# CDDM: Channel Denoising Diffusion Models for Wireless Communications

Tong Wu, Zhiyong Chen, Dazhi He, Liang Qian, Yin Xu, Meixia Tao, Wenjun Zhang
Cooperative Medianet Innovation Center, Shanghai Jiao Tong University,
Shanghai, China
Email: {wu_tong, zhiyongchen, hedazhi, lqian, xuyin, mxtao,
<EMAIL_ADDRESS>
###### Abstract

Diffusion models (DM) can gradually learn to remove noise, which have been
widely used in artificial intelligence generated content (AIGC) in recent
years. The property of DM for removing noise leads us to wonder whether DM can
be applied to wireless communications to help the receiver eliminate the
channel noise. To address this, we propose channel denoising diffusion models
(CDDM) for wireless communications in this paper. CDDM can be applied as a new
physical layer module after the channel equalization to learn the distribution
of the channel input signal, and then utilizes this learned knowledge to
remove the channel noise. We design corresponding training and sampling
algorithms for the forward diffusion process and the reverse sampling process
of CDDM. Moreover, we apply CDDM to a semantic communications system based on
joint source-channel coding (JSCC). Experimental results demonstrate that CDDM
can further reduce the mean square error (MSE) after minimum mean square error
(MMSE) equalizer, and the joint CDDM and JSCC system achieves better
performance than the JSCC system and the traditional JPEG2000 with low-density
parity-check (LDPC) code approach.

##  I Introduction

In machine learning, diffusion models (DM)[1, 2, 3] have achieved
unprecedented success in artificial intelligence generated content (AIGC)
recently, including multimodal image generation and edition [4, 5], text, and
video generation[6, 7]. DM gradually adds Gaussian noise to the available
training data in the forward diffusion process until the data becomes all
noise. Then, in the reverse sampling process, it learns to recover the data
from the noise, as shown in Fig. 1. Generally, given a data distribution
$\mathbf{x}_{0}\sim q(\mathbf{x}_{0})$, the forward diffusion process can
generate the $t$-th sample of $\mathbf{x}_{t}$ by sampling a Gaussian vector
$\epsilon\sim\mathcal{N}(0,\mathbf{I})$ as following

$\displaystyle\mathbf{x}_{t}=\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0}+\sqrt{1-\bar{\alpha}_{t}}\epsilon,$
(1)

where $\bar{\alpha}_{t}={\textstyle\prod_{i=1}^{t}}\alpha_{i}$ and
$\alpha_{i}\in(0,1)$ is a hyperparameter.

In wireless communications, it is well known that the received signal $y$ is a
noisy and distorted version of the transmitted signal $x$, e.g., we have the
following for the additive white Gaussian noise (AWGN) channel

$\displaystyle y=x+n,$ (2)

where $n$ is a white Gaussian noise.

Interestingly, compared to (1) and (2), we can find that the design approach
of DM and wireless communications systems are similar. DM can gradually learn
to remove noise, while the receiver in the wireless communications system is
to recover the transmitted signal from the received signal. Clearly, can DM be
applied to the wireless communications system to help the receiver remove
noise? To the best of our knowledge, there have been no related works in the
literature that address this question.

Figure 1: The forward diffusion process with transition kernel
$q(\mathbf{x}_{t}|\mathbf{x}_{t-1})$ and the reverse sampling process with
learnable transition kernel $p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t})$ of
diffusion model in [2].

Motivated by this, we propose channel denoising diffusion models (CDDM) for
wireless communications in this paper. CDDM can be applied as a new module
after channel equalization to predict the channel noise and eliminate it,
thereby enhancing the performance. We design the forward diffusion process
based on the conditional distribution of the received signal after channel
equalization (or without channel equalization) under Rayleigh fading channel
(or AWGN channel). We design the corresponding training algorithm that solely
relies on the forward diffusion process without any requirement of the
received signal. The forward diffusion process also prompts us to design a
sampling algorithm to achieve channel noise elimination.

Furthermore, we apply the CDDM to a semantic communications system based on
joint source-channel coding (JSCC) technique for wireless image transmission,
where the signal after CDDM is fed into the JSCC decoder to recover the image.
We test the mean square error (MSE) between the transmitted signal and the
received signal after CDDM, and find that compared to the system without CDDM,
the system with CDDM has smaller MSE performance both for Rayleigh fading
channel and AWGN channel. This fact indicates that the proposed CDDM can
effectively reduce the impact of channel noise through learning. The
experimental results show that the joint CDDM and JSCC method outperforms both
the JSCC method and the traditional JPEG2000 with low-density parity-check
(LDPC) code approach in terms of the peak signal-to-noise ratio (PSNR) of the
images.

###  I-A Related Works

Compared to the prosperous researches of DM in AIGC, there are few works of DM
in wireless communications so far. In [8], DM is used to generate the wireless
channel for a end-to-end communications system, and has almost the same
performance as the channel-aware case. In [9], DM with an adapted diffusion
process is proposed for the decoding of algebraic block codes.

In recent years, semantic communications [10, 11] have emerged as a new
paradigmatic approach, characterized by its core idea of JSCC [12, 13, 14,
15], which considers the source and channel processes integrally based on deep
neural network [12]. Most studies on JSCC have designed specific JSCC
frameworks for different data modals and achieved better performance compared
with traditional wireless transmission schemes. In [13], a novel JSCC method
based on attention mechanisms is proposed, which can automatically adapt to
various channel conditions. [14] introduces an adaptive deep learning based
JSCC architecture for semantic communications. In [15], the Swin Transformer
[16] is integrated into the deep JSCC framework to improve the performance of
wireless image transmission. In summary, there have been no publications in
literature regarding the joint design of DM and JSCC over wireless
communications.

##  II Channel Denoising Diffusion Model

In this section, we describe the proposed CDDM which is placed after the
channel equalization as shown in Fig. 2. CDDM is trained using a specialized
noise schedule adapted to the wireless channel, which enables it to
effectively eliminate channel noise through a designed sampling algorithm.

###  II-A Conditional Distribution of the received signals

Figure 2: Architecture of the joint CDDM and JSCC system.

Let $\mathbf{x}\in\mathbb{R}^{2k}$ be the real-valued symbols. Here, $k$ is
the number of channel uses. $\mathbf{x_{c}}\in\mathbb{C}^{k}$ are the complex-
valued symbols which can be transmitted through the wireless channel, and the
$i$-th transmitted symbol of $\mathbf{x_{c}}$ can be expressed as
${x_{c,i}}={x_{i}}+jx_{i+k}$, for $i=1,...,k.$

Thus, the $i$-th received symbol of the received signal $\mathbf{y_{c}}$ is

$\displaystyle y_{c,i}=h_{c,i}x_{c,i}+n_{c,i}$ (3)

where $h_{c,i}\in\mathbb{CN}(0,1)$ are independent and identically distributed
(i.i.d.) Rayleigh fading gains, $x_{c,i}$ has a power constraint
$\mathbb{E}[|x_{c,i}|^{2}]\leq 1$, and $n_{c,i}\in\mathbb{CN}(0,2\sigma^{2})$
are i.i.d. AWGN samples.

In this paper, we use minimum mean square error (MMSE) as an equalizer.
$\mathbf{y_{c}}$ is then addressed by equalization as
$\mathbf{y_{eq}}\in\mathbb{C}^{k}$, following a normalization-reshape module
outputing a real vector $\mathbf{y_{r}}\in\mathbb{R}^{2k}$. We consider that
the receiver can obtain the channel state
$\mathbf{h_{c}}=[h_{c,1},...,h_{c,k}]$ through channel estimation. Therefore,
we can have the conditional distribution of $\mathbf{y_{r}}$ with known
$\mathbf{x}$ and $\mathbf{h_{c}}$, which can be formulated to instruct the
forward diffusion and reverse sampling processes of CDDM.

######  Proposition 1.

With MMSE, the conditional distribution of $\mathbf{y_{r}}$ with known
$\mathbf{x}$ and $\mathbf{h_{c}}$ under Rayleigh fading channel is

$\displaystyle
p(\mathbf{y_{r}}|\mathbf{x},\mathbf{h_{c}})\sim\mathcal{N}(\mathbf{y_{r}};\frac{1}{\sqrt{1+\sigma^{2}}}\mathbf{W_{s}}\mathbf{x},\frac{\sigma^{2}}{{1+\sigma^{2}}}\mathbf{W}^{2}_{\mathbf{n}})$
(4)

where $\mathbf{H_{r}}=diag({\mathbf{h_{r}}})$,
$\mathbf{h}_{\mathbf{r}}=\begin{bmatrix}|\mathbf{h_{c}}|\\\
|\mathbf{h_{c}}|\end{bmatrix}\in\mathbb{R}^{2k}$, and

$\displaystyle\mathbf{W_{s}}=\mathbf{H}^{2}_{\mathbf{r}}(\mathbf{H}^{2}_{\mathbf{r}}+2\sigma^{2}\mathbf{I})^{-1},\mathbf{W_{n}}=\mathbf{H_{r}}(\mathbf{H}^{2}_{\mathbf{r}}+2\sigma^{2}\mathbf{I})^{-1}.$
(5)

###### Proof:

Based on the defination, $\mathbf{W_{s}}$ and $\mathbf{W_{n}}$ are diagonal
matrix, where the $i$-th and ($i+k$)-th diagonal element are

