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Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 20
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 21590)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 20
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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\section{Introduction}
The precise segmentation of blood vessel plays an important role in the diagnosis of related diseases. For example, the morphological changes of retinal vessel may indicate some relevant diseases (eg. glaucoma, diabetes and hypertension), and the quantitative analysis of coronary arteries in digital subtraction angiography (DSA) is commonly used in the assessment of myocardial infarction and coronary atherosclerotic disease. Moreover, some pathophysiological changes of retinal vessel caused by prolonged hyperglycemia are related to the smallest vessels \cite{archer1999diabetic}, so it is vital to extract the thin vessels. In the past decades, many automatic approaches have been proposed to segment the blood vessel, such as tracking based models \cite{chutatape1998retinal}, filtering based models \cite{chaudhuri1989detection} and deformable models \cite{dizdaro2012level}.
Recently, deep learning based methods have significantly improved the performance of vessel segmentation. For example, Hu \textit{et al.} \cite{hu2018retinal} presented a multi-scale convolutional neural network (CNN) with an improved cross-entropy loss for vessel segmentation, outperforming the traditional unsupervised algorithms. However, although CNN has tremendous power to extract high-level representative features, it is inevitable that spatial information is partly lost in the downsampling layers. Further, the lost information mainly belongs to the high-frequency components in the image, such as object boundary which contains essential cues for localization. This is consistent with the observation in \cite{zhang2018deep}, where they found that most mispredicted pixels by UNet are located in the edges of blood vessels. Therefore, it is very necessary to strengthen the network capability to capture edge information in vessel segmentation.
There have been presented a variety of related methods, which can be roughly divided into four categories: 1) Many works reused the low-level features with rich spatial information by a ``skip connection" structure to maintain more edge details, such as SegNet \cite{badrinarayanan2017segnet}, UNet \cite{ronneberger2015u} and DeeplabV3+ \cite{chen2018encoder}; 2) Some approaches employed post-processing techniques (eg. conditional random fields (CRFs) \cite{chen2017deeplab}, active contour models \cite{sun2018extracting}) to refine the segmentation results; 3) Some researchers exploited an additional branch for edge detection to enhance the segmentation task, which can be regarded as a multi-task framework \cite{hatamizadeh2019end,qin2019transfer,zhu2019boundary}; 4) Some studies proposed a boundary-aware loss function \cite{zhen2019learning} or labeled the object contours as an independent class \cite{zhang2018deep}, thus paying more attention to the border pixels.
In this paper, we provide a novel solution to boost the accurate delineation of borders in blood vessel segmentation, which serves as an insertable module called Boundary Enhancement and Feature Denoising (BEFD). By applying the traditional edge detection operator to the raw image, BEFD can gain an edge attention map in an unsupervised way. This pixel-wise attention map assigns higher weights to the pixels around the object boundary, which would be multiplied (element-wise product) by certain intermediate feature maps in the encoder part. In this way, the prior knowledge about vascular edges are transferred to the neural networks, enhancing the boundary information in semantic segmentation. In addition, in order to avoid amplifying noise at the same time, BEFD also conducts feature denoising in the process of skip connection via a denoising block.
We evaluate the proposed BEFD module on both retinal vessel images and angiocarpy images. The experimental results demonstrate the effectiveness of the BEFD module. Compared to existing state-of-the-art models for vessel segmentation, UNet integrated with BEFD module achieves the highest score in accuracy, AUC, sensitivity and specificity on DRIVE dataset.
The contributions of this work are summarized as follows:\\
1) We propose an innovative module (BEFD) to boost boundary feature extraction in semantic segmentation, which can be easily incorporated into any encoder-decoder framework in an end-to-end manner. Moreover, the proposed BEFD module applies the unsupervised edge detector to provide edge prior knowledge for CNNs in medical image segmentation.\\
2) We use the denoising block to eliminate the noise existing in the low-level feature maps, preserving the indeed required spatial information for segmentation.\\
3) Our approach obtains remarkable performance for blood vessel segmentation, actually, it can also be easily generalized to deal with other segmentation tasks.
\section{Method}
\begin{figure}[t]
\centering
\includegraphics[width =\textwidth]{./FIGURE/network.pdf}
\caption{Architecture of BEFD-UNet. It consists of three parts: 1) the basic UNet; 2) the boundary enhancement (BE) part in the gray triangle, which employs an edge detector to provide boundary localization for the encoder path of UNet; 3) the feature denoising (FD) part, formed by three denoising blocks in the phase of skip connection.}
\label{fig:network}
\end{figure}
\subsection{The architecture of BEFD-UNet}
In this section, we choose U-Net as baseline, to elaborate how the BEFD module can fit into it. The incorporated model is named as BEFD-UNet, and its structure is illustrated in Fig.~\ref{fig:network}. The baseline UNet consists of an encoder path and a decoder path. In the encoder path, each step includes two $3 \times 3$ convolutions followed by a $2 \times 2$ max pooling with stride of 2 for downsampling, which doubles the number of feature map channels. In the decoder path, each layer contains a deconvolution for upsampling followed by two $3 \times 3$ convolutions, where the channel number is halved. Additionally, the skip connection between encoder and decoder combines the low-level features in the shallow layers with the high-level abstract features, preserving more spatial information for better localization.
As for the BEFD module, it is composed of two parts: the Boundary Enhancement (BE) part and Feature Denoising (FD) part. In the BE part, edge detection is firstly conducted on the raw image to get a pixel-wise edge attention map, whose values indicate the importance of the corresponding pixels, in this setting, edge pixels are assigned with greater weights. Then, this attention map is incorporated into the last three layers in the encoder path via element-multiplication, before which the attention map has been resized to be consistent with the corresponding feature maps by bilinear interpolation. Benefiting from the edge prior produced by the unsupervised edge detection, the boundary information is enhanced with little extra computation.
In the FD part, the skip connection structure is modified by adding a denoising block before the simple concatenation. The low-level features in the encoder path contain not only rich spatial details, but also background noise which is undesirable for the decoder path, so it is necessary to make feature denoising in the process of skip connection. On the other hand, edge detector may misidentify image noise as boundary, thus enhancing the unexpected noise simultaneously, in this view, feature denoising is also indispensable. Furthermore, the denoising block \cite{xie2019feature} is a typical method to restrain the noise in the intermediate feature maps, so we apply it to the low-level features. In addition, more details about the edge detection and feature denoising are described in the next sections.
\subsection{Unsupervised edge detection}
Edge detection aims to locate the object boundaries in the image, which is an important step towards extracting image features. In this paper, we utilize Sobel edge detector \cite{kittler1983accuracy} to obtain this goal. Sobel edge detector calculates the first derivatives of the raw image for the horizontal direction and vertical direction separately, then combines these two components together via the absolute magnitude of gradient. It can be expressed as follow:
\begin{equation}
G_{x}=\left[\begin{array}{lll}
-1 & 0 & 1 \\
-2 & 0 & 2 \\
-1 & 0 & 1
\end{array}\right] * I,
G_{y}=\left[\begin{array}{ccc}
-1 & -2 & -1 \\
0 & 0 & 0 \\
1 & 2 & 1
\end{array}\right] * I,
G=\sqrt{G_{x}^{2}+G_{y}^{2}}
\end{equation}
where $I$ is the raw image, $*$ denotes the convolution operator, $G_x,G_y$ are the gradient components of $x$ axis and $y$ axis respectively, and $G$ denotes the Sobel edge map. In order to obtain the final edge attention map which emphasizes the boundary pixels in a suitable and adjustable way, we apply the thresholding method as well as linear transformation to the Sobel edge map as follows:
\begin{equation}\begin{aligned}
G_{final}(x,y) &=\left\{\begin{array}{ll}
{1,} & {\text { if } G(x,y) > \lambda_{max} \text { or } < \lambda_{min}} \\
{(1-\frac{G(x,y)-\lambda_{min}}{\lambda_{max}-\lambda_{min}})\cdot \alpha + \beta,} & {\text {otherwise}}
\end{array}\right.
