# GaitMixer: Skeleton-based Gait Representation Learning via Wide-spectrum
Multi-axial Mixer

###### Abstract

Most existing gait recognition methods are appearance-based, which rely on the
silhouettes extracted from the video data of human walking activities. The
less-investigated skeleton-based gait recognition methods directly learn the
gait dynamics from 2D/3D human skeleton sequences, which are theoretically
more robust solutions in the presence of appearance changes caused by clothes,
hairstyles, and carrying objects. However, the performance of skeleton-based
solutions is still largely behind the appearance-based ones. This paper aims
to close such performance gap by proposing a novel network model, GaitMixer,
to learn more discriminative gait representation from skeleton sequence data.
In particular, GaitMixer follows a heterogeneous multi-axial mixer
architecture, which exploits the spatial self-attention mixer followed by the
temporal large-kernel convolution mixer to learn rich multi-frequency signals
in the gait feature maps. Experiments on the widely used gait database,
CASIA-B, demonstrate that GaitMixer outperforms the previous SOTA skeleton-
based methods by a large margin while achieving a competitive performance
compared with the representative appearance-based solutions. Code will be
available at https://github.com/exitudio/gaitmixer

Index Terms—  Gait Recognition, Self-Attention, Large-kernel Convolution,
Multi-axial Mixer

##  1 Introduction

Unlike short-distance biometrics (e.g., fingerprints, facial, iris, palm, and
finger vein patterns), gait can be recognized from a distance without the
subject’s cooperation or interference. Such long-distance biometrics has a
huge potential to extend its applications to forensic identification, access
control, and social security. The gait recognition methods are generally
either appearance-based or skeleton-based. Appearance-based approaches
[1][2][3][4] utilize background subtraction to obtain silhouettes from a video
sequence, which are further analyzed using carefully-designed network models
for gait representation learning. On the other hand, skeleton-based approaches
[5][6][7] utilize the skeleton sequences extracted from 2D/3D pose estimators
as the inputs to learn effective gait representations. Theoretically,
skeleton-based methods are more robust to appearance variations caused by
hairstyles, carrying objects, and clothes. However, the skeleton-based
approaches, which still do not receive sufficient attention, yield a large
performance gap compared with the appearance-based counterparts.

Fig. 1: (a) Global self-attention token mixing [8]. (b) Self-attention mixing
along H and W axes [9]. (c) Convolution-mixing along H and W axes [6] (d)
Heterogeneous multi-axial mixer (ours).  Fig. 2: (top) GaitMixer consists of a
spatial mixer followed by a temporal mixer. (bottom) The detailed network
architecture of the spatial self-attention mixer and large-kernel convolution
mixer

To close this gap, this paper tries to exploit more effective gait feature
encoders by proposing the multi-axial mixer, which is a generic transformer-
like architecture that mixes the feature patches (i.e., tokens) along each
axis of the feature space, respectively, e.g., width-wise, height-wise, and
channel-wise axes in image feature space. Many recent high-performance network
backbones can be considered as the special cases of multi-axial mixer based on
what types of mixing functions are applied, which mainly include convolution
[6][10][11], self-attention [12][13], and multi-layer perceptrons (MLP) [14]
[15] as shown in Fig. 1. Multi-axial mixers have been demonstrated to achieve
SOTA performance in image classification and video recognition tasks, while
significantly reducing computation complexities compared with other
competitive network models, such as vision transformers. Despite their
promising features, current multi-axial mixers generally exploit the
homogeneous architecture design, where the same type of mixing functions
(e.g., either convolution, self-attention, or MLP) is applied along each
feature space axis. Such design, however, has limited capacity to learn multi-
frequency features. In particular, it has been established that convolutions
focus more on local information and therefore are good learners for high-
frequency features [16]. Self-attentions, on the contrary, are designed to
model long-range interactions and are more capable to capture low-frequency
signals (global information) in feature map [17].

In this paper, we propose GaitMixer, a novel _heterogeneous_ spatial-temporal
axial mixer, which can effectively learn the discriminative gait
representation by capturing both high-frequency and low-frequency features. In
particular, GaitMixer consists of a spatial self-attention mixer and a
temporal large-kernel axial mixer (Fig. 2). The spatial axial mixer learns
interactions among the joints within each skeleton frame. The temporal axial
mixer models the interactions among the temporal tokens of each single joint
at different time indices. Experiments on the widely used gait database,
CASIA-B, demonstrate that GaitMixer outperforms the previous SOTA skeleton-
based [6] methods by 12% on average, while achieving a competitive performance
compared with the representative appearance-based solutions.

##  2 RELATED WORK

Appearance-based approaches extract binary images of a human silhouette from
the source images by subtracting static background [18]. GaitNet [1]
integrates silhouette extraction into the model as an end-to-end network for
gait recognition. GaitSet [2] decouples the temporal continuous sequence by
learning identity information from the set of independent frames to be immune
to permutation of frames and be able to integrate frames from different
videos. While the majority of methods [1][2][4] take the entire shape as
input, more recent approaches GaitPart [3] focus on each part of the body
individually assuming that each part of human body needs its own spatial-
temporal learning by separating silhouette into several parts horizontally.
Skeleton-based approaches (i.e., model-based approaches) use skeleton data as
the model inputs. In the early work, pose-based temporal-spatial network
(PTSN) [5] utilizes a long-short term memory (LSTM) to capture the dynamic
information and CNN to learn static information of a gait sequence in
parallel. PoseGait [19] utilizes 3D pose estimated from images in order to be
invariant to view changes, along with hand-crafted features including joint
angle, limb length, and joint motion. The most recent methods, GaitGraph [6]
and GaitGraph2 [7], adopt graph convolution neural networks (GCNs) for gait
recognition, inspired by the successes of GCNs in action recognition tasks.

##  3 GaitMixer: Heterogeneous Wide-Spectrum Spatial-Temporal Mixer

To effectively learn both high-frequency and low-frequency gait features, we
introduce the GaitMixer, a heterogeneous spatial-temporal axial mixer
architecture. As shown in Fig. 2, GaitMixer consists of a spatial self-
attention mixer and a temporal large-kernel axial mixer (Fig. 2). The spatial
axial mixer only learns interactions among the joints within each skeleton
frame. The $d_{y}$-dimensional spatial representation
$y^{t}\in\mathbb{R}^{|J|\times d_{y}}$ for each skeleton frame $t$ with $|J|$
joints is learned after $B_{S}$ self-attention blocks. Then, the
representations of $T$ skeleton frames within a gait sequence are concatenated
into $z\in\mathbb{R}^{|J|\times T\times d_{y}}$, which is then forwarded to a
temporal axial mixer to capture the interactions among the tokens of each
single joint at different temporal indices. The temporal axial mixer consists
of $B_{T}$ one-dimensional large-kernel convolution blocks. Moreover, to
simplify our GaitMixer architecture, both spatial and temporal mixers adopt
the isotropic design, which does not perform feature downsampling and
maintains the same feature resolutions at all layers.

###  3.1 Spatial Mixer with Axial Self-attention

The spatial mixer module aims to learn a high dimensional representation
embedding from each skeleton frame. Although self-attention tends to capture
low-frequency (or global) features, our experiments demonstrate that self-
attention is sufficient to learn both high-frequency and low-frequency signals
in the feature map along spatial axis. This also indicates that self-attention
can effectively model both short-range and long-range inter-joint
dependencies. Given a 2D skeleton with joints $J$, we consider each joint
(_i.e_., $x$ and $y$ coordinates) as a spatial token (with 2 channels) and
perform the feature extraction among all $|J|$ spatial tokens by following the
isotropic transformer pipeline. Specifically, the spatial taken
$x_{i}\in\mathbb{R}^{2}$ is passed through a trainable linear projection,
which maps each token to a high dimension embedding
${\mathbf{x}}_{i}\in\mathbb{R}^{d_{x}}$. Then, the spatial token embeddings of
each skeleton frame
${\mathbf{x}}=({\mathbf{x}}_{1},{\mathbf{x}}_{2},\dots,{\mathbf{x}}_{|J|})$
are mixed by inter-token dot product attentions to generate an output sequence
${\mathbf{y}}=({\mathbf{y}}_{1},{\mathbf{y}}_{2},\dots,{\mathbf{y}}_{|j|})$
where ${\mathbf{y}}_{i}\in\mathbb{R}^{d_{y}}$. Running $h$ self-attentions in
parallel leads to the multihead self-attention with $h$ heads, where the
outputs of the attention heads are concatenated and projected into the
expected dimensions.

###  3.2 Temporal Mixer with Large-kernel Convolution

The essence of a walking sequence is composed of multiple short repeated
cycles. In the temporal axis, self-attention may not be able to capture wide-
band multi-frequency features, considering that the global receptive field of
self-attention is much easier to capture low-frequency features. It demands a
large amount of data for self-attention to establish the desirable locality
inductive bias that is the key to learn high-frequency features. To learn both
high-frequency and low-frequency temporal data, we utilize large kernel depth-
wise separable convolution in the temporal mixer as illustrated in Fig. 2. In
general, convolution neural networks tend to capture high-frequency (local)
features, however, the large kernel allows the model to also learn low-
frequency features. 60 frames are used in the temporal model which covers
around 4 cycles of walking. A large one-dimension kernel with size of
$31\times 1$ is used to capture mid-range information (around 2 walking
cycles). A reverse padding with size of 30 is applied to keep the temporal
dimension the same. The temporal mixer only communicates with all frames along
the temporal axis of the same joints. A temporal mixer is composed of two
types of axial mixers. First, a token mixer is implemented by a depth-wise
convolution that learns all embeddings only in the same channel. Next, a
channel mixer is a $1\times 1$ convolution that learns only specific embedding
along all channels.

###  3.3 Representation head and Loss Function

We apply average pooling along spatial and temporal dimensions to reduce the
output from ${\mathbf{x}}_{temp}\in\mathbb{R}^{F\cdot(J\times c)}$ to
${\mathbf{x}}_{hidden}\in\mathbb{R}^{c}$, where $F$ and $J$ are number of
frames and joints respectively. The number of channels $c$ is set to 256.
Finally, layer norm, fully connected layer, and $l^{2}$-norm are applied
respectively and return the feature embedding in 128 dimensions. To learn the
discriminative gait representation, we apply the triplet loss with multi-
similarity miner.

Table 1: Averaged Rank-1 accuracies on CASIA-B per probe angle excluding
identical-view cases.

