# A No-Reference Quality Assessment Method for Digital Human Head

###### Abstract

In recent years, digital humans have been widely applied in augmented/virtual
reality (A/VR), where viewers are allowed to freely observe and interact with
the volumetric content. However, the digital humans may be degraded with
various distortions during the procedure of generation and transmission.
Moreover, little effort has been put into the perceptual quality assessment of
digital humans. Therefore, it is urgent to carry out objective quality
assessment methods to tackle the challenge of digital human quality assessment
(DHQA). In this paper, we develop a novel no-reference (NR) method based on
Transformer to deal with DHQA in a multi-task manner. Specifically, the front
2D projections of the digital humans are rendered as inputs and the vision
transformer (ViT) is employed for the feature extraction. Then we design a
multi-task module to jointly classify the distortion types and predict the
perceptual quality levels of digital humans. The experimental results show
that the proposed method well correlates with the subjective ratings and
outperforms the state-of-the-art quality assessment methods.

Index Terms—  Digital human head, Textured mesh, Quality assessment, No-
reference, Multi-task learning

##  1 Introduction

Digital humans are virtual characters with human images, human characters, and
behavioral characteristics simulated by computers, which are regarded as the
entrance to the metaverse. Thanks to the rapid development of computer
graphics, digital humans have been gradually integrated into people’s lives in
many fields such as film industry, social media, tourism, etc. ††footnotetext:
This work was supported in part by NSFC (No.62225112, No.61831015), the
Fundamental Research Funds for the Central Universities, National Key R&D
Program of China 2021YFE0206700, Shanghai Municipal Science and Technology
Major Project (2021SHZDZX0102), and STCSM 22DZ2229005.

Fig. 1: The generation of distortions. During the generation process of the
digital humans, the jitter of the laser sensor and the thermal noise of
devices can introduce geometry shift and noise to the digital human models. To
meet the real-time needs of practical application scenarios where the
transmission bandwidth is usually limited, the models are inevitably processed
with compression, downsampling or simplification.

Unfortunately, as shown in Fig. 1, because the digital human models presented
to viewers are often stored as textured meshes, whose geometry and color
information are separately saved, many procedures may lead to the generation
of distortions, which may severely damage the visual quality of digital
humans. Therefore, it is important to carry out objective DHQA methods to
optimize the transmission system and improve the quality of experience (QoE).

Fig. 2: The framework of the proposed NR DHHQA model. Fig. 3: Illustration of
the projection process.

Reviewing the development of 3D quality assessment (3DQA), many quality
assessment metrics [1, 2, 3, 4, 5, 6, 7, 8] as well as 3DQA databases [1, 4,
2, 9] have been proposed. For 3DQA, the mainstream quality assessment methods
can be divided into two categories: model-based methods[1, 2, 3] and
projection-based methods[4, 5]. Specifically, the model-based method gives a
quality score by calculating the distance between discrete points as a
similarity measure. The projection-based method, on the other hand, projects
the 3D model from different viewpoints and obtains the quality score of the
model by evaluating the quality of the projection.
In this paper, we mainly focus on the evaluation of digital human head (DHH),
which is the most concerned part of digital humans. In most practical
occasions, DHHs are essential for representing facial features and
expressions, the perceptual quality of which directly affects the perceptual
quality of digital humans. In addition, NR method is more valuable for
practical applications and its research is more challenging because in many
practical situations, we do not have access to the original reference model.
Therefore, we propose a Transformer-based NR digital human head quality
assessment (DHHQA) method in a multi-task manner. The proposed NR DHHQA method
includes three modules: the projection module, feature extraction module and
multi-task module. First, the projection module renders the front-side
projections of the DHHs as inputs. Then, we select ViT[10] as the feature
extraction backbone and realize the effective extraction of quality-aware
features. Finally, we get the quality-aware embedding by average pooling. In
the design of the multi-task module, we divide the quality assessment problem
into two sub-tasks to promote the model to enhance understanding, focus
attention and share feature information. Sub-task I classifies a DHH model
into a specific distortion category. Sub-task II predicts the perceptual
quality. The two sub-tasks share image features and together constitute the
multi-task module. To analyze the performance of the proposed method, we
select the DHHQA database [11] for validation. The experimental results show
that the proposed method achieves the best performance in the prediction
accuracy of the quality scores, benefiting from the introduction of the multi-
task module.

##  2 Proposed Method

In this section, we describe our proposed NR DHHQA method in detail. The
framework of the method is shown in Fig. 2, which contains three modules: the
projection module, feature extraction module and multi-task module. The
projection module captures front-side projection of the DHH. Then the ViT is
employed to extract the quality-aware features from the projection. Finally,
the features are regressed into quality values and distortion categories by
the multi-task module.

###  2.1 Projection Module

The projection module selects the front-side projection of the DHH because the
front projection contains more facial features and details that most affect
the perceptual quality. Given a DHH represented by a textured mesh, the
geometry mesh structure $M$ can be defined as:

$M\in\\{\\{v|v\in V\\},\\{n|n\in N\\},\\{e|e\in E\\}\\},$ (1)

where $V$, $N$, $E$ represent the set of the vertices, normal vectors and
edges of the mesh respectively. The geometry center of the mesh can be
calculated as:

$\begin{array}[]{l}{C_{M}}=\frac{1}{N}\sum\limits_{i=1}^{N}{{v_{i}}},\\\
\end{array}$ (2)

where $N$ denotes the total number of vertices and ${C_{M}}$ denotes the
center coordinates of DHH model. As shown in Fig. 3, we use random region
proposal to get input patches as $P$.

###  2.2 Feature Extraction Module

In view of the popularity and success of the Transformer applied to computer
vision tasks, the proposed method uses ViT [10] as the feature extraction
backbone. The architecture of ViT we use is a Transformer Encoder structure
consisting of $n$ blocks stacked on top of each other. Before entering the
Encoder, ViT first takes the ${P}$ and further divides them into smaller
patches. Next, we extract the feature maps of the ViT encoder output, and the
process can be described as:

${F}=ViT({P}),$ (3)

where $ViT(\cdot)$ denotes the process of feature extraction using ViT as the
backbone and ${F}$ represents the feature maps output from the ViT network.
Next, the extracted feature maps are computed by average pooling to obtain a
quality-aware embedding, and the process can be expressed as follow:

$\overline{F}=Avg({F}),$ (4)

where $\overline{F}$ denotes the final extracted features of the feature
module, and $Avg(\cdot)$ denotes the operation of average pooling.