$\displaystyle{W_{s,i}}={W_{s,i+k}}=\frac{|h_{c,i}|^{2}}{|h_{c,i}|^{2}+2\sigma^{2}},$
$\displaystyle{W_{n,i}}={W_{n,i+k}}=\frac{|h_{c,i}|}{|h_{c,i}|^{2}+2\sigma^{2}}.$
(6)

The $i$-th output of MMSE ${y_{eq,i}}$ can be expressed as

$\displaystyle{y_{eq,i}}=\frac{|h_{c,i}|^{2}x_{c,i}+h_{c,i}^{H}n_{c,i}}{|h_{c,i}|^{2}+2\sigma^{2}}.$
(7)

Based on (II-A), we have

$\displaystyle\frac{|h_{c,i}|^{2}x_{c,i}}{|h_{c,i}|^{2}+2\sigma^{2}}={W_{s,i}}x_{c,i}.$
(8)

With the resampling trick, the conditional distributions of real part and
imaginary part of $\frac{h_{c,i}^{H}n_{c,i}}{|h_{c,i}|^{2}+2\sigma^{2}}$ are

$\displaystyle
p(Re(\frac{h_{c,i}^{H}n_{c,i}}{|h_{c,i}|^{2}+2\sigma^{2}})|h_{c,i})$
$\displaystyle\sim\mathcal{N}(0,{\sigma^{2}}(\frac{|h_{c,i}|}{|h_{c,i}|^{2}+2\sigma^{2}})^{2})$
$\displaystyle=\mathcal{N}(0,{\sigma^{2}}W_{n,i}^{2}),$ (9) $\displaystyle
p(Im(\frac{h_{c,i}^{H}n_{c,i}}{|h_{c,i}|^{2}+2\sigma^{2}})|h_{c,i})\sim\mathcal{N}(0,{\sigma^{2}}W_{n,i}^{2}).$
(10)

Accordingly, we can rewrite $\mathbf{y_{r}}$ as

$\displaystyle\mathbf{y_{r}}=\frac{1}{\sqrt{1+\sigma^{2}}}(\mathbf{W_{s}x}+\mathbf{n_{r}}),$
(11)

and the distribution $p(\mathbf{n_{r}}|\mathbf{h_{c}})$ is
$\mathcal{N}(0,\sigma^{2}\mathbf{W}^{2}_{\mathbf{n}})$.

Therefore, we have

$\displaystyle
p(\mathbf{y_{r}}|\mathbf{x},\mathbf{h_{c}})\sim\mathcal{N}(\mathbf{y_{r}};\frac{1}{\sqrt{1+\sigma^{2}}}\mathbf{W_{s}}\mathbf{x},\frac{\sigma^{2}}{{1+\sigma^{2}}}\mathbf{W}^{2}_{\mathbf{n}}).$
(12)

∎

Similarly, we have the following proposition for AWGN channel.

######  Proposition 2.

Under AWGN channel, the conditional distribution of $\mathbf{y_{r}}$ with
known $\mathbf{x}$ is

$\displaystyle
p(\mathbf{y_{r}}|\mathbf{x})\sim\mathcal{N}(\mathbf{y_{r}};\frac{1}{\sqrt{1+\sigma^{2}}}\mathbf{W_{s}}\mathbf{x},\frac{\sigma^{2}}{{1+\sigma^{2}}}\mathbf{W}^{2}_{\mathbf{n}})$
(13)

where $\mathbf{W_{s}}$ becomes $\mathbf{I}_{2k}$ and $\mathbf{W_{n}}$ becomes
$\mathbf{I}_{2k}$ under AWGN channel.

Proposition 1 an Proposition 2 demonstrate that the channel noise after
equalization and normalization-reshape can be re-sampled using
$\mathbf{\epsilon}\sim\mathcal{N}(0,\mathbf{I}_{2k})$. Additionally, the noise
coefficient matrix $\mathbf{W_{n}}$ is related to the modulo form of
$\mathbf{h_{c}}$. As a result, $\mathbf{y_{r}}$ can be re-parametered as

$\displaystyle\mathbf{y_{r}}=\frac{1}{\sqrt{1+\sigma^{2}}}\mathbf{W_{s}}\mathbf{x}+\frac{\sigma}{\sqrt{1+\sigma^{2}}}\mathbf{W_{n}}\epsilon.$
(14)

Therefore, the proposed CDDM is trained to obtain
$\mathbf{\epsilon_{\theta}}(\cdot)$, which is an estimation of
$\mathbf{\epsilon}$. Here, $\mathbf{\theta}$ is model parameters. By using
$\mathbf{\epsilon_{\theta}}(\cdot)$ and $\mathbf{W_{n}}$, a sampling algorithm
is proposed to obtain $\mathbf{y}$ with the aim to recover $\mathbf{W_{s}x}$,
which will be described in Section II-C. The whole strcuture of the CDDM
forward diffusion and reverse sampling process is illustrated in Fig. 3.

Figure 3: The forward diffusion process and reverse sampling process of the
proposed CDDM.

###  II-B Training Algorithm of CDDM

For the forward process of the proposed CDDM, the original source
$\mathbf{x}_{0}$ is

$\displaystyle\mathbf{x}_{0}=\mathbf{W_{s}x}.$ (15)

Let $T$ be the hyperparameter. Similar to (1), for all $t\in\\{1,2,...,T\\}$,
we define

$\displaystyle\mathbf{x}_{t}=\sqrt{\alpha_{t}}\mathbf{x}_{t-1}+\sqrt{1-\alpha_{t}}\mathbf{W_{n}}\mathbf{\epsilon},$
(16)

and then it can be re-parametered as

$\displaystyle\mathbf{x}_{t}=\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0}+\sqrt{1-\bar{\alpha}_{t}}\mathbf{W_{n}}\mathbf{\epsilon}$
(17)

such that the distribution $q(\mathbf{x}_{t}|\mathbf{x}_{0},\mathbf{h_{r}})$
is

$\displaystyle{q(\mathbf{x}_{t}|\mathbf{x}_{0},\mathbf{h_{r}})\sim\mathcal{N}(\mathbf{x}_{t};\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0},({1-\bar{\alpha}_{t}})\mathbf{W}^{2}_{\mathbf{n}})}.$
(18)

Based on (4) and (18), if $\bar{\alpha}_{m}=\frac{1}{1+\sigma^{2}}$, the
Kullback-Leibler (KL) divergence is

$\displaystyle
D_{KL}(q(\mathbf{x}_{m}|\mathbf{x}_{0},\mathbf{h_{r}})||p(\mathbf{y_{r}}|\mathbf{x}_{0},\mathbf{h_{c}}))=0,$
(19)

for $t=m$. This indicates that CDDM can be trained on $\mathbf{x}_{m}$ instead
of $\mathbf{y_{r}}$. $\mathbf{x}_{m}$ is defined by $m$ steps as (16) such
that the predicted distribution by CDDM in reverse process can be decomposed
into $m$ small steps and each of them is
$p_{\mathbf{\theta}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{h_{r}})$ for
$t\in\\{1,2,...,m\\}$.

The goal of CDDM is to recover $\mathbf{x}_{0}$ by learning the distribution
of $\mathbf{x}_{0}$ and removing the channel noise. Therefore, the training of
CDDM is performed by optimizing the variational bound on negative log likehood
$L$. The variational bound of $L$ is form by $\mathbf{x}_{0:m}$ and
$\mathbf{y_{r}}$, which is given by

$\displaystyle L=\mathbb{E}\ [-\log\
p_{\mathbf{\theta}}(\mathbf{x}_{0}|\mathbf{h_{r}})]\leq\mathbb{E}_{q}[-\log(\frac{p_{\mathbf{\theta}}(\mathbf{x}_{0:m},\mathbf{y_{r}}|\mathbf{h_{r}})}{q(\mathbf{x}_{1:m},\mathbf{y_{r}}|\mathbf{x}_{0},\mathbf{h_{r}})})]$
$\displaystyle=\mathbb{E}_{q}\
\underbrace{[D_{KL}(q(\mathbf{y_{r}}|\mathbf{x}_{0},\mathbf{h_{r}})||p(\mathbf{y_{r}}|\mathbf{h_{r}}))}_{L_{y}}-\underbrace{\log
p_{\mathbf{\theta}}(\mathbf{x}_{0}|\mathbf{x}_{1},\mathbf{h_{r}})}_{L_{0}}$
$\displaystyle+\underbrace{D_{KL}(q(\mathbf{x}_{m}|\mathbf{y_{r}},\mathbf{x}_{0},\mathbf{h_{r}})||p_{\mathbf{\theta}}(\mathbf{x}_{m}|\mathbf{y_{r}},\mathbf{h_{r}}))}_{L_{m}}$
$\displaystyle+{\sum_{t=1}^{m}\underbrace{D_{KL}(q(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{x}_{0},\mathbf{h_{r}})||p_{\mathbf{\theta}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{h_{r}}))}_{L_{t-1}}}],$
(20)

where $L_{m}$ instructs to select the hyperparameter $m$. In this paper, we
select $m$ by

$\displaystyle arg\min_{m}\
2\sigma^{2}-\frac{1-\bar{\alpha}_{m}}{\bar{\alpha}_{m}}.$ (21)

Similar to the process in [2], $L_{t-1}$ can be calculated in closed-form
using the Rao-Blackwellized method. The optimization object of $L_{t-1}$ can
be simplified by adopting re-parameterization and re-weighting methods as
following

$\displaystyle\mathbb{E}_{\mathbf{x}_{0},\mathbf{\epsilon}}(||\mathbf{W_{n}\epsilon}-\mathbf{W_{n}\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)||^{2}_{2}),$
(22)

where $\mathbf{\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)$ is the
output of CDDM. Moreover, (22) can be re-weighted by ignoring the noise
coefficient matrix $\mathbf{W_{n}}$ as following

$\displaystyle\mathbb{E}_{\mathbf{x}_{0},\mathbf{\epsilon}}(||\mathbf{\epsilon}-\mathbf{\epsilon_{\theta}}(\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0}+\sqrt{1-\bar{\alpha}_{t}}\mathbf{W_{n}}\mathbf{\epsilon})||^{2}_{2}).$
(23)

Finally, to optimize (23) for all $t\in\\{1,2,...,T\\}$, the loss function of
the proposed CDDM is expressed as follows

$\displaystyle
L_{CDDM}(\mathbf{\theta})=\mathbb{E}_{\mathbf{x}_{0},\mathbf{\epsilon},t}(||\mathbf{\epsilon}-\mathbf{\epsilon_{\theta}}(\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0}+\sqrt{1-\bar{\alpha}_{t}}\mathbf{W_{n}}\mathbf{\epsilon})||^{2}_{2}).$
(24)

The training procedures of the proposed CDDM are summarized in Algorithm 1.