Gallery NM#1-4 | 0°-180° | mean
---|---|---
Probe | 0° | 18° | 36° | 54° | 72° | 90° | 108° | 126° | 144° | 162° | 180° |
NM#5-6 | PoseGait [19] | 55.3 | 69.6 | 73.9 | 75.0 | 68.0 | 68.2 | 71.1 | 72.9 | 76.1 | 70.4 | 55.4 | 68.7
GaitGraph [6] | 85.3 | 88.5 | 91.0 | 92.5 | 87.2 | 86.5 | 88.4 | 89.2 | 87.9 | 85.9 | 81.9 | 87.7
GaitGraph2 [7] | 78.5 | 82.9 | 85.8 | 85.6 | 83.1 | 81.5 | 84.3 | 83.2 | 84.2 | 81.6 | 71.8 | 82.0
GaitFormer (ours) | 90.9 | 91.2 | 93.7 | 91.9 | 91.9 | 92.7 | 93.3 | 91.8 | 92.5 | 90.5 | 85.5 | 91.5
| GaitMixer(ours) | 94.4 | 94.9 | 94.6 | 96.3 | 95.3 | 96.3 | 95.3 | 94.7 | 95.3 | 94.7 | 92.2 | 94.9
BG#1-2 | PoseGait [19] | 35.3 | 47.2 | 52.4 | 46.9 | 45.5 | 43.9 | 46.1 | 48.1 | 49.4 | 43.6 | 31.1 | 44.5
GaitGraph [6] | 75.8 | 76.7 | 75.9 | 76.1 | 71.4 | 73.9 | 78.0 | 74.7 | 75.4 | 75.4 | 69.2 | 74.8
GaitGraph2 [7] | 69.9 | 75.9 | 78.1 | 79.3 | 71.4 | 71.7 | 74.3 | 76.2 | 73.2 | 73.4 | 61.7 | 73.2
GaitFormer (ours) | 82.5 | 83.2 | 85.7 | 85.7 | 84.2 | 80.2 | 78.9 | 82.6 | 82.2 | 78.6 | 71.3 | 81.4
| GaitMixer(ours) | 83.5 | 85.6 | 88.1 | 89.7 | 85.2 | 87.4 | 84.0 | 84.7 | 84.6 | 87.0 | 81.4 | 85.6
CL#1-2 | PoseGait [19] | 24.3 | 29.7 | 41.3 | 38.8 | 38.2 | 38.5 | 41.6 | 44.9 | 42.2 | 33.4 | 22.5 | 36.0
GaitGraph [6] | 69.6 | 66.1 | 68.8 | 67.2 | 64.5 | 62.0 | 69.5 | 65.6 | 65.7 | 66.1 | 64.3 | 66.3
GaitGraph2 [7] | 57.1 | 61.1 | 68.9 | 66 | 67.8 | 65.4 | 68.1 | 67.2 | 63.7 | 63.6 | 50.4 | 63.6
GaitFormer (ours) | 76.1 | 80.3 | 81.0 | 78.2 | 77.7 | 76.6 | 77.4 | 75.8 | 76.5 | 75.7 | 77.2 | 77.2
| GaitMixer(ours) | 81.2 | 83.6 | 82.3 | 83.5 | 84.5 | 84.8 | 86.9 | 88.9 | 87.0 | 85.7 | 81.6 | 84.5

Table 2: Averaged Rank-1 accuracies on CASIA-B comparison with both
appearance-based and skeleton-based methods

| Method | Probe
---|---|---
NM | BG | CL | Mean
Appearance based | GaitNet [1] | 91.6 | 85.7 | 58.9 | 78.7
GaitSet [2] | 95.0 | 87.2 | 70.4 | 84.2
GaitPart [3] | 96.2 | 91.5 | 78.7 | 88.8
Skeleton based | PoseGait | 68.7 | 44.5 | 36.0 | 49.7
GaitGraph | 87.7 | 74.8 | 66.3 | 76.3
GaitGraph2 | 82.0 | 73.2 | 63.6 | 72.9
GaitFormer (ours) | 91.5 | 81.4 | 77.2 | 83.4
| GaitMixer (ours) | 94.9 | 85.6 | 84.5 | 88.3

Fig. 3: 2D FFT of GaitMixer and GaitFormer feature maps of 4 channels. Higher
temperature indicates larger magnitude. Pixels closer to the center represent
lower frequencies

##  4 EXPERIMENTS

###  4.1 Dataset

CASIA-B [20] has been widely adopted as a multi-view, RGB, and silhouette gait
dataset. The data acquisition is performed by 124 individuals from 11 viewing
angles ranging from 0 to 180 with 18 angle differences. To mimic typical daily
walking conditions, each subject performs six sequences of normal walking
(NM), two sequences of walking with a coat (CL), and two sequences of walking
with carrying a bag (BG). For each individual, ten sequences are captured from
each view angle. This paper follows a widely-used test protocol
[1][2][3][6][7][19], which uses the data of the first 74 subjects’ sequences
for the training and the remaining 50 subjects’ sequences for testing.
Furthermore, the test dataset is divided into gallery and probe sets. The
gallery set includes the first four sequences of the normal walking condition.
The probe set consists of the last two sequences of normal walking, two
walking with a coat on, and walking with carrying a bag. Finally, the results
are reported for all viewing angles.

###  4.2 Implementation Details

Training Details HRNet [21] is used as a 2D human pose estimator. We follow
data augmentations from GaitGraph[6] and add normalization of the joint
position in $(x,y)$-coordinates by dividing 320 which is the width of the
original videos to input data while keeping the aspect ratio. Adam optimizer
is used with $6\mathrm{e}{-3}$ learning rate with 1-cycle learning rate and
$1\mathrm{e}{-5}$ weight decay. We are using a balanced batch sampler to
sample the number of walking data per person equally. The batch size is (74,
4), denoting 74 people and 4 walking samples per person. Testing. Each gait
testing sample contains 60 frames selected from the middle of the sequence
data. The test set is separated into probe and gallery. Both are fed into the
model to obtain the feature representations. The ID of the gallery
representation that has the smallest cosine distance from the probe will be
the predicted ID of the probe.

###  4.3 Comparison with the SOTA Methods

To demonstrate the superior performance of GaitMixer as a heterogeneous multi-
axial mixer model, we also build GaitFormer, which is a homogeneous multi-
axial mixer that adopts self-attention for both spatial and temporal axes. In
Fig. 3, we visualize the frequency magnitude of the output feature maps from
GaitMixer and GaitFormer, respectively. It can be observed that GaitMixer
concentrates on both high-frequency and low-frequency components along both
temporal and spatial axes in feature maps. This confirms the superior capacity
of GaitMixer to capture features in wide-spectrum bands. GaitFormer, however,
cannot effectively model the high-frequency feature components. The
performance comparisons between our approaches and the SOTA skeleton-based
methods are shown in Table 1. It is shown that our multi-axial mixer models
outperform the existing solutions by a large margin in both cross-view and
cross-walking-condition cases. Moreover, GaitMixer achieves better recognition
accuracy than GaitFormer because GaitMixer can jointly exploit the
heterogeneous mixing at different feature space dimensions. Table 2 shows a
competitive performance of our skeleton-based methods, compared with the
representative appearance-based methods. GaitMixer achieves much higher
accuracy than all appearance-based approaches in wearing coat condition. It is
due to the inherent robustness of skeleton data against large appearance
changes.

Fig. 4: Grad-CAM [22] visualizations. Top-left: GaitGraph [6]. Top-right:
GaitFormer. Bottom: GaitMixer. X-axis represents frames 1 to 60 and Y-axis
represents 17 joints. Features with higher contributions have higher heat
temperatures.

###  4.4 Visualization

We use class activation map (Grad-CAM) [22] to show which parts of the input
gait sequence contribute most to the final recognition result. As shown in
Fig. 4 (bottom), GaitMixer focuses on continuous joint sequences with a
variety of different temporal windows, thus capturing short-, mid, and- long-
range temporal feature interactions. Moreover, GaitMixer also pays attentions
to a diverse set of joints except ears, eyes, and nose, which is also as
expected because the landmarks on face are not relevant to the gait dynamics.
As shown in Fig. 4 (top-left), GaitGraph tends to focus on some specific
joints over a large temporal window and it also exploits the features from
face landmarks for gait recognition. Both limitations could degrade the
performance of GaitGraph. GaitFormer (Fig. 4 (top-right)) pays more attention
to certain skeleton frameworks without capturing rich spatial-temporal feature
interactions. This can be the key contributing factor that affects its
performance.

##  5 Conclusion

In this paper, we present GaitMixer model, a novel heterogeneous multi-axial
architecture combining a spatial self-attention mixer and a large kernel
temporal convolution mixer to capture both high-frequency and low-frequency
dynamics of gait data. Our approach achieves the best accuracy on the well-
known CASIA-B gait dataset for all conditions when compared to previous
skeleton-based methods and is superior to appearance-based approaches with
coats conditions.

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# Optimal design of a superconducting transmon qubit with tapered wiring

John M. Martinis  Quantala, Santa Barbara, Ca 93105, USA
<EMAIL_ADDRESS>
(August 29, 2024)

###### Abstract

Analytical formulas are presented for simplified but useful qubit geometries
that predict surface dielectric loss when its thickness is much less than the
metal thickness, the limiting case needed for real devices. These formulas can
thus be used to precisely predict loss and optimize the qubit layout.
Surprisingly, a significant fraction of surface loss comes from the small wire
that connects the Josephson junction to the qubit capacitor. Tapering this
wire is shown to significantly lower its loss. Also predicted are the size and
density of the two-level state (TLS) spectrum from individual surface
dissipation sites.

Quantum computers are made from quantum bits, which have natural sources of
noise and dissipation that produce errors in quantum gates. Decreasing these
errors increases the size and complexity of quantum algorithms that can be run
on a quantum computer. When errors are reduced to about 0.1% per gate
operation, then quantum error correction may be used on a large array of
qubits in order to lower logical errors and execute vastly more complex
quantum algorithms [1]. Qubit errors are often limited by the rate of energy
decay from loss mechanisms.

Superconducting qubits can be thought of as an inductor-capacitor resonator,
with the superconducting Josephson junction giving a non-linear inductance
that allows the two lowest energy levels to behave as a qubit. The Josephson
junction and the capacitance are designed to be separate physical entities, as
illustrated in Fig. 1, and thus can be separately optimized.. The size of the
Josephson junction is about 100 nm. Its natural capacitance is negligible and
junction defects are statistically unlikely because of its small size; the
junction can thus typically be modeled as bringing no energy loss. The
capacitance is made from superconducting pads with a relatively large
millimeter size and about $100\,\mu\textrm{m}$ spacing, producing a
capacitance of about 100 fF for the transmon qubit [2]. When the capacitor is
designed properly with control lines weakly coupled to an external circuit,
dielectric surface loss from the superconductor and substrate is the dominant
mechanism of energy loss. As for any surface loss mechanism, it has been found
experimentally that increasing the size of this capacitor lowers the net
effect of the surface loss on the qubit device [3].