###  2.3 Multi-Task Module

Specifically, the quality assessment problem is divided into two sub-tasks
with the aim of enabling the method to perform the classification task while
introducing additional quality information to improve the prediction accuracy
of the quality scores. Sub-task I classifies a DHH model into a specific
distortion category. Sub-task II predicts the perceptual quality. The two sub-
tasks share image features and jointly optimize the network parameters to
improve the performance of the visual perception task.
After extracting features with perceptual quality significance, we use fully
connected (FC) layers to aggregate them into quality scores for DHHs. For the
classification task, we obtain the category probabilities by a two-layer FC
layer consisting of 128 neurons and $N_{d}$ ($N_{d}$ denotes the number of
distortion categories) neurons which represent the possibilities belonging to
each category. For the regression task, the quality scores are derived from
two FC layers consisting of 1024 neurons and 1 neuron respectively.
Specifically, the process can be described as:

$\begin{gathered}\widehat{c}=\alpha(\overline{F}),\hfill\\\
\widehat{q}=\beta(\overline{F}),\hfill\\\ \end{gathered}$ (5)

where $\widehat{c}$, $\widehat{q}$ are the distortion categories and quality
scores predicted by the model respectively, $\alpha(\cdot)$, $\beta(\cdot)$
denote the regression of Sub-task I and Sub-task II respectively. In both
tasks, we use the mean squared error (MSE) as the loss function. Thus, for the
whole network, the loss function can be expressed as:

$Loss=\frac{1}{{Ns}}\sum\limits_{i=1}^{Ns}{(\lambda{{\left\|{{{\widehat{c}}_{{\text{i
}}}}-{c_{{\text{i }}}}}\right\|}^{2}}+{{\left\|{{{\widehat{q}}_{{\text{i
}}}}-{q_{{\text{i }}}}}\right\|}^{2}})},$ (6)

where $N_{\text{s}}$ is the number of samples in the mini-batch, ${c_{{\text{i
}}}}$, ${q_{{\text{i }}}}$ denote the ground truth distortion categories and
quality scores respectively, and $\lambda$ is a coefficient factor that is
used to subjectively adjust the relative importance of the classification task
and the regression task.

##  3 Experiment Evaluation

###  3.1 Experiment Details.

The proposed method is validated on DHHQA database consisting of 1,540 digital
human head distortion projections. The database collects 55 scanned models of
digital human heads and introduces seven distortions on the 3D models. Each
distortion provides four different distortion levels and corresponding mean
opinion scores (MOS). To test the generalization ability of the model, five-
fold cross-validation strategy is employed. Specifically, we split the
database into 5 folds and each fold contains 11 groups of DHHs. Then we select
4 folds for training and leave 1 fold for testing. Such process is repeated 5
times so that every fold has been used for testing. The average performance of
the 5 folds is recorded as the final results. It’s worth mentioning that there
is no content overlap between the training and testing sets. The selected ViT
backbone which contains 12 Transformer Blocks is pre-trained on
ImageNet-21k[12]. The Adam optimizer[13] with an initial learning rate of 1e-5
is utilized. The default number of training epochs and batch size are set as
50 and 30 respectively. We split the dataset into five folds, and calculate
the average of the above evaluation criteria as the final performance.

###  3.2 Experiment Criteria.

We compared the proposed method with ten NR IQA methods, including
BRISQUE[14], CPBD[15], IL-NIQE[16], NFERM[17], NFSDM[18], NIQE[19], DBCNN[20],
StairIQA[21]. Among these ten NR IQA methods, CPBD, IL-NIQE, and NIQE do not
require training. Besides, we also include the performance of some classic
full-reference point cloud quality assessment (FR PCQA) methods such as
p2point[22], p2plane[23] and psnr-yuv[24] through converting the DHHs from
textured meshes to point clouds. The default experimental setup is maintained
for the rest of the compared methods.
To compare the performance of these methods in the assessment of the visual
perceptual quality of the digital human head, we use four commonly used
evaluation criteria to evaluate different NR IQA methods, namely, Spearman
rank order correlation coefficient (SRCC), Pearson linear correlation
coefficient (PLCC), Kendall rank order correlation coefficient (KLCC), and
root mean square error (RMSE).

###  3.3 Performance Comparison with Other Methods

The experimental results on the DHHQA database are exhibited in Table 1. From
Table 1, we can see that the proposed method outperforms the other NR IQA
metrics and leads by a significant margin (better than the second-place method
StairIQA by 12.20%). Firstly, this indicates that the proposed method is much
more effective than the NR IQA based on handcrafted extracted features in
assessing the quality of DHH models. It can be explained that most methods
based on handcrafted extracted features are mainly designed for natural scene
statistics (NSS). However, the rendered images are not necessarily subject to
the prior distribution of NSS. Besides, in allusion to the fact that the
proposed method outperforms some deep-learning based NR IQA methods, we
speculate that the multi-task module can obtain more quality-aware information
derived from the classification of distortion, which promotes the improvement
of the performance. Finally, the classic FR PCQA methods are not effective for
DHHQA task. It can be explained that these classic FR PCQA methods are not
suitable for dealing with relatively more complicated models.

###  3.4 Ablation Experiments

In this section, to further validate the effectiveness of multi-task learning,
we conduct the following ablation experiments. We set up two groups of models
for comparison. One group is the baseline model based on ViT which only
includes the Sub-task II, and the other group is the ViT model with multi-task
module which can realize the Sub-task I and Sub-task II. The default
experiment setup is maintained and the results of the two groups of models are
shown in Table 2. As can be seen from Table 2, the multi-task module enables
the model to obtain better performance than the baseline model in visual
quality perception, indicating that the involvement of the Sub-task I can help
improve the model’s understanding of visual quality for DHHs.

Table 1: Performance comparison of different methods. The best performance results are marked in RED and the second performance results are marked in BLUE. Type | Method | SRCC$\uparrow$ | PLCC$\uparrow$ | KRCC$\uparrow$ | RMSE$\downarrow$
---|---|---|---|---|---
FR | MSE-p2ponit | 0.2891 | 0.29116 | 0.2359 | 21.0813
MSE-p2plane | 0.2698 | 0.2961 | 0.2250 | 21.0502
psnr-yuv | 0.1761 | 0.2272 | 0.1369 | 21.4299
NR | BRISQUE | 0.6008 | 0.5958 | 0.4214 | 17.3669
CPBD | 0.2621 | 0.2599 | 0.1718 | 20.6701
IL-NIQE | 0.5419 | 0.6232 | 0.3825 | 17.4119
NFERM | 0.6414 | 0.6876 | 0.4610 | 15.4528
NFSDM | 0.5921 | 0.6598 | 0.4214 | 16.1233
NIQE | 0.4232 | 0.4410 | 0.2784 | 18.9814
DBCNN | 0.7575 | 0.8238 | 0.5723 | 12.0087
StairIQA | 0.8052 | 0.8271 | 0.6123 | 11.8403
Proposed | 0.9272 | 0.9275 | 0.7711 | 7.4662
Table 2: Performance of two models for ablation study. I indicates the Sub-task I and II indicates the Sub-task II. The best performance results are marked in BOLD. I | II | SRCC$\uparrow$ | PLCC$\uparrow$ | KRCC$\uparrow$ | RMSE$\downarrow$ | ACC$\uparrow$
---|---|---|---|---|---|---
✕ | ✓ | 0.9167 | 0.9173 | 0.7451 | 7.7209 | -
✓ | ✓ | 0.9272 | 0.9275 | 0.7711 | 7.4662 | 0.9732

###  3.5 Performance Comparison of Different Backbones

We compare the performance of three popular backbones, ResNet50[25],
MobileNetV2[26], and ViT[10]. The performance of the three backbones is
exhibited in Table 3. From Table 3, we can see that MobileNetV2 yields similar
performance compared with ViT and conclude that firstly, our proposed method
is still valid on a lightweight platform. Second, ViT achieves the best
performance on the regression task of visual perception while maintaining
about the same accuracy of distortion classification as ResNet50, further
demonstrating the rationality of selecting ViT as the backbone.

