# TEDGE-Caching: Transformer-based Edge Caching Towards 6G Networks
††thanks: This Project was partially supported by Department of National
Defence’s Innovation for Defence Excellence & Security (IDEaS), Canada.

1st Zohreh Hajiakhondi Meybodi  Electrical and Computer Engineering
Concordia University
Montreal, Canada  2nd Arash Mohammadi  Concordia Ins. for Inf. Systems Eng.
Concordia University
Montreal, Canada  3nd Elahe Rahimian  Concordia Ins. for Inf. Systems Eng.
Concordia University
Montreal, Canada  4nd Shahin Heidarian  Electrical and Computer Engineering
Concordia University
Montreal, Canada  5nd Jamshid Abouei  Department of Electrical Engineering
Yazd University
Yazd, Iran  6nd Konstantinos N. Plataniotis  Electrical & Computer Engineering
University of Toronto,
Toronto, Canada

###### Abstract

As a consequence of the COVID-19 pandemic, the demand for telecommunication
for remote learning/working and telemedicine has significantly increased.
Mobile Edge Caching (MEC) in the 6G networks has been evolved as an efficient
solution to meet the phenomenal growth of the global mobile data traffic by
bringing multimedia content closer to the users. Although massive connectivity
enabled by MEC networks will significantly increase the quality of
communications, there are several key challenges ahead. The limited storage of
edge nodes, the large size of multimedia content, and the time-variant users’
preferences make it critical to efficiently and dynamically predict the
popularity of content to store the most upcoming requested ones before being
requested. Recent advancements in Deep Neural Networks (DNNs) have drawn much
research attention to predict the content popularity in proactive caching
schemes. Existing DNN models in this context, however, suffer from long-term
dependencies, computational complexity, and unsuitability for parallel
computing. To tackle these challenges, we propose an edge caching framework
incorporated with the attention-based Vision Transformer (ViT) neural network,
referred to as the Transformer-based Edge (TEDGE) caching, which to the best
of our knowledge, is being studied for the first time. Moreover, the TEDGE
caching framework requires no data pre-processing and additional contextual
information. Simulation results corroborate the effectiveness of the proposed
TEDGE caching framework in comparison to its counterparts.

###### Index Terms:

Mobile Edge Caching (MEC), Popularity Prediction, Deep Neural Network (DNN),
Vision Transformer.

##  I Introduction

Mobile Edge Caching (MEC) [1, 2] is an emerging technology in the Beyond Fifth
Generation (5G) communication networks (also referred to as 6G) developed to
meet the phenomenal growth of the global mobile data traffic. Enabling caching
at the edge of the network provides the opportunity to store popular content
at the storage of the heterogeneous next generation Node B (hgNB) during the
off-peak intervals [3, 4]. After requesting a content by an edge (e.g.,
Internet of Thing (IoT)) device, this request is directly served by the
neighboring hgNBs, having the requested content. In such scenarios, the cache-
hit occurs; otherwise, it is known as a cache-miss and the requested content
is sent from the content server to the hgNB to serve the request [5, 6].
Integrating Unmanned Aerial Vehicles (UAVs) as the flying hgNBs into the
terrestrial MEC networks [7] extends the service coverage and improves the
Quality of Service (QoS) of User Equipment (UE) in the Beyond 5G networks. Due
to the limited local storage capacity of cache-enabled hgNBs, it is of
significant importance to identify/store the most popular content to enhance
the cache efficiency of the network. In the MEC networks, there are two types
of caching strategies, i.e., Reactive caching and proactive caching.
Conventional reactive caching schemes [8], such as First-In-First-Out (FIFO),
Least Recently Used (LRU), and Least Frequently Used (LFU) frameworks,
identify the most popular content based on the underlying pattern of observed
users’ requests. A critical drawback of reactive caching is that popular
content can only be identified after being requested. As a consequence, they
are not robust to the dynamically changing behavior of the content popularity.
Therefore, the main focus of recent researches has been shifted to use
proactive caching, e.g., using Deep Neural Networks (DNN) models to predict
the Content Popularity (CP) from the request patterns. In this context,
popular content can be dynamically allocated in the storage of hgNBs before
being requested. The paper aims to further advance this emerging field.

Literature Review: Generally speaking, both temporal and spatial correlations
exist within the time-variant request pattern of multimedia content. While
spatial correlation reflects different users’ preferences, depending on the
geographical location and users’ contextual information, the temporal
correlation represents the variation of content popularity over time. In this
context, several DNN models [9, 10, 11, 12, 13, 14, 15, 16, 17] are introduced
to capture the temporal and/or spatial features of user preferences in
proactive caching schemes. For instance, Yu et al. [18] used an auto-encoder
model to predict users’ preferences in the future by learning the latent
representation of raw data in an unsupervised fashion. Auto-encoder models,
however, suffer from training complexity. Tsai et al. [19] used Convolutional
Neural Network (CNN) for predicting users’ interests based on sentence
analysis. Ndikumana et al. [20] introduced a DNN-based caching framework,
compromising of Multi-Layer Perceptron (MLP) and CNN models, where contextual
information such as age, emotion, and gender are utilized for making caching
decisions. Although CNN-based proactive caching schemes have local spatial
feature awareness, they are inefficient for extracting temporal features from
the patterns of sequential requests. Furthermore, such models require multi-
source input such as regional information, and contextual information of users
to improve the cache performance. Therefore, they need an efficient data pre-
processing model to extract this information.

To deal with the time-varying behavior of request patterns, Recurrent Neural
Networks (RNNs), such as Long Short Term Memory (LSTM) [12, 21], are
introduced to use historical information of the content. To extract both
spatial and temporal features of CP data, Ale et al. [10] used a combination
of LSTM and CNN models. LSTM-based caching frameworks, however, suffer from
long-term dependencies, computation complexity, and unsuitability for parallel
computing. To address challenges associated with RNN architectures, the
Transformer neural network [22] has been designed to handle sequential input
data, which is purely reliant on attention mechanisms with no recurrence or
convolutions. One of the most important advantages of Transformers over RNN
models is the attention mechanism, which eliminates the need to analyze data
in the same order. Consequently, Transformers have higher parallelization
capabilities than RNNs, implying reduced training time.

Contributions: Motivated by the above discussion, we introduce a Vision
Transformer-based Edge (TEDGE) caching framework with the application to the
MEC networks. The proposed TEDGE framework learns the real-time caching
strategy from sequential requests of multimedia content. The main objective of
several recent time-series prediction models that have been applied to the
multimedia content caching [23, 24] is to predict the underlying patterns of
the future multimedia content requests, i.e., the number of content requests
using historical information. Considering the fact that the users’ preferences
remain unchanged for a while [14], it is sufficient to predict the potential
Top-$K$ popular content using the learned patterns from historical requests.
The main focus of this study, therefore, is to predict the Top-$K$ popular
content using historical information instead of predicting the number of
upcoming requests. In summary, the paper makes the following key
contributions:

  * •

The TEDGE caching framework is an edge-assisted intelligent caching framework
that learns the caching strategy from the historical request patterns without
relying on data pre-processing or feature engineering. More precisely, the
TEDGE caching framework is a multi-label classification model with the aim of
minimizing the difference between the actual Top-$K$ popular content and the
predicted ones.

  * •

To simultaneously analyze the sequential pattern of all content, the TEDGE
caching framework employs a ViT architecture instead of using conventional
Transformer models. The input of the ViT model is an image, where each pixel
indicates the number of requests of each content in a specific time.

Simulation results based on real-trace of multimedia requests illustrate that
the proposed TEDGE caching framework outperforms its state-of-the-art
counterparts in cache-hit-ratio. The remainder of the paper is organized as
follows: In Section II, the system model is described and the main assumptions
required for implementation of the proposed TEDGE framework are introduced.
Section III presents the proposed TEDGE caching framework. Simulation results
are presented in Section IV. Finally, Section V concludes the paper.

##  II System Model and Problem Description

Figure 1: A typical structure of the UAV-aided cellular network.