Input: Training set $S$, hyper-parameter $T$ and $\bar{\alpha}_{t}$.
Output: The trained CDDM.

Algorithm 1 Training algorithm of CDDM

1: while the training stop condition is not met do

2: Randomly sample $\mathbf{x}$ from $S$

3: Randomly sample $t$ from $Uniform(\\{1,...,T\\})$

4: Sapmle $|\mathbf{h_{c}}|$ and compute $\mathbf{H_{r}}$, $\mathbf{W_{s}}$
and $\mathbf{W_{n}}$

5: Randomly sample $\mathbf{\epsilon}$ from $\mathcal{N}(0,\mathbf{I}_{2k})$

6: Take gradient descent step according to (16) and
(24)$\nabla_{\mathbf{\theta}}(||\mathbf{\epsilon}-\mathbf{\epsilon_{\theta}}(\sqrt{\bar{\alpha}_{t}}\mathbf{W_{s}x}+\sqrt{1-\bar{\alpha}_{t}}\mathbf{W_{n}}\mathbf{\epsilon})||^{2}_{2})$

7: end while

###  II-C Sampling Algorithm of CDDM

To reduce the time consumption of sampling process, (II-B) implies that
selecting $m$ according to (21) and setting $\mathbf{x}_{m}=\mathbf{y_{r}}$ is
a promising way. By utilizing the received signal $\mathbf{y_{r}}$, only $m$
steps are needed to be excuted. For each time step $t\in\\{1,2,...,m\\}$, the
trained CDDM outputs
$\mathbf{\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)$, which attempts
to predict $\mathbf{\epsilon}$ from $\mathbf{x}_{t}$ without knowledge of
$\mathbf{x}_{0}$. A sampling algorithm is required to sample
$\mathbf{x}_{t-1}$. The process is excuted for $m$ times such that
$\mathbf{x}_{0}$ can be computed out finally.

We first define the sampling process $f(\mathbf{x}_{t-1})$ with the knowledge
of $\mathbf{\epsilon}$ as following

$\displaystyle
f(\mathbf{x}_{t-1})=q(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{x}_{0},\mathbf{h_{r}}).$
(25)

Applying Bayes rule, the distribution can be expressed as a Gaussian
distribution

$\displaystyle
q(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{x}_{0},\mathbf{h_{r}})$
$\displaystyle\sim\mathcal{N}(\mathbf{x}_{t-1};\sqrt{\bar{\alpha}_{t-1}}\mathbf{x}_{0}+\sqrt{1-\bar{\alpha}_{t-1}}\frac{\mathbf{x}_{t}-\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0}}{\sqrt{1-\bar{\alpha}_{t}}},0),$
(26)

where $\mathbf{x}_{0}$ is acquired by re-writing (17) as following

$\displaystyle\mathbf{x}_{0}=\frac{1}{\sqrt{\bar{\alpha}_{t}}}(\mathbf{x}_{t}-\sqrt{1-\bar{\alpha}_{t}}\mathbf{W_{n}}\mathbf{\epsilon}).$
(27)

However, only $\mathbf{\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)$ is
available for sampling. $\mathbf{x}_{0}$ is derived through an estimation
process by replacing $\mathbf{\epsilon}$ with
$\mathbf{\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)$ as following

$\displaystyle{\hat{\mathbf{x}}_{0}}=\frac{1}{\sqrt{\bar{\alpha}_{t}}}(\mathbf{x}_{t}-\sqrt{1-\bar{\alpha}_{t}}\mathbf{W_{n}}\mathbf{\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)).$
(28)

As a result, the sampling process is replaced with

$\displaystyle
f_{\mathbf{\theta}}(\mathbf{x}_{t-1})=p_{\mathbf{\theta}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\hat{\mathbf{x}}_{0},\mathbf{h_{r}}).$
(29)

Without the knowledge of $\mathbf{\epsilon}$, a sample of $\mathbf{x}_{t-1}$
is

$\displaystyle e\mathbf{x}_{t-1}=$
$\displaystyle\sqrt{\bar{\alpha}_{t-1}}\underbrace{(\frac{1}{\sqrt{\bar{\alpha}_{t}}}(\mathbf{x}_{t}-\sqrt{1-\bar{\alpha}_{t}}\mathbf{W_{n}}\mathbf{\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)))}_{estimate\
\mathbf{x}_{0}}$
$\displaystyle+\underbrace{\sqrt{1-\bar{\alpha}_{t-1}}\mathbf{W_{n}\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)}_{sample\
\mathbf{x}_{t-1}}.$ (30)

Note that for the last step $t=1$, we only predict $\mathbf{x_{0}}$ such that
sampling is taken as

$\displaystyle\mathbf{y}=\frac{1}{\sqrt{\bar{\alpha}_{1}}}(\mathbf{x}_{1}-\sqrt{1-\bar{\alpha}_{1}}\mathbf{W_{n}\epsilon_{\theta}}(\mathbf{x}_{1},\mathbf{h_{r}},1)).$
(31)

The sampling method is summarized in Algorithm 2.

Input: $\mathbf{y_{r}}$,$\mathbf{h_{r}}$,hyperparameter $m$
Output: $\mathbf{y}$

Algorithm 2 Sampling algorithm of CDDM

1: $\mathbf{x}_{m}=\mathbf{y_{r}}$

2: for $t=m,...,2$ do

3:
$\mathbf{z}=\mathbf{W_{n}}\mathbf{\epsilon_{\theta}}(\mathbf{x}_{t},\mathbf{h_{r}},t)$

4:
$\mathbf{x}_{t-1}=\sqrt{\bar{\alpha}_{t-1}}(\frac{\mathbf{x}_{t}-\sqrt{1-\bar{\alpha}_{t}}\mathbf{z}}{\sqrt{\bar{\alpha}_{t}}})+\sqrt{1-\bar{\alpha}_{t-1}}\mathbf{z}$

5: end for

6: $t=1$

7:
$\mathbf{z}=\mathbf{W_{n}}\mathbf{\epsilon_{\theta}}(\mathbf{x}_{1},\mathbf{h_{r}},1)$

8:
$\mathbf{y}=\frac{\mathbf{x}_{1}-\sqrt{1-\bar{\alpha}_{1}}\mathbf{z}}{\sqrt{\bar{\alpha}_{1}}}$.

##  III Application of CDDM in Semantic Communications System Based on JSCC

In this section, the proposed CDDM is applied into a semantic communications
system based on JSCC for wireless image transmission.

###  III-A System Structure

An overview architecture of the joint CDDM and JSCC system is shown in Fig. 2.
An RGB source image $\mathbf{s}$ is encoded by a JSCC encoder. In this paper,
the JSCC is built upon the Swin Transformer[16] backbone, which has a more
powerful expression ability than vision transformer by replacing the standard
multi-head self attention in vision transformer with a shift window multi-head
self attention. Two convolution layers are adopted as the output layer of the
JSCC encoder, constituting variational auto-encoder (VAE)[17] structure. The
JSCC encoder computes the source image $\mathbf{s}$ as
$\mathbf{\mu_{\phi}}\in\mathbb{R}^{2k}$ and
$\mathbf{\sigma_{\phi}}\in\mathbb{R}^{2k}$. Finally, the JSCC encoder samples
the transmitted signal $\mathbf{x}$ as

$\displaystyle\mathbf{x}=\mathbf{\mu_{\phi}+\sigma_{\phi}\xi},$ (32)

where $\mathbf{\phi}$ encapsulates all parameters of the JSCC encoder and
$\mathbf{\xi}\sim\mathcal{N}(0,\mathbf{I}_{2k})$. $\mathbf{x}$ is then
tranmitted and processed into $\mathbf{y_{r}}$ at the receiver, as described
in Section II. At the receiver, the proposed CDDM removes the channal noise
from $\mathbf{y_{r}}$ using Algorithm 2. Following this, the output of CDDM is
fed into the JSCC decoder to reconstruct the source image $\mathbf{\hat{s}}$.