Table 3: Performance comparison of different backbones. MNV2 indicates the MobileNetV2. The best performance results are marked in BOLD. Backbone | SRCC$\uparrow$ | PLCC$\uparrow$ | KRCC$\uparrow$ | RMSE$\downarrow$ | ACC$\uparrow$
---|---|---|---|---|---
MNV2 | 0.9086 | 0.9043 | 0.7331 | 9.0336 | 0.9442
ResNet50 | 0.9210 | 0.9165 | 0.7519 | 8.4616 | 0.9669
ViT | 0.9272 | 0.9275 | 0.7711 | 7.4662 | 0.9732

(a) SRCC performance

(b) ACC performance

Fig. 4: Different backbone training process.

To further discuss the performance of different backbones. We record the
process of training the same fold time for three backbones and the result is
shown in Fig. 4. From Fig. 4(a) we can conclude that our proposed method
converges faster and better when dealing with the regression task of visual
perception compared to ResNet50 and MobileNetV2. Moreover, according to Fig.
4(b), it can be seen that ViT converges slower in the first few training
rounds when dealing with the classification task, but with the increase of the
num of epochs, it also eventually gets about the same or even better
performance than ResNet50. In response to this result, we speculate that: CNN
has a stronger feature extraction ability and is more likely to capture the
difference between various distortions; while attention-based ViT is better at
global visual quality perception from a broader view.

##  4 Conclusion

In this paper, we propose a novel Transformer-based NR 3DQA multi-task method
for DHHQA. In order to reduce the computational cost of the method, we first
record the front-side projection of the DHH model and crop the projection into
patches. Then, we use ViT to perform feature extraction on the patches to
obtain quality-awareness features. Specifically, we use the average pooling to
process the output feature maps and then obtain the final quality-awareness
embedding. Finally, the multi-task module is involved to regress quality-
awareness embedding into quality scores as well as distortion classification
respectively. The experimental results show that the proposed method
significantly outperforms other methods on DHHQA and achieves competitive
performance on the classification task, which validates the effectiveness of
the proposed method for predicting the visual quality of DHHs.

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# Dipole coupling of a bilayer graphene quantum dot to a high-impedance
microwave resonator

Max J. Ruckriegel  Lisa M. Gächter  David Kealhofer  Laboratory for Solid
State Physics, ETH Zürich, CH-8093 Zürich, Switzerland  Mohsen Bahrami Panah
Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
Quantum Center, ETH Zürich, CH-8093 Zürich, Switzerland  Chuyao Tong
Christoph Adam  Michele Masseroni  Hadrien Duprez  Rebekka Garreis  Laboratory
for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland  Kenji
Watanabe  Research Center for Electronic and Optical Materials, National
Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Takashi
Taniguchi  Research Center for Materials Nanoarchitectonics, National
Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Andreas
Wallraff  Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich,
Switzerland  Quantum Center, ETH Zürich, CH-8093 Zürich, Switzerland  Thomas
Ihn  Klaus Ensslin  Laboratory for Solid State Physics, ETH Zürich, CH-8093
Zürich, Switzerland  Quantum Center, ETH Zürich, CH-8093 Zürich, Switzerland
Wei Wister Huang  Laboratory for Solid State Physics, ETH Zürich, CH-8093
Zürich, Switzerland

###### Abstract

We implement circuit quantum electrodynamics (cQED) with quantum dots in
bilayer graphene, a maturing material platform for semiconductor qubits that
can host long-lived spin and valley states. The presented device combines a
high-impedance ($Z_{\mathrm{r}}\approx$1\text{\,}\mathrm{k\SIUnitSymbolOhm}$$)
superconducting microwave resonator with a double quantum dot
electrostatically defined in a graphene-based van der Waals heterostructure.
Electric dipole coupling between the subsystems allows the resonator to sense
the electric susceptibility of the double quantum dot from which we
reconstruct its charge stability diagram. We achieve sensitive and fast
detection with a signal-to-noise ratio of 3.5 within
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ integration time. The charge-photon
interaction is quantified in the dispersive and resonant regimes by comparing
the coupling-induced change in the resonator response to input-output theory,
yielding a maximal coupling strength of $g/2\pi=$49.7\text{\,}\mathrm{MHz}$$.
Our results introduce cQED as a probe for quantum dots in van der Waals
materials and indicate a path toward coherent charge-photon coupling with
bilayer graphene quantum dots.

##  I Introduction

The canonical circuit quantum electrodynamics (cQED) system is a solid-state
qubit coupled to photons in a superconducting microwave resonator [1]. While
the large dipole moments common to many superconducting qubits enable strong
coupling [2], even the minuscule dipole of an individual electron isolated in
a semiconductor quantum dot (QD) can interact coherently with microwave
photons [3, 4]. Strong coupling enables applications such as fast charge and
spin readout [5], high-resolution state spectroscopy [6], and photon-mediated
long-range interactions [7, 8, 9]. Combining cQED with semiconductor QDs is
therefore of practical relevance to a possible spin qubit device architecture
[10]. Hybrid cQED has been realized in semiconductor materials like GaAs [11,
3], silicon [12, 13], germanium [14, 15], and InAs [16, 17]. In this work, we
extend this list to van der Waals (vdW) heterostructures by demonstrating
hybrid cQED experiments with QDs in bilayer graphene.

Making use of the variety of vdW materials in cQED is a promising route for
quantum technologies. For example, highly coherent quantum circuits built with
hexagonal boron nitride (hBN) and $\mathrm{NbSe_{2}}$ have been reported [18,
19]. Also monolayer graphene has been integrated with microwave circuits in
the form of etched nanostructures [20] or gate-tunable Josephson junctions
[21, 22, 23]. Here, we choose as our material bilayer graphene encapsulated
with hBN, whose voltage-induced bandgap and atomically clean interfaces make
it an excellent 2D material for electrostatically defined QDs [24]. Moreover,
as a host for spin qubits, bilayer graphene also promises low spin decoherence
rates due to its weak spin-orbit coupling [25, 26] and weak hyperfine
interaction [27]. Because of these properties, bilayer graphene has become a
rapidly developing material platform for spin and valley qubits. Crucial
ingredients for quantum computing applications, including time-resolved charge
detection [28] and switchable Pauli spin and valley blockade [29, 30, 31],
have been demonstrated in electrostatically defined bilayer graphene QDs. The
spin lifetime of up to $50\text{\,}\mathrm{ms}$ in these systems [32] is
comparable to that of semiconductor QD systems such as 3-5 [33, 34, 35],
silicon- [36, 37, 38, 39] and germanium- [40] based heterostructures. Bilayer
graphene also offers a unique, controllable valley degree of freedom, which
exhibits exceptionally long lifetimes approaching $1\text{\,}\mathrm{s}$ [41].
The long relaxation times allow for high-fidelity spin and valley qubit
readout [32, 41]. However, the detection bandwidth of the dc charge sensors
has so far been limited to around $2\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$
[42]. These characteristics make gate-based dispersive charge readout [43, 44]
with superconducting microwave resonators attractive for bilayer graphene,
where the fundamental limit to the bandwidth is the resonator linewidth.