We consider a UAV-aided cellular network with heterogeneous Radio Access
Technologies (RATs) as the $6$G network model. There are $N_{h}$ number of
hgNBs, consisting of $N_{u}$ number of UAVs, denoted by $u_{k}$, for ($1\leq
k\leq N_{u}$), along with $N_{f}$ number of FAPs, denoted by $f_{i}$, for
($1\leq i\leq N_{f}$), serving requests of $N_{g}$ number of UEs, denoted by
$U_{j}$, for ($1\leq j\leq N_{g}$). The hgNBs are equipped with a limited
cache size, denoted by $K$. As shown in Fig. 1, FAPs are independently and
randomly distributed in the environment following a Poisson Point Process
(PPP) [25]. We also consider a Gaussian mixture distribution for UEs, leading
to a dense population in some environments. Due to the movement of UEs, the
population is changed over time, therefore, the location of UAVs are
determined by the $K$-means clustering algorithm [26], where UAVs remain
hovering at their locations while serving a request for data delivery [2]. A
Software Defined Network (SDN) controller is used to manage the aerial and
terrestrial connections and control the link quality and topology of UAVs [2].
We denote a library of content $\mathcal{C}=\\{c_{1},\ldots,c_{N_{c}}\\}$,
where $N_{c}=|\mathcal{C}|$ is the cardinality of contents in the network. For
simplicity, it is assumed that the size of all contents $c_{l}$, for ($1\leq
l\leq N_{c}$), are the same [2], and UEs request at most one content in each
time slot.

CP in multimedia services follows Mandelbrot–Zipf (M-Zipf) distribution [27],
where the global probability of requesting content $c_{l}$ by all UEs, denoted
by $p_{l}$, is given by

$p_{l}=\dfrac{(l+\zeta)^{-\gamma}}{\sum\limits_{r=1}^{N_{c}}{(r+\zeta)^{-\gamma}}},$
(1)

where $\gamma$ and $\zeta$ represent the skewness and plateau factors,
respectively, and term $r$ is the rank of content $c_{r}$, when all contents
are sorted in descending order of their popularity. In addition, $p_{l}^{(b)}$
represents the local probability of requesting content $c_{l}$ by IoT devices
located in the coverage area of hgNB $b$, where $b$ could be $u_{k}$, for
($1\leq k\leq N_{u}$), or $f_{i}$, for ($1\leq i\leq N_{f}$), where
$p_{l}=\sum\limits_{b=1}^{N_{h}}p_{l}^{(b)}$.

##  III TEDGE Caching Framework

In this section, we present the TEDGE caching framework, which is designed to
predict the Top-$K$ popular content. To be specific, we first briefly
introduce the dataset used in this study, and also present the preparation
phase to adopt the dataset to the TEDGE caching framework. Then, we explain
different blocks of the ViT architecture, which is used as the multi-label
classification within the TEDGE caching framework.

###  III-A Dataset

In this study, we use MovieLens Dataset [28], which is one of the well-known
movie recommendation services. In this dataset, movies with related
information such as movie titles, release date, and genre are provided. Each
content is requested by several users in different timestamps, where the
contextual information of users such as age, gender, occupation, and their ZIP
codes are also released. With the assumption that users leave a comment after
watching a movie [12, 29, 30] and in order to extract the content request
pattern, commenting on a content is considered as a request. Moreover, to
identify the users’ location in each timestamp, ZIP codes are converted to
longitude and latitude coordinates [12]. Considering the limited transmission
range of hgNBs, hgNBs’ locations, and users locations, the available hgNBs for
serving requests of all users will be determined. Our main goal in the TEDGE
caching framework is to monitor the historical requests pattern of each
content to predict the Top-$K$ popular content in an upcoming time period.
Therefore, the preparation of the dataset is performed in the following four
steps:

Step 1 (Request Matrix Formation): In the first step, the dataset is sorted
for each content $c_{l}$, for ($1\leq l\leq N_{c}$), in the ascending order of
time. Therefore, we form an ($T\times N_{c}$) indicator request matrix for
each hgNB, denoted by R, where $T$ and $N_{c}$ represent the total number of
timestamps and the total number of distinct content, respectively. In the
request matrix, $r_{t,l}=1$ illustrates that content $c_{l}$ is requested at
time $t$; otherwise, $r_{t,l}=0$.

Step 2 (Time Windowing): Considering the fact that the most popular content
should be cached at the storage of hgNBs during the off-peak time [3], there
is no need to predict the content popularity at each timestamp. We, therefore,
define the updating time $t_{u}$ (i.e., the off-peak time), as the timestamp
that the storage of hgNBs is updated by the new popular content. In this case,
we will have a time window with the length of $\mathcal{W}$, where
$\mathcal{W}$ is associated to the time duration between two updating times,
and the number of time windows is represented by
$N_{\mathcal{W}}=\frac{T}{\mathcal{W}}$. Therefore, we have a
($N_{\mathcal{W}}\times N_{c}$) window-based request matrix, denoted by
$\textbf{R}^{(\mathcal{W})}$, where
$r^{(w)}_{t_{u},l}=\sum\limits_{t=(t_{u}-1)\mathcal{W}+1}^{t_{u}\mathcal{W}}r_{t,l}$
illustrates the total number of requests of content $c_{l}$ between updating
time $t_{u}-1$ and $t_{u}$.

Step 3 (Data Segmentation): As mentioned previously, the main target of the
TEDGE caching framework is to use the historical information of content to
predict the Top-$K$ popular content in the next updating time. Given the
window-based request matrix $\textbf{R}^{(\mathcal{W})}$, the collected
request pattern data is segmented via an overlapping sliding window of length
$l$. As it can be seen from Fig. 2, the window-based request matrix
$\textbf{R}^{(\mathcal{W})}$ is converted into
$\mathcal{D}=\\{(\textbf{X}_{u},\textbf{y}_{u})\\}_{u=1}^{M}$, where $M$
represents the total number of segments. Moreover, terms
$\textbf{X}_{u}\in\mathbb{R}^{M\times N_{c}}$ and
$\textbf{y}_{u}\in\mathbb{R}^{N_{c}\times 1}$ represent the request pattern of
all content before updating time $t_{u}$ with the length of $l$, and its
corresponding label, respectively. Considering the fact that there are $N_{c}$
number of content through the network, and our objective is to predict the
Top-$K$ popular content in the next updating time, the problem at hand is a
multi-label classification, where ${y_{u}}_{l}=1$ illustrates that content
$c_{l}$ would be popular at $t_{u+1}$. Therefore, $c_{l}$ should be stored at
the storage of hgNB to increase the cache-hit-ratio.

Figure 2: Left: Pipeline of the ViT Architecture, Right: Architecture of the
Transformer Encoder.

Step 4 (Data Labeling): Due to the limited storage of hgNBs, it is sufficient
to identify the Top-$K$ popular content, instead of predicting the popularity
of all content at each updating time. According to the request pattern of
multimedia content, we calculate the probability of requesting content
$c_{l}$, for ($1\leq l\leq N_{c}$), which is obtained as follows

$p_{l}^{(t,b)}=\dfrac{r^{(w)}_{t_{u},l}}{\sum\limits_{l=1}^{N_{c}}r^{(w)}_{t_{u},l}}.$
(2)

Note that, relying on the probability of content as a single criteria for
identifying the popularity of content has the following disadvantages: (i)
Popular content with a high number of requests will be identified as the
Top-$K$ popular content for a long time, even if they are becoming unpopular,
and; (ii) The popularity of new/unknown coming content (first appearance)
would be predicted with a considerable delay, because the cumulative number of
requests of such content is less than other content that are existing for a
long time. To tackle with this issue, we use the skewness of the request
pattern as another metric, which is a widely used indicator in time-series
forecasting models [31]. The skewness of content $c_{l}$ is denoted by
$\zeta_{l}$, where $\zeta_{l}<0$ shows that the number of requests of content
$c_{l}$ increases over time. Finally, the Top-$K$ content with the highest
probability and the negative skew will be labeled as the Top-$K$ popular
content. This completes presentation of the data preparation for training the
TEDGE caching framework. Next, we present the ViT architecture.