\end{aligned}\end{equation}
Here $G(x,y)$ denotes the Sobel gradient value at pixel $(x,y)$, and $\lambda_{min},\lambda_{max},\alpha,\beta$ are tunable parameters to regulate the scope of weights. Note that in the setting of $1-\frac{G(x,y)-\lambda_{min}}{\lambda_{max}-\lambda_{min}}$, weak edges are emphasized with higher weights than the strong ones, as the key challenge in vessel segmentation is the detection of micro vessels with low contrast.
\subsection{Feature denoising}
\begin{figure}[!t]
\centering
\includegraphics[width =0.5\textwidth]{./FIGURE/nonlocal.pdf}
\caption{Architecture of the feature denoising block \cite{xie2019feature}.}
\label{fig:nonlocal}
\end{figure}
The concept of feature denoising \cite{xie2019feature} is proposed to improve adversarial robustness in image classification, based on the observation that adversarial perturbations on images result in noise in the features. At the same time, segmentation networks always make efforts to maintain more high-frequency components for sake of restoring more details (eg. skip connection in UNet \cite{ronneberger2015u}), providing an opportunity for the introduction of noise in the feature maps. Hence, it is also necessary to conduct feature denoising in image segmentation as we do in this work.
Non-local means \cite{buades2005non} is widely used for the task of image denoising, which calculates a weighted mean of all locations in the image. It is defined as:
\begin{equation}
\mathbf{y}_{i}=\frac{1}{\mathcal{C}(\mathbf{x})} \sum_{\forall j} f\left(\mathbf{x}_{i}, \mathbf{x}_{j}\right) \cdot \mathbf{x}_{j},
\end{equation}
where $\mathbf{y}_{i}$ is the output value at location $i$, $\mathbf{x}_{i}$ is the input value at location $i$, and $j$ denotes all possible positions. $f$ is a weighting function and $\mathcal{C}(\mathbf{x})$ is a normalization factor. Following the idea of non-local means \cite{buades2005non} and non-local neural networks \cite{wang2018non}, the work \cite{xie2019feature} presented a denoising block to make feature denoising. As shown in Fig.2, except for the non-local means part, the denoising block also contains a $1\times 1$ convolution and residual connection for feature fusion. In this work, we use this denoising block to eliminate the feature noise in segmentation networks, where the pairwise function $f$ is set to dot product $f\left(\mathbf{x}_{i},\mathbf{x}_{j}\right)=\mathbf{x}_{i}^{\mathrm{T}} \mathbf{x}_{j}$ and $\mathcal{C}(x)=N$ ($N$ is the number of locations in $\mathbf{x}$). More information about the denoising block can be found in \cite{xie2019feature,wang2018non}.
\section{Experiments}
To evaluate the effectiveness of the BEFD module, we compare two models: 1) the baseline UNet; 2) BEFD-UNet, which integrates BEFD module into UNet.\\
\\
\textbf{Data Description.}
We evaluate the new approach on two datasets of blood vessel: DRIVE and HEART. The DRIVE dataset is a public resource including 40 fundus images and corresponding labels of retinal vessel (the first manual annotation). The resolution of each image is $565\times 584$, and the whole set has originally been divided into two parts: 20 images for training and the rest 20 images for testing. Apart from the fundus images, we also collect 1092 digital subtraction angiographies (DSA) of coronary (546 for train, 218 for valid and 328 for test). All images are resized to the same dimension $256\times 256$, and this dataset is named as HEART. Moreover, both datasets are preprocessed by Contrast Limited Adaptive Histogram Equalization (CLAHE) and normalization. To avoid overfitting, data augmentation is used by horizontal and vertical flip.\\
\\
\textbf{Evaluation Metrics.}
To quantify the performance of our approach, we use several metrics consisting of sensitivity (Sen), specificity (Spe), accuracy (Acc), and F1-score (F1), the formulas are shown below:
\begin{equation}\begin{aligned}
&Acc=\frac{TP+TN}{TP+TN+FP+FN},Sen=\frac{TP}{TP+FN},\\
&Spe=\frac{TN}{TN+FP},F1=\frac{2TP}{2TP+FP+FN},
\end{aligned}\end{equation}
where TP,TN,FP,FN denote the numbers of true positives, true negatives, false positives and false negatives respectively. We also calculate the AUC metric (the area under the ROC curve) for evaluation. Note that these metrics are calculated over the whole image. In addition, the DRIVE dataset provides masks for labels, and the performance evaluated within the masks is shown in Table 1 of the supplementary material. \\
\\
\begin{table}[!b]
\begin{center}
\caption{Quantitative analysis of different methods on DRIVE dataset.}
\begin{tabular*}{0.70\linewidth}{c|c|ccccc}
\hline
Methods & Year & Sen & Spe & F1 & Acc & Auc \\
\hline
Hu\cite{hu2018retinal} & 2018 & 0.7772 & 0.9793 & - & 0.9533 & 0.9759 \\
Zhuang\cite{zhuang2018laddernet} & 2018 & 0.7856 & 0.9810 & 0.8202 & 0.9561 & 0.9793 \\
Alom\cite{alom2019recurrent} & 2019 & 0.7792 & 0.9813 & 0.8171 & 0.9556 & 0.9784 \\
Jin\cite{jin2019dunet} & 2019 & 0.7963 & 0.9800 & 0.8237 & 0.9566 & 0.9802 \\
Mou\cite{mou2019dense} & 2019 & 0.8126 & 0.9788 & - & 0.9594 & 0.9796 \\
Wu\cite{wu2019vessel} & 2019 & 0.8038 & 0.9802 & - & 0.9578 & 0.9821 \\
Lyu\cite{lyu2019fundus} & 2019 & 0.7940 & 0.9820 & - & 0.9579 & 0.9826 \\
Wang\cite{wang2019dual}& 2019 & 0.7940 & 0.9816 & \textbf{0.8270} & 0.9567 & 0.9772 \\
Zhou\cite{zhou2019symmetric}& 2019 & 0.8135 & 0.9768 & 0.8249 & 0.9560 & 0.9739 \\
Li\cite{li2020iternet}& 2020 & 0.7791 & 0.9831 & 0.8218 & 0.9574 & 0.9813 \\
\hline
UNet & 2020 & 0.7887 & \textbf{0.9861} & 0.8140 & 0.9686 & 0.9836\\
\textbf{BEFD-UNet} & 2020 & \textbf{0.8215} & 0.9845 & 0.8267 & \textbf{0.9701} & \textbf{0.9867} \\
\hline
\end{tabular*}
\label{tab:table1}
\end{center}
\end{table}
\begin{table}[!b]
\begin{center}
\caption{Quantitative analysis of different methods on HEART dataset.}
\begin{tabular*}{0.64\linewidth}{c|ccccc}
\hline
Methods & Sen & Spe & F1 & Acc & Auc \\
\hline
UNet & 0.9186 & 0.9839 & 0.9073 & 0.9750 & 0.9938\\
BE-UNet & \textbf{0.9411} & 0.9779 & 0.9030 & 0.9730 & 0.9935\\
FD-UNet & 0.9234 & \textbf{0.9849} & 0.9140 & 0.9767 & 0.9939\\
\textbf{BEFD-UNet} & 0.9333 & 0.9835 & \textbf{0.9141} & \textbf{0.9767} & \textbf{0.9947} \\
\hline
\end{tabular*}
\label{tab:table2}
\end{center}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width =1\textwidth]{./FIGURE/compare.pdf}
\caption{Segmentation results of different models. The top two rows: prediction maps on DRIVE dataset; The bottom row: prediction maps on HEART dataset.}
\label{fig:compare}
\end{figure}
\textbf{Implementation Details.}
All networks are implemented in TensorFlow 1.2.1
(https://www.tensorflow.org) on a single NVIDIA GTX1080ti GPU. We use the Adam algorithm to minimize the cross entropy loss, which is trained for 30k iterations with batch size of 8. In order to accelerate convergence, batch normalization is followed by each convolutional layers. In the UNet architecture, we set the channel number in the first layer to 64. The parameters $\lambda_{min},\lambda_{max},\alpha,\beta$ in edge detection are set to 0.8, 5.0, 2.0 and 1.0 respectively.