Figure 1:  Bilayer graphene double quantum dot (DQD) integrated with a high-
impedance microwave resonator. (a) Optical image of the device showing the
microwave circuit fabricated from $15\text{\,}\mathrm{nm}$ thick NbTiN with
the vdW material stack on the right (black rectangle). Inset: Scanning
electron micrograph (SEM) of the coplanar waveguide (CPW) at the midpoint of
the half-wavelength resonator. (b) Top view and cross-section schematic of the
DQD device. The vdW material stack with bilayer graphene, encapsulated in hBN
and back-gated with graphite, is deposited on a $\mathrm{Si}/\mathrm{SiO_{2}}$
substrate. Two layers of metal top gates define the DQD device. (c) A
simplified circuit schematic of the device in (a). The resonator is probed via
a feedline (left). The voltage $V_{\mathrm{R}}$ is applied to the plunger gate
through the center conductor of the CPW via a LC-filter and quarter-wavelength
tap. The Fermi energy $E_{\mathrm{F}}$ of the bilayer graphene along the
channel is tuned by plunger and barrier gates to form a double-well potential
for holes. (d) SEM of the gates on top of the vdW material stack, with
voltages applied to the barrier and plunger gates labeled. Crossed boxes
indicate ohmic contacts to the bilayer graphene. (e) Microwave transmission
spectrum through the feedline around the resonator frequency $f_{\mathrm{r}}$
(dashed line).

In this paper, we present a device that integrates an on-chip superconducting
resonator with a bilayer graphene double QD (DQD). We use the kinetic
inductance of NbTiN to realize a high-impedance resonator whose zero-point
voltage fluctuations are more than four times larger than for a conventional
$50\text{\,}\mathrm{\SIUnitSymbolOhm}$ resonator. The larger voltage
fluctuations enhance the coupling strength $g$ to the electric dipole moment
of DQD state transitions. We demonstrate resonator-based sensing of the DQD
electric susceptibility that complements transport measurements. The high
sensitivity of the measurement allows us to detect an interdot transition with
a signal-to-noise ratio (SNR) of $3.5$ within
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ integration time. We estimate the
minimum time
$t_{\mathrm{min}}\approx$1\text{\times}{10}^{-7}\text{\,}\mathrm{s}$$ needed
to achieve a SNR larger than one, four orders of magnitude faster than current
dispersive gate-based sensing of QDs in bilayer graphene [44] and comparable
to state-of-the-art readout with on-chip resonators in silicon [5].
Furthermore, we measure the resonator response to the DQD dipole moment in
both the dispersive and resonant cases, and extract the relevant system
parameters from a least-squares fitting routine. We achieve a bare resonator
coupling strength of up to $g/2\pi=$49.7\text{\,}\mathrm{MHz}$$ that we assume
to be limited by the small lever arm difference $\beta$ of the coupling gate.
The charge decoherence rate $\gamma=$643\text{\,}\mathrm{MHz}$$ is likely
caused by high tunneling rates of electrons to the leads, effectively making
this value an upper bound for $\gamma$ [45]. This work constitutes the next
technological step for bilayer graphene as a spin or valley qubit platform and
marks the necessary advances to achieve coherent charge-photon coupling in vdW
materials.

##  II Hybrid device architecture

Figure 1 shows the hybrid device. We sputter $15\text{\,}\mathrm{nm}$ of NbTiN
on an intrinsic silicon wafer with $100\text{\,}\mathrm{nm}$ thermal
$\mathrm{SiO_{2}}$. A half-wavelength resonator is patterned into the
superconducting film as a coplanar waveguide with a center conductor width of
$1.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a length of
$900\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [panel (a)]. Based on the
resonator geometry and its center frequency
$f_{\mathrm{r}}=$6.033\text{\,}\mathrm{GHz}$$ we estimate the resonator
impedance
$Z_{\mathrm{r}}=\sqrt{L_{\mathrm{l}}/C_{\mathrm{l}}}\approx$1\text{\,}\mathrm{k\SIUnitSymbolOhm}$$.
The inductance per unit length $L_{\mathrm{l}}$ is dominated by the large
contribution of the sheet kinetic inductance of the NbTiN film which we
estimate to
$L_{\square,\mathrm{kin}}\approx$150\text{\,}\mathrm{pH}$\mathrm{/}\square$.
At one end, the resonator is capacitively coupled to a
$50\text{\,}\mathrm{\SIUnitSymbolOhm}$ feedline. Close to the other end of the
resonator, we fabricate the bilayer graphene DQD device, similar to previously
reported samples [46].

Figure 1 (b) shows a schematic of the device. The vdW material stack consists
of Bernal-stacked bilayer graphene, encapsulated in hexagonal boron nitride
(hBN) and back-gated with graphite. Two layers of gold gates are fabricated on
top of the vdW material stack with ALD-deposited $\mathrm{Al_{2}O_{3}}$ as the
intergate dielectric. One of the gates connects to the resonator center
conductor. We apply a voltage to this gate through a quarter-wavelength bias
tap at the midpoint of the resonator. Electron transport through the quantum
dot is probed via ohmic contacts to the graphene, achieved by etching through
the top layer of hBN before metal deposition. To prevent microwave leakage
through parasitic gate capacitances [47, 48], all gate lines are biased
through on-chip low-pass filters consisting of a shunt capacitance to ground
in series with the large inductance of a narrow NbTiN wire.

The sample is mounted in a dilution refrigerator that reaches the base
temperature $T=$10\text{\,}\mathrm{mK}$$. For resonator spectroscopy, we probe
the microwave transmission through the feedline as a function of frequency
$f_{\mathrm{p}}$. The transmitted microwave signal undergoes a heterodyne
detection scheme that allows us to measure the complex transmission amplitude
$S_{21}(f_{\mathrm{p}})=Ae^{i\phi}=I+iQ$. Transmission is reduced around the
resonator frequency [see Figure 1 (e)] as expected for the notch-type coupling
to the feedline. From a fit to the spectrum around $f_{\mathrm{r}}$ [49] we
extract the internal resonator decay rate
$\kappa_{\mathrm{int}}/2\pi=$1.9\text{\,}\mathrm{MHz}$$ and the total
linewidth $\kappa/2\pi=$8.8\text{\,}\mathrm{MHz}$$. The internal quality
factor $Q_{\mathrm{int}}=2\pi f_{\mathrm{r}}/\kappa_{\mathrm{int}}=3250$ is
limited by the dielectric environment of $\mathrm{SiO_{2}}$ and
$\mathrm{Al_{2}O_{3}}$ [50]. However, we have observed that the
$\mathrm{SiO_{2}}$ layer is necessary to improve device yield.