###  III-B ViT Architecture

Generally speaking, the main characteristics of Transformers are as follows:
(i) Non-sequential: Unlike RNN, the Transformer’s attention mechanism makes it
unnecessary to process data in the same order. As a result, Transformer is
more parallelization than RNNs, which means it takes less time to train; (ii)
Self Attention, indicating the similarity scores between different elements of
a sequential data, and; (iii) Positional Embeddings: Since Transformers are
non-sequential learning models, the order of information in a sequential data
is missing. Therefore, Positional embeddings is introduced for recovering
position information. As it can be seen from Fig. 2, the TEDGE caching
framework consists of the following three modules: (a) Patch and Position
Embeddings; (b) Transformer encoder, and; (c) MLP head, which are described as
follows:

Patch and Position Embeddings: As it can be seen from Fig. 2, the segmented CP
data $\textbf{X}_{u}$ is split into $N$ sequence of non-overlapping patches
with the fixed-size of $(S\times S)$, where the total number of patches is
$N=w/S$. After this step, each patch is flattened into a vector
${\mathbf{x}}^{p}_{u,_{j}}\in\mathbb{R}^{S^{2}}$ for ($1\leq j\leq N$). To
embed vector ${\mathbf{x}}^{p}_{u,_{j}}\in\mathbb{R}^{S^{2}}$ into the model’s
dimension $d$, a linear projection $\bm{E}\in\mathbb{R}^{S^{2}\times d}$ is
used, which is shared among all patches, where the output of this projection
is referred to as the patch embeddings. We append a learnable embedding token
${\mathbf{x}}^{cls}$ to the beginning of the sequence of embedded patches
[32]. Finally, the position embeddings $\bm{E}^{pos}\in\mathbb{R}^{(N+1)\times
d}$, is added to the patch embeddings to explicitly encode the order of the
input sequence. The output of the patch and position embeddings $\bm{Z}_{0}$
is given by

$\displaystyle\bm{Z}_{0}=[{\mathbf{x}}^{cls};{\mathbf{x}}^{p}_{u,1}\bm{E};{\mathbf{x}}^{p}_{u,2}\bm{E};\dots;{\mathbf{x}}^{p}_{u,N}\bm{E}]+\bm{E}^{pos}.$
(3)

Transformer Encoder: Given the output of the linear projection, the sequence
of vectors $\bm{Z}_{0}$ is fed to the transformer encoder [22]. As it can be
seen from Fig. 2, the transformer encoder consists of $L$ layers, with two
modules, i.e., the Multihead Self-Attention (MSA) mechanism, and the MLP
modules, where MLP module consists of two linear layers with Gaussian Error
Linear Unit (GELU) activation function. The output of the MSA and MLP modules
of layer $l$, for ($1\leq l\leq L$) are given by

$\displaystyle\bm{Z}^{{}^{\prime}}_{l}$ $\displaystyle=$ $\displaystyle
MSA(LayerNorm(\bm{Z}_{l-1}))+\bm{Z}_{l-1},$ (4) $\displaystyle\bm{Z}_{l}$
$\displaystyle=$ $\displaystyle
MLP(LayerNorm(\bm{Z}^{{}^{\prime}}_{l}))+\bm{Z}^{{}^{\prime}}_{l},$ (5)

where a layer-normalization is used to avoid the degradation problem [33].
Finally, the output of the Transformer is

$\displaystyle\bm{Z}_{L}=[{\mathbf{z}}_{L0};{\mathbf{z}}_{L1};\dots;{\mathbf{z}}_{LN}],$
(6)

where ${\mathbf{z}}_{L0}$ is used for classification purposes, which is passed
to a Linear Layer (LL), i.e.,

$\displaystyle\mathbf{y}=LL(LayerNorm({\mathbf{z}}_{L0})).$ (7)

This completes the description of the Transformer autoencoder. Next, we
present the description of the SA and the MSA, respectively.

1\. Self-Attention (SA): The SA module [22] is used in the Transformer
architecture to focus on significant parts of a given input by capturing the
interaction between different vectors in $\bm{Z}\in\mathbb{R}^{N\times d}$,
where $\bm{Z}$ consists of $N$ vectors, each with an embedding dimension of
$d$. Towards this goal, three different matrices are defined, named Queries
$\bm{Q}$, Keys $\bm{K}$, and Values $\bm{V}$, computed by a linear
transformation as follows

$\displaystyle[\bm{Q},\bm{K},\bm{V}]=\bm{Z}\mathit{W}^{QKV},$ (8)

where $\mathit{W}^{QKV}\in\mathbb{R}^{d\times 3d_{h}}$ represents the
trainable weight matrix, and $d_{h}$ is the dimension of $\bm{Q}$, $\bm{K}$,
and $\bm{V}$. The SA block measures the pairwise similarity between each query
and all keys. The output of the SA block $SA(\bm{Z})\in\mathbb{R}^{N\times
d_{h}}$, which is the weighted sum over all values $\bm{V}$, is given by

$\displaystyle
SA(\bm{Z})=\text{softmax}(\frac{\bm{Q}\bm{K}^{T}}{\sqrt{d_{h}}})\bm{V},$ (9)

where term $\dfrac{\bm{Q}\bm{K}^{T}}{\sqrt{d_{h}}}$ is the scaled dot-product
of $\bm{Q}$ and $\bm{K}$ by $\sqrt{d_{h}}$ and softmax is used to convert the
scaled similarity to the probability.

TABLE I: Variants of the TEDGE caching framework. Model ID | Layers | Model dimension $d$ | MLP layers | MLP size | Heads | Params | Accuracy | Loss
---|---|---|---|---|---|---|---|---
1 | 1 | 32 | 1 | 256 | 8 | 887,140 | 91.09 $\%$ | 0.2023
2 | 1 | 64 | 1 | 256 | 8 | 1,822,055 | 91.70 $\%$ | 0.1888
3 | 1 | 128 | 1 | 256 | 8 | 3,913,063 | 92.61 $\%$ | 0.1645
4 | 2 | 128 | 1 | 256 | 8 | 4,506,983 | 91.68 $\%$ | 0.1961
5 | 1 | 128 | 2 | 256 | 8 | 3,978,852 | 93.19 $\%$ | 0.1495
6 | 1 | 128 | 3 | 256 | 6 | 3,912,807 | 92.89 $\%$ | 0.1600
7 | 1 | 128 | 3 | 256 | 8 | 4,044,644 |  93.72 $\%$ | 0.1391
8 | 1 | 128 | 3 | 256 | 10 | 4,176,487 | 92.61 $\%$ | 0.1650
9 | 1 | 128 | 1 | 512 | 8 | 7,215,719 | 91.48 $\%$ | 0.2060

2\. Multihead Self-Attention (MSA): In the MSA, the SA block is performed $h$
times in parallel, which results in attending to information from different
representation subspaces at different positions. More precisely, MSA module
consists of $h$ heads with different trainable weight matrix for each head
$\\{\mathit{W}^{QKV}_{i}\\}^{h}_{i=1}$. After applying the SA mechanism on
input $\bm{Z}$ for each head (Eqs. (8)-(9)), the outputs of $h$ heads are
concatenated into a single matrix
$[SA_{1}(\bm{Z});SA_{2}(\bm{Z});\dots;SA_{h}(\bm{Z})]\in\mathbb{R}^{N\times
h.d_{h}}$. Finally, the output of the MSA module is given by

$\displaystyle
MSA(\bm{Z})=[SA_{1}(\bm{Z});SA_{2}(\bm{Z});\dots;SA_{h}(\bm{Z})]W^{MSA},$ (10)

where $\mathit{W}^{MSA}\in\mathbb{R}^{hd_{h}\times d}$ and $d_{h}$ is set to
$d/h$.

##  IV Simulation Results

In this Section, we first evaluate different variants of the proposed TEDGE
caching framework to obtain the best architecture through the process of trial
and error. Considering the location of UE, which is obtained from the ZIP code
and followed by Reference [20], six hgNBs are employed in different areas,
where the classification accuracy is averaged over all hgNBs. In all
experiments, the one-dimensional time-series content’s request data is
converted to a sequential set of images, which is known as the Gramian Angular
Field (GAF) technique [34]. Utilizing GAF method, not only the temporal
characteristics of the data is preserved, but also the temporal correlations
of data are included. In Back Propagation (BP) training, Adam optimizer is
employed, where the weight decay and betas are set to $0.001$ and
($0.9,0.999$). The size of the input image, the size of input patches, and the
batch size are ($25\times 25$), ($5\times 5$), and $256$, respectively.
Finally, we use binary cross-entropy as the loss function for our multi-label
classification problem. According to the results in Table I, increasing the
model dimension from $32$ to $128$ (Model $1$ to Model $3$), and also the
number of MLP layers from $1$ to $3$ (Models $3$, $5$, and $6$) increase the
classification accuracy, while increasing the number of trainable parameters.
Moreover, we evaluate the effect of MLP size on the classification accuracy.
As it can be seen from Table I, increasing the MLP size from $256$ to $512$
(Model $3$ and Model $9$) decreases the classification accuracy. Similarly,
there is no improvement in the classification accuracy by increasing the
number of transformer layers (see Models $3$ and $4$). Furthermore,
considering Model $6$ to Model $8$, the best number of heads in this
architecture is equal to $8$. Note that we also evaluated the effect of window
length on the classification accuracy, while no improvement has been achieved
by changing the window length.