Calculating the surface loss is difficult because of the divergence of the
electric fields at the metal edges, which has pushed researchers to solve the
problem numerically with finite-element models [4, 5, 6, 7, 8]. More recently,
an analytical result was obtained using solutions of conformal mapping, which
describes the electric fields of ribbon and coplanar geometries [9].
Unfortunately, this result is only approximate since it assumed the dielectric
is thicker than the metal, opposite of the real design. Here, a more practical
solution is presented that is valid for a few nanometer lossy dielectric
surrounding a much thicker metal layer about $0.1\,\mu$m. Changes to the
conformal predictions are calculated using the scaling of corner fields and
numerical simulation, and are simple to use and understand.

For planar transmons, ribbon capacitors are typically embedded in a ground
plane, and thus the analytical results are not valid. Numerical simulations
and fit functions give capacitance and surface loss for this important
practical case.

New surface loss predictions are also given for the wires that connect the
Josephson junction to the capacitor pads. They typically have a width
approximately the size of the junction, about $0.1\,\mu$m, and extend in
length from the junction to the pads, about $50\,\mu\textrm{m}$. They are
typically narrow compared to their length and thus have large electric fields
at its edges. Conformal and numerical solutions are used to give analytic
predictions of surface loss. Their long length produces significant surface
loss, which can be reduced by tapering the wires, as as illustrated in Fig. 1.

Figure 1: Qubit design with taper. Drawing of prototypical qubit device, with
relative dimensions approximately to scale. The qubit capacitance $C\simeq
100\,\textrm{fF}$ is made from two ribbons (shaded gray) of width
$b-a=100\,\mu\textrm{m}$ and length $\ell=1300\,\mu\textrm{m}$, separated by
distance $2a=100\,\mu\textrm{m}$. Wires (drawn in red) connect to the
sub-$\mu$m Josephson junction in a conventional design. Surprisingly, the loss
from the small junction wires is about equal to that coming from the large
ribbons. This paper proposes tapering these wires (shaded blue) to reduce
their loss.

For numerical solutions, meshing is always a concern given the range of size
scales, from nanometer thick oxides to millimeter sized capacitor pads.
Meshing is particularly important for 2-D and 3-D numerical solvers where the
large grid makes it more difficult to calculate edge fields accurately. In
this paper, the assumption of flat substrates (no trenching) allows solutions
based on surface charges that are effectively 1-D, so that fine meshing at
corners enables accurate checking of formulas. These formulas are thus a
useful “gold standard” reference for verifying numerical methods. This is
especially needed for experiments where surface loss parameters want to be
accurately extracted [10, 11].

Expressing loss with formulas is also useful since the designer can separate
out all the loss mechanisms, instead of modeling the entire device at once
using numerical solvers. Optimization is more transparent, for example trading
off the surface loss of the qubit capacitance pads and the junction wires. In
order to give useful design formulas for various geometries of the qubit
capacitor, the surface loss is analyzed for 3 cases: a parallel plate, a
ribbon capacitor where electric fields are between the two electrodes, and a
coplanar capacitor where the fields connect through a ground plane. A typical
design should be able to be modeled as a combination of these geometries, thus
enabling surface loss predictions from formulas derived here.

##  I Qubit Model and Participation Ratios

Figure 2 shows an example design of a full differential qubit, include the
qubit electrodes and an shielding ground plane (gray). The full design can be
broken up into the junction and tapered wires (red), a ribbon capacitor (blue)
and a coplanar capacitor (green). The capacitances and losses from the
combination of the structures would then be used to optimize the design.

Figure 2: Example of full transmon design. Example design of a differential
transmon qubit that incorporates several of the capacitance structures
described here. Red shows the junction and tapered wire, blue is a ribbon
capacitor, and green is a coplanar capacitor. The outer ground plan is gray.
The capacitances and losses would be added to give a good approximation for
the entire design.

We are interested here in calculating the loss from dielectrics, since the
loss from the metallic structures are typically negligible for
superconductors. For a crystalline substrate such as silicon or sapphire, the
loss is dominated by the thin surface layers of the films [4]: the metal-air
(MA), metal-substrate (MS) and substrate-air (SA), typically coming from
amorphous oxides.

The total loss tangent for these thin layers is given by $\Sigma
p_{i}\tan\delta_{i}$, where surface interface type $i$ has loss tangent
$\tan\delta_{i}$ and participation ratio of the stored energy

$\displaystyle p_{i}=\frac{\epsilon_{i}/2}{W}\int dA\,t_{i}\,|E_{i}|^{2}$ (1)

where the normal volume integral is replaced by a surface integral $dA$ for a
thin dielectric layer with thickness $t_{i}$, dielectric constant
$\epsilon_{i}$, and a surface electric field $E_{i}$. The participation ratio
is normalized by the total total capacitor energy $W=CV^{2}/2$, where $C$ is
the total capacitance and $V$ the voltage.

When designing the qubit, the qubit capacitance $C$ is usually fixed to a
desired parameter. Because the results are more easy to interpret in terms of
design distances, it is convenient to describe the qubit capacitance in terms
of a length using

$\displaystyle C\equiv\epsilon_{0}L\ .$ (2)

For $C=100\,\textrm{fF}$, a value used for a qubit non-linearity of about 200
MHz, one finds $L=11.3\,\textrm{mm}$.

For thin films, the electric fields of the top and bottom dielectrics can be
considered separately. Thus the electric fields $E_{0}$ can be solved for
$\epsilon=\epsilon_{0}$, the free space value, and then multiplied by 1 for
the solution on the air side, and by $\epsilon_{s}$ for the substrate side.
The surface dielectrics can be taken into account with the three participation
ratios [4]

$\displaystyle p_{\textrm{MA}}=$
$\displaystyle\frac{1}{\epsilon_{\textrm{MA}}}\frac{t_{\textrm{MA}}}{L\epsilon_{0}/2}\Big{[}\frac{\epsilon_{0}}{2}\int_{\textrm{MA}}dA\,\,|E_{0}/V|^{2}\Big{]}\
,$ (3) $\displaystyle p_{\textrm{MS}}=$
$\displaystyle\frac{\epsilon_{s}^{2}}{\epsilon_{\textrm{MS}}}\,\frac{t_{\textrm{MS}}}{L\epsilon_{0}/2}\Big{[}\frac{\epsilon_{0}}{2}\int_{\textrm{MS}}dA\,\,|E_{0}/V|^{2}\Big{]}\
,$ (4) $\displaystyle p_{\textrm{SA}}=$ $\displaystyle\
\epsilon_{\textrm{SA}}\,\frac{t_{\textrm{SA}}}{L\epsilon_{0}/2}\Big{[}\frac{\epsilon_{0}}{2}\int_{\textrm{SA}}dA\,\,|E_{0}/V|^{2}\Big{]}\
,$ (5)

where the area integrals correspond to the appropriate surfaces for each type,
and the bracketed terms are called surface energies. Tangential fields are
only included for the SA formula, as appropriate for thin films [4]. For the
above MA and MS surface energies, the electric field is for only one side of
the metal. Thus for the total surface energy $U$ calculated in the next
sections, the above MA and MS energies in brackets should be $U/2$.

Dielectric constants are for a silicon substrate $\epsilon_{s}=11.7$, aluminum
oxide $\epsilon_{\textrm{MA}}=\epsilon_{\textrm{MS}}=9.8$, and silicon dioxide
$\epsilon_{SA}=3.8$; the relative weights of the MA:MS:SA dielectric terms are
$0.10:14:3.8$.

##  II Differential Parallel Plate Capacitor

The simplest geometry is a parallel plate capacitor. This contribution is
typically needed when transmon qubits are made using bump-bonded substrates,
where the second substrate acts as a ground plane above the qubit metal pads
and thus adds capacitance to the qubit. This structure can be treated as
parallel plate with each plate having width $w$ and length $\ell$ and a
separation $s$ to the ground plane, with capacitance

$\displaystyle C_{\textrm{p}}=(1/2)\epsilon_{0}\ell w/s\ ,$ (6)

where the $1/2$ factor coming from the differential design of the qubit, where
the capacitance of each parallel plate is in series.

The electric field in each differentially-driven capacitor is
$E_{\textrm{p}}=\tfrac{1}{2}V/s$, and the total surface energy is

$\displaystyle U_{\textrm{p}}$ $\displaystyle=(\epsilon/2)2(2\ell
w)(V/2s)^{2}\ ,$ (7)

where a factor of 2 comes from the two parallel plates, and another from
surface loss at the 2 plates of the capacitor.

The participation ratio for the parallel-plate capacitor is

$\displaystyle p_{\textrm{MA}}^{\textrm{p}}=$
$\displaystyle\frac{1}{\epsilon_{\textrm{MA}}}\frac{t_{\textrm{MA}}}{L}\frac{\ell
w}{s^{2}}\ .$ (8)

Participation ratios are written with first the dielectric factor, then the
dielectric thickness, and finally the geometric factors for the design.

##  III Thickness correction

The finite thickness of the metal film changes the surface electric fields
mostly at the edges of the film. Since the edge fields will be similar for
different geometries, their effect will be calculated here for a simple flat
coaxial film. The resulting simple correction to the surface energy can then
be applied to different geometries.

It is useful to start with a 2-D solution of a coax line, with an inner
conductor of radius $r$ and an outer conductor of radius $R$ as illustrated in
Fig. 3a. The solution for the radial electric field on the surface of the
inner conductor is

$\displaystyle E_{\textrm{c}}$ $\displaystyle=\frac{V}{r\ln(R/r)},$ (9)

with its strength decreasing with radius $x$ as

$\displaystyle E_{\textrm{c}}(x)=(r/x)E_{\textrm{c}}$ (10)

Figure 3: Design geometries. Drawings of capacitor geometries considered here.
a) Cross section of coax, with inner radius $r$ and outer radius $R$. b) Cross
section of flat coax, with width $2\overline{r}$ of inner thin film. c) Top
view of coplanar capacitor, with width $2a$ and length $\ell$ for each
section, separated from the ground plane by gap $b-a$. d) Enlarged view of
junction wiring, with length $d$ for each wire from junction to capacitor
film. The width of the wire is $2\overline{r}$ for a straight wire, drawn in
red. A tapered wire is drawn in blue.

The electric field energy $(\epsilon/2)\int E^{2}dv$ is calculated from a
volume integral $dv$ of the electric field $E$. Because the interest here is
for the surface energy in a 2-D geometry, we compute the surface energy
$U/\ell$ for a line length $\ell$ so that the full energy will be multiplied
by the surface thickness and length $\ell$. For the coax geometry, the surface
energy of the inner metal at radius $r$ is

$\displaystyle U_{\textrm{c}}^{\textrm{m}}/\ell$
$\displaystyle=(\epsilon/2)2\pi rE_{\textrm{c}}^{2}$ (11)
$\displaystyle=\epsilon E_{c}^{2}r\,\pi.$ (12)

The surface energy corresponding to a substrate surface along a cut through
the middle of the coax is

$\displaystyle U_{\textrm{c}}^{\textrm{s}}/\ell$ $\displaystyle=(\epsilon/2)\
2\int_{r}^{R}E_{\textrm{c}}(x)^{2}dx$ (13) $\displaystyle=\epsilon
E_{c}^{2}r\,[1-r/R],$ (14)

where the factor of 2 before the integral comes from the left and right
substrate sides.