\section{Results}
\textbf{Evaluation of the performance of the BEFD module.}
Table~\ref{tab:table1},~\ref{tab:table2} report the qualitative results of retinal vessel segmentation on DRIVE dataset and cardiac vessel segmentation on HEART dataset respectively. Compared to the baseline UNet, BEFD-UNet obtains better performance in four of the total five metrics with a considerable margin on both datasets, only Spe (specificity) is a bit lower. Moreover, BEFD-UNet provides a significant improvement in Sen (sensitivity) from 0.7887 (0.9186) to 0.8215 (0.9333) on DRIVE dataset (HEART dataset), indicating that our approach yields a lower false negative (FN) ratio. This can also be demonstrated by the segmentation results shown in Fig.~\ref{fig:compare}, where BEFD-UNet successfully captures the extremely thin retina vessels and identifies the coronary arteries with low contrast to background. Besides the evaluation of the whole image, we also evaluate the models on small vessels (with a width of 1 or 2 pixels) only. In such a setting, BEFD-UNet makes a significant improvement in F1 score from 0.7163 (UNet) to 0.7628 (BEFD-UNet) on DRIVE dataset, indicating that our method has strong ability to capture the thin vessels.
On the other hand, we also compare the results with recent state-of-the-art models on DRIVE dataset as listed in Table~\ref{tab:table1}. It shows that the proposed BEFD-UNet ranks first in four metrics except for F1 score. More specifically, it achieves the highest accuracy (1.05\% higher than the second best \cite{mou2019dense}), the optimal AUC (0.41\% higher than the suboptimal result \cite{lyu2019fundus}) as well as the best sensitivity (0.80\% higher than the previous highest score \cite{zhou2019symmetric}). Additionally, it is worth mentioning that the F1 score (0.8267) of our method is relatively close to the best score (0.8270) \cite{wang2019dual}. The above comparisons demonstrate the strong capacity of the new BEFD module to tackle semantic segmentation.\\
\\
\textbf{Discussion about the mechanism of the BEFD module.}
\begin{figure}[!t]
\centering
\includegraphics[width =0.8\textwidth]{./FIGURE/edgemap.pdf}
\caption{Example of edge attention map on DRIVE dataset (left) and HEART dataset (right). Boundary pixels (mostly white) have higher values than background pixels (blue), while noise (red points) is widely distributed on the map, requiring feature denoising to restrain it.}
\label{fig:edgemap}
\end{figure}
To explore the effect of the BEFD module, we show an example of edge attention map on both datasets in Fig.~\ref{fig:edgemap}. It can be observed that boundary pixels have higher values, making it possible to pay more attention to the object contours. When the feature maps in the encoder path are multiplied (element-wise) by such attention map, the object boundaries are enhanced accordingly. At the same time, maps contain lots of noise due to the emphasis on weak vessels, for which it requires introducing the feature denoising block. In addition, we also conduct experiments using the boundary enhancement (BE) part or feature denoising (FD) part independently, which are shown in Table~\ref{tab:table2}. The
results indicate that using either BE or FD part would produce an unbalanced result. BE-UNet tends to generate high Sensitivity with low Specificity, because the BE part may misidentify image noise as boundaries, thus amplifying the undesirable noise. On the other hand, FD-UNet performs oppositely (high Specificity with low Sensitivity), since the FD block may make excessive denoising, which eliminates some fine structures. Therefore, we apply both BE and FD blocks jointly in BEFD-UNet, which leverages the advantages of both the BE and FD blocks.
\section{Conclusion}
In this paper, we propose a novel BEFD module to boost the boundary localization in the encoder-decoder framework for blood vessel segmentation. The integrated BEFD-UNet outperforms the baseline UNet as well as most of state-of-the-art approaches, resulting from its powerful ability to detect extremely tiny vessels. More broadly, the BEFD module provides a novel solution to leverage the advantage of traditional image processing algorithm, which can compensate for the defects of CNNs in an unsupervised way. This mechanism is worth investigating further in the future work.
\subsubsection*{Acknowledgments.} This work was supported by Natural Science Foundation of China (NSFC) under Grants 81801778, 71704024, 11831002; National Key R\&D Program of China (No. 2018YFC0910700); Beijing Natural Science Foundation (Z180001).
\bibliographystyle{splncs04}
|
{
"timestamp": "2021-04-09T02:18:22",
"yymm": "2104",
"arxiv_id": "2104.03768",
"language": "en",
"url": "https://arxiv.org/abs/2104.03768"
}
|
\section{Introduction}
\subsection{Motivation}
Color codes are interesting from the point of view of fault tolerant quantum computation.
In two dimensions they allow transversal implementation of Clifford gates \cite{Bombin2006}.
There are various ways of doing encoded computation using 2D color codes.
One of the early approaches was due to Fowler \cite{Fowler2011}.
In this information was encoded using holes and gates realized using code deformation.
Independently, Landahl et al.~\cite{Landahl2011} also proposed alternate protocols for fault tolerant quantum computation with color codes.
They also used holes for encoding information,
and code deformation for implementing encoded gates.
Lattice surgery~\cite{Horsman2012,Landahl2014, Litinski2019} is an alternative to quantum computation with holes.
Lattice surgery was used to show universal computation in color codes and the analysis of the resources required for the same was studied by Landahl and Ryan-Anderson~\cite{Landahl2014}.
Twists are yet another method to encode information into a 2D lattice.
Furthermore, they can also be used to perform encoded gates.
Twists are defects in the lattice that permute the label of anyons.
Loosely speaking, an anyon is the syndrome resulting from a violated stabilizer.
Anyons in color codes have far richer structure compared to surface codes thus allowing for more permutations.
Anyons in color codes are characterized by two labels, namely, charge and color.
Twists in color code can permute charge, color or both.
Kesselring et al. \cite {Kesselring2018} classified the various types of twists possible in color codes.
However, certain aspects of color codes with twists were left unexplored.
This motivates us to undertake a study of twists in color codes and their application to fault-tolerant quantum computation.
Another reason for our study comes from the fact that twists can potentially lead to lower complexity for quantum computation.
For instance, twists in surface codes are shown to provide gains in space time complexity of computation~\cite{Hastings2015}.
Braiding in surface codes needs both lattice modification as well as stabilizer modification.
A particular type of twist in color codes, viz. charge permuting twist, requires no lattice modification for movement.
Only the stabilizers of the twist face and the faces in the path of the twist movement need to be changed.
This can be accomplished without lattice modification for charge permuting twists.
Therefore, braiding procedure is simpler in comparison to surface codes with twists.
\subsection{Previous work and contributions}
The use of dislocations for encoding was suggested by Kitaev~\cite{Kitaev2003}.
The ends of a dislocation are called twists.
Twists in toric code model were first studied by Bombin~\cite{Bombin2010} where twists were shown to behave like non-Abelian anyons.
Twists were studied from the perspective of unitary tensor categories by Kitaev and Kong~\cite{Kitaev2012}.
Yu and Wen~\cite{Yu-Wen2012} studied twists in qudit systems where the authors showed that twists exhibit projective non-Abelian statistics.
Hastings and Geller~\cite{Hastings2015} showed that by using twists in surface code along with arbitrary state injection leads to reduction in amortized time overhead.
Twists in surface codes were identified with corners of lattice by Brown et al.~\cite{Brown2017} which helps in performing single qubit Clifford gates.
Further, the authors also proposed a hybrid encoding scheme with holes and twists in surface codes that was made use of to implement CNOT gate.
Surface codes with twists in odd prime dimension and braiding protocols to implement generalized Clifford gates were studied in Ref.~\cite{GowdaSarvepalli2020}.
Twists have also been studied in other class of topological codes.
Bombin~\cite{Bombin2011} studied twists in topological subsystem color codes and showed that Clifford gates can be implemented by braiding twists.
Litinski and von Oppen~\cite{Litinski2018} studied twists in Majorana surface codes and have shown that all logical Clifford gates can be done with zero time overhead.