Finally, we compare the performance of the proposed TEDGE caching framework
with five state-of-the-art caching schemes on the Movielens dataset, including
LRU, LFU, PopCaching [30], LSTM-C [12], and the TRansformer (TR) caching,
which is an upgraded version of the attention-based neural network in
Reference [35]. While the attention-based model in Reference [35] is used for
predicting the request pattern of online content, we adopt it to predict the
Top-$K$ popular ones. Fig. 3 compares the performance of the proposed TEDGE
scheme with other baselines mentioned above from the aspect of the cache-hit
ratio, when the DNN models reach the steady state. In the content caching
context, cache-hit-ratio is a widely used metric, illustrating the ratio of
requests served by hgNBs versus total requests. Considering the Zipf
distribution for the content popularity profile, we set the storage capacity
of hgNBs to $10\%$ of the total content [5]. As shown in Fig. 3, the optimal
strategy [12] is a caching scheme, where all requests are served through
hgNBs, which cannot be obtained in reality. According to the results in Fig.
3, the proposed TEDGE caching framework obtains the highest cache-hit-ratio in
comparison to its state-of-the-art counterparts.

##  V Conclusion

Figure 3: A comparison with state-of-the-arts based on the cache-hit-ratio.

In this paper, we presented a Transformer-based Edge (TEDGE) caching framework
with the application to the Mobile Edge Caching (MEC) networks. In order to
efficiently learn the real-time caching strategy from the time-series request
pattern of multimedia content, we employed a Vision Transformer (ViT)
architecture. To the best of our knowledge, this is the first time that a ViT
architecture is used in MEC networks to increase the cache-hit-ratio by
simultaneously identifying the Top-$K$ popular content with high accuracy.
Simulation results showed that the proposed TEDGE caching-CS scheme improves
the cache-hit ratio when compared to its state-of-the-art counterparts.

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# Integer Fluxonium Qubit

Raymond A. Mencia1,2 Wei-Ju Lin2 Hyunheung Cho2 Maxim G. Vavilov3 Vladimir E.
Manucharyan1,2 1Institute of Physics, Ecole Polytechnique Federale de
Lausanne, 1015 Lausanne, Switzerland  2Department of Physics, Joint Quantum
Institute, and Quantum Materials Center, University of Maryland, College Park,
MD 20742, USA  3Department of Physics, University of Wisconsin-Madison,
Madison, WI 53562, USA

###### Abstract

We describe a superconducting qubit derived from operating a properly designed
fluxonium circuit in a zero magnetic field. The qubit has a frequency of about
4 GHz and the energy relaxation quality factor $Q\approx 0.7\times 10^{7}$,
even though the dielectric loss quality factor of the circuit components is in
the low $10^{5}$ range. The Ramsey coherence time exceeds 100 us, and the
average fidelity of Clifford gates is benchmarked to $\mathcal{F}>0.999$.
These figures are likely to improve by an order of magnitude with optimized
fabrication and measurement procedures. Our work establishes a ready-to-use
“partially protected” superconducting qubit with the error rate comparable to
the best transmons.

###### pacs:

††preprint: APS/123-QED

##  I  Introduction

In recent years, superconducting fluxonium qubits have reached [1, 2, 3, 4, 5,
6, 7] and may have even exceeded [8, 9] the state-of-the-art coherence time
and gate error, defined by the industry-standard transmons [10, 11, 12].
Circuit-wise, the difference between the two devices seems minimal. A transmon
is fundamentally a weakly anharmonic electromagnetic oscillator defined by a
Josephson junction’s inductance and a shunting capacitance [10], while a
fluxonium contains an additional high-inductance (superinductance) shunt,
which nevertheless dramatically increases the qubit anharmonicity without
introducing new decoherence channels [13, 14, 15]. The strongly anharmonic
spectrum of fluxoniums is an important new resource for superconducting
quantum processors, as it can help mitigate the propagation of coherent errors
[16]. Another notable distinction is that fluxoniums operate at a much lower
frequency, usually ranging around $100-1000$ MHz, as opposed to the typically
4-6 GHz frequency range for transmons. In fact, given that the dielectric loss
[17, 18, 19, 20] is the primary decoherence mechanism in both devices, it is
essentially the lower frequency of fluxoniums that allows matching the best
transmons with far less sophisticated material science research and
fabrication procedures [21, 22, 23, 24, 25, 26].

Other ideas for post-transmon qubits are being actively explored. One
direction is developing control techniques for even lower frequency fluxoniums
[27, 2, 28] which are expected to have even longer coherence time, albeit not
necessarily higher quality factor. The more sophisticated “protected” qubits
can presumably have arbitrarily strong protection from any local decoherence
source by operating at a nearly zero frequency [29, 30, 31]. However, these
elegant ideas rely on a high degree of symmetry in the circuit Hamiltonians,
which is challenging to implement with the present level of fabrication
disorder [32, 33]. A more practical direction is exploring “partially
protected” qubits [34], for example, qubits in the transmon frequency range
that are to some degree decoupled from dielectric loss, such as the recently
demonstrated “soft $0-\pi$” [35] and bifluxon qubits [36]. Finally, a
significant effort is dedicated to hardware-efficient quantum error correction
with bosonic codes, based on using transmons as switches rather than qubits,
to control the quantum states of radiation in linear microwave cavities [37,
38]. Such schemes would likely benfit from substituting a transmon with a
lower-error rate control device.

Here, we circle back to conventional fluxoniums and point out that they can
equally well operate as relatively high-frequency qubits when biased at the
less explored integer flux quantum sweet spot, as opposed to the usual half-
integer flux quantum sweet spot. Circuit parameters must be adjusted such that
the low-energy dynamics is governed entirely by flux quantization in the loop
and is well described by the dual of the Cooper pair box Hamiltonian [39]. In
this case, the lowest excited state is a doublet, originating from the
classical degeneracy of charging an inductance $L$ with a positive vs. a
negative flux quantum at energy $(h/2e)^{2}/2L$. Tunneling splits the doublet.
The lower doublet and the ground states define the integer fluxonium qubit,
while the parity selection rule forbids the transition to the higher doublet
state from the ground state. Such a doublet has been spectroscopically
observed [13] and characterized in the time domain [15] in the very first
fluxonium device. A more recently published study reports the observation of
comparably long relaxation and coherence times, well in excess of $100$
$\mu$s, at both sweet spots [40]. This work demonstrates that the doublet
nature of the fluxonium transition at zero flux is compatible with precision
single-qubit gates, the average fidelity of which is benchmarked to higher
than $0.999$. This benchmark is expected to improve by at least an order of
magnitude if an active magnetic flux control is performed and with appropriate
optimization of fabrication and measurement procedures. In the meantime, this
result adds a ready-to-use device, nicknamed “integer fluxonium” qubit, to the
relatively short list of high-performance superconducting qubits.

Figure 1:  (a) Fluxonium circuit model (see text). (b) An example of the
effective potential energy profile in zero field superimposed on the lowest
four energy states is Transition $|0\rangle$-$|1\rangle$ originates from
tunneling between the local minima, and it defines the integer fluxonium
qubit. The parity selection rule forbids the nearly degenerate transition
$|0\rangle$-$|2\rangle$. Finally, transition $|0\rangle$-$|3\rangle$
corresponds to anharmonic oscillations in the lowest minima, and it is
sufficiently detuned from the qubit. (c,d) Frequencies and charge matrix
elements of the two lowest transitions calculated near zero field. (e)
Illustration of the required balance between energy relaxation quality factor
$Q_{1}$ and $1/f$ flux noise dephasing quality factor $Q_{1/f}$ for finding
optimal integer fluxonium parameters. Given the experimental values of the
$1/f$ flux noise amplitude $A_{1/f}=1.4\times 10^{-6}$ and the base dielectric
loss quality factor $Q_{\textrm{diel}}=2.5\times 10^{5}$, there is an optimal
value $E_{J}/E_{C}\approx 4$ that maximizes the qubit’s total decoherence
quality factor $Q_{2}=(2Q_{1}^{-1}+Q_{1/f}^{-1})^{-1}$

The paper is organized as follows. In section II we demonstrate the
equivalence of the integer and the half-integer flux sweet spots with respect
to the energy relaxation rate using an elementary model of flux quantization
in a superconducting loop. In section III we describe devices and data,
including a complete case study for device D in Table 1. In section IV we
share concluding remarks.