Fig. 3b shows a flat coax, where the circular inner conductor is replaced by a
thin film of width $2\overline{r}$. The electric field magnitude along the
coordinate $x$ for this flat is found from numerical solutions for all $x$ to
be given by a conformal-mapping solution

$\displaystyle E_{\textrm{f}}$
$\displaystyle=\frac{V}{\overline{r}\ln(2R/\overline{r})}\ ,$ (15)
$\displaystyle E_{\textrm{f}}(x)$
$\displaystyle=E_{\textrm{f}}\sqrt{\frac{\overline{r}}{|\overline{r}+x|}}\sqrt{\frac{\overline{r}}{|\overline{r}-x|}}\
,$ (16)

which fits well for $R>2\overline{r}$. The electric field is perpendicular to
the metal surface but parallel to the substrate surface. The voltage integral
checks properly

$\displaystyle\int_{\overline{r}}^{R}E_{\textrm{f}}(x)\,dx=V[1+O(\overline{r}^{2}/R^{2})]\
.$ (17)

Figure 4 shows a comparison between the numerical solution and the formula of
Eqs. (15)-(16), showing excellent agreement. The square-root divergence at the
metal edge is characteristic of the electric fields of thin metal films.

Figure 4: Electric field of flat coax. Plot of the surface electric field for
a flat coax for both the metal surface (black) and substrate (blue), obtained
by numerical simulation (dots). The solid lines (green, red) are predictions
from Eqs. (15)-(16) and fit well the numerics. Parameters are
$\overline{r}=10\,\mu\textrm{m}$ and $R=100\,\mu\textrm{m}$

The surface energy for the metal surface ($|x|<\overline{r}$) is

$\displaystyle U_{\textrm{f}}^{\textrm{m1}}/\ell$
$\displaystyle=(\epsilon/2)E_{f}^{2}\
4\int_{0}^{\overline{r}-t/2}\frac{\overline{r}^{2}}{\overline{r}^{2}-x^{2}}\
dx$ (18) $\displaystyle\simeq\epsilon
E_{\textrm{f}}^{2}\overline{r}\ln(4\overline{r}/t)$ (19)

where the factor of 4 is for the top/bottom and left/right parts of the metal,
and the logarithmic divergence in the integral at the edge is cut-off at half
the thickness $t$ of the film.

Numerical simulation for a film with a rectangular cross-section of thickness
$t$ shows that the electric fields within $t/2$ of the outside corner have a
power law behavior with exponent $p=-1/3$, as appropriate for a 90 degree
corner [4, 12]. As an initial approximate solution, this power law dependence
of the corner field is then matched to the computed field
$E_{\textrm{f}}(\overline{r}-t/2)$ at a distance $t/2$ from the corner. At a
distance $r_{c}$ from the corner, the corner field is

$\displaystyle E_{\textrm{c}}$
$\displaystyle=E_{\textrm{f}}(\overline{r}-t/2)\ [r_{c}/(t/2)]^{p}$ (20)
$\displaystyle=E_{f}\sqrt{\overline{r}/t}\ [2r_{c}/t]^{p}.$ (21)

Including all 4 corners, with 2 sides per corner, the line energy for the
corner is approximately

$\displaystyle U_{\textrm{f}}^{\textrm{m2}}/\ell$
$\displaystyle=(\epsilon/2)E_{f}^{2}(\overline{r}/t)\
8\int_{0}^{t/2}[2r_{c}/t]^{2p}dr_{c}.$ (22) $\displaystyle=4\epsilon
E_{f}^{2}(\overline{r}/t)(t/2)/(1+2p)$ (23) $\displaystyle=\epsilon
E_{f}^{2}\overline{r}\ [2/(1+2p)]\ .$ (24)

With $1+2p=1/3$, the numerical factor in Eq. (24) is 6 and does not depend on
$t$.

The total surface energy for the metal is the sum of the two energies

$\displaystyle U_{\textrm{f}}^{\textrm{m}}/\ell$ $\displaystyle=\epsilon
E_{\textrm{f}}^{2}\overline{r}\ [\ln(4\overline{r}/t)+c_{m}]\ ,$ (25)

where $c_{m}$ is the corner correction for the finite thickness of the metal.
Figure 5 gives $c_{m}$ obtained from numerical integration of the surface
energy. The corner correction is slowly varying with relative film thickness
$t/\overline{r}$ and has a typical value

$\displaystyle c_{m}$ $\displaystyle=5.0$ (26)

close to the value 6 obtained above by scaling of the corner fields. It is
useful that the surface energy is predicted well even for thick film, with
thickness as much as one-half the width. The edges typically contribute about
1/3 of the total surface energy. Also shown is the case of a semicircular
edge, which lowers the surface energy a non-negligible but small amount,
providing a lower bound for the correction of a rounded edge.

Figure 5: Corner corrections. Plot of corner corrections obtained by numerical
simulation of films with finite thickness $t/\overline{r}$ and a straight
vertical edge, for the metal $c_{m}$ (black) and substrate $c_{s}$ (blue).
Data in red is for a semi-circular edge, showing that sharp corners have a
non-negligible but non-dominant effect. The corrections vary slowly with
thickness and are taken as $c_{m}=5.0$ and $c_{s}=1.6$

This result shows that a constant term added to the logarithmic cut-off term
well represents the corner fields. Note the similarity to Eq. (12). Here, the
bracket term in Eq. (25) is slightly larger than the corresponding $\pi$
constant in Eq. (12), as expected since the flat coax has large edge fields.
The correction factor
$\ln(4\overline{r}/t)\rightarrow\ln(4\overline{r}/t)+c_{m}$ will be used in
all formulas for the metal edge.

The surface energy for the substrate surface ($\overline{r}<x<R$) is

$\displaystyle U_{\textrm{f}}^{\textrm{s1}}/\ell$
$\displaystyle=(\epsilon/2)E_{f}^{2}\
2\int_{\overline{r}+t/2}^{R}\frac{\overline{r}^{2}}{x^{2}-\overline{r}^{2}}\
dx$ (27) $\displaystyle=\epsilon
E_{\textrm{f}}^{2}\,(\overline{r}/2)\Big{[}\ln(4\overline{r}/t)-\ln\frac{R+\overline{r}}{R-\overline{r}}\,\Big{]}\
,$ (28)

where the factor of 2 is for the left/right parts of the substrate. Like found
for the metal surface, numerical integration for finite thickness gives a
corner correction 1.6. Since typically $\overline{r}\ll R$, the total
substrate surface energy is

$\displaystyle U_{\textrm{f}}^{\textrm{s}}/\ell$ $\displaystyle=\epsilon
E_{\textrm{f}}^{2}\,(\overline{r}/2)[\ln(4\overline{r}/t)+c_{s}-2\overline{r}/R]\
,$ (29) $\displaystyle c_{s}$ $\displaystyle=1.6\ .$ (30)

This is smaller than the surface energy for the metal surface since it does
not include a sharp edge. The correction factor
$\ln(4\overline{r}/t)\rightarrow\ln(4\overline{r}/t)+c_{s}$ will be used in
all formulas for the substrate edge.

##  IV Differential Ribbon Capacitor

Considered next is the capacitance between the two leads of the qubit, modeled
as two long and straight ribbons as illustrated in Fig. 1. Each ribbon has
metal spanning a distance $a$ to $b$ from the centerline, with length $\ell\gg
b$ and a metal thickness $t$. A conformal-mapping solution from Ref. [9] is
used for the electric fields. The ribbon capacitance for a differential
voltage $V$ is

$\displaystyle C_{\textrm{r}}$
$\displaystyle=[(\epsilon_{s}+1)/2]\,\epsilon_{0}\ell/C_{K}(a/b)\ ,$ (31)
$\displaystyle C_{K}$ $\displaystyle=K(a/b)/K^{\prime}(a/b)$ (32)
$\displaystyle\simeq(1/\pi)\ln[2(1+\sqrt{a/b})/(1-\sqrt{a/b})]\ ,$ (33)
$\displaystyle K^{\prime}(k)$ $\displaystyle=K(\sqrt{1-k^{2}})$ (34)

where $K(k)$ is the complete elliptic integral of the first kind. Equation
(33) is an excellent approximation to Eq. (32). The effective dielectric
constant has contribution from both the air ($\epsilon_{0}/2$) and substrate
($\epsilon_{s}\epsilon_{0}/2$).

From the conformal-mapping solution Eq. (5) of Ref. [9], the surface fields
are

$\displaystyle|E_{r}(x)|^{2}=\Big{(}\frac{V/2}{K(a/b)}\Big{)}^{2}\frac{b^{2}}{|(x^{2}-a^{2})(x^{2}-b^{2})|}\
,$ (35)

where the $E$ field is parallel to the surface on the substrate and
perpendicular on the metal. The surface integral is evaluated in three
sections:

$\displaystyle\textrm{inner}\ \ \ \ \ \ \ \ \ \ \ \ \ \ 0<x<a-t/2\ ,$ (36)
$\displaystyle\textrm{center}\ \ \ \ \ a+t/2<x<b-t/2\ ,$ (37)
$\displaystyle\textrm{outer}\ \ \ \ \ \ \,b+t/2<x<\infty\ ,$ (38)

giving

$\displaystyle S_{i}$
$\displaystyle=\int_{0}^{a-t/2}dx\,\frac{b^{2}}{|(x^{2}-a^{2})(x^{2}-b^{2})|}$
(39)
$\displaystyle=\frac{\frac{1}{a}\ln\frac{a-x}{a+x}+\frac{1}{b}\ln\frac{b-x}{b+x}}{2(1-a^{2}/b^{2})}\
\Big{|}_{0}^{a-t/2}$ (40)
$\displaystyle\simeq\frac{\frac{1}{a}\ln\frac{4a}{t}+\frac{1}{b}\ln\frac{b-a}{b+a}}{2(1-a^{2}/b^{2})}\
,$ (41) $\displaystyle S_{c}$
$\displaystyle\simeq\frac{\frac{1}{a}(\ln\frac{4a}{t}+\ln\frac{b-a}{b+a})+\frac{1}{b}(\ln\frac{4b}{t}+\ln\frac{b-a}{b+a})}{2(1-a^{2}/b^{2})}\
,$ (42) $\displaystyle S_{o}$
$\displaystyle\simeq\frac{\frac{1}{a}\ln\frac{b-a}{b+a}+\frac{1}{b}\ln\frac{4b}{t}}{2(1-a^{2}/b^{2})}\
.$ (43)

Note that $S_{c}=S_{i}+S_{o}$.