Kesselring et al.~\cite{Kesselring2018} have cataloged the twists in color codes.
They have presented the basic twists with which all twists in color codes can be realized.
In this paper we undertake a systematic study of color codes with twists from the perspective of coding theory with a view to quantum computation.
Our contributions are listed below.
\begin{compactenum}[(i)]
\item Systematic construction of color codes with charge permuting twists and color permuting twists from a $2$-colex.
\item Implementation of Clifford gates by braiding charge permuting twists.
\item Implementation of Clifford gates by Pauli frame update and by braiding holes around color permuting twists.
\item For the charge permuting twists and the color permuting twists, we also give implementations of a non-Clifford gate by using magic state distillation.
Our protocol is a variation of the one presented in Ref.~\cite{Bravyi2006}.
\end{compactenum}
Our treatment of the color codes with twists makes extensive use of mappings between Pauli operators and strings on the code lattices.
This approach brings out the geometry of the stabilizer generators and the logical operators in a very transparent fashion.
Furthermore, it allows for a consistent and simplified analysis of the encoded gates using these codes.
\subsection{Overview}
In Section~\ref{sec:background}, we briefly discuss color codes and review the work presented in Ref.~\cite{Kesselring2018}.
In Section~\ref{sec:charge}, we study charge permuting twists.
while Section~\ref{sec:color} is devoted to color permuting twists.
We discuss code parameters, logical operators and string representation for Pauli operators in the presence of color permuting twists.
In Section~\ref{sec:gates-charge-permuting}, we show that encoded Clifford gates can be realized by braiding charge permuting twists.
In Section~\ref{sec:gates-color-permuting}, we show that Clifford gates can be realized using charge permuting twists by Pauli frame update and joint parity measurements.
\noindent \emph{Notation.}
The set of faces with color $c \in \{r,g,b \}$ is denoted by $F_c$.
We use the notation $A_2(f)$ to denote the set of faces that share an edge with the face $f$.
The color of a face $f$ is denoted by $c(f)$.
The vertices of a face is denoted by $V(f)$.
The set of faces of color $c$ is denoted by $\mathsf{F}_c$.
The set of faces with colors $c$ and $c^\prime$ is denoted by $\mathsf{F}_{c c^\prime}$.
The string of color $c$ encircling twists $t_i$ and $t_j$ is indicated by $\mathcal{W}_{i,j}^{c}$.
After braiding twists $t_i$ and $t_j$, the string encircling them is indicated as $\mathcal{W}_{i,j}^{c}$.
However, a string encircling twists $t_i$ and other twist $t_k$ after braiding is denoted as $\mathcal{W}_{j,k}^c$ i.e after braiding $\mathcal{W}_{i,k}^c \mapsto \mathcal{W}_{j,k}^c$.
\section{Background}
\label{sec:background}
Color codes are defined on trivalent and three-face-colorable lattices (also called $2$-colexes).
A well known example of trivalent and three colorable lattice is the honeycomb lattice.
Two stabilizers are defined on every face of the lattice:
\begin{equation}
B_f^{X} = \prod_{v \in V(f)}X_v, \text{ and } B_f^{Z} = \prod_{v \in V(f)} Z_v.
\label{eqn:cc-stabilizers}
\end{equation}
The stabilizers defined in Equation~\eqref{eqn:cc-stabilizers} apply to faces of all colors.
However, not all stabilizers are independent.
They satisfy the relation~\cite{Bombin2006},
\begin{subequations}
\begin{eqnarray}
\prod_{f \in \mathsf{F}_r} B_f^X &=& \prod_{f \in \mathsf{F}_g} B_f^X = \prod_{f \in \mathsf{F}_b} B_f^X, \\
\prod_{f \in \mathsf{F}_r} B_f^Z &=& \prod_{f \in \mathsf{F}_g} B_f^Z = \prod_{f \in \mathsf{F}_b} B_f^Z.
\end{eqnarray}
\label{eqn:cc-stab-constraint}
\end{subequations}
Equations~\eqref{eqn:cc-stab-constraint} indicate that there are four dependent stabilizers.
The number of logical qubits depends on the topology of surface on which graph is embedded.
If the surface has genus $g$, then the number of encoded qubits is $4g$.
Of particular interest to us in this paper is the case $g = 0$.
Examples of such lattices are shown in Fig.~\ref{fig:lattice-boundary}.
Two stabilizers, $X$ type and $Z$ type, are defined for each face of the lattice.
All vertices in the lattice are trivalent and hence we have $3v = 2e$, where $e$ and $v$ are the number of edges and vertices of the lattice respectively.
Using this in the Euler characteristic equation $v + f - e = 2$, where $f$ is the number of faces, we get $2f = v + 4$.
It is easy to verify that this lattice also obeys the constraints given in Equations~\eqref{eqn:cc-stab-constraint}.
Therefore, the number of independent stabilizers is $s = 2f-4 = v$, thus giving zero encoded qubits.
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1.25]{fig-sq-octagon-lattice.pdf}
\subcaption{Square octagon lattice that does not encode any logical qubits. The outer unbounded face has blue color.}
\label{fig:sq-octagon-lattice}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = .5]{fig-hexagon-lattice-boundary.pdf}
\subcaption{Hexagon lattice that does not encode any logical qubits. The outer unbounded face has blue color. Stabilizers along the boundary are of weight two.}
\label{fig:hexagon-lattice-boundary}
\end{subfigure}
\caption{Trivalent and three colorable lattices with boundary.}
\label{fig:lattice-boundary}
\end{figure}
\subsection{Previous work on twists in color codes}
We now review some relevant material from Ref.~\cite{Kesselring2018}.
The syndromes in color codes can be written as $cp$ where $c \in \{r,g,b\}$ denotes the color of face on which the syndrome is present and $p \in \{x,y,z\}$ denotes the type of stabilizer violated.
If a $Y$ error occurs, then both $X$ and $Z$ stabilizers are violated which is indicated as $cy$.
The syndromes present in the color code are given in Table~\ref{tab:bosons}.
\begin{table}[htb]
\centering
\begin{tabular}{c|c|c}
\hline
$rx$ & $gx$ & $bx$ \\ \hline
$ry$ & $gy$ & $by$ \\ \hline
$rz$ & $gz$ & $bz$\\\hline
\end{tabular}
\caption{Types of syndromes present in color codes.}
\label{tab:bosons}
\end{table}
Twists are defects in the lattice that permute a label of an anyon to another i.e. they permute anyons when anyons go around twist (more precisely, when they cross the domain wall).
Domain wall is a virtual path in the lattice that marks the point where anyons are permuted.
In color codes with twists, there is more than one way to permute anyons and the corresponding twists are given below.
\begin{compactenum}[a)]
\item \textit{Charge permuting twists}.
These twists exchange the Pauli label of a pair of anyons in a column of Table~\ref{tab:bosons} while leaving the third unchanged.
For example, $x$ and $z$ labels are exchanged and $y$ label is left unchanged.
These twists do not permute the color.
For example, $rx$ is permuted to $rz$ and vice versa but $ry$ is unaltered.
\item \textit{Color permuting twists}.
These twists correspond to exchange of the color label of a pair of anyons in a row of
Table~\ref{tab:bosons}.
They leave the color of the third entry in that row unchanged while exchanging the colors of the other two entries.
Note that these twists permute only color but not charge.
For example a twist that leaves red anyons unchanged but permute $b\alpha$ to $g \alpha$ and vice versa.
This corresponds to exchanging the corresponding entries in two rows in Table~\ref{tab:bosons}.
\item \textit{Domino twists}. Apart from exchanging anyons along rows and columns, one can also exchange them across the diagonal in Table~\ref{tab:bosons}.
In this case, the twist acts by transposition i.e. the diagonal entries $rx$, $gy$ and $bz$ are unaltered whereas the following permutations take place: $ry \leftrightarrow gx$, $rz \leftrightarrow bx$ and $gz \leftrightarrow by$.
This transformation can also be seen as permutation of color and charge labels: $r \leftrightarrow x$, $g \leftrightarrow y$ and $b \leftrightarrow z$.