##  II Theory of integer fluxonium

The fluxonium circuit (see Fig. 1a) is defined by three parameters: the
Josephson energy $E_{J}$ of the Josephson junction, the charging energy
$E_{C}=e^{2}/2C$ of the total capacitance $C$ and the inductive energy
$E_{L}=(\hbar/2e)^{2}/L$ of the superinductance $L$. The junction and the
inductance form a superconducting loop, which can be biased by an external
magnetic flux $\varphi_{\textrm{ext}}\times(\hbar/2e)$. Quantum mechanics of
fluxoniums is usually described by a pair of conjugate macroscopic variables,
the superconducting phase-difference $\varphi$ across the Josephson junction
and the charge $2e\times n$. The two observables, the phase, $\hat{\varphi}$,
and the Cooper pair number operator, $\hat{n}$, displaced at the capacitor
satisfy the standard position-momentum-like commutation relation
$[\hat{\varphi},\hat{n}]=i$ and the Hamiltonian reads

$H=4E_{C}\hat{n}^{2}+\frac{1}{2}E_{L}\hat{\varphi}^{2}-E_{J}\cos(\hat{\varphi}-\varphi_{\rm{ext}}).$
(1)

The parameter regime for integer fluxonium is derived from the following
arguments. First, as with all fluxoniums, it is essential to have $E_{L}\ll
E_{J}$, in which case the potential energy (the $\varphi$-dependent term of
Eq. (1)) has multiple local minima (Fig. 1b). The charging energy term is
equivalent to the kinetic energy of a particle with a mass $C$ moving in such
a potential. There are two types of excitations: oscillations in a single well
(plasmon) and tunneling between the wells (fluxon). One must decouple these
excitations to implement integer fluxonium, which is achieved by setting the
plasmon frequency sufficiently above the fluxon frequency. In the asymptotic
limit of weak tunneling, that is $E_{J}/E_{C}\gg 1$, the plasmon energy is
given approximately by $\sqrt{8E_{J}E_{C}}$ while the fluxon energy is
approximately $2\pi^{2}E_{L}$, which reduces the parameter requirement for the
integer fluxonium to $\sqrt{8E_{J}E_{C}}\gg 2\pi^{2}E_{L}$, with qubit
frequency given approximately by $\hbar\omega_{01}\approx 2\pi^{2}E_{L}$, the
offset between the lowest two potential wells in Fig. 1b.

###  II.1 Flux-dual Cooper pair box model

Although spectral properties of integer fluxonium can be calculated
numerically from the Hamiltonian (1) for arbitrary circuit parameters and flux
bias (see Fig. 1c,d), it is insightful to consider an effective model defined
entirely in terms of flux quantization and tunneling [41, 14]. We start with
the basis of fluxon states $|m\rangle$ corresponding to a particle resting at
the bottom of the m-th Josephson well (in reality, we are only interested in
the lowest three wells shown in Fig. 1a). Introducing the phenomenological
tunneling energy amplitudes $\epsilon_{1}$ to change $m$ by $\pm 1$ and
$\epsilon_{2}$ to change $m$ by $\pm 2$ we can write the following effective
Hamiltonian

$\displaystyle H_{\textrm{eff}}=\sum_{m}\frac{E_{L}^{\Sigma}}{2}(2\pi
m-\varphi_{\textrm{ext}})^{2}|m\rangle\langle m|+$ (2)
$\displaystyle-\frac{\epsilon_{1}}{2}\sum_{m}|m\rangle\langle m\pm
1|+\frac{\epsilon_{2}}{2}\sum_{m}|m\rangle\langle m\pm 2|,$

where $E_{L}^{\Sigma}=(E_{L}^{-1}+E_{J}^{-1})^{-1}$ and is the total
linearized loop inductance. The values of the tunneling amplitudes
$\epsilon_{1}$ and $\epsilon_{2}\ll\epsilon_{1}$ depend on the three circuit
parameters $E_{J}$, $E_{C}$, and $E_{L}$, but it is safe to assume that both
amplitudes are exponentially suppressed with the parameter
$\sqrt{8E_{J}/E_{C}}$. The double-fluxon tunneling amplitude $\epsilon_{2}$ is
not necessary to derive the main properties of integer fluxonium. Still, it is
necessary to obtain an accurate value of $\epsilon_{1}$ by matching the low-
energy spectra of the two Hamiltonians (1,2). In fact, for $\epsilon_{2}=0$,
the Hamiltonian (2) describes a Cooper pair box (a charge qubit) in the regime
of strong charge quantization, where charging energy is replaced by the
inductive energy, Cooper pairs with fluxons, and offset charge with the flux
bias.

The spectrum of the model Hamiltonian (2) near zero bias can be obtained by
diagonalizing the corresponding $3\times 3$ matrix in the basis of three
fluxon states $|m=-1\rangle$, $|m=0\rangle$, and $|m=+1\rangle$. Setting
$\varphi_{\textrm{ext}}=0$ in Eq. 2 and introducing a small parameter
$\alpha=\epsilon_{1}/4\pi^{2}E_{L}^{\Sigma}$ we get the following perturbative
expressions for the three lowest-energy eigenstates:

$\displaystyle|0\rangle$
$\displaystyle=|m=0\rangle+\alpha\big{(}|m=-1\rangle+|m=+1\rangle\big{)},$ (3)
$\displaystyle|1\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}|m=-1\rangle-|m=+1\rangle\big{)},$
$\displaystyle|2\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}|m=-1\rangle+|m=+1\rangle)-\sqrt{2}\alpha|m=0\rangle.$

The eigenstates $|1\rangle$ and $|2\rangle$ define the excited-state doublet
of integer fluxonium. The two transition frequencies are given by
$\hbar\omega_{01}=2\pi^{2}E_{L}^{\Sigma}+\alpha\epsilon_{1}-\epsilon_{2}/2$
and
$\hbar\omega_{02}=2\pi^{2}E_{L}^{\Sigma}+2\alpha\epsilon_{1}+\epsilon_{2}/2$.
Thus, the doublet is split by
$\hbar\omega_{12}=\alpha\epsilon_{1}+\epsilon_{2}$. Note that $\epsilon_{2}$
parameter does not affect the eigenstates but merely increases the doublet
splitting $\omega_{12}$. In the qualitative analysis below, we will set
$\epsilon_{2}$ to zero to simplify the expressions without losing the essence.

The matrix elements of $\hat{\varphi}$ and $\hat{n}$ operators can be obtained
by noting the relation $\hat{\varphi}=2\pi\hat{m}$ and using the identity
$\langle i|\hat{n}|j\rangle=\langle
i|\hat{\varphi}|j\rangle\times(\hbar\omega_{ij}/8E_{C}$), which follows
directly from the commutation rule and Eq. (1). For the phase-diference matrix
elements, we get $\langle 0|\hat{\varphi}|2\rangle=0$ and $\langle
0|\hat{\varphi}|1\rangle\approx 2\pi\alpha\ll 1$. Note, in case of a
conventional fluxonium operation at $\varphi_{\textrm{ext}}=\pi$, within the
model of Eq. (2), we have $\hbar\omega_{01}=\epsilon_{1}$ and $\langle
0|\hat{\varphi}|1\rangle\approx\pi$, which is a much larger number than at the
integer flux bias. Yet, the matrix elements of the conjugate Cooper pair
number operator $\hat{n}$ come out essentially the same at the two sweet
spots:

$\langle 0|\hat{n}|1\rangle_{\varphi_{\textrm{ext}}=0}\approx\sqrt{2}\langle
0|\hat{n}|1\rangle_{\varphi_{\textrm{ext}}=\pi}\approx\frac{\sqrt{2}\pi}{8}\frac{\epsilon_{1}}{E_{C}}.$
(4)

The degree of protection of integer fluxonium from energy relaxation can be
judged by comparing this matrix element estimate with the typical transmon
value, $\sqrt[4]{E_{J}/8E_{C}}\simeq 1$ [42].

The above qualitative analysis based on Eq. (2) compliments accurate numerical
calculations based on Eq. 1. Examples of calculated wave functions, transition
frequencies, and matrix elements are shown in Fig. (1b-d). More importantly,
the plasmon transition enables a dispersive qubit-cavity interaction used for
the qubit readout, as long as the cavity frequency is not too far detuned from
the plasmon [43].