A negative voltage applied to the back-gate opens a bandgap in bilayer
graphene and together with a pair of split-gates confines valence band holes
to a narrow channel. The second set of gates across the channel defines
plunger and barrier gates. Figure 1 (c) shows a device schematic and Figure 1
(d) a scanning electron micrograph of the device with the plunger and barrier
gates labeled. Each of these gates varies the local potential along the
channel, placing the Fermi energy in the valence band, the bandgap, or the
conduction band. With appropriate positive voltages applied, they realize a
double-well potential to host the DQD. The plunger gate voltages
$V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ change the electrochemical potentials of
electrons in the left and right QD. The gate voltages $V_{\mathrm{BL}}$ and
$V_{\mathrm{BR}}$ are set to place the Fermi energy in the bandgap such that
these regions act as tunnel barriers between the dots and their leads.
Similarly, the voltage at the central barrier, $V_{\mathrm{BC}}$, changes the
coupling between the QDs. The gate connected to the resonator is the right QD
plunger gate, biased through the resonator and low-pass filter with
$V_{\mathrm{R}}$. Its capacitance to the right QD couples the DQD to the
electric field fluctuations at the voltage antinode of the microwave
resonator. This makes the resonator susceptible to the dipole moment of
electrons in the QDs and thereby acts as a sensor.

##  III Dispersive sensing

Figure 2:  DQD charge stability diagram. (a) Current $I_{\mathrm{SD}}$
measured through the channel at a $200\text{\,}\mathrm{\SIUnitSymbolMicro V}$
bias voltage as a function of plunger gate voltages. White dashed lines mark
the boundaries of one hexagonal region with stable charge configuration.
Circles mark the triple points. (b) Change in the quadrature component $Q$ of
the microwave signal transmitted through the feedline at fixed probe frequency
$f_{\mathrm{p}}=f_{\mathrm{r}}$. The charge stability diagram is visible
(marked in black) with interdot transitions connecting the triple points.
Figure 3:  Dispersive resonator shift and SNR analysis. (a) Microwave response
of one example interdot transition. The black arrow indicates the detuning
axis. The inset is a schematic energy diagram of the charge qubit states as a
function of QD energy detuning $\delta$. Tunnel coupling leads to
hybridization at $\delta=0$ with an energy splitting $2t_{\mathrm{c}}$. (b)
Phase response of the resonator as a function of probe tone $f_{\mathrm{p}}$
at the two detunings marked by symbols in (a). Dispersive interaction with the
charge qubit shifts the resonator to lower frequency at $\delta=0$ (purple
squares) compared to $\delta=$4\text{\,}\mathrm{mV}$$ (blue circles). (c)
Three example traces across the transition for different integration times
$t_{\mathrm{int}}$. A fit of a Gaussian peak to the data (black solid lines)
determines the SNR as a function of $t_{\mathrm{int}}$. For
$t_{\mathrm{int}}=$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ the phase signal
is still clearly discernible with a SNR of 3.5. (d) The SNR reduces with
shorter $t_{\mathrm{int}}$ but stays well above unity even for integration
times below $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$. A linear extrapolation
to SNR = 1 (orange solid line) gives
$t_{\mathrm{min}}\approx$1\text{\times}{10}^{-7}\text{\,}\mathrm{s}$$. (e) SNR
and width $\sigma$ of the fitted Gaussian peak as a function of resonator
probe power. Error bars for SNR in (d) and (e) are smaller than the marker
size.

Figure 2 (a) and (b) depict the DQD charge stability diagram measured
simultaneously in dc current and resonator response. The source-drain current
$I_{\mathrm{SD}}$ through the device as a function of plunger gate voltages
$V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ shows the hexagonal pattern typical for
transport through a DQD. Within one hexagon, the number of charges in each QD
is constant, and the current is suppressed due to Coulomb blockade. At the
four boundaries of a stable charge configuration, exemplarily indicated by
dashed white lines in (a), the electrochemical potential of one QD is resonant
with the Fermi energy of the respective lead while the other dot is Coulomb
blockaded, and transport can occur via co-tunneling processes through the
other dot. High current at the triple points, marked with white circles,
corresponds to the QD electrochemical potentials in both dots being aligned
inside the bias window.

Simultaneous with the current, we monitor the complex microwave transmission
through the feedline with the probe frequency $f_{\mathrm{p}}=f_{\mathrm{r}}$.
Figure 2 (b) shows the change in the quadrature component $Q$ of the complex
signal as a function of $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$. The observed
pattern matches the current data with a large change in $Q$ along the lines of
positive slope connecting adjacent triple points and lines of negative slope
coinciding with the co-tunneling current visible in (a). Along these co-
tunneling lines, the resonator is susceptible to tunneling of electrons
between the right QD and its lead. On the other hand, the signal between
adjacent triple points corresponds to transitions between charge
configurations with the same total number of electrons. The charge states
hybridize along interdot transitions due to the tunnel coupling
$t_{\mathrm{c}}$ between the QDs. These hybridized states form a two-level
system whose electric susceptibility changes the resonator response and
therefore the transmission through the feedline at $f_{\mathrm{p}}$.

We observe this sensing signal over a wide range of gate voltages and QD
configurations. Figure 3 (a) shows an interdot transition at higher gate
voltages with a strong phase response. In section IV we show that this sensing
signal of the DQD electric susceptibility is in the dispersive regime.
Sweeping $V_{\mathrm{L}}$ across the transition (black arrow) changes the
energy detuning $\delta$ between the QD electrochemical potentials in the left
and right dot. Microwave spectroscopy at two different QD detunings [Figure 3
(b)] reveals a clear shift of the resonator to lower frequencies for
$\delta=0$ (purple squares) compared to non-zero detuning (blue circles).
Measured at a fixed-frequency probe tone $f_{\mathrm{p}}$, the shift of the
resonator frequency results in a phase change $\Delta\phi$ of more than
$26\text{\,}\mathrm{\SIUnitSymbolDegree}$.