The anyons $rx$, $gy$ and $bz$ contain the charges that are mutually permuted and hence are left invariant whereas the other bosons are permuted.
For example, under this permutation $ry$ will be permuted to $bx$ which is the diagonally opposite entry in Table~\ref{tab:bosons}.
\end{compactenum}
\begin{remark}
\label{rmk:other-twists}
It is to be noted that these are fundamental twist types.
One can combine two or more of these twist types to obtain other twists.
\end{remark}
\begin{table}[htb]
\centering
\begin{tabular}{p{2cm}|p{2cm}|p{2cm}|p{2cm}}
\hline
\hline
Features / Twist type & Charge permuting & Color permuting & Domino\\
\hline
Lattice modification & Not needed & Needed & Needed \\
\hline
Physical qubits added or removed from lattice & No & Removed & Added\\
\hline
Type of code & non-CSS & non-CSS${}^\ast$ & non-CSS \\
\hline
Geometry of logical operators & Closed strings encircling a pair of twists & Closed strings encircling a pair of twists & Closed strings encircling a pair of twists\\
\hline
$T$-line & Absent & Present & Absent \\
\hline
\end{tabular}
\caption{Comparison of twists in color codes. ${}^\ast$One can also obtain a obtain CSS code by choosing $Z$ or $X$ type stabilizer on twist face.}
\label{tab:comp_twists}
\end{table}
A comparison of various twist types is given in Table~\ref{tab:comp_twists}.
For completeness we have also included all the three fundamental types of twists in color codes. For the rest of the paper we only consider charge and color permuting twists.
In the following sections we assume that the lattices are obtained by embedding graphs on a two dimensional plane.
We also assume that the boundary of lattices have the same color as given in Fig.~\ref{fig:sq-octagon-lattice} and Fig.~\ref{fig:hexagon-lattice-boundary}.
\subsection{Pauli operators as strings in color codes without twists}
In this section, we give a mapping between Pauli operators and strings on color code lattices without twists.
This correspondence is useful in abstracting away the lattice information and studying the code properties using the string algebra.
The three single qubit Pauli operators, namely, $X$, $Y$ and $Z$ are represented using three different types of strings.
A solid string of any color represents Pauli operator $Z$, a dashed string of any color represents Pauli operator $X$ and a dash-dotted string of any color represents Pauli operator $Y$, see Fig.~\ref{fig:cc-pauli-string-map}.
\begin{figure}[htb]
\centering
\includegraphics[scale = 1.5]{fig-Qb-cc-pauli-string-map.pdf}
\caption{String to Pauli operator correspondence: solid string of any color is mapped to $Z$, dashed string of any color is mapped to $X$ and dash dotted string of any color is mapped to $Y$.}
\label{fig:cc-pauli-string-map}
\end{figure}
We begin by representing a Pauli operator on a vertex in a color code without twists.
When a Pauli error occurs on a vertex, syndromes are created on all the three faces incident on that vertex, see Fig.~\ref{fig:Qb-cc-pauli-string-1}.
This is because, all the faces have both $Z$ and $X$ type stabilizers defined on them and an error violated at least one of these stabilizers.
A single Pauli is represented by a string with three terminals as shown in Fig.~\ref{fig:Qb-cc-pauli-string-5}.
The end point of the strings are interpreted as syndromes.
Now consider applying the same error operator on two adjacent vertices as shown in Fig.~\ref{fig:Qb-cc-pauli-string-3}.
This can be understood from stabilizer viewpoint.
The error operator commutes with the stabilizer of the blue face and hence no syndrome should be observed on this face.
Similarly, on the green face, the syndromes vanish.
(In effect two $bz$ syndromes fuse to vacuum.)
The string representation of Fig.~\ref{fig:Qb-cc-pauli-string-3} is shown in Fig.~\ref{fig:Qb-cc-pauli-string-7}.
This representation is obtained from the single qubit representation shown in Fig.~\ref{fig:Qb-cc-pauli-string-5}.
The two stings on blue and green faces are connected as these faces do not host syndromes.
On the other hand, on the red faces the string is terminated indicating the presence of syndromes.
Note that if one of the red faces hosts a syndrome $rz$, then applying the operator shown in Fig.~\ref{fig:Qb-cc-pauli-string-4} annihilates the syndrome on the face with prior syndrome and creates one on other without any prior syndrome.
This process can be seen as moving the syndrome between adjacent faces of the same color.
The operators that move syndromes are known as hopping operators~\cite{BhagojiSarvepalli2015}.
A compact representation of the string in Fig.~\ref{fig:Qb-cc-pauli-string-7} is given in Fig.~\ref{fig:Qb-cc-pauli-string-8}.
Note that the string is drawn dashed indicating that is represents Pauli operator $X$ and its color is red indicating that red edges are in its support.
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-1.pdf}
\subcaption
An $X$ error on the vertex violates $Z$ type stabilizers on the three faces creating three nonzero syndromes.}
\label{fig:Qb-cc-pauli-string-1}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-5.pdf}
\subcaption{String representation of the single qubit $X$ error. The end points of strings indicate syndromes.}
\label{fig:Qb-cc-pauli-string-5}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1.15]{fig-Qb-cc-pauli-string-3.pdf}
\subcaption{Syndromes of a two qubit $X$ error on adjacent qubits.
}
\label{fig:Qb-cc-pauli-string-3}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1.15]{fig-Qb-cc-pauli-string-4.pdf}
\subcaption{Syndromes on blue and green faces are annihilated since the two qubit $X$ operator commutes with stabilizers of those faces.}
\label{fig:Qb-cc-pauli-string-4}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1.15]{fig-Qb-cc-pauli-string-7.pdf}
\subcaption{String representation for the two qubit error in Fig.~\ref{fig:Qb-cc-pauli-string-3}.
String terminates in faces with syndrome and is continuous in faces without syndrome.}
\label{fig:Qb-cc-pauli-string-7}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-8.pdf}
\subcaption{An alternative representation for the two qubit $X$ error in Fig.~\ref{fig:Qb-cc-pauli-string-3}.
}
\label{fig:Qb-cc-pauli-string-8}
\end{subfigure}
\caption{Mapping Pauli operators to strings. Syndromes introduced as a result of an error and their string representation is shown in Fig.~\ref{fig:Qb-cc-pauli-string-1} and Fig.~\ref{fig:Qb-cc-pauli-string-5} respectively.
The situation when the same error happens on adjacent qubits and its string representation is shown in Fig.~\ref{fig:Qb-cc-pauli-string-3} and Fig.~\ref{fig:Qb-cc-pauli-string-7} respectively. Finally, the same Pauli error on adjacent qubits and their string representation is shown in Fig.~~\ref{fig:Qb-cc-pauli-string-4} and Fig.~\ref{fig:Qb-cc-pauli-string-8} respectively.}
\label{fig:str_pauli}
\end{figure}
\section{Charge permuting twists}
\label{sec:charge}
In this section, we discuss charge permuting twists.
The process of twist creation and movement corresponds to code deformation~\cite{Bombin2009}.
After reviewing the procedure to create charge permuting twists,
we derive the number of encoded logical qubits by stabilizer counting.
We also give logical operators for the encoded qubits.
We give a mapping between Pauli operators and strings on the lattice.
Charge permuting twists are faces in the lattice that permute the Pauli labels of anyons that encircle them.
A $p$ type charge permuting twist acts as follows:
\begin{subequations}
\begin{eqnarray}
cp &\mapsto& cp \\
cp^\prime &\mapsto& cp^{\prime \prime}\\
cp^{\prime \prime} &\mapsto& cp^\prime
\end{eqnarray}
\label{eqn:p-twist}
\end{subequations}
where $p$, $p^\prime$ and $p^{\prime \prime}$ are distinct Pauli labels.
An example is the $Y$ twist~\cite{Kesselring2018} that exchanges the $X$ and $Z$ labels of anyons while leaving the $Y$ label unchanged.