###  II.2 Dielectric loss vs. $1/f$ flux noise

An optimal design for an integer fluxonium must simultaneously suppress the
effect of the two prevailing decoherence mechanisms: the dielectric loss and
the $1/f$ flux noise. Both channels originate from defects in the surface
oxide layers of superconducting leads and thus cannot be readily eliminated,
at least not in the near future. Let us define the dielectric loss via the
effective loss tangent $\tan\delta_{C}$ of the circuit’s total capacitance $C$
[44, 45, 46]. For both transmon and fluxonium qubits, the energy relaxation
quality factor $Q_{1}$ is related to $Q_{\textrm{diel}}$ according to the
expression of Fermi’s Golden rule $Q_{1}=\omega_{01}T_{1}$, and
$1/T_{1}=E_{C}\times 32\pi|\langle 0|\hat{n}|1\rangle|^{2}/Q_{\textrm{diel}}$,
where we define $Q_{\textrm{diel}}=1/\tan\delta_{C}$ as the effective
dielectric loss quality factor. For a transmon, $Q_{1}=Q_{\textrm{diel}}$. For
the best transmons, $Q_{\textrm{diel}}\approx 1.5\times 10^{7}$[47, 48, 49].
For integer fluxonium, we can qualitatively estimate the parameter dependence
of $Q_{1}$ as

$Q_{1}\propto\frac{E_{C}}{\hbar\omega_{12}}\times Q_{\textrm{diel}}\,.$ (5)

To estimate the dephasing rate due to the $1/f$ flux noise, we define the
noise spectral density as $S_{1/f}(\omega)=2\pi
A_{1/f}^{2}\times(h/2e)^{2}/\omega$, where the quantity $A_{1/f}$
characterizes the strength of the flux noise. Slow variation in the value of
$\varphi_{\textrm{ext}}$ induces a slow variation of the qubit frequency to
second order in the value of $A_{1/f}$ and proportionally to the curvature of
the qubit transition with respect to the flux bias, that is
$d^{2}\omega_{01}/d\varphi_{\textrm{ext}}^{2}$ at $\varphi_{\textrm{ext}}=0$.
The resulting pure dephasing quality factor $Q_{1/f}=\omega_{01}T_{1/f}$ is
given up to a numerical factor:

$Q_{1/f}\propto\frac{\hbar\omega_{12}}{E_{L}}\times\frac{1}{A_{1/f}^{2}}\,.$
(6)

Reducing the frequency $\omega_{12}$ would increase $Q_{1}$ but reduce
$Q_{1/f}$ by the same amount, without changing the qubit transition frequency
$\hbar\omega_{01}\approx 2\pi^{2}E_{L}$. The combined decoherence quality
factor $Q_{2}=1/(1/2Q_{1}+1/Q_{1/f})$ reaches its maximum for a specific value
of $E_{J}/E_{C}$, which depends on the experimental values of
$Q_{\textrm{diel}}$ and $A_{1/f}$. In other words, the strength of flux noise
$A_{1/f}$ determines how much the qubit’s quality factor can be increased over
the dielectric loss limit $Q_{\textrm{diel}}$. Fig. 1e illustrates the
described balance using experimentally motivated values of $A_{1/f}=10^{-6}$
and $Q_{\textrm{diel}}=2.5\times 10^{5}$, resulting in an enhancement factor
as large as $Q_{2}/Q_{\textrm{diel}}\approx 40$.

###  II.3 Leakage to state $|2\rangle$

The more integer fluxonium is made protected from energy relaxation, the
smaller the splitting between the computational level $|1\rangle$ and the non-
computational level $|2\rangle$. As long as $\varphi_{\textrm{ext}}=0$, no
leakage to state $|2\rangle$ is expected during qubit manipulations. Indeed, a
resonant qubit drive at a frequency around $\omega_{01}$ does not couple to
transition $|0\rangle$-$|2\rangle$ because $\langle 0|\hat{n}|2\rangle=0$, nor
does it couple to transition $|1\rangle$-$|2\rangle$ because of the large
frequency detuning. Thus, in theory, integer fluxonium is effectively a
2-level system: even transition $|0\rangle$-$|3\rangle$ would not be coupled
by the drive because of its large detuning.

In practice, though, there is always a small deviation
$\delta\varphi_{\textrm{ext}}\ll 1$ of the value $\varphi_{\textrm{ext}}$ from
an integer value, either due to an instrument imprecision, or a background
field, or at the very least due to the same $1/f$ flux noise that causes pure
dephasing. Consequently, the qubit drive acquires a small coupling to state
$|2\rangle$, proportional to the value of $(d\langle
0|\hat{n}|2\rangle/d\varphi_{\textrm{ext}})\delta\varphi_{\textrm{ext}}$ near
$\varphi_{\textrm{ext}}=0$. The frequencies $\omega_{01}$ and $\omega_{02}$
would also be slightly shifted to the second order in
$\delta\varphi_{\textrm{ext}}$. For a given $\delta\varphi_{\textrm{ext}}$, we
can model the resulting coherent errors in our qubit using the following
elementary 3-level Hamiltonian

$\displaystyle
H_{\textrm{drive}}(\delta\varphi_{\textrm{ext}})/\hbar=\omega_{01}|1\rangle\langle
1|+\omega_{02}|2\rangle\langle 2|+$ (7)
$\displaystyle+\hat{n}\times\big{(}I(t)\cos\omega t+Q(t)\sin\omega t\big{)},$

where $\omega\approx\omega_{01}$ and $I(t)$ and $Q(t)$ are slowly varying
envelopes defining the qubit pulse shape. We use a numerical solution of this
Hamiltonian to interpret the results of randomized benchmarking of
experimental devices.

Figure 2: (a) Optical image of integer fluxonium device. (b) Two-tone
spectroscopy signal (arbitrary units) as a function of frequency and flux bias
near $\varphi_{\textrm{ext}}=0$. Inset shows a cross-section at
$\varphi_{\textrm{ext}}=0$. Note the absence of the $|0\rangle$-$|2\rangle$
within the given signal-to-noise ratio.

##  III Qubit Characterization

###  III.1 Spectroscopy and decoherence

Spectroscopy and time-domain data were collected on four different devices.
The device parameters are extracted from fitting the model in Eqs. (1) and (2)
and are summarized in Table 1. In comparison to Ref. [1], we increased the
charging energy $E_{C}$ and the Josephson energy $E_{J}$ but reduced the
inductive energy $E_{L}$. Similarly to Ref. [8], the readout is performed
using a capacitive coupling to a 3D cavity for samples A, B, and D. Sample C
was coupled inductively to an on-chip resonator and read out using a wireless
coupling to a 3D waveguide, similarly to Ref. [50, 51, 52]. Fabrication and
measurement procedures are similar to those in Ref. [1].

Figure 2a shows an optical image of device A, reminiscent of fluxonium devices
from Ref. [53] except with a smaller antenna part. Fig. 2b shows the spectrum
of device D near $\varphi_{\textrm{ext}}=0$. The data reveals a doublet
analogous to the one first observed in Refs. [13, 54], except the splitting at
$\varphi_{\textrm{ext}}=0$ is reduced to a mere 11 MHz. The inhibition of
transition $|0\rangle$-$|2\rangle$ is illustrated in the inset of Fig. 1b. The
transition $|0\rangle$-$|3\rangle$ is above 5 GHz. This device’s calculated
Cooper pair number transition matrix element is $\langle
0|\hat{n}|1\rangle\approx 0.056$ and is at least an order of magnitude below
the typical transmon value.

Figure 3:  (a) The experimentally measured transition frequencies (top) and
energy relaxation times (bottom) of sample C as a function of external flux
bias. The average values of $T_{1}$ are shown in green. (b) The quality factor
of the measured qubit transitions due to energy relaxation, $Q_{1}$, at each
flux sweet spot along with the extracted value of $Q_{\text{diel}}$.  Figure
4:  (a) The Ramsey coherence time $T_{2}^{*}$ of device D as a function of
external flux. Dashed line is the first-order theoretical value with $A_{1/f}$
$\approx 1.4\times 10^{-6}$. (b) Example Ramsey fringe measured at
$\varphi_{\textrm{ext}}=0$; corresponding to the orange data point in (a).

Energy relaxation time $T_{1}$ shows a negligible flux dependence (Fig. 3c) as
the qubit frequency is increased by an order of magnitude from its half-
integer flux value of $\omega_{01}/2\pi=190$ MHz to the integer flux bias
value of $\omega_{01}/2\pi=3450$ MHz. Interleaved measurements of $T_{1}$ and
$T_{2}^{E}$ were done extensively over many hours to probe the repeatability
of each characteristic time. We obtain the respective average times shown in
Table 1 as $\bar{T}_{1}$ and $\bar{T}_{2}^{E}$ by averaging each individual
fit of the respective measurements. The energy relaxation quality factors
$Q_{1}=\omega_{01}T_{1}$ are compared for each device at
$\varphi_{\textrm{ext}}=0,\pi$ as well as with the extracted value of
$Q_{\textrm{diel}}$ (Fig. 3b). We highlight that the measured quality factor
in devices C and D are in the $Q_{1}\simeq(6.5-7.0)\times 10^{6}$ range. Not
only is this number within a factor of two from the highest Q reported for
optimally fabricated transmons [26], but the dielectric quality factor for
these devices is in the low $10^{5}$ range. Frequency dependence of $T_{1}$ in
multiple devices seems consistent with the dielectric loss model (Fig. 7).
Nevertheless, our experiment cannot exclude other energy relaxation channels,
so the estimation of the $Q_{\textrm{diel}}$ should be taken as an upper
bound.