###  III-B Training algorithm

The entire training algorithm of the joint CDDM and JSCC system consists of
three stages. In the first stage, the JSCC encoder and decoder are trained
jointly through the channel shown in Fig. 2, except for the CDDM module, to
minimize the distance $d(\mathbf{s,\hat{s}})$. MSE is used as the performance
metric and a slight KL divergence punishment with normal distribution is
exerted on the JSCC encoder. The slight punishment does not reduce the final
performance but it can constraint $\mathbf{x}$ in a more structured way,
thereby enhancing the convergence of CDDM in the second stage. Therefore, the
loss function for this stage is given by

$\displaystyle L_{1}(\mathbf{\phi,\varphi})$
$\displaystyle=\mathbb{E}_{\mathbf{s}\sim
p_{\mathbf{s}}}\mathbb{E}_{\mathbf{y_{r}}\sim
p_{\mathbf{y_{r}|s}}}||\mathbf{s}-\mathbf{\hat{s}}||^{2}_{2}$
$\displaystyle+\lambda D_{KL}(p(\mathbf{x|s})||N(0,\mathbf{I}_{2k})),$ (33)

where $\mathbf{\varphi}$ encapsulate all parameters of JSCC decoder and
$\lambda$ is the punishment weight.

In the second stage, the parameters of the JSCC encoder are fixed such that
CDDM can learn the distribution of $\mathbf{x}_{0}$ via Algorithm 1. The
training process is not affected by the channel noise power because Algorithm
1 has a special noise schedule, and the noise has been designed specially to
simulate the distribution of channel noise. Benefitting from this, CDDM is
designed for handling various channel conditions and requires only one
training process.

In the third stage, the JSCC decoder is re-trained jointly with the trained
JSCC encoder and CDDM to minimize $d(\mathbf{s,\hat{s}})$. The entire joint
CDDM and JSCC system is performed through the real channel, while only the
parameters of the decoder are updated. The loss function is derived as

$\displaystyle L_{3}(\mathbf{\varphi})=\mathbb{E}_{\mathbf{y}\sim
p_{\mathbf{y|s}}}||\mathbf{s}-\mathbf{\hat{s}}||^{2}_{2}.$ (34)

The training algorithm is summarized in Algorithm 3.

##  IV EXPERIMENTS RESULTS

Input: Training set $p(\mathbf{s})$, hyper-parameter $T$, $\bar{\alpha_{t}}$,
and the channel estimation result $\mathbf{h_{c}}$ and $\sigma^{2}$.
Output: The trained joint CDDM and JSCC system.

Algorithm 3 Training algorithm of the joint CDDM and JSCC system

1: while the training stop condition of stage one is not met do

2: Randomly sample $\mathbf{s}$ from $S$

3: Perform forward propagation through channel without CDDM.

4: Compute $L_{1}(\mathbf{\mathbf{\phi,\varphi}})$ and update
$\mathbf{\phi,\varphi}$

5: end while

6: while the training stop condition of stage two is not met do

7: Randomly sample $\mathbf{s}$ from $S$

8: Compute $\mathbf{s}$ as $\mathbf{x}$

9: Train CDDM with Algorithm 1.

10: end while

11: while the training stop condition of stage three is not met do

12: Randomly sample $\mathbf{s}$ from $S$

13: Perform forward propagation through channel with noise power $\sigma^{2}$
with the trained CDDM

14: Compute $L_{3}(\mathbf{\varphi})$ and update $\mathbf{\varphi}$

15: end while

In this section, we provide experiments results to verify the effectiveness of
the proposed CDDM. In the experiments, the CDDM is established on U-Net
architecture similar to [2], which accommodates $\mathbf{x}$ and
$\mathbf{h_{r}}$ as input components. We use CIFAR10[18] dataset for training
and testing. We set $T=1000$ and $\lambda=5\times 10^{-5}$. We set
$\alpha_{t}$ to constants decreasing linearly from $\alpha_{1}=0.9999$ to
$\alpha_{T}=0.98$.

We adopt the JSCC system and classical separation-based source and channel
coding scheme as the benchmarks for our performance comparison. It should be
noted that in both the joint CDDM and JSCC system, as well as the JSCC system,
we have used the same structure for JSCC. For the JSCC system, each SNR
requires its corresponding model to be trained. The channel bandwith ratio is
set as $\frac{1}{8}$. For the classical scheme, we employ the JPEG2000 codec
for compression and LDPC[19] codec for channel coding, marking as
“JPEG2000+LDPC”.

Fig. 4 illustrates the MSE performance of CDDM in different signal-to-noise
ratio (SNR) regimes. In the case of using CDDM, we caculate the MSE between
$\mathbf{x}$ and $\mathbf{y}$, while in the case of not using CDDM, we
calculate the MSE between $\mathbf{x}$ and $\mathbf{y}_{r}$. As shown in Fig.
1, $\mathbf{y}_{r}$ and $\mathbf{y}$ are the input and output of CDDM,
respectively. We can see that the system with CDDM performs much better than
the system without CDDM in all SNR regimes under both AWGN and Rayleigh fading
channels. For example, for AWGN channel, the proposed CDDM has a $0.49$ dB
gain in MSE at SNR=$20$ dB. Meanwhile, it can be seen that as the SNR
decreases, the gain of CDDM in MSE increases. This indicates that as the SNR
decreases, i.e., the channel noise increases, the proposed CDDM is easier to
remove more noise, e.g. $3.55$ dB gain at SNR=$5$ dB for AWGN channel.
Moreover, it is important to note that under Rayleigh fading channel, MMSE has
theoretically minimized the MSE, but CDDM can further reduce the MSE after
MMSE. The reason for this fact is that CDDM can learn the distribution of
$\mathbf{x}_{0}=\mathbf{W_{s}x}$, and utilizes this learned knowledge to
remove the noise, improving the effective SNR and thereby further reducing the
MSE.

Fig. 5 and Fig. 6 show the PSNR performance versus SNR under AWGN channel and
Rayleigh fading channel, respectively. Given a SNR, both the joint CDDM and
JSCC system and the JSCC system need to be retrained to achieve the best
performance. Under both Rayleigh fading and AWGN channels, the joint CDDM and
JSCC system achieves better PSNR performance compared to the JSCC system at
SNR ranging from $5$ dB to $20$ dB. For example, compared to the JSCC system,
the joint CDDM and JSCC system achieves $1.06$ dB gain at SNR=$20$ dB over
Rayleigh fading channel. Moreover, we also can observe that the CDDM and JSCC
system significantly outperforms the “JPEG2000+LDPC” scheme over both Rayleigh
fading and AWGN channels.

Figure 4: The MSE performance of the proposed CDDM versus SNR over different
channels. Figure 5: The PSNR performance versus SNR over AWGN channel. Figure
6: The PSNR performance versus SNR over Rayleigh fading channel.

##  V CONCLUSION

In this paper, we have proposed the channel denoising diffusion models to
eliminate the channel nosie under Rayleigh fading channel and AWGN channel.
CDDM is trained utilizing a specialized noise schedule adapted to the wireless
channel, which permits effective elimination of the channel noise via a
suitable sampling algorithm in the reverse sampling process. CDDM is then
applied into the semantic communications system based on JSCC. Experimental
results show that under both AWGN and Rayleigh fading channels, the system
with CDDM performs much better than the system without CDDM in terms of MSE
and PSNR.

## References

  * [1] J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli, “Deep unsupervised learning using non-equilibrium thermodynamics,” in _Proc. Int. Conf. Mach. Learn._ , 2015, pp. 2256–2265.
  * [2] J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilistic models,” in _Proc. Adv. Neural Inf. Process. Syst._ , vol. 33, 2020, pp. 6840–6851.
  * [3] J. Song, C. Meng, and S. Ermon, “Denoising diffusion implicit models,” in _Proc. International Conference on Learning Representations_ , 2021.
  * [4] C. Meng, Y. He, Y. Song, J. Song, J. Wu, J.-Y. Zhu, and S. Ermon, “SDEdit: Guided image synthesis and editing with stochastic differential equations,” in _Proc. International Conference on Learning Representations_ , 2022.
  * [5] J. Choi, S. Kim, Y. Jeong, Y. Gwon, and S. Yoon, “ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models,” in _Proc. IEEE/CVF ICCV_ , 2021, pp. 14 347–14 356.
  * [6] L. Zheng, J. Yuan, L. Yu, and L. Kong, “A reparameterized discrete diffusion model for text generation,” _https://arxiv.org/abs/2302.05737_ , 2023.
  * [7] S. Yu, K. Sohn, S. Kim, and J. Shin, “Video probabilistic diffusion models in projected latent space,” _https://arxiv.org/abs/2302.07685_ , 2023.
  * [8] M. Kim, R. Fritschek, and R. F. Schaefer, “Learning end-to-end channel coding with diffusion models,” in _Proc. WSA & SCC 2023_, 2023, pp. 1–6.
  * [9] Y. Choukroun and L. Wolf, “Denoising diffusion error correction codes,” in _Proc. the Eleventh International Conference on Learning Representations_ , 2023.
  * [10] Q. Lan, D. Wen, Z. Zhang, Q. Zeng, X. Chen, P. Popovski, and K. Huang, “What is semantic communication? a view on conveying meaning in the era of machine intelligence,” _Journal of Communications and Information Networks_ , vol. 6, no. 4, pp. 336–371, 2021.
  * [11] J. Choi and J. Park, “Semantic communication as a signaling game with correlated knowledge bases,” in _Proc. IEEE VTC 2022-Fall_ , 2022, pp. 1–5.
  * [12] E. Bourtsoulatze, D. Burth Kurka, and D. Gündüz, “Deep joint source-channel coding for wireless image transmission,” _IEEE Transactions on Cognitive Communications and Networking_ , vol. 5, no. 3, pp. 567–579, 2019.
  * [13] J. Xu, B. Ai, W. Chen, A. Yang, P. Sun, and M. Rodrigues, “Wireless image transmission using deep source channel coding with attention modules,” _IEEE Transactions on Circuits and Systems for Video Technology_ , vol. 32, no. 4, pp. 2315–2328, 2022.
  * [14] J. Xu, T.-Y. Tung, B. Ai, W. Chen, Y. Sun, and D. Gunduz, “Deep Joint Source-Channel Coding for Semantic Communications,” _https://arxiv.org/abs/2211.08747_ , 2022.
  * [15] K. P. Yang, S. Wang, J. Dai, K. Tan, K. Niu, and P. Zhang, “WITT: A Wireless Image Transmission Transformer for Semantic Communications,” _https://arxiv.org/abs/2211.00937_ , 2022.
  * [16] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, and B. Guo, “Swin Transformer: Hierarchical Vision Transformer using Shifted Windows,” in _Proc. IEEE/CVF ICCV_ , 2021, pp. 9992–10 002.
  * [17] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” in _Proc. International Conference on Learning Representations_ , 2014.
  * [18] A. Krizhevsky, “Learning multiple layers of features from tiny images,” 2009.
  * [19] “Frame stucture channel coding and modulation for the second generation digital terrestrial television broadcasting system (DVB-T2),” _DVB Document A122_ , 2008.