The sensitivity of the charge detection signal is investigated in more detail
by determining the signal-to-noise ratio (SNR) as a function of integration
time and probe power. We sweep the gate voltage $V_{\mathrm{L}}$ across the
interdot transition and reduce the measurement integration time
$t_{\mathrm{int}}$ from $100\text{\,}\mathrm{ms}$ to $100\text{\,}\mathrm{ns}$
at a probe power of $P=$-90\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$. Figure 3
(c) shows three examples of measured traces for different integration times. A
fit of a Gaussian function to the data (solid black lines) determines the peak
height $A$ and the peak width $\sigma$. Subtracting the fit from the data and
calculating the standard deviation gives the noise $B$ and the
$\mathrm{SNR}=A/B$. The SNR decreases with shorter $t_{\mathrm{int}}$ [Figure
3 (d)] from $\mathrm{SNR}>20$ to a value of 3.5 at
$t_{\mathrm{int}}=$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. We extrapolate
the $\propto\sqrt{t_{\mathrm{int}}}$ trend for short $t_{\mathrm{int}}$
(orange solid line) and estimate the minimum integration time
$t_{\mathrm{min}}\approx$1\text{\times}{10}^{-7}\text{\,}\mathrm{s}$$ needed
to achieve $\mathrm{SNR}=1$. This value compares well with state of the art
dispersive readout of silicon QDs [5] and is four orders of magnitude faster
than previous dispersive charge sensing in bilayer graphene [44].

Figure 3 (e) shows the dependence of the SNR on resonator probe power (black
dots). The signal strength grows with an increase in probe power up to a power
of $P=$-115\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$. Beyond this power level,
the SNR remains the same. Additionally, when the power exceeds
$P=$-95\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$, thermal broadening of the
peak becomes evident, marked by a significant increase in the peak width
$\sigma$ (orange squares).

##  IV Charge-photon coupling

Beyond charge sensing, we characterize the dipole coupling of the DQD to
microwave photons in the resonator. The gate voltage $V_{\mathrm{L}}$ varies
the energy detuning $\delta$ between charge states of an electron being in the
left or in the right QD, forming the two-level system of a canonical charge
qubit with energy
$E_{\mathrm{q}}(\delta)=\sqrt{4t_{\mathrm{c}}^{2}+\delta^{2}}$. At $\delta=0$,
the states hybridize due to tunnel coupling with amplitude $t_{\mathrm{c}}$
and are separated by the minimal charge qubit energy $2t_{\mathrm{c}}$. Its
interaction with the resonator depends on two properties of the charge qubit:
The coupling strength is proportional to its electric dipole moment which is
maximized when the electron wavefunction is completely delocalized between the
QDs at $\delta=0$. Furthermore, the charge qubit needs to be resonant with the
resonator ($E_{\mathrm{q}}/h=f_{\mathrm{r}}$) for the two systems to exchange
energy, otherwise the interaction is dispersive.

Figure 4:  Comparison of charge-photon coupling for two different interdot
transitions with $2t/h>f_{\mathrm{r}}$ (left column) and $2t/h<f_{\mathrm{r}}$
(right column). (a) Resonator phase signal as a function of DQD detuning
$\delta$ and probe frequency $f_{\mathrm{p}}$. Dispersive interaction with the
charge qubit shifts the resonator frequency around $\delta=0$. Purple and blue
marks indicate the line traces shown in Figure 3 (b). (c) Line cut of
$\Delta\phi$ along detuning for fixed $f_{\mathrm{p}}$ indicated by the dashed
black line in panel (a). (e) Extracted resonator frequency shift $\Delta
f_{\mathrm{r}}$ as a function of detuning. Solid black lines overlay input-
output theory calculations for best fit parameters
$(g/2\pi,\gamma/2\pi,2t/h)=($49.7\text{\,}\mathrm{MHz}$,$643\text{\,}\mathrm{GHz}$,$10.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$)$.
The spectroscopy data in (b) shows a different interdot transition with
$2t/h<f_{\mathrm{r}}$. A larger charge rearrangement was corrected in this
data set. (d) Phase shift along the dashed black line in (b) and (f) resonator
frequency shift extracted from (b). Calculations with best fit parameters
$(g/2\pi,\gamma/2\pi,2t/h)=($37.5\text{\,}\mathrm{MHz}$,$1.11\text{\,}\mathrm{GHz}$,$2.99\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$)$
are shown as solid black lines in (d) and (f).

Input-output theory [51] describes photons in a resonator interacting with a
two-level system, both in the resonant and in the dispersive regimes. A
comparison between experimental resonator spectroscopy data and input-output
theory characterizes the hybrid system and allows us to extract its coupling
and decoherence properties. We determine some system parameters from
independent measurements, such as the frequency $f_{\mathrm{r}}$ and the
linewidth $\kappa$ of the unperturbed resonator. Parameters to be derived from
input-output theory are the charge qubit decoherence rate $\gamma$ and the DQD
tunnel coupling $t_{\mathrm{c}}$. We also leave the coupling strength $g$ as a
fitting parameter that we can compare to the estimate
$g/2\pi=\frac{\sqrt{\pi}}{2}\beta
f_{\mathrm{r}}\sqrt{\frac{Z_{\mathrm{r}}}{h/e^{2}}}\approx$38\text{\,}\mathrm{MHz}$$.
This value for $g$ uses the lever arm difference $\beta\approx 0.037\pm 0.005$
of the resonator gate $V_{\mathrm{R}}$ to the DQD that we estimate from finite
bias measurements.

Figure 4(a) and (b) contrast $\Delta\phi$ as a function of probe frequency
$f_{\mathrm{p}}$ and detuning $\delta$ for two different interdot transitions
that demonstrate qualitatively different behavior. The data displayed in
Figure 4(a) is from the transition in Figure 3(a). Purple and blue marks
indicate the line traces from Figure 3(b) taken at two different detunings. We
interpret the observed resonator shift to lower frequencies around $\delta=0$
as the effect of the charge qubit with its minimal excitation frequency
$2t_{\mathrm{c}}/h$ being larger than the resonator frequency
$f_{\mathrm{r}}$. For $2t_{\mathrm{c}}/h\gg f_{\mathrm{r}}$, the interaction
is dispersive and acts as a perturbation on the two subsystems. Dipole
coupling to the charge qubit shifts the resonator frequency by $2\pi\Delta
f_{\mathrm{r}}\approx-g^{2}/\left(E_{\mathrm{q}}/\hbar-2\pi
f_{\mathrm{r}}\right)$. The black dashed line in Figure 4(a) indicates the cut
for fixed probe frequency shown in panel (c). From fits of the transmission
spectrum at each value of $\delta$ we extract the frequency shift $\Delta
f_{\mathrm{r}}$ as a function of detuning [panel (e)]. The largest shift is
$\Delta f_{\mathrm{r}}=$-0.65\text{\,}\mathrm{MHz}$$ at $\delta=0$. A least-
squares fit of input-output theory to the full data gives best estimates of
the free parameters (see Appendix C for details). For the dispersive case with
$2t_{\mathrm{c}}/h\gg f_{\mathrm{r}}$ the best fit is for
$g/2\pi=$49.7\text{\,}\mathrm{MHz}$$, $\gamma/2\pi=$643\text{\,}\mathrm{MHz}$$
and $2t_{\mathrm{c}}/h=$10.2\text{\,}\mathrm{GHz}$$. Solid lines in panels (c)
and (e) overlay the data with $\Delta\phi$ and $\Delta f_{\mathrm{r}}$
calculated using this set of parameters, demonstrating reasonable agreement.