The coherence in samples A and B was limited by energy relaxation, as
indicated by $T_{2}^{E}\approx 2T_{1}\approx 200$ $\mu$s. The coherence time
in devices C, D is most likely limited by cavity photon shot noise. In case of
sample D we describe a more accurate measurement of the uncorrected Ramsey
coherence time $T_{2}^{*}$ in the vicinity of the sweet spot
$\varphi_{\textrm{ext}}=0$ (see Fig. 4). We find that the flux dependence away
from the sweet spot is consistent with the 1/f flux noise with the amplitude
$A_{1/f}\approx 1.4\times 10^{-6}$, which is similar to what was measured in
previous fluxonium experiments. The value of $T_{2}^{*}$ saturates at about
$T_{2}^{*}\approx 100$ $\mu$s in an interval
$|\varphi_{\textrm{ext}}|/2\pi<10^{-5}$. From this measurement, we conclude
that the short-term stability of the flux bias in an integer fluxonium device
can be on the order of $\delta\varphi_{\textrm{ext}}/2\pi<10^{-5}$.

Figure 5: (a) The RB trace for optimal fidelity $\mathcal{F}=99.92\pm.01\%$
measured at a Rabi frequency of 6 MHz. (b) RB measurements for different Rabi
frequencies (black diamonds) superimposed with the gate errors expected based
on small flux offsets, $\delta\varphi_{\text{ext}}/2\pi$. The gate error due
to flux offsets does not dominate until about
$\delta\varphi_{\text{ext}}/2\pi=10^{-4}$. (c) The Rabi-phase error (dashed
line) and the $|2\rangle$ leakage (dotted line) combine to produce the error
fidelity. Here, each separate contribution is shown.

###  III.2 Gates benchmarking

We arrive at the most important demonstration of the potential usefulness of
integer fluxonium, benchmarking of the single-qubit Clifford-group gates. This
test is especially critical due to the possibility of the state-$|2\rangle$
leakage under an imperfect integer flux quantum bias. For benchmarking, we
select device D as the most protected and, hence, most potentially sensitive
to the leakage.

The experimentally performed RB sequence consists of randomly chosen Clifford
gates applied to the qubit before applying a single inversion gate; bringing
the state back to the initial. The number of Clifford gates before the final
state inversion gate is increased, resulting in a decaying probability of
properly recovering the initial state. Measurements are fit according to the
polynomial $A+Bp^{m}$; where p is the depolarization parameter while A and B
are constants that account for state preparation and measurement (SPAM) errors
[55]. The average error rate $1-\mathcal{F}$ for Clifford group gates is
defined as $(1-\mathcal{F}_{\textrm{Cliff}})=(1-p)/2$. Since there are on
average 1.833 physical gates per Clifford the average physical gate fidelity
is $(1-\mathcal{F})=(1-\mathcal{F}_{\textrm{Cliff}})/1.833$ [8, 7]. The best
average physical gate fidelity measured was $\mathcal{F}=0.9992(1)$ (Fig. 5a)
with a Rabi frequency of $\Omega_{\text{Rabi}}=6.2$ MHz.

###  III.3 Leading errors

Randomized benchmarking measurements for sample D are then simulated in QuTip
[56] to quantify different contributing sources of error. The simulation uses
smooth-edged pulses with a switching on/off time of 5 ns to evaluate the full
evolution operator $U$ at the end of the pulse. The theoretical estimate of
the fidelity is obtained from [57]

$\mathcal{F}=\frac{\text{Tr}(U^{\dagger}_{\rm comp}U_{\rm
comp})+|\text{Tr}(U_{\rm comp}^{\dagger}U_{\text{ideal}}))|^{2}}{6},$

where $U_{\rm comp}$ is the evolution operator projected to the computational
subspace. We find the primary source limiting the experimental gate fidelity
is best described by a model assuming that a small static flux detuning (on
the order of $\Phi_{0}\times 10^{-5}$) away from $\varphi_{\text{ext}}=0$ is
developed after the gates were calibrated. The origin of this offset is
probably related to an imperfect shielding of our experiment from external
stray magnetic fields. The flux offset produces a non-zero
$|0\rangle-|2\rangle$ matrix element and can therefore enhance leakage for
large Rabi frequencies. We evaluate the leakage error to state $|2\rangle$ in
terms of the matrix elements of the full evolution operator as
$\varepsilon_{\rm leak}=(|U_{02}|^{2}+|U_{12}|^{2})/2$. The results of this
estimation for the flux detuning $\delta\varphi_{\rm ext}/2\pi=10^{-4}$ and
$10^{-5}$ are shown in Fig. 5c.

Furthermore, if flux-detuning occurs after excitation pulse calibration, the
pulse no longer provides a perfect rotation. We focus on the $X$ gate when
$U_{\text{ideal}}=\sigma_{x}$. In this case, the gate error without leakage
can be estimated as $\varepsilon_{\rm det}=1-\mathcal{F}=2(1-|U_{01}|^{2})/3$.
We estimate the magnitude of $|U_{01}|^{2}$ using the expression for the
amplitude of the Rabi oscillations with detuning
$\delta\omega_{01}=\omega_{01}(\delta\varphi_{\textrm{ext}})-\omega_{01}(0)$
and obtain the error associated with the detuning of the transition frequency
$\varepsilon_{\rm det}=(8/3)(\delta\omega_{01}/2\pi)^{2}t_{g}^{2}$ for the
gate time $t_{g}$. The change in qubit frequency due to the flux offset can be
approximated by $\delta\omega_{01}/2\pi=8\pi^{2}\langle
0|\hat{\varphi}|1\rangle E_{L}\delta\varphi_{\text{ext}}$ [2] which is on the
order of hundreds of kHz for sample D with
$\delta\varphi_{\text{ext}}/2\pi=10^{-4}$. The summation of the two error
sources for each static flux offset is shown in Fig. 5b. These two errors
scale inversely to one another as a function of Rabi frequency as illustrated
in Fig. 5c for flux offset $\delta\varphi_{\text{ext}}/2\pi$.

| Sample | A | B | C | D
---|---|---|---|---|---
Fluxonium model fit |  $E_{J}$ (GHz) | 4.12 | 3.84 | 7.20 | 6.86
|  $E_{C}$ (GHz) | 1.64 | 1.75 | 2.04 | 1.46
|  $E_{L}$ (GHz) | 0.18 | 0.14 | 0.18 | 0.23
Flux tunneling model fit |  $E_{L}^{\Sigma}$ (GHz) | 0.172 | 0.135 | 0.175 | 0.222
|  $\epsilon_{1}$ (MHz) | 308 | 362 | 190 | 91
|  $\epsilon_{2}$ (MHz) | 73 | 81 | 6 | 9
Measured parameters |  $\omega_{01}/2\pi$ (GHz) | 3.20 | 2.51 | 3.45 | 4.14
|  $\omega_{12}/2\pi$ (MHz) | 103 | 106 | 24 | 11
|  $\bar{T}_{1}$ ($\mu$s) | 109 | 101 | 328 | 255
|  $\bar{T}_{2}^{E}$ ($\mu$s) | 175 | 201 | 81 | 185
|  $\bar{T}_{2}^{*}$ ($\mu$s) | 38 | 61 | 57 | 118
Inferred parameters | $Q_{\text{diel}}$ | $3.5\times 10^{5}$ | $3.6\times 10^{5}$ | $2.6\times 10^{5}$ | $1.0\times 10^{5}$
|  $|n_{01}|^{2}$ (IFQ) | $2.12\times 10^{-2}$ | $1.85\times 10^{-2}$ | $4.4\times 10^{-3}$ | $3.2\times 10^{-3}$
Table 1: A summary of integer fluxonium device parameters. All measured
parameters were performed while $\varphi_{\textrm{ext}}=0$.