# Minimal setup for non-Abelian braiding of Majorana zero modes

Jie Liu  Department of Applied Physics, School of Science, Xian Jiaotong
University, Xian 710049, China  Wenqin Chen  Department of Applied Physics,
School of Science, Xian Jiaotong University, Xian 710049, China  Ming Gong
International Center for Quantum Materials, School of Physics, Peking
University, Beijing 100871, China  Yijia Wu  International Center for Quantum
Materials, School of Physics, Peking University, Beijing 100871, China  X. C.
Xie<EMAIL_ADDRESS>International Center for Quantum Materials, School of
Physics, Peking University, Beijing 100871, China  Beijing Academy of Quantum
Information Sciences, Beijing 100193, China  CAS Center for Excellence in
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China

###### Abstract

Braiding Majorana zero modes (MZMs) is the key procedure toward topological
quantum computation. However, the complexity of the braiding manipulation
hinders its experimental realization. Here we propose an experimental setup
composing of MZMs and a quantum dot state which can substantially simplify the
braiding protocol of MZMs. Such braiding scheme, which corresponds to a
specific closed loop in the parameter space, is quite universal and can be
realized in various platforms. Moreover, the braiding results can be directly
measured and manifested through electric current, which provides a simple and
novel way to detect the non-Abelian statistics of MZMs.

###### pacs:

74.45.+c, 74.20.Mn, 74.78.-w

_Introduction._ — Owing to the charming non-Abelian statistics kitaev ; nayak
, Majorana zero mode (MZM) is deemed as the most promising candidate for
topological quantum computation (TQC). To date, topological superconductors
(TSCs) that support MZMs have been proposed and realized in various
experimental platforms Fu ; sau ; fujimoto ; sato ; alicea2 ; lut ; 2DEG1 ;
2DEG2 ; kou ; deng ; das1 ; Marcus ; hao ; perge ; Yaz2 ; Jia ; Fes1 ; Fes2 ;
Fes3 ; PJJ1 ; PJJ2 . Nevertheless, most of the experimental signals so far are
based on the zero-bias conductance peaks, which are insufficient to manifest
the existence of MZMs Jie1 ; brouwer ; Aguado ; ChunXiao1 ; Moore ; Wimmer ;
Aguado2 ; Pan . The most convincing way demonstrating the presence of MZMs is
certainly detecting the signals of its non-Abelian statistics Ivanov ; alicea3
; NQP ; TQC1 , however, only a few studies have paid attention to this topic
thus far TQC2 ; TQC3 ; TQC4 . The main obstruction is that the braiding
protocols which have been proposed MSQ ; net1 ; net2 ; Yu1 ; QD1  so far are
quite complicated and hard to be realized experimentally.

The braiding operations usually require a complex 2D network structure to
avoid the collision of MZMs during the braiding, where the latter will
inevitably happen in a 1D structure alicea3 . However, as previous work has
demonstrated ABSJie , MZM is a Hermitian particle possessing half degree of
freedom compared with an ordinary fermion. When an MZM meets an ordinary
fermion, it only couples with half of the fermion while leaves another half
intact. This means that the moving and swapping of the MZMs through an
ordinary fermion could be possible with a proper manipulation. Thus, the non-
Abelian statistics may still be manifested in a quasi-1D structure, though
strictly speaking, the swapping of two MZMs is not allowed in a strict 1D
system. Such an idea will greatly facilitate the braiding operation so that it
can be realized within the current technology. The starting point of our
proposal follows the traditional braiding protocol of the Y-junction TQC ,
once the simplest proposed platform for carrying out the braiding operations.
Moreover, in our proposal, the experimental setup is further simplified by
replacing two MZMs with a normal fermionic state. Hence, the platform here is
essentially a 1D Josephson junction (JJ) connected through a quantum dot (QD)
[Fig. 1(a)]. By tuning the coupling strengths between the MZMs and the QD
through gate voltages, the MZMs could be moved and swapped after specific
manipulations. Moreover, the braiding results are solely determined by the
geometric phase corresponding to a specific closed loop in the parameter
space, that is time independent and rather robust.

Based on such braiding protocol, we further investigate the electric current
signals induced by the braiding of the MZMs. The non-Abelian statistics
suggests that the local parity will be reversed during the braiding,
corresponding to a local charge transfer process PTC1 ; PTC2 . This means that
in principle, we can observe a charge transfer from one TSC nanowire to
another TSC nanowire. To be explicit, there are two braiding-induced charge
transfer processes: the direct electron transition and the Andreev reflection.
The total probability of these two processes is exactly equal to the
probability of the non-Abelian-braiding-induced parity inversion.
Consequently, although the electric current can not directly manifest the non-
Abelian statistics, it does relate to the non-Abelian braiding of MZMs and can
be used as an indicator of the non-Abelian statistics. We further numerically
examine all these proposals in a concrete specific model as the semiconductor-
superconductor (SS) nanowire, which is one of the most promising TSC
platforms. The numerical simulations are fully consistent with the theoretical
predictions. This suggests that our proposal is experimentally feasible and
hopefully can be performed in the future experimental studies.

Figure 1:  (a) A JJ structure for demonstrating the non-Abelian statistics of
MZM. (b) A minimal sketch consisting of two MZMS and one QD. Through
manipulating the coupling strengths, MZMs can be moved and then swapped in the
assistance of the QD. (c) The Bloch sphere of the coupling vector
$\vec{\delta}$. A non-trivial Berry phase is accumulated after a solid angle
is spanned by the closed path of $\vec{\delta}$. (d) The non-trivial Berry
phase induced by tuning the coupling strengths is only picked up by
$\gamma_{2}$ other than by $\gamma_{1}$. Since two MZMs $\gamma_{1}$ and
$\gamma_{2}$ together form an ordinary fermionic state as
$\psi_{1}=\gamma_{1}+i\gamma_{2}$, such a fermionic state will not come back
to its initial state after braiding. (e) If the pair of MZMs in (d) are
replaced by an ordinary QD state, then although the QD can be decomposed into
two MZMs, the non-trivial Berry phase will be picked up by both these two MZMs
since they both couple with the other QD with the same strength. Hence, only a
global phase will be added to the initial state after braiding.

_Simplified setup for non-Abelian braiding of MZMs._ —Our setup can be
straightforwardly constructed by adding an additional QD to the JJ [Fig.
1(a)]. Such structure actually has been fabricated in a recent experiment JJ .
On each TSC nanowire, a pair of MZMs are non-locally distributed at the two
ends of the nanowire. Since the MZMs $\gamma_{1}$ and $\gamma_{4}$ are far
from the interfaces of the JJ due to the length of the TSC nanowire, we can
safely neglect them during the braiding. Thus, the simplest Hamiltonian
describing such setup is

$H_{s}={2E_{d}}{d^{\dagger}}d+i({t_{1}}d+t_{1}^{*}{d^{\dagger}}){\gamma_{2}}+({t_{2}}d-t_{2}^{*}{d^{\dagger}}){\gamma_{3}},$
(1)

where $d$ is the annihilation operator for the fermioinic state in the QD, and
$E_{d}$ is the on-site energy of this QD state. $\gamma_{2}$ and $\gamma_{3}$
are the operators for the MZMs near the interfaces of the JJ, the coupling
strengths between the QD and the MZMs $\gamma_{2},\gamma_{3}$ are
$t_{1}=|t_{1}|e^{i\phi_{1}/2}$ and $t_{2}=|t_{2}|e^{i\phi_{2}/2}$,
respectively, in which the coupling phases $\phi_{1}$ and $\phi_{2}$ are the
superconducting phase of two TSC. The crucial feature of this Hamiltonian is
that these MZMs couple to the QD in a unique way, because the MZM is Hermitian
and possesses half degree of freedom compared with a normal fermionic mode.
This feature can be manifested by composing the QD into two intrinsically-
fused MZMs $\gamma_{A}$ and $\gamma_{B}$ as
$d=\frac{1}{2}e^{-i\frac{\phi_{1}}{2}}(\gamma_{A}+i\gamma_{B})$. In such way,
the Hamiltonian $H_{s}$ can be written in the representation of the Majorana
operators as:

$\displaystyle H_{M}$
$\displaystyle=i{E_{d}}\gamma_{A}\gamma_{B}+i|t_{1}|\gamma_{A}{\gamma_{2}}$
(2)
$\displaystyle+i|t_{2}|\left[-\gamma_{A}\sin\left(\frac{\phi_{1}-\phi_{2}}{2}\right)+\gamma_{B}\cos\left(\frac{\phi_{1}-\phi_{2}}{2}\right)\right]\gamma_{3}.$

Such Hamiltonian unveils that the coupling between an MZM and the QD can be
treated as the coupling between different MZMs. This leads to an idea that the
braiding can be performed by individually tuning these two coupling terms.