The surface energy of the center metal section is

$\displaystyle U_{\textrm{r}}^{\textrm{m}}/\ell$ $\displaystyle=(\epsilon/2)\
4\,[V/2K(a/b)]^{2}S_{c}$ (44) $\displaystyle=\frac{\epsilon
V^{2}}{2\,K^{2}(a/b)}\frac{S_{a}(c_{m})}{a}\ ,$ (45) $\displaystyle
S_{a}(c_{m})$
$\displaystyle\equiv\frac{(\ln\frac{4a}{t}+c_{m}+\ln\frac{b-a}{b+a})+\frac{a}{b}(\ln\frac{4b}{t}+c_{m}+\ln\frac{b-a}{b+a})}{2(1-a^{2}/b^{2})}\
.$ (46)

where the factor of 4 comes from the two ribbons and the top/bottom surfaces.
The dimensionless surface integral is obtained from $S_{a}/a=S_{c}$, along
with adding the corner correction $c_{m}$ for a finite thickness

The surface energy of the inner and outer substrate sections is

$\displaystyle U_{\textrm{r}}^{\textrm{s}}/\ell$ $\displaystyle=(\epsilon/2)\
2\,[V/2K(a/b)]^{2}(S_{i}+S_{0})$ (47) $\displaystyle=\frac{\epsilon
V^{2}}{4\,K^{2}(a/b)}\frac{S_{a}(c_{s})}{a}\ ,$ (48)

where the factor of 2 is from the two sides of the ribbon. The substrate
surface energy is smaller than the metal by approximately a factor of 2.

The ribbon capacitor has participation ratios coming from the surface-air,
metal-substrate, and substrate air interfaces, calculated using Eqs. (3)-(5),
(45) and (48)

$\displaystyle p_{\textrm{MA}}^{\textrm{r}}=$
$\displaystyle\frac{1}{\epsilon_{\textrm{MA}}}\frac{t_{\textrm{MA}}}{L}\frac{\ell}{a}\
\frac{S_{a}(c_{m})}{2\,K^{2}(a/b)}\ ,$ (49) $\displaystyle
p_{\textrm{MS}}^{\textrm{r}}=$
$\displaystyle\frac{\epsilon_{s}^{2}}{\epsilon_{\textrm{MS}}}\frac{t_{\textrm{MS}}}{L}\,\frac{\ell}{a}\
\frac{S_{a}(c_{m})}{2\,K^{2}(a/b)}\ ,$ (50) $\displaystyle
p_{\textrm{SA}}^{\textrm{r}}=$
$\displaystyle\,\epsilon_{\textrm{SA}}\,\frac{t_{\textrm{SA}}}{L}\frac{\ell}{a}\
\frac{S_{a}(c_{s})}{2\,K^{2}(a/b)}\ .$ (51)

The only difference in the 3 participation ratios is the dielectric factors
and the small changed from the corner constant.

Figure 6: Surface energies. Plot of the normalized surface energies
$S_{a}(c_{m})/K^{2}$ for ribbon (black) and $S_{a}(c_{s})/K^{\prime 2}$ for
coplanar (blue) geometries, versus the normalized distance $(b-a)/a$. The plot
uses the normalized thickness $a/t=1000$. The solid lines are for the metal
surface with $c_{m}=5.0$, whereas dashes are for the substrate with
$c_{s}=1.6$. Also plotted in red is $S_{a}/KK^{\prime}$ for the case of all
capacitance coming from the ribbon or coplanar geometry.

The black lines in Fig. 6 is a plot of the dimensionless surface energy
$S_{a}(c_{m})/K^{2}$ for the metal (solid) and $S_{a}(c_{s})/K^{\prime 2}$ for
the substrate (dashed) as a function of the normalized distance $(b-a)/a$. The
metal surface energy is greater because of the higher corner constant
$c_{m}>c_{s}$. As the distance $b-a$ increases, the surface energy decreases.
Typical designs use $(b-a)/a\sim 1$. Note that the surface energy drops by a
non-negligible amount with the lower corner constant, showing that the edge
fields from the finite thickness are important.

For the case where all of the capacitance comes from the ribbon
$C_{r}=\epsilon_{0}L$, the participation for the metal-substrate interface is

$\displaystyle p_{\textrm{MS}}^{\textrm{r}}(C_{r})=$
$\displaystyle\frac{\epsilon_{s}^{2}}{\epsilon_{\textrm{MS}}(\epsilon_{s}+1)/2}\frac{t_{\textrm{MS}}}{a}\,\frac{1}{2}\
\frac{S_{a}(c_{m})}{K(a/b)K^{\prime}(a/b)}\ .$ (52)

The last factor is the geometric mean of the ribbon and coplanar curves of
Fig. 6, shown in red.

##  V Differential Coplanar Capacitor

The qubit can also have capacitance to ground. This can be modeled as a
coplanar structure as shown in Fig. 3c, where each side of the qubit has a pad
with width $2a$ and length $\ell\gg a$, with a ground plane at a distance $b$
from the centerline. As this is the “dual” of the ribbon capacitor, with metal
and substrate switched, similar conformal solutions can be used with minor
modifications. The differential capacitance is

$\displaystyle C_{\textrm{c}}=$
$\displaystyle(1/2)[(\epsilon_{s}+1)/2]\,\epsilon_{0}\ell\,4C_{K}(a/b)\ ,$
(53)

where $C_{K}$ is defined in Eq. (32), and the initial factor of 1/2 comes from
the two coplanar capacitors in series. From Eq. (25) of Ref. [9], the electric
field for each coplanar capacitor is

$\displaystyle|E_{c}(x)|^{2}=\Big{(}\frac{V/2}{K^{\prime}(a/b)}\Big{)}^{2}\frac{b^{2}}{|(x^{2}-a^{2})(x^{2}-b^{2})|}\
,$ (54)

where now the field is perpendicular to the substrate in the inner and outer
sections, and parallel in the center.

The surface energy of the metal sections is similar to the ribbon case except
for an extra factor of 2 to account for the series capacitors, as can be seen
from Fig. 3c since $\ell$ only accounts for half of the total length. The
metal and substrate surface energies are

$\displaystyle U_{\textrm{c}}^{\textrm{m}}/\ell$ $\displaystyle=\frac{\epsilon
V^{2}}{K^{\prime 2}(a/b)}\frac{S_{a}(c_{m})}{a}\ ,$ (55) $\displaystyle
U_{\textrm{c}}^{\textrm{s}}/\ell$ $\displaystyle=\frac{\epsilon
V^{2}}{2\,K^{\prime 2}(a/b)}\frac{S_{a}(c_{s})}{a}\ ,$ (56)

Note the similarities to the ribbon formulas. The participation ratios are
twice as large as the ribbon and with $K$ replaced by $K^{\prime}$

$\displaystyle p_{\textrm{MA}}^{\textrm{c}}=$
$\displaystyle\frac{1}{\epsilon_{\textrm{MA}}}\frac{t_{\textrm{MA}}}{L}\frac{2\,\ell}{a}\
\frac{S_{a}(c_{m})}{2\,K^{\prime 2}(a/b)}\ ,$ (57) $\displaystyle
p_{\textrm{MS}}^{\textrm{c}}=$
$\displaystyle\frac{\epsilon_{s}^{2}}{\epsilon_{\textrm{MS}}}\frac{t_{\textrm{MS}}}{L}\,\frac{2\,\ell}{a}\
\frac{S_{a}(c_{m})}{2\,K^{\prime 2}(a/b)}\ ,$ (58) $\displaystyle
p_{\textrm{SA}}^{\textrm{c}}=$
$\displaystyle\,\epsilon_{\textrm{SA}}\,\frac{t_{\textrm{SA}}}{L}\frac{2\,\ell}{a}\
\frac{S_{a}(c_{s})}{2\,K^{\prime 2}(a/b)}\ .$ (59)

For the case where all the capacitance comes from the coplanar structure, the
participation is the same as the ribbon design, for example

$\displaystyle
p_{\textrm{MS}}^{\textrm{c}}(C_{c})=p_{\textrm{MS}}^{\textrm{r}}(C_{r})\ .$
(60)

A single-ended coplanar design is used to test resonators. In this case, the
coplanar capacitance of Eq. (53) does not have the initial 1/2 term. The
surface energy $U$ and participation ratios are a factor of 2 larger. The
participation $p_{\textrm{MS}}^{\textrm{r}}(C_{c})$ includes these two
factors, so Eq. (60) is unchanged for the single-ended design.

##  VI Differential Ribbon Capacitor
With Ground

Planar transmons are typically designed to have a ground plane surrounding the
qubit capacitor, as shown in Fig. 2. For the ribbon capacitor considered
previously, a ground plane is included here from $c$ to infinity and $-c$ to
minus infinity, where $c>b$. The capacitance and surface loss is computed
numerically and then fit to functions based on the previous ribbon formulas.

The surface electric field is well described a simple modification to the
ribbon case of Eq. (35)

$\displaystyle|E_{rg}(x)|^{2}=|E_{r}(x)|^{2}\frac{c^{2}}{|x^{2}-c^{2}|}\ .$
(61)

Integration of surface charge from the numerical solutions gives a simple
modification to the ribbon differential capacitance of Eq. (31)

$\displaystyle C_{\textrm{rg}}$ $\displaystyle\simeq
C_{\textrm{r}}/[1-(x_{e}/c)^{2}]^{0.23}\ ,$ (62) $\displaystyle x_{e}$
$\displaystyle=b-0.15(b-1.2a)\ .$ (63)

Figure 7 shows the numerical results (points) and the fit function (line)
versus ground plane separation $(c-b)/b$ for three values of $a$,
representative of $a/b$ ratios that would commonly be used. The fit function
represents the numerical results well.

Figure 7: Ribbon capacitor with ground. Plot of the capacitance (top panel)
and surface loss (bottom panel) versus the ground plane separation $(c-b)/b$,
for $a=(25,50,70)\,\mu$m (top to bottom) and $b=100\,\mu$m. Points are
numerical simulations and lines are fit formulas. The metal and substrate
surface loss is colored black and blue, respectively. Numerical simulations
are for a infinitely thin metal with an integration cutoff $t/2=0.05\,\mu$m,
with the fit also using $c_{m}=c_{s}=0$. Dashed lines include corner
correction $c_{m}=5.0$ and $c_{s}=1.6$.