The $X$ twist leaves the $X$ label unchanged and exchanges the $Y$ and $Z$ labels, see Fig.~\ref{fig:charge-permuting-twists-action}, and the $Z$ twist exchanges $X$ and $Y$ labels leaving the $Z$ label unchanged.
Lattice modification is not required for creating charge permuting twists but only stabilizers of certain faces need to be modified~\cite{Kesselring2018}.
Before moving on to twist creation, we need a notion of how far apart the twist are as the code distance depends on it.
The notion of how far apart two charge permuting twists are is captured in the definition below.
\begin{definition}[Twist separation]
Separation between twists is the distance between the vertices corresponding to twists in the dual lattice.
\label{def:twist-separation}
\end{definition}
The domain wall in Fig.~\ref{fig:charge-permuting-twists} between $t_3$ and $t_4$ has been drawn to illustrate that a domain wall need not be the shortest path between twist faces.
The twist separation in that case still conforms to the above definition.
Twist separation determines the code distance, for this reason we include this as a design parameter while constructing color codes with twists.
\subsection{Creation of charge permuting twists}
We now summarize the procedure to create $k$ pairs of charge permuting twists in a $2$-colex.
In case of charge permuting twists, there are no physical dislocations in the lattice.
Twists are introduced by breaking the uniformity of stabilizer assignment for each face of the 2-colex.
To introduce $k$ pairs of charge permuting twists, we first need to choose $2k$ vertices in the dual lattice such that any pair of them are at least a distance $\ell \ge 2$ apart.
Now, connect a disjoint pair of vertices by a path so that no two paths crossover.
The faces corresponding to the terminal points of a path become the twists while the path connecting them becomes a domain wall.
The faces which correspond to the intermediate vertices of the path are the ones through which the domain wall passes through, see Fig.~\ref{fig:charge-permuting-twists}.
Once the twists and the faces through which domain wall passes through are chosen, assign the stabilizers on them as discussed next.
\begin{figure}[htb]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = 1.15]{fig-charge-permuting-twists-action.pdf}
\subcaption{Action of $X$ charge permuting twists on anyons. These twists permute only the charge of the anyons. An anyon with Pauli label $X$ is left unchanged whereas the anyons with $Z$ and $Y$ label are exchanged as shown.}
\label{fig:charge-permuting-twists-action}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .9]{fig-charge-permuting-twists.pdf}
\subcaption{Charge permuting twists. In the figure, faces marked $t_1$, $t_2$, $t_3$ and $t_4$ are twists and the dashed line is the domain wall.
Charge permuting twists need not be of the same color, see twist pair $t_1$ and $t_2$ which are faces of different color.
The domain wall need not be the shortest path between the twist faces, see twist pair $t_3$ and $t_4$.}
\label{fig:charge-permuting-twists}
\end{subfigure}
\caption{Charge permuting twists and their action on the anyons.}
\end{figure}
Faces in color code lattices with charge permuting twists can be grouped into three categories: i) twists, ii) faces through which domain wall passes through and iii) faces that are neither twists nor faces through which domain wall passes through.
Stabilizers for the $Y$ twist are given in Ref.~\cite{Kesselring2018}.
We give the stabilizer assignment for the $X$ twist.
Two stabilizers are defined on faces that are neither twists nor through which domain wall passes through:
\begin{equation}
B_f^Z = \prod_{v \in V(f)} Z_v, \text{ and } B_f^Y = \prod_{v \in V(f)} Y_v. \label{eqn:stab-unmodified-faces}
\end{equation}
The faces through which domain wall passes through has to permute the labels $Y$ and $Z$ of anyons while leaving the $X$ label unchanged.
This will happen if the Pauli operators of stabilizers on one side of the domain wall are of $Y$ type ($Z$ type) and on the other side they are of $Z$ type ($Y$ type).
Let the domain wall partition the vertices of a face $f$ into two sets $M_1$ and $M_2$.
Two stabilizers are assigned to such faces as defined below:
\begin{subequations}
\begin{eqnarray}
B_{f,1} &=& \prod_{v \in M_1} Y_v \prod_{v \in M_2} Z_v\\
B_{f,2} &=& \prod_{v \in M_1} Z_v \prod_{v \in M_2} Y_v.
\end{eqnarray}
\label{eqn:charge-domain-stab}
\end{subequations}
\begin{figure}[htb]
\centering
\includegraphics[scale = .9]{fig-charge-domain-wall-stab.pdf}
\caption{Vertex partition for faces comprising domain walls in charge permuting twists.
Dashed line is the domain wall and it does not pass through any vertex.
The vertices indicated as dark and white circles form the partition of the vertices of the face.}
\label{fig:domain-wall-stab}
\end{figure}
The stabilizers defined in Equations~\eqref{eqn:charge-domain-stab} are shown in Fig.~\ref{fig:domain-wall-stab}.
The stabilizer assigned to the twist face has to commute with the stabilizer defined on the face adjacent to it through which domain wall passes through.
These two faces share exactly an edge.
Given the assignment in Eq.~\eqref{eqn:charge-domain-stab}~and~\eqref{eqn:stab-unmodified-faces},
the only possible stabilizer that can be assigned to the twist face is that of $X$ type,
\begin{equation}
B_{\tau} = \prod_{v \in V(\tau)} X_v.
\label{eqn:charge-twist-stab}
\end{equation}
\noindent \emph{Stabilizer dependency.} The stabilizer assignment in Eq.~\eqref{eqn:stab-unmodified-faces}--\eqref{eqn:charge-twist-stab} is constrained by the following relations:
\begin{subequations}
\begin{eqnarray}
\prod_{f \in T\cap \mathsf{F}_{rb}} B_f \prod_{f \in \mathsf{F}_{rb} \cap M} B_{f,1}B_{f,2} \prod_{f \in \mathsf{F}_{rb} \cap U} B_{f}^ZB_{f}^Y&= &I\\
\prod_{f \in T\cap \mathsf{F}_{rg}} B_f \prod_{f \in \mathsf{F}_{rg} \cap M} B_{f,1}B_{f,2} \prod_{f \in \mathsf{F}_{rg} \cap U} B_{f}^ZB_{f}^Y&= &I
\end{eqnarray}
\label{eqn:charge-perm-stab-constraint}
\end{subequations}
where $T$, $M$ and $U$ denote the set of twist faces, modified faces and unmodified faces respectively.
From Equations~\eqref{eqn:charge-perm-stab-constraint}, we can see that one of the green and blue face stabilizers is dependent.
We take the outer unbounded blue face stabilizer to be the dependent one.
Since two stabilizers are defined on the unbounded blue face and only one of them is dependent, we have to measure the other non-local stabilizer which is undesirable.
We therefore discard one of the stabilizers and redefine the unbounded blue face stabilizer as
\begin{equation}
B_{f_e} = \prod_{v \in V(f_e)} X_v.
\end{equation}
This stabilizer still satisfies the constraint in Equations~\eqref{eqn:charge-perm-stab-constraint}.
\noindent \emph{Stabilizer commutation.}
Note that any two adjacent faces share two common vertices.
The Pauli operators corresponding to stabilizers are either same or different on the common vertices.
Hence, stabilizer commutation follows.
A more detailed analysis is given in Appendix~\ref{sec:charge-stabilizer}.
\begin{remark}
The stabilizers of color code with $X$ twist can be obtained from that of the $Y$ twist by applying phase gate on all qubits.
\end{remark}
We summarize with the following result on the parameters of color codes with
charge permuting twists.
\begin{lemma}[Encoded qubits for charge permuting twists]
A color code lattice with $t$ charge permuting twists encodes
$\left(t - 1\right)$ logical qubits.
\label{lm:encoded-qubits-charge}
\end{lemma}
\begin{proof}
In a color code lattice with charge permuting twists, all vertices are trivalent.
Therefore, we have $2e = 3v$, where $e$ and $v$ are the number of edges and vertices of the lattice respectively.
Using this in the Euler's formula, $v + f - e = 2$, where $f$ is the number of faces, we get, $2f = v + 4$.
Only one stabilizer is defined on twist faces and the unbounded face.