Figure 2:  (a) Evolution of the wavefunction $\psi_{1}^{-}(t)$ during the
braiding. After swapping MZMs $\gamma_{2}$ and $\gamma_{3}$ twice in
succession, $\psi_{1}^{-}$ evolves into $\psi_{1}^{+}$ due to the non-Abelian
statistics of the MZMs. Here, $E_{d}=0.04\Delta$, $t_{c}=0.2\Delta$, and
$T=1000/\Delta$. (b) Braiding results as functions of $E_{d}$. (c) Braiding
results as functions of the braiding time-cost $T=(1000+\delta T)/\Delta$ with
$\phi_{1}-\phi_{2}=\pi+0.05\pi$. Due to the deviation of phase difference
$\delta\phi=0.05\pi$, $\gamma_{2}$ and $\gamma_{3}$ will couple with each
other and induce a dynamic phase. Such a dynamic phase can be eliminated by
the spin-echo technology. (d) Braiding results as functions of the phase
difference $\phi$ when the spin-echo technology is adopted. The non-Abelian
braiding properties recovers for almost all the $\phi_{1}-\phi_{2}$ values
except for $\phi_{1}-\phi_{2}=0$.

The braiding protocol could be greatly simplified by tuning the flux to keep
$\phi_{1}-\phi_{2}=\pi$. In this case, the Hamiltonian [Eq. (2)] reduces to an
elegant form as
$\widetilde{H}_{M}=i\gamma_{A}(\vec{\delta}\cdot\vec{\gamma})$, where the
Majorana vector
$\vec{\gamma}\equiv(\gamma_{x}(\gamma_{2}),-\gamma_{y}(\gamma_{3}),\gamma_{z}(\gamma_{B}))$,
and the coupling vector $\vec{\delta}\equiv(|t_{1}|,|t_{2}|,E_{d})$. Such a
mathematical form is in parallel with the traditional Y-junction composed of
four MZMs TQC , which has been deemed as the simplest model for non-Abelian
braiding of MZMs. Accordingly, our proposal follows the braiding scheme
adopted in the Y-junction but with a simpler structure and simpler operations
[Fig. 1(b)]. To initialize the braiding, infinitesimal coupling strengths
$|t_{1}|,|t_{2}|\ll E_{d}$ are used to decouple $\gamma_{2},\gamma_{3}$. After
that, the braiding protocol takes four steps (the time-cost for each step is
$T$) to swap $\gamma_{2}$ and $\gamma_{3}$ spatially. In step 1, $|t_{1}|$ is
increased from $0$ to $t_{c}$ so that $|t_{1}|\gg{}E_{d}$, hence $\gamma_{2}$
is teleported to the original position of $\gamma_{B}$. In step 2, $|t_{2}|$
is increased from $0$ to $t_{c}$ so that $|t_{2}|\approx|t_{1}|\gg E_{d}$. In
step 3, $|t_{1}|$ is decreased to $0$ so that $|t_{1}|\ll E_{d}$ and
$\gamma_{3}$ is teleported to the original position of $\gamma_{2}$. Finally
in step 4, $|t_{1}|$ is decreased to $0$ so that the spatial positions of
$\gamma_{2}$ and $\gamma_{3}$ are swapped, although $\vec{\delta}$ comes back
to its initial form.

Such a braiding operation, represented by the operator
$B(\gamma_{i},\gamma_{j})=\exp(\frac{\pi}{4}\gamma_{i}\gamma_{j})$, transforms
the MZMs as $\gamma_{i}\rightarrow\gamma_{j}$ and
$\gamma_{j}\rightarrow-\gamma_{i}$ Ivanov . For two pairs of MZMs whose
eigenstates are in the forms of $\psi_{j}^{\pm}(0)=(\gamma_{2j-1}\pm
i\gamma_{2j})/\sqrt{2}$, if $\gamma_{2}$ and $\gamma_{3}$ are swapped twice in
succession, then the wavefunctions will evolve into
$\psi_{1}^{\pm}(8T)=(\gamma_{1}\mp i\gamma_{2})/\sqrt{2}=\psi_{1}^{\mp}(0)$
and $\psi_{2}^{\pm}(8T)=(-\gamma_{3}\pm
i\gamma_{4})/\sqrt{2}=-\psi_{2}^{\mp}(0)$. We calculate the wavefunction
evolution during the braiding as
$|\psi_{j}^{\pm}(t)\rangle=U(t)|\psi_{j}^{\pm}(0)\rangle$, where
$U(t)=\hat{T}\exp[i\int_{0}^{t}\mathrm{d}\tau H_{s}(\tau)]$ is the time-
evolution operator and $\hat{T}$ is the time-ordering operator Jie2 ; Jie3 ;
Dirac . The results confirm that $\psi_{j}^{+}$ evolves into $\psi_{j}^{-}$
after adiabatically swapping $\gamma_{2}$ and $\gamma_{3}$ twice in succession
[Fig. 2(a)], which demonstrates the non-Abelian braiding properties discussed
above.

To clearly formulate the braiding process, we can visualize $\vec{\delta}$ as
a radius vector in a 3D parameter space in analogy with the Bloch sphere [Fig.
1(c)]. During the braiding, $\vec{\delta}$ go along a closed loop on the Bloch
sphere that first rotates around the $y$-axis from $(0,0,1)$ to $(1,0,0)$, and
then rotates around the $z$-axis from $(1,0,0)$ to $(0,1,0)$. Finally,
$\vec{\delta}$ rotates around the $x$-axis back to its start point $(0,0,1)$.
Such a closed loop spans an octant of the Bloch sphere and consequently, a
geometric phase of $\pi/4$ is accumulated as half of the solid angle of an
octant $\Omega_{c}=\pi/2$. Furthermore, there is a general relation between
the path on the Bloch sphere and the braiding results. The rotation of the
coupling vector $\vec{\delta}$ will bring one MZM, defined as the vector along
the polar direction ($\theta$-direction) perpendicular to $\vec{\delta}$, away
from its initial direction towards another MZM which is represented as the
vector along the azimuthal direction ($\phi$-direction) perpendicular to
$\vec{\delta}$. More specifically, such a path corresponding to the swapping
of $\gamma_{2}$ and $\gamma_{3}$ can be characterized by the braiding operator
$U(\gamma_{2},\gamma_{3})=\exp(\frac{\Omega_{c}}{2}\gamma_{2}\gamma_{3})$ S1 ,
which transforms $\gamma_{2}$ and $\gamma_{3}$ as
$\gamma_{2}\rightarrow\cos(\Omega_{c})\gamma_{2}+\sin(\Omega_{c})\gamma_{3}$,
and
$\gamma_{3}\rightarrow-\sin(\Omega_{c})\gamma_{2}+\cos(\Omega_{c})\gamma_{3}$.
Therefore, the braiding results are determined by the solid angle $\Omega_{c}$
spanned by the loop of $\vec{\delta}$. In Fig. 2(b), we show the weight of
$\psi_{i}^{\pm}(8T)$ on $\psi_{i}^{\pm}(0)$ after two successive braiding
operations, in which $t_{1}$ and $t_{2}$ varies in the range of
$\left[0,t_{c}\right]$, while $E_{d}$ remains unchanged. The solid angle is in
the general form of $\Omega_{c}=\arccos(E_{d}/\sqrt{E_{d}^{2}+t_{c}^{2}})$,
thus the weight of $\psi_{2}^{\pm}(8T)$ on $\psi_{i}^{\mp}(0)$ is
$\frac{1-\cos(2\Omega_{c})}{2}=\frac{t_{c}^{2}}{E_{d}^{2}+t_{c}^{2}}$. The
numerical results in Fig. 2(b) are fully consistent with the analysis above
except for $E_{d}=0$.

Now we emphasis that the non-Abelian braiding of MZMs relies on the non-
trivial Berry phase as well as the MZM’s half fermionic degree of freedom.
Since two MZMs $\gamma_{1}$ and $\gamma_{2}$ together form a non-local state
$\psi=\gamma_{1}+i\gamma_{2}$, the non-trivial Berry phase accumulated by
tuning the coupling strength (or spatially moving the MZMs) will only be
picked up by $\gamma_{2}$. Thus, the final state cannot return to its original
form after braiding. This is completely different from the ordinary fermionic
state. If we replace the pair of the MZMs with another QD, then the coupling
term between these two QDs is $t_{1}d_{1}^{\dagger}d_{1}+h.c.$, where
$d_{1}^{\dagger}$ is the creation operator of the state in the new QD. In the
Majorana representation, such coupling term reads as
$i|t_{1}|(\gamma_{1}\gamma_{A}-\gamma_{B}\gamma_{2})$, indicating that there
are two coupled pairs of MZMs between these two QDs with the same coupling
strength [Fig. 1(e)]. Then even if a non-trivial Berry phase is accumulated,
the initial state will not evolve into another state since both $\gamma_{1}$
and $\gamma_{2}$ picks up the same Berry phase. Such an analysis shows the
necessity of the half fermionic degree of freedom. Moreover, recent progress
ChunXiao1 ; Aguado3  suggests that before the system enters the topological
region, an inhomogeneous interface potential could induce a quasi-MZM state
comprising of two weakly-coupled MZMs separated in a short distance. Though
such state could exhibit non-Abelian braiding properties in specific
parameters, such braiding results are extremely fragile and can be
experimentally distinguished from the case of MZM S1 .