For the surface loss of the metal, the fit function for the numerical results
are

$\displaystyle U_{\textrm{rg}}^{\textrm{m}}/\ell$ $\displaystyle=\epsilon
V^{2}\Bigg{[}\frac{0.98}{2\,K^{2}(a/b)}\frac{S_{a}(a,b,t,c_{m})}{a}$ (64)
$\displaystyle\ \ \ \ \ \ +\frac{1.70}{2\,K^{\prime
2}(b/c)}\frac{S_{ao}(b,c,t,c_{m})}{b}\Bigg{]}\ ,$ (65)

where the contribution of $S_{ao}$ corresponds to the outer metal of a
coplanar capacitor between $b$ and $c$

$\displaystyle S_{ao}(b,c,t,c_{m})$
$\displaystyle=\frac{\ln\frac{c-b}{c+b}+\frac{b}{c}(\ln\frac{4c}{t}+c_{m})}{2(1-b^{2}/c^{2})}\
.$ (66)

The surface loss for both the inner and outer substrate gaps gives the fit
function

$\displaystyle U_{\textrm{rg}}^{\textrm{s}}/\ell$ $\displaystyle=\epsilon
V^{2}\Bigg{[}\frac{0.95}{4\,K^{2}(a/b)}\frac{S_{a}(a,b,t,c_{s})}{a}$ (67)
$\displaystyle\ \ \ \ \ \ +\frac{0.80}{4\,K^{\prime
2}(b/c)}\frac{S_{a}(b,c,t,c_{s})}{b}\Bigg{]}\ ,$ (68)

where the second contribution of $S_{a}$ corresponds to the substrate of a
coplanar capacitor between $b$ and $c$. Figure 7 shows the numerical results
are well represented by the fit functions.

##  VII Differential Junction Wires

The connections between the Josephson junction and the capacitor electrodes
are made through two junction wires of total length $2d$, as shown in Fig. 3d.
Treating these wires as round with radius $r$ and placed end-to-end each with
length $d$, the surface field can be calculated numerically using the
potential matrix Eq. (96) for a 2-D geometry with cylindrical symmetry. For a
differential voltage $V$, the surface electric field as a function of distance
$y$ from the junction is well described by

$\displaystyle E_{\textrm{cw}}(y)=\frac{1}{2}\ \frac{V}{r\ln(2y/r)}\ ,$ (69)

as shown in Fig. 8. In comparison with Eq. (9), the second term is equivalent
to a coax of inner radius $r$ and outer radius $2y$. The first term 1/2
represents this coax placed in series with a second coax of the same
dimensions, which represents the fields emanating from one circular wire,
expanding to a distance $2y$, and then converging in to the other circular
wire.

This 2-D numerical calculation also allows the radius $r$ to change with
distance $y$. Modeling a linear taper with $r=S\,y$, the electric field is
found to be well described by Eq. (69) with $r$ replaced by $r(y)$, as long as
the taper is not too large $S<0.4$; larger slopes are found not to reduce the
electric field significantly. Figure 8 shows numerical results for both a
straight and tapered cylindrical wire, with good agreement to the
approximation formula Eq. (69).

Figure 8: Electric field of cylindrical wire. Plot of the surface electric
field for a cylindrical wire for both straight ($S=0$) and tapered ($S=0.2$)
radius. Numerical solutions (dots) match well with the approximation formulas
(lines) of Eq. (69). Parameters are $r=0.1\,\mu\textrm{m}$ and
$d=100\,\mu\textrm{m}$. The uptick of the field at the end of the wire is
expected for an edge field.

A solution for a flat wire can be obtained by assuming the wire has the
$x$-dependence of the electric field as given in Eq. (16), but with an overall
dependence of $E_{\textrm{fw}}$ with $y$ that is determined numerically. The
appendix shows how to solve this problem with a potential matrix. Numerical
solutions for both straight and tapered flat wires show that a good fitting
function is

$\displaystyle E_{\textrm{fw}}(y)=\frac{1}{2}\
\frac{V}{\overline{r}\,\ln(4y/\overline{r})}\ ,$ (70)

which has the form of $E_{\textrm{f}}$ in Eq. (15) but with $2R$ replaced by
$4y$. It is again valid for small slope $S<0.4$. The factor of 4 in the
logarithm can be understood as a factor of 2 from the coax to the wire
geometry, and another factor of 2 from the circular to flat coax formula in
Eqs. (9) to (15).

The metal surface energy for a straight wire of constant width can thus be
found by integrating this surface field, which is equivalent to integrating
one-fourth of the line energy Eq. (25) over the wire length

$\displaystyle U_{\textrm{sw}}^{\textrm{m}}$
$\displaystyle=2\int_{2\overline{r}}^{d}[\tfrac{1}{4}U_{\textrm{f}}^{m}(R=2y)/\ell]\,dy$
(71) $\displaystyle=2\epsilon
V^{2}\int_{2\overline{r}}^{d}\frac{\ln(4\overline{r}/t)+c_{m}}{4\
\overline{r}\ln^{2}(4y/\overline{r})}\,dy$ (72)
$\displaystyle\simeq\frac{\epsilon
V^{2}}{2}\frac{\ln(4\overline{r}/t)+c_{m}}{\ln^{2}(d/\overline{r})}\
\frac{d}{\overline{r}}\ ,$ (73)

where the factor of 2 before the integral accounts for both junction wires.
The last formula was fit to numerical integration. Because of the
$d/\overline{r}$ factor, this surface energy can be large, so a more optimal
solution is to taper the wire as explained below.

Similarly from Eq. (29), the surface energy of the substrate for a straight
wire of constant width is

$\displaystyle U_{\textrm{sw}}^{\textrm{s}}$
$\displaystyle=2\int_{2\overline{r}}^{d}[\tfrac{1}{4}U_{\textrm{f}}^{s}(R=2y)/\ell]\,dy$
(74) $\displaystyle\simeq\frac{\epsilon
V^{2}}{4}\frac{\ln(4\overline{r}/t)+c_{s}}{\ln^{2}(d/\overline{r})}\
\frac{d}{\overline{r}}\ ,$ (75)

The participation ratios for straight wires are

$\displaystyle p_{\textrm{MA}}^{\textrm{sw}}=$
$\displaystyle\frac{1}{\epsilon_{\textrm{MA}}}\frac{t_{\textrm{MA}}}{L}\frac{d}{\overline{r}}\,p_{\textrm{M}}^{\textrm{sw}\prime}\
,$ (76) $\displaystyle p_{\textrm{MS}}^{\textrm{sw}}=$
$\displaystyle\frac{\epsilon_{s}^{2}}{\epsilon_{\textrm{MS}}}\frac{t_{\textrm{MS}}}{L}\frac{d}{\overline{r}}\,p_{\textrm{M}}^{\textrm{sw}\prime}\
,$ (77) $\displaystyle p_{\textrm{SA}}^{\textrm{sw}}=$
$\displaystyle\,\epsilon_{\textrm{SA}}\,\frac{t_{\textrm{SA}}}{L}\frac{d}{\overline{r}}\,p_{\textrm{S}}^{\textrm{sw}\prime}\
,$ (78)

where multiplicative factors are given by

$\displaystyle p_{\textrm{M}}^{\textrm{sw}\prime}=$
$\displaystyle\frac{1}{2}\frac{\ln(4\overline{r}/t)+c_{m}}{\ln^{2}(d/\overline{r})}\
,$ (79) $\displaystyle p_{\textrm{S}}^{\textrm{sw}\prime}=$
$\displaystyle\frac{1}{2}\frac{\ln(4\overline{r}/t)+c_{s}}{\ln^{2}(d/\overline{r})}\
.$ (80)

As found previously, the equations differ only in the epsilon factors and the
corner constant.

The capacitance of the straight junction wires is found from numerical
simulation

$\displaystyle C_{\textrm{sw}}\simeq
4.1\,[(\epsilon_{s}+1)/2]\,\epsilon_{0}\,d/\ln(d/\overline{r})\ .$ (81)

##  VIII Tapered Junction Wires

The large $d/\overline{r}$ ratio in the above participation ratios contributes
to a large surface energy, since the small width of the wires produce large
electric fields at its surface. As surface loss decreases with increasing
size, it is natural to increase the width of the wire to lower loss. A
solution to minimize surface energy is to taper the wire, increasing the wire
width with increasing distance $y$ from the junction as shown in Fig. 3d. The
contribution to the line energy, the surface energy per line length $dy$, is
the integrand of Eq. (72), where $\overline{r}$ is now a function of $y$. The
integrand is is minimized at distances $y/t=(10,100,1000)$ for a half-width
$\overline{r}/y=(0.363,0.402,0.425)$, respectively.

An effective solution is to taper the wire according to
$\overline{r}(y)=\textrm{max}(\overline{r}_{0},(y-5t)S)$ with the taper
starting at $y=5t$, optimizing the slope $S$ for lowest energy. Numerical
integration of the line energy gives the metal surface energy for a tapered
wire that is fit by

$\displaystyle U_{\textrm{tw}}^{\textrm{m}}$ $\displaystyle\simeq
0.68\,\epsilon V^{2}\frac{\ln(d/\overline{r}_{0})}{S}\
\frac{\ln(4Sd/t)+c_{m}}{\ln^{2}(4/S)}\,\ ,$ (82)

Although this has a minimum energy at slope $S=0.45$, it is a broad minimum
increasing by 2% at $S=0.28$ and only 10% at $S=0.16$. Note that this formula
is similar to Eq. (73) for a constant width wire, except for the logarithm
dependence on $d$.

Similarly, the substrate surface energy for a tapered wire is fit by

$\displaystyle U_{\textrm{tw}}^{\textrm{s}}$ $\displaystyle\simeq
0.29\,\epsilon V^{2}\frac{\ln(d/\overline{r}_{0})}{S}\
\frac{\ln(4Sd/t)+c_{s}}{\ln^{2}(4/S)}\,.$ (83)

Figure 9: Junction wire surface energy. Metal surface energy (and loss) of the
junction wire for straight and tapered designs versus wire length per side
$d$. The tapered wire shows significantly lower energy for wire lengths
$d\gtrsim 10\,\mu\textrm{m}$. At large distances the straight and tapered
energy scale with $d$ approximately linearly and logarithmicly, respectively.
Parameters are $t=\overline{r}_{0}=0.1\,\mu\textrm{m}$ and $S=0.4$.

The metal surface loss for the junction wires is plotted in Fig. 9 for the
straight and tapered cases, obtained by numerical integration of Eq. (72). At
small distances $d\lesssim 5\,\mu\textrm{m}$, the two results are similar, but
at large distances the logarithmic scaling makes the tapered loss
significantly lower. It is standard practice to increase the overall size of
the qubit capacitor to lower its loss. When using a large $d$, it is thus
increasing important to optimally design the junction wires with a taper.