Therefore, the number of stabilizers is $s = 2f - t - 1 = v + 3 - t$.
Ignoring the dependent stabilizers (see Equations~\eqref{eqn:charge-perm-stab-constraint}), we get, $s = v - t + 1 = v - 2 (t /2 - 1) - 1$.
Since all stabilizers are independent, the number of encoded qubits is $k = t - 1 = 2 (t /2 - 1) + 1$.
\end{proof}
\subsection{Pauli operators as strings on a color code lattice with charge permuting twists}
The representation of the Pauli operators by strings as given in Fig.~\ref{fig:str_pauli} does not work in the presence of charge permuting twists as these twists permute the charge label of anyons.
Consider the operator shown in Fig.~\ref{fig:Qb-cc-pauli-string-6}.
The upper blue face is twist, domain wall passes through the green face and blue faces are neither twists nor faces through which domain wall passes through.
Red faces are normal faces.
The Pauli operator $Z_u Y_v$ creates two $bx$ syndromes on twist face and syndromes $gy$ and $gz$ on the green face.
Also, $ry$ and $rz$ syndromes are created on red faces.
Syndromes on the twist face annihilate each other and hence the string is continuous in twist face.
However, the Pauli operator $Z_u Y_v$ commute with the stabilizers of the green face.
Therefore, string is continuous in green face too, see Fig.~\ref{fig:Qb-cc-pauli-string-2}.
The red faces host anyons and hence the string terminates there.
An alternate representation for the operator described in Fig.~\ref{fig:Qb-cc-pauli-string-6} is given in Fig.~\ref{fig:Qb-cc-pauli-string-9}.
Note that in Fig.~\ref{fig:Qb-cc-pauli-string-9}, the string changes midway as it crosses the domain wall.
\noindent
{\em String representation of stabilizers.}
We represent the stabilizers using the string representation just developed.
Twist stabilizer in string notation is given in Fig.~\ref{fig:Qb-cc-charge-twist-stab-string}.
Note that the twist is assigned $X$ type stabilizer and hence the string is dashed.
Also, the $X$ twist does not permute $X$ label and hence the string is not changed as it crosses the domain wall.
An alternate representation of twist stabilizer in Fig.~\ref{fig:Qb-cc-charge-twist-stab-string} would be to take green sting instead of blue.
Stabilizers of faces through which domain wall passes through is given in Fig.~\ref{fig:Qb-cc-charge-modified-stab-string-1} and Fig.~\ref{fig:Qb-cc-charge-modified-stab-string-2}.
These stabilizers are of mixed type with $Z$ ($Y$) on vertices on one side of the domain wall and $Y$ ($Z$) on vertices on the other side of domain wall.
Note that the strings change as they cross domain wall.
For other faces that are not twists and through which domain wall does not pass through, the strings are of $Z$ type and $Y$ type.
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-6.pdf}
\subcaption{Syndromes introduced on upper blue twist face and lower green face. Note that the syndromes on twist face vanish and on the green face the anyon $gx$ is present. }
\label{fig:Qb-cc-pauli-string-6}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-2.pdf}
\subcaption{Strings are continuous in blue and green faces indicating no syndromes are present on them. Open string in red faces indicate syndrome on them.}
\label{fig:Qb-cc-pauli-string-2}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-9.pdf}
\subcaption{String representation of Fig.~\ref{fig:Qb-cc-pauli-string-6}. Note that syndrome changes its charge while retaining color.}
\label{fig:Qb-cc-pauli-string-9}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1.15]{fig-Qb-cc-charge-twist-stab-string.pdf}
\subcaption{String representation of twist stabilizer. The string does not change charge as the twist does not permute $X$ label.}
\label{fig:Qb-cc-charge-twist-stab-string}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1.15]{fig-Qb-cc-charge-modified-stab-string-1.pdf}
\subcaption{String representation of a stabilizer on the face through which domain wall passes through.}
\label{fig:Qb-cc-charge-modified-stab-string-1}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1.15]{fig-Qb-cc-charge-modified-stab-string-2.pdf}
\subcaption{String representation of another stabilizer on face through domain wall passes through. }
\label{fig:Qb-cc-charge-modified-stab-string-2}
\end{subfigure}
\caption{Strings and stabilizers in the presence of charge permuting twists. Change in the charge (or the Pauli operator on the support of the string) is shown in Fig.~\ref{fig:Qb-cc-pauli-string-6} to Fig.~\ref{fig:Qb-cc-pauli-string-9}. Stabilizers of the faces through which domain wall passes through is shown in Fig.~\ref{fig:Qb-cc-charge-modified-stab-string-1} and Fig.~\ref{fig:Qb-cc-charge-modified-stab-string-2}. Twist stabilizer string is shown in Fig.~\ref{fig:Qb-cc-charge-twist-stab-string}.}
\label{fig:Qb-cc-charge-modified-stab-string}
\end{figure}
\noindent
\emph{String representation of logical operators.}
Logical operators (including stabilizers) are closed strings encircling an even number of twists.
A closed string enclosing exactly a pair of twists is a nontrivial logical operator.
Further, we also need to show that these operators are not generated by stabilizers.
This is done by showing the existence of a string whose operator anticommutes with that of the given string, see Fig.~\ref{fig:LO-X-charge-permuting-lattice} and Fig.~\ref{fig:LO-Z-charge-permuting-lattice}.
These operators are logical $X$ and $Z$ operators respectively.
An important property of a logical operator is that it commutes with all the stabilizers.
Every closed string must enter and exit a face an even number of times.
Therefore, every face shares an even number of common vertices with the logical operator.
This also holds for faces through which domain wall passes through.
Taking into account the stabilizer structure of domain wall faces, we conclude that the operators shown in Fig.~\ref{fig:LO-charge-permuting-lattice} commute with all stabilizer generators.
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = .9]{fig-LO-X-charge-permuting-lattice.pdf}
\subcaption{Logical $X$ operator depicted on lattice.}
\label{fig:LO-X-charge-permuting-lattice}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = .9]{fig-LO-Z-charge-permuting-lattice.pdf}
\subcaption{Logical $Z$ operator depicted on lattice.}
\label{fig:LO-Z-charge-permuting-lattice}
\end{subfigure}
\caption{Logical operators for one encoded qubit in color code lattice with $X$ type charge permuting twists. Note that both operators have one vertex in common where the Pauli operator differs.}
\label{fig:LO-charge-permuting-lattice}
\end{figure}
Note that the product of twist stabilizers $t_1$, $t_2$ and the stabilizers defined on faces through which domain wall passes through (except the blue face) has the same support as the operator shown in Fig.~\ref{fig:LO-Z-charge-permuting-lattice} with $X$ on vertices in the support.
The product of these stabilizers with the logical $Z$ operator in Fig.~\ref{fig:LO-Z-charge-permuting-lattice} flips the Pauli operator on all vertices in its support to $Y$.
Therefore, the two operators are equivalent up to stabilizers.
The same holds true for logical $X$ operator shown in Fig.~\ref{fig:LO-X-charge-permuting-lattice}.
Pauli operators $Y$ and $Z$ can be flipped by taking the product of twist stabilizers on $t_1$ and $t_3$ and the blue face on which the operator has support.
\begin{figure}[htb]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-LO-charge.pdf}
\subcaption{Logical operators for encoded qubits in lattice having six $X$ type charge permuting twists.}
\label{fig:lo_charge}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .75]{fig-LO-charge-canonical.pdf}
\subcaption{Canonical logical operators for encoded qubits in lattice having six $X$ type charge permuting twists for $\lfloor t/3 \rfloor$ encoding.}
\label{fig:lo_charge_canonical}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .75]{fig-LO-charge-perm-extra-qubit.pdf}
\subcaption{Logical operators for the the logical qubit in the presence of a pair of twists. The green string has support on the physical qubits on the boundary.}
\label{fig:LO-charge-perm-extra-qubit}
\end{subfigure}
\caption{Logical operators for encoded qubits in lattices with $X$ type charge permuting twists.}
\label{fig:charge_LO}
\end{figure}
For purposes of simple representation of twist and logical operators, we hide the lattice information and represent only twists, domain walls and logical operators.