We have proposed a minimal setup supporting non-Abelian braiding of the MZMs
and discussed the underlying physics. However, it should be noted that our
proposal requires a $\pi$-junction to facilitate the manipulation. To show
what will happen when $\phi_{1}-\phi_{2}$ deviates from $\pi$, we calculate
the braiding results versus the braiding time-cost $T$ with the phase
deviation $\delta\phi=0.05\pi$ [Fig. 2(c)]. The braiding results sinusoidally
oscillate with $T$, which means that a dynamic phase is induced by the phase
deviation. Indeed, since our setup is essentially a JJ with Josephson energy
$E_{J}=\Gamma\cos(\frac{\phi}{2}\gamma_{2}\gamma_{3})$ where $\Gamma$ is the
hybridization energy of the junction, an additional dynamic phase will be
induced by $\Gamma$. In general, the dynamic phase can be eliminated by the
traditional spin-echo technique or parity echo technique, which reverses the
sign of the eigenenergy at the middle of the symmetric braiding protocol GQC .
Indeed, since this dynamic phase is induced by the Josephson energy, it can be
eliminated by reversing the sign of the phase deviation as
$\delta\phi=-0.05\pi$ in the last two steps of the braiding so that the non-
Abelian geometric phase can be retrieved [the dashed lines in Fig. 2(c)].
Figure 2(d) exhibits the braiding results as functions of $\phi_{1}-\phi_{2}$
when the spin-echo technique is adopted. The non-Abelian braiding results
remain intact except for $\phi_{1}-\phi_{2}=0$, which is quite stable when the
spin-echo technique is involved.

_Non-Abelian-braiding-induced charge transfer._ —The detection of the non-
Abelian braiding result is another significant problem in the TQC. In the
many-body basis
$(|0\rangle,\Psi_{1}^{\dagger}|0\rangle,\Psi_{2}^{\dagger}|0\rangle,\Psi_{2}^{\dagger}\Psi_{1}^{\dagger}|0\rangle)$,
$U(\gamma_{2},\gamma_{3})$ has the matrix form of

$\begin{bmatrix}\cos(\frac{\Omega_{c}}{2})&0&0&-i\sin(\frac{\Omega_{c}}{2})\\\
0&\cos(\frac{\Omega_{c}}{2})&-i\sin(\frac{\Omega_{c}}{2})&0\\\
0&-i\sin(\frac{\Omega_{c}}{2})&\cos(\frac{\Omega_{c}}{2})&0\\\
-i\sin(\frac{\Omega_{c}}{2})&0&0&\cos(\frac{\Omega_{c}}{2})\\\ \end{bmatrix}.$
(3)

Figure 3:  (a) Inversion of the local parity will induce the charge transfer.
(b) Electric-current-induced charge transfer during the braiding (blue line)
as a function of $E_{d}$ with $t_{c}=0.2\Delta$. The probability of electron
tunneling $p_{e}$ dominates at $E_{d}<0$ (green dashed line) and the
probability of electron tunneling $p_{h}$ dominates at $E_{d}>0$.
$2(p_{e}+p_{h})$ is equal to the weight of
$|\langle\psi_{1}^{\mp}(0)|\psi_{1}^{\pm}(8T)\rangle|$ (purple line). Since
$\delta N=(p_{e}+2p_{h})e$, the charge transfer is about
$t_{c}^{2}/2(E_{d}^{2}+t_{c}^{2})$ at $E_{d}<0$ and about
$t_{c}^{2}/({E_{d}^{2}+t_{c}^{2}})$ at $E_{d}>0$. Resonant charge transfer
happens near $E_{d}=0$ in which both these two processes matter. (c) The
charge transfer as a function of $t_{c}$ with $E_{d}=-0.6\Delta$. The charge
transfer is about half of
$|\langle\psi_{1}^{\mp}(0)|\psi_{1}^{\pm}(8T)\rangle|$.

The non-zero off-diagonal elements in the braiding matrix demonstrate that an
initial state $a|0\rangle+b\Psi_{2}^{\dagger}\Psi_{1}^{\dagger}|0\rangle$ will
evolve into
$\left[a\cos(\frac{\Omega_{c}}{2})-ib\sin(\frac{\Omega_{c}}{2})\right]|0\rangle+\left[b\cos(\frac{\Omega_{c}}{2})-ia\sin(\frac{\Omega_{c}}{2})\right]\Psi_{2}^{\dagger}\Psi_{1}^{\dagger}|0\rangle$
after braiding. Since the inversion of the local parity of $\Psi_{1}$ or
$\Psi_{2}$ corresponds to a charge transfer from the left nanowire to the
right one [Fig. 3(a)], the charge transfer in an isolated system is equal to
$(|a|^{2}-|b|^{2})\sin^{2}(\Omega/2)$ and can be detected via charge sensing
PTC1 ; PTC2 . Remarkably, we find that the electric current can also manifest
the non-Abelian braiding in an open system. The charge transfer here can be
calculated through the integration of the time-dependent current as $\delta
N=\int_{0}^{8T}\langle\hat{J}(t)\rangle\mathrm{d}t$, where the current follows
the definition of Lei

$\langle\hat{J}(t)\rangle=-\frac{i}{2}\left[\langle\psi_{L}(t)|H_{c}(t)|\psi_{R}(t)\rangle-h.c.\right].$
(4)

Here, $|\psi_{L}(t)\rangle$ is the wavefunction at the right end of the left
nanowire at time $t$, while $|\psi_{R}(t)\rangle$ is the wavefunction at the
left end of the right nanowire at time $t$. Since
$H_{c}(t)=i({t_{1}}d+t_{1}^{*}{d^{\dagger}}){\gamma_{2}}$, the current is
contributed from two possible processes. The first process (with probability
denoted by $p_{e}$) is that an electron in the left nanowire directly
transmits into the right nanowire. The other one is an electron in the left
nanowire transforms into a hole and then transmits into the right nanowire
(Andreev reflection, whose probability is denoted as $p_{h}$). Since a
positive $E_{d}$ indicates a barrier for electron transmission, we can see
that $p_{h}$ dominates for $E_{d}>0$ while $p_{e}$ dominates for $E_{d}<0$
[the dashed line in Fig. 3(b)]. Moreover, it can be proved that
$2(p_{e}+p_{h})=\frac{t_{c}^{2}}{E_{d}^{2}+t_{c}^{2}}$, which is exactly the
population of the charge occupied after braiding
$|\langle\psi_{i}^{\pm}(8T)|\psi_{i}^{\mp}(0)\rangle|$. In comparison, the
actual charge transfer is $\delta N=p_{e}+2p_{h}$ [Fig. 3(b)], hence the
charge transfer is highly related to the non-Abelian geometric phase that
$\delta N\approx|\langle\psi_{i}^{\pm}(8T)|\psi_{i}^{\mp}(0)\rangle|/2$ for
$E_{d}<0$, while $\delta
N\approx|\langle\psi_{i}^{\pm}(8T)|\psi_{i}^{\mp}(0)\rangle|$ for $E_{d}>0$.
Figure 3(c) shows the charge transfer as a function of $t_{c}$ with
$E_{d}=-0.6\Delta$, in which $\delta
N\approx|\langle\psi_{i}^{\pm}(8T)|\psi_{i}^{\mp}(0)\rangle|/2$ in a large
region of $t_{c}$. These results indicate that the non-Abelian statistics may
be directly detected through the electric current.

Figure 4:  (a) Braiding operation steps in the SS nanowire system. (b) The
braiding result (purple line) and the charge transfer (blue line) in the SS
nanowire versus $t_{c}$ with $E_{d}=-0.6\Delta$ and braiding time-cost
$T=80/\Delta$. (c) The braiding result in the SS nanowire as a function of $T$
with $E_{d}=-0.4\Delta$ and $t_{c}=1.5\Delta$. The charge transfer (blue line)
significantly increases when $T>150/\Delta$ due to the finite size effect.

_Numerical simulation in the semiconductor superconducting nanowires._ —We
have demonstrated that the non-Abelian braiding of MZMs can be well performed
through the protocol we proposed. Here, we would like to numerically examine
it in a specific system. Since the semiconductor superconducting (SS) nanowire
is deemed as the most promising platform for TQC, we numerically simulate the
non-Abelian braiding in a system composed of two SS nanowires [Fig. 4(a)], in
which the coupling strengths between the MZMs and the QD can be manipulated
through two voltage gates located near the intersection of the nanowires and
the QD. The whole braiding process is quite simple and can be performed by
first turning off the gate voltages in $G1$ and $G2$ simultaneously, and then
turning on $G1$ and $G2$ in succession [Fig. 4(a)]. The wavefunction evolution
and the charge transfer during the braiding can be numerically demonstrated as
shown in Fig. 4(b), (c). We can see that the weight of of $\psi_{i}^{\pm}(8T)$
on $\psi_{i}^{\mp}(0)$ is still
$\frac{1-\cos(2\Omega_{c})}{2}=\frac{t_{c}^{2}}{E_{d}^{2}+t_{c}^{2}}$, though
the on-site energy of the QD state shifts due to the influence of background.
Besides, the electron tunneling dominates due to the negative shift of the on-
site energy of the QD state. Finally, the charge transfer significantly
increases when $T>150/\Delta$ due to the finite size effect of the nanowire S1
, implying that its non-Abelian braiding properties will be better exhibited
in a longer nanowire.