The formulas for the participation ratios for a tapered wire are

$\displaystyle p_{\textrm{MA}}^{\textrm{tw}}=$
$\displaystyle\frac{1}{\epsilon_{\textrm{MA}}}\frac{t_{\textrm{MA}}}{L}\frac{\ln(d/\overline{r}_{0})}{S}\,p_{\textrm{M}}^{\textrm{tw}\prime}\
,$ (84) $\displaystyle p_{\textrm{MS}}^{\textrm{tw}}=$
$\displaystyle\frac{\epsilon_{s}^{2}}{\epsilon_{\textrm{MS}}}\frac{t_{\textrm{MS}}}{L}\frac{\ln(d/\overline{r}_{0})}{S}\,p_{\textrm{M}}^{\textrm{tw}\prime}\
,$ (85) $\displaystyle p_{\textrm{SA}}^{\textrm{tw}}=$
$\displaystyle\,\epsilon_{\textrm{SA}}\,\frac{t_{\textrm{SA}}}{L}\frac{\ln(d/\overline{r}_{0})}{S}\,p_{\textrm{S}}^{\textrm{tw}\prime}\
,$ (86)

with multiplicative factors

$\displaystyle p_{\textrm{M}}^{\textrm{tw}\prime}=$ $\displaystyle
0.68\frac{\ln(4Sd/t)+c_{m}}{\ln^{2}(4/S)}\ ,$ (87) $\displaystyle
p_{\textrm{S}}^{\textrm{tw}\prime}=$ $\displaystyle
0.58\frac{\ln(4Sd/t)+c_{s}}{\ln^{2}(4/S)}\ .$ (88)

It is recommended using a continuous taper as described above, not a stepped
taper, since the continuous taper is optimal at every distance from the
junction $x$. Also, the sharp corners of the steps will produce large electric
fields and increase the surface energy.

The capacitance of the tapered junction wires is found from numerical
simulation

$\displaystyle C_{\textrm{tw}}\simeq
3.5\,[(\epsilon_{s}+1)/2]\,\epsilon_{0}\,\sqrt{S}\,d\ .$ (89)

##  IX Discussion

loss$\times$106 |  MA |  MS |  SA
---|---|---|---
Ref. [4] | 0.10 | 6.13 | 4.02
This work | 0.060 | 5.93 | 3.57
Participation ratio (%) | MA | MS | SA
Ref. [11] ($d=0.28\,\mu$m) | 0.017 | 0.297 | 0.156
This work ($d=0$) |  0.0012 | 0.139 | 0.027
Table 1: Check data. Comparison of prior numerical results with formulas from
this paper. The first example shows good agreement for
$(a,b,t)=(2.5,4.5,0.1)\,\mu$m and surface parameters
($\epsilon_{s},\epsilon_{\textrm{MA}},\epsilon_{\textrm{MS}},\epsilon_{SA})=(10,10,10,10)$,
thickness 3 nm and loss tangent 0.002. The second example does not agree well
and uses $(a,b,t)=(3,6,0.25)\,\mu$m and surface parameters
($\epsilon_{s},\epsilon_{\textrm{MA}},\epsilon_{\textrm{MS}},\epsilon_{SA})=(11.7,11.4,10,4)$
and thickness 2 nm, but different trench depths $d$.

These formulas agree well with a prior numerical simulation, as detailed in
Table 1. The first example shows good agreement with numerical results from
Ref. [4]. Note the only difference in the MS and SA formulas are from the
corner constants $c_{m}$ and $c_{s}$. The second example does not agree well
with the MS geometry of Ref. [11], although the results here are for a flat
substrate with no trenching. It is unexpected that the prior numerical results
with trenching gives a higher participation ratio for all surfaces.

Surface loss closely scales as the inverse of the system size, as described
previously in Ref. [4]. However, the calculation for the participation ratio
from the junction wires have the opposite effect, as its participation
increases with length. Thus there is a crossover in distance $d$ where the
surface loss of the wire goes from relatively unimportant to dominant.
Formulas for predicting this crossover is an important result of this work.

Table 2 shows the participation ratios for the 3 interfaces and 5 qubit
capacitance types, for an example geometry of size scale of $\sim
100\,\mu\textrm{m}$ that is appropriate for current devices. Here a constant
thickness 2 nm of the surface oxides is assumed. The ribbon has the same
participation as the coplanar geometry, as expected. Of course, predictions
depend on actual device parameters, which can be readily made with these
formulas.

For the qubit capacitance, the metal-substrate (MS) interface dominates the
surface participation. For the ribbon design, the substrate-air (SA) is about
10 times smaller due to the dielectric factors, half the surface, and a lower
corner constant $c_{s}$. However, the wire loss is not much smaller and
clearly indicates that for present designs this contribution should be
carefully considered. Importantly, the tapering of the wire will produce a
significant improvement in qubit performance, about a factor of 2.

interface |  Eqs. |  MA |  MS |  SA
---|---|---|---|---
parallel plate | (8) | 8.16e-5 |  |
ribbon | (49)-(51) | 1.04e-6 | 1.42e-4 | 2.74e-5
coplanar | (57)-(59) | 1.04e-6 | 1.42e-4 | 2.74e-5
straight wires | (76)-(80) | 7.47e-7 | 1.02e-4 | 1.30e-5
tapered wires | (84)-(88) | 4.21e-7 | 5.76e-5 | 9.47e-6
Table 2: Participation ratios for various qubit structures. The top three are
for the primary qubit capacitance; for comparison purpose, each uses a length
$\ell$ such that its capacitance is 100 fF. The bottom two are for straight
and tapered wires that connect the junction with the qubit capacitance.
Geometry parameters are thickness $t=0.1\,\,\mu\textrm{m}$; parallel plate
$(s,w,\ell_{p})=(5,100,1130)\,\mu\textrm{m}$; ribbon and coplanar
$(a,b,\ell_{r},\ell_{c})=(50,100,1391,1138)\,\mu\textrm{m}$; junction wires
$(2d,\overline{r},\overline{r}_{0})=(100,0.1,0.1)\,\mu\textrm{m}$ and $S=0.4$.
Dielectric parameters are
($\epsilon_{s},\epsilon_{\textrm{MA}},\epsilon_{\textrm{MS}},\epsilon_{SA})=(11.7,9.8,9.8,3.8)$.
For simplicity, here the oxide thickness is assumed to be
$t_{\textrm{MA}}=t_{\textrm{MS}}=t_{\textrm{SA}}=2\,\textrm{nm}$; results can
be simply scaled with expected thickness. Total loss can be estimated by
multiplying the surface loss tangents [11]; typical values for amorphous
insulators are 0.005 [15].

When qubit designs use multiple chips that are bump-bonded together, a
parallel plate capacitance is often formed between the qubit chip and ground.
Table 2 shows that the participation ratio of this structure needs to be
considered even for a plate separation of $s=5\,\mu\textrm{m}$, especially
since the thickness of the other surfaces are likely less than 3 nm.

Although the formulas predict surface energy will decrease slightly with taper
slopes greater than 0.4, doing so is not recommended since numerical
simulations show that electric fields do not decrease in this range. Besides,
the surface energy only slightly decreases above a slope of 0.2.

An interesting question is how much more surface loss is there for thin films,
arising from the large fields at the edges. It is possible to compare surface
energy for a round coax and flat coax of the same width using Eqs. (12) and
(25), which shows that the ratio of the metal surface energy is

$\displaystyle\frac{U_{\textrm{f}}^{\textrm{m}}}{U_{\textrm{c}}^{\textrm{m}}}$
$\displaystyle\simeq\frac{\ln(4\overline{r}/t)+c_{m}}{\pi}\simeq 4.0\ $ (90)

for $\overline{r}=50\,\mu\textrm{m}$ and $t=0.1\,\mu\textrm{m}$, typical
dimensions considered here. Although the metal-film edges produce more loss,
the increase is still acceptable. Note that the logarithm factor is 7.6, so
that about 1/3 of the surface energy comes from the corners within $t/2$ of
the edges.

The MA and MS participation formulas in Eqs. (3)-(4) use surface energies
$U/2$ for the two sides of the film, which are then multiplied by dielectric
constant factors. However, since the metal film usually sits on top of the
substrate, this splitting of surface energy should change somewhat. One
expects the air side of the surface energy to include both sides of the top
corner and the outside of the bottom corner, while the substrate side only
includes the film edge of the bottom corner. Since these two sides of the
corner contributes similarly, one expects the constant factor added to the
logarithm to be about $1.5\,c_{m}$ for the air side and and $0.5\,c_{m}$ for
the substrate side. For example, this modification changes the MA prediction
of Table 1 from 0.060 to 0.077, closer to the numerical result 0.010.

Since the MS interface clearly dominates in the participation ratio, there has
been effort to minimize this oxide layer by surface treating the silicon wafer
before depositing the metal film [14]. The MS thickness and loss tangent are
thus parameters that should be measured carefully to optimize a design. The
qubit capacitor is much larger size than the junction and its wires are often
made in a separate step patterned with optical lithography, while the junction
and wire is patterned with electron-beam lithography. If the surface treatment
is easier or even possible with the optical lithography step, it is then
recommended that the taper is brought down to within $1\,\mu\textrm{m}$ or so
of the junction to minimize its loss. In this case the data in Fig. 9 would be
used to estimate the loss from both sections of wire; because of the
logarithmic dependence, there would still be some contribution from even the
short junction section.

##  X Two-level States

Surface loss typically comes from two-level states (TLS) [15], which saturate
and produce less loss at high excitation fields. Using the numerically
computed surface fields, the dependence on power can be found by scaling the
reduction in loss from the local electric field $E$ with

$\displaystyle E^{2}$ $\displaystyle\rightarrow
E^{2}/\sqrt{1+E^{2}/E_{s}^{2}}$ (91) $\displaystyle=E\,E_{s}\ \ \
\textrm{for}\ E\gg E_{s}\ ,$ (92)

where the saturation electric field $E_{s}$ depends on microscopic parameters
of the TLS [15].

Since saturation measurements are typically made with coplanar resonators,
numerical integration of the surface loss is shown in Fig. 10 for three values
of $a$, each with the gap equal to the inner metal width $b=2a$. As expected,
for large saturation fields (loss at low power) the largest resonator gives
lowest loss. At large fields, the loss of all three resonators converges. This
behavior can be understood using dimensional analysis: scaling all the lengths
by $D$ decrease the electric field by $E\sim 1/D$, but increases the surface
integration by $D$. For loss at low power, the integral scales as $E^{2}D\sim
1/D$. But when saturated, $EE_{s}D\sim E_{s}$ gives constant scaling. Figure
10 also shows volume saturation, for example coming from TLS in the substrate.

Figure 10: Saturation loss of a coplanar resonator. Top panel: Plot of surface
energy versus saturation electric field $E_{s}$, using surface loss scaled for
saturation according to Eq. (91). The single-ended coplanar resonators have
voltage $V=1\,$Volt and parameters $a=(2,10,50)\,\mu$m, $b=2a$ and
$t=0.1\,\mu$m. In the low power limit (high saturation $E_{s}$), the loss is
inversely proportional to $1/a$, whereas at high power the curves merge
together. The plus symbol $(+)$ is the characteristic crossover point, given
here by the single-ended prediction of the surface energy Eq. (55) and
$3E_{c}(0)$, where $E_{c}(0)$ is the center $x=0$ electric field of Eq. (54).
Bottom panel: Volume energy versus saturation field for the same geometric
parameters and colors. At low fields the volume energy is equivalent to the
capacitance per unit length for all curves, as expected. At left, the
characteristic saturation field scales inversely with metal dimension.