The second data set [Figure 4(b)] was taken for a different interdot
transition at different plunger gate voltages. The black dashed line indicates
where the trace of $\Delta\phi$ shown in panel (d) was taken, and we again
extract the resonator shift $\Delta f_{\mathrm{r}}$ as a function of detuning
[panel (f)]. Both $\Delta\phi$ and $\Delta f_{\mathrm{r}}$ showcase the
qualitative differences to the interdot transition in the left column of
Figure 4. In contrast to the dispersive case, the charge qubit frequency
crosses the resonator at
$\delta=\pm$25\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ and
reaches a minimal frequency $2t_{\mathrm{c}}/h<f_{\mathrm{r}}$. Within this
range of detuning, $E_{\mathrm{q}}(\delta)/h$ is below the resonator frequency
$f_{\mathrm{r}}$ so that interactions with the charge qubit shift the
resonator to higher frequencies by $\Delta
f_{\mathrm{r}}=$0.39\text{\,}\mathrm{MHz}$$. At the detuning energy where
$E_{\mathrm{q}}(\delta)/h$ is close to $f_{\mathrm{r}}$, the interaction is
resonant and the high decoherence rate $\gamma$ of the charge qubit reduces
the resonator quality factor. We determine the best fitting parameters for the
$2t_{\mathrm{c}}/h<f_{\mathrm{r}}$ case to be
$g/2\pi=$37.5\text{\,}\mathrm{MHz}$$,
$\gamma/2\pi=$1.11\text{\,}\mathrm{GHz}$$ and
$2t_{\mathrm{c}}/h=$2.99\text{\,}\mathrm{GHz}$$.

In both the dispersive and the resonant case we obtain a value for $g/2\pi$
that is close to or even higher than our first estimate of
$38\text{\,}\mathrm{MHz}$. This confirms that there is no significant loss in
coupling strength from impedance mismatches along the gate connected to the
resonator. Differences to the estimated coupling strength are likely
introduced through the high uncertainty in estimating the lever arm difference
$\beta$ from finite bias measurements. While the gate capacitances that
determine the lever arms depend on the exact tuning of the DQD, they are
mostly determined by the implemented gate layout. Generally, we also note that
the charge qubit decoherence rate $\gamma$ is large compared to the coupling
strength $g$ and resonator linewidth $\kappa$, preventing the system from
reaching the strong coupling regime. We assume that the observed $\gamma$ is
large because the DQD is not sufficiently isolated from the leads [52]. High
tunneling rates to the leads limit the average time charges spend in the DQD
and act as a loss of charge qubit coherence. Effectively, the value for
$\gamma$ estimated from our measurements therefore sets an upper bound for the
decoherence rate of a charge qubit in bilayer graphene.

##  V Conclusion & Outlook

We demonstrated electric dipole coupling of a bilayer graphene DQD to a high-
impedance superconducting microwave resonator. We used the on-chip resonator
with low internal losses for dispersive charge sensing and observed a signal-
to-noise ratio well above unity even for integration times shorter than
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$. Furthermore, we investigated the
charge-photon coupling in the dispersive and resonant cases. By comparing
spectroscopy data to input-output theory, we estimated the relevant coupling
parameters. Notably, the achieved coupling strengths are comparable to those
in early-stage silicon quantum dots [53], despite the small gate lever arms in
our device. Our analysis shows that, with $g\gg\kappa$ already satisfied, only
the charge qubit decoherence $\gamma$ prevents the system from reaching the
anticipated strong charge-photon coupling regime that holds the full potential
of hybrid cQED. Better tunability of tunneling barriers and a reduction in
charge decoherence should be achievable by an improved gate design with
overlapping barrier and plunger gates [4]. Additionally, refining the gate
layout to increase the gate lever arms would enhance the charge-photon
coupling rates. Already with the current coupling strength, one
straightforward optimization is a reduction of the resonator coupling to the
feedline that dominates the total linewidth $\kappa$. In particular, a
critically coupled resonator would bring the system closer to
$\kappa/2\pi\approx\Delta f$ for maximal signal visibility while maintaining a
high detection bandwidth.

Our results demonstrate how hybrid cQED techniques can be used to probe
bilayer graphene QDs and lead the way to coherent charge-photon coupling. To
go further, hybridization of charge and spin degrees of freedom, achieved by
similar means as in silicon [54, 12], can enable strong spin-photon coupling
also with bilayer graphene QDs. Looking ahead to spin and valley qubits, cQED
applications such as fast state readout or long range interactions may become
essential for bilayer graphene as a material platform for qubits.

## Acknowledgements

We thank P. Märki and T. Bähler as well as the FIRST staff for their technical
support. We thank Ekatarina Al-Tavil and Andrés Rosario Hamann for valuable
discussions. We thank A. Denisov for valuable input during writing of this
manuscript. We acknowledge financial support from the European Graphene
Flagship, the ERC Synergy Grant Quantropy, and the European Union’s Horizon
2020 research and innovation program under grant agreement number
862660/QUANTUM E LEAPS and NCCR QSIT (Swiss National Science Foundation). K.W.
and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers 21H05233 and
23H02052) and World Premier International Research Center Initiative (WPI),
MEXT, Japan for the growth of h-BN crystals.

##  Appendix A Device Fabrication

The sputtering process was carried out in an Orion 8 magnetron sputtering
system (AJA International, Inc.) in reactive dc mode from a NbTi (Nb/Ti 70/30
wt%) target (ACI Alloys, Inc.) with 99.99% purity at
$1.5\text{\times}{10}^{-8}\text{\,}$ mTorr base pressure. We initiate the
process with a target cleaning step (5 min exposure to argon (Ar) plasma at 50
sccm flow rate and 100 W) and a conditioning phase (4 sccm nitrogen
($\mathrm{N_{2}}$) for 1 min with shutter closed) . The subsequent deposition
stage consists of Ar/$\mathrm{N_{2}}$ flows at 50 sccm and 4 sccm,
respectively, maintaining a power of 100 W, a pressure of 3.5 mTorr, and a
working distance of 10 cm. The parameters were chosen to optimize the growth
conditions for NbTiN films, film quality, thickness uniformity, and structural
properties.