_Conclusion._ —With the assistance of a QD, we show that the non-Abelian
braiding operations can be manifested in a simple 1D structure. Such braiding
operation is quite simple and feasible in the state-of-art technology. We want
to stress that though we take the SS nanowire as a demonstration, our minimal
model is quite general. Such proposed minimal model can be realized in various
TSC platforms such like ferromagnetic atomic chains and vortices on iron-based
TSC. Moreover, in the SS nanowire system we have demonstrated, in addition to
tuning the coupling strengths between the MZMs and the QD, the non-Abelian
braiding operations corresponding to the closed loop on the Bloch sphere in
Fig. 1(c) can also be realized through manipulating the on-site energy of the
QD state $E_{d}$.

_Acknowledgement._ — We acknowledge Lei Wang for the numerical support. This
work is financially supported by NSFC (Grants No. 11974271) and NBRPC (Grants
No. 2017YFA0303301, and No. 2019YFA0308403).

## References

  * (1) A. Kitaev, Phys. Usp. 44, 131 (2000).
  * (2) C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * (3) L. Fu, and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
  * (4) J. D. Sau, R. M. Lutchyn, S. Tewari, S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010).
  * (5) S. Fujimoto, Phys. Rev. B. 77, 220501(R) (2008).
  * (6) M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. B 82,134521 (2010).
  * (7) J. Alicea, Phys. Rev. B 81, 125318 (2010).
  * (8) R.M. Lutchyn, J.D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010).
  * (9) M. Hell, M. Leijnse, K. Flensberg, Phys. Rev. Lett. 118, 107701 (2017).
  * (10) F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, Phys. Rev. X 7, 021032 (2017).
  * (11) V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven, Science 336, 1003 (2012).
  * (12) M. T. Deng, C.L. Yu, G. Y. Huang, M. Larsson, P. Caroff, H. Q. Xu, Nano Lett. 12, 6414 (2012).
  * (13) A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, H.Shtrikman, Nat. Phys. 8, 887 (2012).
  * (14) S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nygard, P. Krogstrup and C. M. Marcus, Nature 531, 206 (2016).
  * (15) Hao Zhang, Michiel W.A. de Moor, Jouri D.S. Bommer, Di Xu, Guanzhong Wang, Nick van Loo, Chun-Xiao Liu, Sasa Gazibegovic, John A. Logan, Diana Car, Roy L. M. Op het Veld, Petrus J. van Veldhoven, Sebastian Koellinga, Marcel A. Verheijen, Mihir Pendharkar, Daniel J. Pennachio, Borzoyeh Shojaei, Joon Sue Leeb, Chris J. Palmstrom, Erik P.A.M. Bakkers, S. Das Sarma, Leo P. Kouwenhoven, arXiv: 2101.11456 (2021).
  * (16) S. Nadj-perge, llya K. Drozdov, Jian Li, Hua Chen, Sangjun Jeon, J. Seo, Allan H. MacDonald B. Andrei Bernevig, Ali Yazdani, Science 346, 6209 (2014).
  * (17) B. E. Feldman, M. T. Randeria, Jian Li, Sangjun Jeon, Yonglong Xie, Zhijun Wang, Ilya K. Drozdov, B. Andrei Bernevig, Ali Yazdani, Nat. Phys. 13, 286 (2017).
  * (18) H.H. Sun, K.W. Zhang, L.H. Hu,C. Li, G.Y. Wang, H.Y. Ma, Z.A. Xu, C.L. Gao, D.D. Guan, Y.Y. Li, C. Liu, D. Qian, Y. Zhou, L. Fu, S.C. Li, F.C. Zhang, J.F. Jia, Phys. Rev. Lett. 116, 257003 (2016).
  * (19) Dongfei Wang, Lingyuan Kong, Peng Fan, Hui Chen, Shiyu Zhu, Wenyao Liu, Lu Cao, Yujie Sun, Shixuan Du, J. Schneeloch, Ruidan Zhong, Genda Gu, Liang Fu, Hong Ding, Hong-Jun Gao, Science 362, 333 (2018).
  * (20) Qin Liu, Chen Chen, Tong Zhang et al., Phys. Rev. X 8, 041056 (2018).
  * (21) Peng Zhang, Koichiro Yaji, Takahiro Hashimoto, Yuichi Ota, Takeshi Kondo, Kozo Okazaki, Zhijun Wang, Jinsheng Wen, G. D. Gu, Hong Ding, Shik Shin, Science 360, 182 (2018).
  * (22) Antonio Fornieri, Alexander M. Whiticar, F. Setiawan, Elias Portolés, Asbjørn C. C. Drachmann, Anna Keselman, Sergei Gronin, Candice Thomas, Tian Wang, Ray Kallaher, Geoffrey C. Gardner, Erez Berg, Michael J. Manfra, Ady Stern, Charles M. Marcus, and Fabrizio Nichele, Nature 569, 89 (2019).
  * (23) H. Ren, F. Pientka, S. Hart, A. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. Molenkamp, B. I. Halperin, and A. Yacoby, Nature 569, 93 (2019).
  * (24) Jie Liu, Andrew C. Potter, K. T. Law, Patrick A. Lee, Phys. Rev. Lett. 109, 267002 (2012).
  * (25) G. Kells, D. Meidan, and P. W. Brouwer, Phys. Rev. B 86, 100503(R) (2012).
  * (26) E. Prada, P. San-Jose, and R. Aguado, Phys. Rev. B 86, 180503(R) (2012).
  * (27) Chun-Xiao Liu, Jay D. Sau, and S. Das Sarma, Phys. Rev. B 97, 214502 (2018).
  * (28) C. Moore, T. D. Stanescu, and S. Tewari, Phys. Rev. B 97, 165302 (2018).
  * (29) A. Vuik, B. Nijholt, A. R. Akhmerov and M. Wimmer, arXiv: 1806.02801 (2018).
  * (30) F. Penaranda, R. Aguado, P. San-Jose, and E. Prada, Phys. Rev. B 98, 235406 (2018).
  * (31) Haining Pan, and S. Das Sarma, Phys. Rev. Research 2, 013377 (2020).
  * (32) D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001).
  * (33) J. Alicea, Y. Oreg, G. Refael, F. von Oppen, M. P. A. Fisher, Nat. Phys. 7, 1915 (2011).
  * (34) van Heck B, Akhmerov AR, Hassler F, Burrello M, Beenakker CWJ. 2012. New J. Phys. 14,035019 (2012).
  * (35) D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham,J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus,K. Flensberg, and J. Alicea, Phys. Rev. X 6, 031016 (2016)
  * (36) Ryan V. Mishmash, Bela Bauer, Felix von Oppen, Jason Alicea, Phys. Rev. B 101, 075404 (2020).
  * (37) I. C. Fulga, B. van Heck, M. Burrello, and T. Hyart, Phys. Rev. B 88, 155435 (2013).
  * (38) J. Manousakis, C. Wille, A. Altland, R. Egger, K. Flensberg, and F. Hassler, Phys. Rev. Lett. 124, 096801 (2020).
  * (39) Sagar Vijay, Liang Fu, Phys. Rev. B 94, 235446 (2016).
  * (40) T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, Phys. Rev. B 95, 235305 (2017).
  * (41) C. Schrade and L. Fu, Phys. Rev. Lett. 121, 267002 (2018).
  * (42) Yuval Oreg, and Felix von Oppen, Annual Review of Condensed Matter Physics 11,397-420 (2020).
  * (43) W. Chen, Y. Wu, J. Liu, and X.C. Xie, arXiv: 2005.00735 (2020).
  * (44) Karsten Flensberg, Phys. Rev. Lett. 106, 090503 (2011).
  * (45) Torsten Karzig, Yuval Oreg, Gil Refael, and Michael H. Freedman, Phys. Rev. X 6, 031019 (2016).
  * (46) Dihao Sun, and Jie Liu, Phys. Rev. B 97, 035311 (2018).
  * (47) Y. Wu, H. Liu, J. Liu, H. Jiang, and X.C. Xie, National Science Review bf 7, 572 (2020).
  * (48) Y. Wu, H. Jiang, J. Liu, H. Liu, and X.C. Xie, Phys. Rev. Lett. 125, 036801 (2020).
  * (49) See Supplementary materials.
  * (50) G. Szechenyi, and Andras Palyi, Phys. Rev. B 101, 235441 (2020).
  * (51) M. I. K. Munk, J. Schulenborg, R. Egger, and Karsten Flensberg, Phys. Rev. Research 2, 033254 (2020).
  * (52) D. Rzmadze, E. C. T. OFarrell, P. Krogstrup, and C. M. Marcus, Phys. Rev. Lett. 125, 116803 (2020).
  * (53) J. Avila, F. Penaranda, E. Prada, P. San-Jose, and R. Aguado, Communications Physics bf 2, 133 (2019)
  * (54) Bao-Jie Liu, Xue-Ke Song, Zheng-Yuan Xue, Xin Wang, and Man-Hong Yung, Phys. Rev. Lett. 123, 100501 (2019).
  * (55) Lei Wang, Matthias Troyer, and Xi Dai, Phys. Rev. Lett. 111, 026802 (2013).