Our RB results indicate that the gate error is not currently limited by the
qubit coherence instead we find that small external flux offsets after
parameter optimization at $\varphi_{\text{ext}}$ produces a Rabi-phase error
due to the nonperfect gate calibration at low Rabi frequency; while at large
Rabi frequency the error is produced by the non-zero matrix element for
$|0\rangle-|2\rangle$. Interestingly, the leakage to the second eigenstate is
negligible for an external flux offset less than
$\delta\varphi_{\textrm{ext}}/2\pi<10^{-5}$ supporting the idea that the
leakage into the closely situated state $|2\rangle$ is not an issue for
perfect flux biasing at $\varphi_{\text{ext}}=0$. In future experiments, the
static offset can be eliminated by carefully removing all components producing
magnetic noise close to the device.

##  IV Conclusions

Integer fluxonium qubit operates in zero magnetic field and in the same
frequency range as transmons, it has a comparably high quality factor and gate
fidelity, despite a much lower quality of the circuit dielectrics. The degree
of decoupling from the dielectric loss and similar relaxation mechanisms is
controlled by the Cooper pair number matrix element $\langle
0|\hat{n}|1\rangle$, the value of which can be user-defined by adjusting the
ratio $E_{J}/E_{C}$, independently of the qubit frequency.

The only catch of the integer fluxonium design is the presence of the second
transition $|0\rangle$-$|2\rangle$ close to the qubit transition
$|0\rangle$-$|1\rangle$, which can get activated by the qubit drive due to a
slight deviation in the flux bias from zero or the integer flux quantum. So
far, we demonstrated that even in a relatively extreme case of
$\omega_{12}/2\pi=11$ MHz (device D) and a relatively large flux offset of
$\delta\varphi_{\textrm{ext}}/2\pi\sim 10^{-4}$, the gate error remains under
$0.1/\%$ with about 10-20 MHz Rabi rate ($\pi$-pulse duration of 50-25 ns).
Future experiments will reveal the practical limits to the flux-bias accuracy.
We envision that the flux offset can be reduced to
$\delta\varphi_{\textrm{ext}}/2\pi\lesssim 10^{-5}$ by either improved
magnetic field shielding or an active feedback. Furthermore, due to the strong
anharmonicity of the non-computational transitions, the leakage to state
$|2\rangle$ may not be as dangerous as in transmons: the system remains
contained to only three states $|0\rangle$, $|1\rangle$, and $|2\rangle$, and
the system can be efficiently reset back to the computational subspace.

Fundamentally, the mechanism of decoupling of the integer fluxonium’s
transition from energy relaxation is precisely the same as in the case of a
regular low-frequency fluxonium, namely the suppression of the matrix element
$\langle 0|\hat{n}|1\rangle$ far below the transmon’s value for the flux
tunneling (fluxon) transition. This observation calls for two important
remarks. First, a Cooper pair box (charge qubit) device operated at zero
offset gate charge could have been operated in the same manner as integer
fluxonium were it not for the excessively high level of background charge
noise felt by a typical superconducting island in comparison to the background
flux noise felt by a typical superconducting loop. Second, integer fluxonium
shares many common characteristics with the recently demonstrated “soft
$0-\pi$” qubit [35]. Both devices have a doublet qubit transition, leading to
suppressed matrix elements of the flux and charge operators, combined with a
first-order insensitivity to flux noise. It has been previously conjectured
that achieving such a portfolio of properties requires a circuit with more
than one degree of freedom [33]. Indeed, the “soft $0-\pi$” qubit is
constructed by a relatively strong coupling of three modes: fluxonium-like,
transmon-like, and harmonic oscillator. By contrast, integer fluxonium is
described by a single degree of freedom. Understanding the profound difference
between the two devices in terms of the origins of the noise decoupling could
lead to new ideas for high-performance qubit designs.

###### Acknowledgements.

We wish to acknowledge fruitful conversations with Chen Wang. This research
was supported by the ARO HiPS (contract No. W911-NF18-1-0146) and GASP
(contract No. W911-NF23-10093) programs.

Figure 6:  (a) The two-tone spectroscopy of sample A superimposed with the
fluxonium Hamiltonian (eq. 1, solid lines) and the effective Hamiltonian (eq.
2, dashed lines) fits. The thin lines correspond to transitions from the first
excited state and are fit using the fluxonium Hamiltonian. (b) A zoom-in
around $\varphi_{\textrm{ext}}=0$ of the first two transitions plus both fits
to demonstrate the high-degree of agreeability between the measured data and
the two models.

##  Appendix A Fluxonium numerics

The general fluxonium Hamiltonian is expressed as:

$H=4E_{C}\hat{n}^{2}+\frac{1}{2}E_{L}\hat{\varphi}^{2}-E_{J}\cos(\hat{\varphi}-\varphi_{\rm{ext}})$
(8)

The phase, $\hat{\varphi}$, and the Cooper pair number, $\hat{n}$, operators
are expressed in the harmonic oscillator basis:

$\displaystyle\hat{\varphi}$
$\displaystyle=\frac{1}{\sqrt{2}}\bigg{\\{}{\frac{8E_{C}}{E_{L}}}\bigg{\\}}^{1/4}({a}^{\dagger}+{a})\
,$ (9) $\displaystyle\hat{n}$
$\displaystyle=\frac{i}{\sqrt{2}}\bigg{\\{}{\frac{E_{L}}{8E_{C}}}\bigg{\\}}^{1/4}({a}^{\dagger}-{a})$

here $a,a^{\dagger}$ are the harmonic oscillator raising and lower operators
each with a Hilbert space dimension exceeding fifty oscillator states.

After numerical diagonalization, the eigenvalues of the fluxonium Hamiltonian
are returned; however, the eigenfunctions are the coefficients necessary to
construct the fluxonium wavefunctions by the linear superposition of harmonic
oscillator wavefunctions defined as:

$\psi_{k}(\varphi)=\frac{1}{(2\pi\varphi_{\rm
zpf}^{2})^{1/4}}\frac{1}{\sqrt{2^{k}k!}}H_{k}(\varphi/\varphi_{\rm
zpf})e^{{-\varphi^{2}/4\varphi_{\rm zpf}^{2}}}\ ,$ (10)

where $k$ is the integer index of the harmonic oscillator state and
$\varphi_{\rm zpf}=(8E_{C}/E_{L})^{1/4}$. Using these wavefunctions with the
coefficients found by diagonalization we write
$\Psi_{j}(\varphi)=\sum_{k=0}^{k=K}c_{k}\psi_{k}(\varphi)$, where $c_{k}$ is
the $k$th component of the eigenvector for the $j$th eigenstate after
diagonalizing and $\sum_{k}|c_{k}|^{2}=1$.

Figure 7: The quantity $32\pi E_{C}|n_{01}|^{2}\times T_{1}$ vs. the qubit
frequency for four devices (see Table 1). The trend for frequency-independence
over several octaves of frequency is consistent with the dielectric loss
model. Data for devices A-C was compiled from Ref. [40]

##  Appendix B Auxiliary data

The experimental spectra for each device was fit using both the conventional
fluxonium Hamiltonian Eq. (1) and the discrete flux Hamiltonian Eq. (2).
Extracted parameters from both fit models for each device is summarized in
Table 1. The spectrum of qubit sample A is shown in Fig. 6. The fits to
spectroscopy data show a near-perfect agreement of the fluxonium to the fluxon
fit up to the vicinity of the plasmon. The comparison of the fits verified
that all samples $f_{01}^{\text{IFQ}}$ were indeed less than the plasmon mode
frequency. The first two transitions show a sharp external flux dependence at
integer flux values (Fig 6b). The larger the $E_{J}/E_{C}$ ratio, the smaller
the $|1\rangle\rightarrow|2\rangle$ frequency. The $|0\rangle-|2\rangle$
transition has a vanishing matrix element at IFQ and hence cannot be excited
and is experimentally revealed by the two-tone signal vanishing at
$\varphi_{\text{ext}}=0$.

To further characterize qubits, $T_{1}$ was measured several times across half
a flux quantum period for samples A-C; sample D was only measured at both
sweet spots. Per sample, the multiple energy relaxation traces were first fit
individually and were then binned together. The average decay constant per bin
is the reported $T_{1}$ while the standard deviation per bin is the error. The
binned statistics of energy relaxation times were then converted into an
effective $Q_{\text{diel}}$ using Fermi’s golden rule at zero temperature
($32\pi E_{C}|n_{01}|^{2}\times T_{1}$). The results are shown in Fig. 7
versus the qubit transition frequency and illustrate the effective
$Q_{\textrm{diel}}$ does not depend explicitly on the qubit transition
frequency. We note that incorporating a finite temperature to the lossy
dielectric would only further increase the quality factor.

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