We sputter a nominally $15\text{\,}\mathrm{nm}$ thick film of NbTiN on a two-
inch wafer of intrinsic silicon
($\rho>$10\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$$) with
$100\text{\,}\mathrm{nm}$ of thermally grown $\mathrm{SiO_{2}}$ (Alineason
Materials Technology GmbH). The microwave circuit is patterned using a direct-
write photo-lithography system (Heidelberg Instruments DWL66+) and reactive
ion etching (RIE) with $\mathrm{SF_{6}}/\mathrm{Ar}$. Afterwards, we deposit
gold markers and bondpads in a lift-off process and dice the wafer into
$5\times$8\text{\,}\mathrm{mm}$$ chips. The vdW material stack is fabricated
on the pre-patterned chips using standard mechanical exfoliation from bulk
crystals and polymer-based dry transfer techniques. We first deposit the
bottom hBN ($\sim$25\text{\,}\mathrm{n}\mathrm{m}$$ thick) and graphite back-
gate onto the chip and clean it from polymer residues. The top hBN
($\sim$35\text{\,}\mathrm{n}\mathrm{m}$$ thick) and bilayer graphene are
placed onto the pre-deposited bottom half in a separate deposition. Metal
gates are fabricated in a lift-off process by electron beam lithography (EBL)
and metal evaporation. For ohmic contacts to the graphene, we etch hBN before
metal deposition using RIE with $\mathrm{CHF_{3}}$. The split-gates are made
from 3/$20\text{\,}\mathrm{nm}$ Cr/Au with a $100\text{\,}\mathrm{nm}$ wide
channel. Plunger and barrier gates are also made from
3/$20\text{\,}\mathrm{nm}$ Cr/Au with $25\text{\,}\mathrm{nm}$ in width and a
gate pitch of $70\text{\,}\mathrm{nm}$. They are separated from the first gate
layer by $20\text{\,}\mathrm{nm}$ of $\mathrm{Al_{2}O_{3}}$ grown by atomic
layer deposition (ALD) at $150\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The
$\mathrm{Al_{2}O_{3}}$ layer is removed by wet etching to establish contact to
the resonator below the dielectric.

##  Appendix B Input-Output Theory

The complex transmission $S_{21}$ through the microwave feedline with a
resonator coupled in a notch-type configuration can be derived from input-
output theory [51] to be

$S_{21}(f_{\mathrm{p}},\delta)=1+\frac{i\kappa_{\mathrm{ext}}}{2\pi(f_{\mathrm{p}}-f_{\mathrm{r}})-i\kappa+g^{*}\chi_{\mathrm{e}}}.$

The resonator frequency $f_{\mathrm{r}}$ and total linewidth
$\kappa=\kappa_{\mathrm{ext}}+\kappa_{\mathrm{int}}$ are determined from a fit
to the spectrum of the unperturbed resonator [49]. For this fit, the external
coupling $\kappa_{\mathrm{ext}}$ is taken as a complex-valued parameter to
account for an asymmetric Fano lineshape of the resonance arising from non-
idealities in the resonator-feedline coupling.

The effect of the charge qubit is described by its electric susceptibility

$\chi_{\mathrm{e}}=\frac{g^{*}}{i\gamma+2\pi(f_{\mathrm{p}}-E_{\mathrm{q}}(\delta)/h)},$

that considers the charge qubit decoherence rate $\gamma$ and probe detuning.
The charge qubit energy
$E_{\mathrm{q}}(\delta)=\sqrt{4t_{\mathrm{c}}^{2}+\delta^{2}}$ depends on the
interdot tunnel coupling $t_{\mathrm{c}}$ and the DQD energy detuning
$\delta$. We calculate $\delta$ from the lever arms
$\alpha^{\mathrm{L}}_{\mathrm{dL(R)}}$ of the left plunger gate to the left
(right) dot as
$\delta=(\alpha^{\mathrm{L}}_{\mathrm{dL}}-\alpha^{\mathrm{L}}_{\mathrm{dR}})\mathrm{e}V_{\mathrm{L}}=$0.024\text{\,}$\times\mathrm{e}V_{\mathrm{L}}$
with the lever arms determined by finite bias measurements of the DQD. The
effective charge qubit coupling strength is
$g^{*}=g\frac{2t_{\mathrm{c}}}{E_{\mathrm{q}}(\delta)}$ using the bare
resonator coupling strength

$g/2\pi=\frac{\sqrt{\pi}}{2}\beta
f_{\mathrm{r}}\sqrt{\frac{Z_{\mathrm{r}}}{h/e^{2}}}.$

It is proportional to the lever arm difference of the coupling gate
$\beta=\alpha^{\mathrm{R}}_{\mathrm{dR}}-\alpha^{\mathrm{R}}_{\mathrm{dL}}$,
the resonator frequency $f_{\mathrm{r}}$, and the square root of the resonator
impedance $Z_{\mathrm{r}}$.

##  Appendix C Fitting procedure

The parameters $g$, $\gamma$ and $2t_{\mathrm{c}}$ are determined by a least-
squares fitting routine. We calculate the complex transmission
$S_{21}(f_{\mathrm{p}},\delta)$ from input-output theory for a given set of
fit parameters $\theta=[g,\gamma,2t_{\mathrm{c}}]$ and compute the mean
squared error $Q(\theta)$ between calculation and measurement data. The
parameters given in the main text for each data set are the parameters
$\hat{\theta}$ that minimize the error, found by sampling all three parameters
over a wide range of values. We numerically calculate the hessian matrix
$H_{i,j}=\left.\frac{N}{Q(\hat{\theta})}\frac{\partial^{2}Q(\theta)}{\partial\theta_{i}\partial\theta_{j}}\right|_{\hat{\theta}}$
around $\hat{\theta}$ with $N$ the size of $S_{21}$. From $H$ we calculate the
variance-covariance matrix $K=H^{-1}$ which contains information about
uncertainties and correlations of our parameter estimates.

For the dispersive case with $2t_{c}/h\gg f_{r}$ the parameters that minimize
the mean squared error are $g/2\pi=49.72\pm$0.27\text{\,}\mathrm{MHz}$$,
$\gamma/2\pi=643\pm$24\text{\,}\mathrm{MHz}$$ and
$2t_{\mathrm{c}}/h=10.193\pm$0.046\text{\,}\mathrm{GHz}$$. The strongest
correlation is observed between $g$ and $2t_{\mathrm{c}}$ as expected from the
definition of the effective coupling strength $g^{*}$ above, while the
estimate of $\gamma$ is mostly uncorrelated to the other two parameters. The
relative uncertainty in our estimate of $\gamma$ is significantly larger
compared to the uncertainty in the other two parameters. This is because the
resonator-qubit detuning is large over the whole range of $\delta$. The
incoherent dispersive interaction dominates the resonator response and does
not depend strongly on the decoherence rate $\gamma$, rendering the mean
squared error insensitive to the estimate of the decoherence rate.

For the resonant case with $2t_{c}/h<f_{r}$ the parameters that minimize the
mean squared error are $g/2\pi=37.49\pm$0.19\text{\,}\mathrm{MHz}$$,
$\gamma/2\pi=1.112\pm$0.008\text{\,}\mathrm{GHz}$$ and
$2t_{\mathrm{c}}/h=2.986\pm$0.015\text{\,}\mathrm{GHz}$$. We observe stronger
correlations between each pair of parameters compared to the dispersive case.
Furthermore, the uncertainty in $\gamma$ is smaller, because the resonant
interaction is limited by charge qubit decoherence. Therefore, the mean
squared error is more sensitive to the estimate of the decoherence rate.

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