The analysis so far has treated the dissipation continuously. However, surface
loss comes from a bath of two-level states, with individual states that are
spectroscopically observable for small-area devices [15]. Simple models
predict both the magnitude and the density of TLS, so its spectrum can be
extremely useful for identifying the physical location of the loss.

The dipole moment of the TLS couples to the electric field of the 0 to 1 qubit
transition. This produces a qubit splitting with random frequency and
splitting size, but with a maximum splitting size $S_{\textrm{max}}$ given by
Eq. (3) of Ref. [15] that is proportional to the qubit electric field and
inversely proportional to the square-root of the qubit capacitance. For a
junction capacitor with parallel-plate separation 2 nm and a qubit capacitance
2 pF, a value $S_{\textrm{max}}=$ 74 MHz was measured.

For a transmon qubit with $C=$ 0.1 pF, the above scaling gives

$\displaystyle S_{\textrm{max}}=(330\,\textrm{MHz}\cdot 2\,\textrm{nm})\ E/V\
,$ (93)

where $E/V$ has a dimension of inverse distance and has been computed here for
the various surface electric fields. For the MS interface of a ribbon
capacitor, Fig. 11 shows a plot of $S_{\textrm{max}}$ versus the distance from
the inner corner $r_{c}$, which includes edge corrections at a distance less
than the half-thickness $t/2$. The size of the largest splittings are in the
few hundred kilohertz range.

Figure 11: TLS for ribbon capacitor. Plot of maximum splitting
$S_{\textrm{max}}$ versus distance from edge of ribbon capacitor $r_{c}$.
Black is ribbon solution of $\epsilon_{s}/\epsilon_{MS}$ multiplied by Eq.
(35) ($\sim r_{c}^{-1/2}$), and blue is edge scaling Eq. (20) ($\sim
r_{c}^{-1/3}$). The integrated area is shown on the top x-axis for $\ell=$1.4
mm and 2 MS electrodes, showing that the effective areas are typically much
greater than $1\,\mu\textrm{m}^{2}$ (arrow).

The number of splittings is proportional to the capacitor volume. Figure 2 of
Ref. [15] shows that the size of the splittings have a log-normal
distribution, so that the largest splittings are between $S_{\textrm{max}}/3$
and $S_{\textrm{max}}$ and have a density $0.5/\mu\textrm{m}^{2}$GHz. The
expected TLS density of the ribbon capacitor can be estimated by the effective
integrated area $A(S)$, obtained by multiplying $r_{c}$ by twice the length of
the ribbon 2.8 mm and a factor 3/2 to account for the thicker 3 nm thick
surface oxide. Since the observed splittings are dominated by those close to
$S_{\textrm{max}}$, the splitting density $\rho_{S}$ between splittings
$S_{1}$ and $S_{2}$ is thus approximately given by

$\displaystyle\rho_{S}\simeq(0.5/\mu\textrm{m}^{2}\textrm{GHz})[A(S_{2})-A(S_{1})]\
.$ (94)

If one assumes the qubit splitting measurements spans a 2 GHz frequency range,
then the first observable splitting should occur on average for an integrated
area $A=1\,\mu\textrm{m}^{2}$. Using Fig. 11, the largest splittings at 3 nm
should have size 300 kHz, with an average spacing of about one per 200 MHz in
the qubit frequency.

For a parallel plate capacitor, the effective distance for the electric field
is the separation multiplied by $\epsilon_{\textrm{MA}}=9.8$. For the example
of Table 2, this gives $49\,\mu\textrm{m}$. One finds a splitting size of 13
kHz, and an effective area of 1.5 times the capacitor area.

Junction wire results are shown in Fig. 12 for the untapered and tapered
cases. These plots were obtained by numerically breaking up the wire into
about 100k sections, then computing $S_{\textrm{max}}$ and the differential
area $dA$ for each section. The curve is obtained by sorting
$S_{\textrm{max}}$ from large to small, and then cumulative summing over the
corresponding $dA$ to obtain the integrated area $A$. The tapered case shows
lower splittings $S_{\textrm{max}}$, consistent with the continuum theory. The
TLS become statistically observable for $A>1\,\mu\textrm{m}^{2}$, which
predicts splittings in the several MHz range. The dependence on $d$ shows that
the dominant contribution to the TLS are for distances greater than about
$10\,\mu\textrm{m}$; shorter distances are unimportant because they have small
areas. The dominance of an intermediate length scale is perhaps a surprising
result, and shows why detailed theory is needed to optimize the wire design.

Figure 12: TLS for junction wire. Plot of $S_{\textrm{max}}$ versus integrated
area for untapered ($S=0$) and tapered ($S=0.2$) wires. The electric fields
are obtained from Eqs. (16) and (70), and numerically breaking up the wire
into small area sections. The tapered wire shows lower maximum splitting,
consistent with the continuum results. For the untapered wire, the observable
areas $A>1\,\mu\textrm{m}^{2}$ (arrow) has $S_{\textrm{max}}$ in the 1-4 MHz
range, whereas for tapered it is below 1.5 MHz. Note that the dominant
contribution comes for wire surfaces at a distance greater than about
$10\,\mu\textrm{m}$.

This result suggests that undercutting the junction wires into the substrate
can be an effective solution to decrease the contribution of the metal-
substrate interface. Since there is little contribution at small distances, it
is not essential to undercut around the junction, which should improve the
reliability of the fabrication and the stability of the junctions.

##  XI Summary

Calculation of participation ratios and surface loss is challenging because of
the divergence of the electric fields at metal edges. Previously, these fields
were solved in the infinitely thin limit using solutions from conformal
mapping. Here, the solutions were extended to the useful limit where the thin
surface oxide (few nm) is less than the metal film thickness ($0.1\,\mu$m),
and less than the typical film size ($100\,\mu$m). The finite thickness
condition was solved via a calculation that matched the conformal fields to
edge fields, then checked and refined with numerical simulation. Going
forward, these formulas are also useful when checking numerical simulations
for systematic errors due to meshing.

Formulas are given for common capacitor structures. By separating out the
geometery of actual designs, participation ratios can be calculated accurately
and then used to optimize the design. This is an important check on numerical
calculations since misleading results can come from finite meshing when
structures range in size from nanometers to millimeters.

For junction wires, a solution for the capacitance and surface loss was
obtained using well formed models, approximations and numerics, which should
give accurate and reliable formulas. A tapered junction wire was shown to have
superior performance compared to straight wires when the wire length is longer
than about $\sim 10\,\mu$m. This design feature is important for the latest
generation of devices that use large capacitor size to lower surface loss. A
further design improvement for the taper was suggested.

These electric field solutions enable a prediction of the TLS spectrum, which
could be invaluable to identify where the TLS comes from in the qubit design.

Finally, it is hoped that these results will encourage researchers to
precisely test surface loss theory, and measure in additional experiments the
various surface loss parameters. By doing so, this should speed the
optimization and development of long coherence time qubits.

*

##  Appendix A Appendix
Numerical Calculation of Surface Electric Fields

For thin films suitable for superconducting qubits, it is useful to
numerically calculate the electric fields, for example for a thin film of
finite thickness. Fortunately, realistic transmon designs are well
approximated by simple geometries with fields that can be well described using
simple fitting functions, so that they can be physically understood and
optimized. Two-dimensional geometries are particularly amenable to efficient
numerical solutions and thus their method of solution will be described here
first. For simplicity, the calculations here will assume a constant dielectric
constant $\epsilon$. Corrections due to the substrate and vacuum are included
in the main text.

Figure 3 shows geometries to be considered here. The first is 2-dimensional,
with solutions given per unit length in the third dimension, with results
typically scaled with length $\ell$. The second solution uses cylindrical
symmetry to turn a 3-dimensional problem to 1-D. The last uses an
approximation to the edge fields so that a thin film wire can be similarly
calculated in 1-D.

Numerical solutions can be obtained through inverting a matrix. For a 2-D
geometry with translational invariance in the z direction, the problem is
first broken into an vector of points in the x-y plane that have line charge
$\vec{q}$. The voltage $\vec{V}$ can be solved with the matrix equation
$\vec{V}=\bar{M}\vec{q}$, where the potential matrix $\bar{M}$ has elements

$\displaystyle M_{ij}$ $\displaystyle=\frac{1}{2\pi\epsilon}\ln(1/\rho_{ij})\
,$ (95)

where $\rho_{ij}$ is the distance between point $i$ and $j$. For metal
electrodes the voltage is set instead, so the charge can be obtained using
$\bar{q}=\bar{M}^{-1}\vec{V}$, where the inverse matrix $\bar{M}^{-1}$ can be
thought of as a capacitance matrix. The time to solution grows as the cube of
the number of points, which can be solved quickly for size 1k - 10k.

For a 3-D geometry with cylindrical symmetry, the potential matrix can be
solved for a circular ring of total charge $q$, giving a potential matrix with
elements

$\displaystyle M_{ij}$
$\displaystyle=\frac{1}{2\pi^{2}\epsilon}\frac{\textrm{ellipk}(-4r_{i}r_{j}/\rho_{ij}^{2})}{\rho_{ij}}\
,$ (96) $\displaystyle=\frac{1}{4\pi\epsilon\rho_{ij}}\ \ \textrm{for}\ \
\rho_{ij}\gg r_{i}r_{j}\ ,$ (97)

where $\rho_{ij}$ is the distance between points in the r-z plane, and $r_{i}$
and $r_{j}$ are the radial components. The Python ellipk($m$) function is
equivalent to $K(k)$ but with $m=k^{2}$, and allows negative $m$ arguments.

For a similar 3-D geometry of a flat coax, the potential matrix for the inner
conductor is

$\displaystyle M_{ij}$
$\displaystyle=\int_{-\overline{r}_{j}}^{\overline{r}_{j}}\frac{1/\pi}{\sqrt{\overline{r}_{j}^{2}-x^{2}}}\frac{1/4\pi\epsilon}{\sqrt{x^{2}+y_{ij}^{2}}}\,dx$
(98) $\displaystyle=\frac{1}{2\pi^{2}\epsilon\
}\,\frac{\textrm{ellipk}(-\overline{r}_{j}^{2}/y_{ij}^{2})}{y_{ij}}\ ,$ (99)
$\displaystyle=\frac{1}{4\pi\epsilon\,y_{ij}}\ \ \textrm{for}\ \
y_{ij}\gg\overline{r}_{j}\ ,$ (100)

where $y_{ij}$ is the distance between points $i$ and $j$ on the centerline of
the wire. In comparison with the matrix for the cylinder geometry Eq. (96),
the difference is the absence of the factor of 4 in the ellipk argument.

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