j-i+1)(j+i+2)}{2}$ is an integer. Therefore, the effective
Hamiltonian $\hat{H}_{\text{eff}}$ can indeed emerge $\mathbb{Z}_{2}$ gauge
invariance at the ground state under proper transverse and longitude field,
although the $\mathbb{Z}_{2}$ gauge invariance is absent in
$\hat{H}_{\text{eff}}$. In Figs. S5(d), we present the expectation values of
$\mathbb{Z}_{2}$ flux and charge for all eigenstates of $\hat{H}_{\text{eff}}$
at half-filling regime and periodic boundary condition. We can find that
$\mathbb{Z}_{2}$ gauge invariance is almost absent in high excited state.

Now we use time evolving block decimation (TEBD) method [9] to study the
quench dynamics of $\hat{H}_{\text{eff}}$. Here, we choose second-order
Suzuki-Trotter decomposition. We also enlarge the maximum bond dimension and
decrease time of single step till the final results converge. We set the total
$s$-charge to $\sum_{j=1}^{L}\hat{s}_{j}^{+}\hat{s}_{j}^{-}=1$. The initial
state is chosen as $|\psi_{0}\rangle=|s\rangle\otimes|\tau\rangle$, where
$|s\rangle$ and $|\tau\rangle$ label the states of $s$ and $\tau$ sectors,
respectively. We let
$|s\rangle=\ket{...\uparrow\uparrow\downarrow\uparrow\uparrow...}$ and
$|\tau\rangle=\ket{...\Phi_{\theta}\Phi_{\theta}\Phi_{\theta}\Phi_{\theta}\Phi_{\theta}...}$,
where$\ket{\Phi_{\theta}}=\cos\frac{\theta}{2}\ket{\uparrow}+\sin\frac{\theta}{2}\ket{\downarrow}$.
Thus, this initial state is closely related to the experiment in maintext. In
addition, we know that, when $\theta=-\pi/2-\beta$, the system is much close
to the ground state.

Firstly, we calculate the time evolution of the extended imbalance of
$s$-spins defined in Eq. (5) of main text. According to Fig. S6(a), we can
find $s$-sector can exhibit a localization in a short time regime when
$h_{x}=6$ and $h_{z}=-4.45$, and localization strength depends on the initial
state. Fig. S6(b) shows that the localization strength of $s$-sector approach
the strongest when $\theta\approx-\pi/2-\beta$. Now we study the time
evolution of $\mathbb{Z}_{2}$ gauge generator $\hat{G}_{\ell}$. Similar to the
main text, we define an ansatz of $\mathbb{Z}_{2}$ gauge generator as

$\displaystyle\hat{G}_{\ell}(\alpha):=\hat{T}_{\ell-\frac{1}{2}}(\alpha)\hat{s}_{\ell}^{z}\hat{T}_{\ell+\frac{1}{2}}(\alpha),$
(S10)

where
$\hat{T}_{\ell-\frac{1}{2}}(\alpha)=\cos(\alpha)\hat{\tau}^{z}_{\ell-\frac{1}{2}}+\sin(\alpha)\hat{\tau}^{x}_{\ell-\frac{1}{2}}$,
so $\hat{T}_{\ell+\frac{1}{2}}(\beta)=\tilde{\tau}^{x}_{\ell-\frac{1}{2}}$ and
$\hat{G}_{\ell}(\beta)=\hat{G}_{\ell}$ is the emergent $\mathbb{Z}_{2}$ gauge
generator. According to Figs. S6(c–d), we can find that the stead value of
$\hat{G}_{\ell}(\alpha)$ indeed approach the minimum when $\alpha=\beta$.
These quench dynamics is completely consistent with our experimental results.
Here we note that there is nearly no oscillation during the dynamics of
$\hat{G}_{\ell}(\beta)$ for $\hat{H}_{\text{eff}}$ [see Fig. S6(d)], which is
distinct to the corresponding experimental results [see Fig. 4(b) in main
text]. Thus, we conjecture that this oscillation of the dynamics for original
Hamiltonian results from the high-order term.

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Figure S1: Diagram of the experimental setup. The left bottom is the
electronic control instrument system.  Figure S2: 10 qubit Readout. The red
dots means the measurement state is in $|1\rangle$ and the bule dots means the
measurement state is in $|0\rangle$ Figure S3: Z pulse calibration. Figure S4:
Diagram of the action of $\tilde{H}_{1}$ and $\tilde{h}_{1,\ell}$. Figure S5:
Expectation values of $\mathbb{Z}_{2}$ charge and the flux at ground state
with $h_{x}=-6$ and (a) $h_{z}=0$, (b) $h_{z}=-4.45$, (c) $h_{z}=-6$. The
results are calculated via density matrix renomalized group (DMRG) method [7,
8] with $L=80$. (d) Expectation values of $\mathbb{Z}_{2}$ charge and the flux
for all eigenstates with $h_{z}=-4.45$. The result is calculated vis exact
diagonalization method with $L=6$. The inset is shows the $\mathbb{Z}_{2}$
charge and the flux for low-energy eigenstates.  Figure S6: Quench dynamics of
$\hat{H}_{\text{eff}}$ with $h_{x}=-6$, $h_{z}=-4.45$ and $L=21$. (a) Time
evolution of the extended imbalance of $s$-spins for different initial states.
(b) The relation between stead values of the extended imbalance
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average of $\mathcal{I}(t)$ for $t\in[4,8]$. (c) The stead values of
$\hat{G}_{11}(\alpha)$, and $\braket{\hat{G}_{11}(\alpha)}_{\infty}$ is also
the average of $\braket{\hat{G}_{11}(\alpha)}(t)$for $t\in[4,8]$. (d) Time
evolution of $\mathbb{Z}_{2}$ gauge generator
$\hat{G}_{\ell}(\alpha=\beta=-6.38)$, and there is nearly no oscillation.
Here, the results are calculated via TEBD method.


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