Logical operators are represented as closed strings.
These strings are edges in the shrunk lattice of appropriate color.
Note that upon crossing the domain wall, color of the string is unaffected.
Logical operators when three pairs of $X$ twists are present in the lattice is given in Fig.~\ref{fig:lo_charge}.
The disadvantage with this choice of logical operators is that whenever a gate is to be performed only on a particular encoded qubit by braiding, say encoded qubit $1$ ($2$), we also end up performing a gate on logical qubit $3$ ($4$) (see Fig.~\ref{fig:lo_charge}) which is undesirable.
Therefore, we treat encoded qubits $3$ and $4$ as a gauge qubits.
The canonical form of logical operators for $\lfloor t/3 \rfloor$ encoding is shown in Fig.~\ref{fig:lo_charge_canonical}.
With $\lfloor t/3 \rfloor$ encoding, a closed string encircling three pairs of twists encoding two logical qubits is a stabilizer.
We use $\lfloor t / 3 \rfloor$ encoding which encodes two qubits per six twists for implementing gates by braiding.
\begin{theorem}
A color code with $t$ charge permuting twists with $ \left\lfloor \frac{t}{3} \right\rfloor$ encoding defines an $\left[ \left[n, \left\lfloor \frac{t}{3} \right\rfloor\right]\right]$
subsystem code
with $\left(t - 1\right) - \left\lfloor \frac{t}{3} \right\rfloor $ gauge qubits.
\end{theorem}
\begin{remark}
Note that equivalent logical $Z$ and $X$ operators can be obtained by adding the gauge operators.
The operators equivalent to logical $Z$ and $X$ are represented by red strings as shown in Fig.~\ref{fig:equivalent_LO}.
\end{remark}
\noindent \emph{Code distance}.
Distance of the code is the minimum weight of the operator in $C(S) \setminus S$ where $S$ is the stabilizer and $C(S)$ is the centralizer of stabilizer.
If the separation between any pair of twists is at least $\ell$, then the smallest weight of logical operator enclosing an even number of twists is $O(\ell)$.
We conjecture that the distance of the code is $O(\ell)$.
\begin{figure}[htb]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .8]{fig-Qb-cc-logY-1.pdf}
\subcaption{Adding a gauge operator (green string) to the logical $Z$ operator. The result is a red string encircling twists $t_1$ and $t_2$.}
\label{fig:Qb-cc-logY-1}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .8]{fig-Qb-cc-logY-2.pdf}
\subcaption{Similarly, logical $X$ operator, shown as green string, is modified to red string by adding a gauge operator shown as blue string.}
\label{fig:Qb-cc-logY-2}
\end{subfigure}
\caption{Equivalent logical $Z$ and $X$ operators.}
\label{fig:equivalent_LO}
\end{figure}
We use the charge permuting twists constructed in this section for realizing encoded Clifford gates by braiding which is presented in Section~\ref{sec:gates-charge-permuting}.
\section{Color permuting twists }
\label{sec:color}
In this section, we present the construction of color permuting twists.
Our construction differs from the one given in Ref.~\cite{Kesselring2018} and is more along the lines of Ref.~\cite{Bombin2011}.
We first address the modification of an arbitrary $2$-colex to create color permuting twists.
While creating twists, we should also ensure that the code resulting from lattice modification has the specified distance.
This is accomplished by tailoring the separation between twists.
We then discuss stabilizer assignment, code parameters and logical operators.
Color permuting twists are faces in the lattice which permute the color label of anyons that encircle them.
A $c$-type color permuting twist acts as follows:
\begin{subequations}
\begin{eqnarray}
cp &\mapsto& cp\\
c^\prime p&\mapsto& c^{\prime \prime} p\\
c^{\prime \prime} p &\mapsto& c^\prime p
\end{eqnarray}
\label{eqn:c-twist}
\end{subequations}
where the colors $c$, $c^\prime$ and $ c^{\prime \prime}$ are all distinct.
Observe that color permuting twists fix one color and exchange the other two.
The Pauli label of an anyon is not altered by color permuting twists.
\subsection{Creation and movement}
\noindent \emph{Intuition.} In a $2$-colex, edges of a specific color connect the faces of the same color, see Fig.~\ref{fig:int-color-perm-0}.
As a result, the syndromes can only be moved between faces of same color.
In order to create color permuting twist, we need to introduce an edge between two faces of different color.
Suppose that we introduce an edge $(u,v)$ between blue and green face, see Fig.~\ref{fig:int-color-perm-1}.
This edge cuts through the common red face to the blue and green faces.
This move also makes the vertices $u$ and $v$ tetravalent and splits the red face into two: the upper pentagon face and the lower triangular face.
It is not possible to assign stabilizers to the new red faces such that they commute with the rest.
Therefore, the vertices $u$ and $v$ have to be made trivalent.
This is done by removing the common neighbor $w_1$ to vertices $u$ and $v$, see Fig~\ref{fig:int-color-perm-2}.
The vertex $w_2$ is rendered two-valent by this move.
The neighbors of the vertex $w_2$ are connected by an edge making them tetravalent.
The tetravalency of the neighbors of $w_2$ is resolved by removing the vertex $w_2$, see Fig~\ref{fig:int-color-perm-3}.
Note that the new edge also connects blue and green faces.
The face in between twists has mixed color.
One can assign either blue or green color to this face.
Similar procedure is followed for moving twists apart.
The twist created in this illustration is red twist since it permutes green and blue labels leaving the red label unchanged.
Other twists like green twist and blue twist are defined similarly.
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-int-color-perm-0.pdf}
\subcaption{In a $2$-colex, edges connect faces of the same color. Red, green and blue edges connect the faces of respective color as shown.}
\label{fig:int-color-perm-0}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-int-color-perm-1.pdf}
\subcaption{To create twist, introduce an edge between two faces of different color. This renders two vertices tetravalent.}
\label{fig:int-color-perm-1}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-int-color-perm-2.pdf}
\subcaption{The vertex $w_1$ is removed to resolve the tetravalency of vertices $u$ and $v$. This move also makes the vertex $w_2$ two-valent.}
\label{fig:int-color-perm-2}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-int-color-perm-3.pdf}
\subcaption{The neighbors of two-valent vertex $w_2$ are connected by an edge making them tetravalent. The tetravalency is resolved by removing the vertex $w_2$.}
\label{fig:int-color-perm-3}
\end{subfigure}
\caption{Process of twist creation in a $2$-colex.}
\end{figure}
\begin{remark}
Note that the twist faces have an odd number of edges.
This breaks the three-face-colorability in the vicinity of such faces of the lattice and creates color permuting twists.
\end{remark}
We would like to introduce $k \ge 1$ pairs of twists of the same color $c$ in a $2$-colex.
To do so, we first choose an even number of vertices of the same color $c$ in the dual lattice such that any pair of them is at least a distance $\ell$ apart.
A path is introduced between disjoint pairs of vertices so that no two paths overlap.
The aforesaid path is not arbitrary but has to satisfy the properties given below.
\begin{compactenum}[C1)]
\item The faces corresponding to the nearest $c$-colored vertices on the path are connected by an edge in the original lattice.
\item Let $u$, $v$ and $w$ are three consecutive $c$-colored vertices on the path and such that distance between $u$ and $v$ is two and distance between $v$ and $w$ is two in the dual lattice.
The vertex $w$ cannot be at a distance of two from $u$.
\end{compactenum}
The first condition ensures that the faces with color $c^\prime$ and $c^{\prime \prime}$ are merged during twist creation.
The second condition ensures that the intermediate faces between twists do not grow in size.
With this, one can create an even number of twists in an arbitrary $2$-colex.
The procedure to create color permuting twists in an arbitrary $2$-colex is given in Algorithm \ref{alg:color-permuting-twists}.
\begin{algorithm}
\caption{Algorithm to create $k$ pairs of $c$-color permuting twists in a $2$-colex.}
\label{alg:color-permuting-twists}
\begin{flushleft}
\algorithmicrequire{ $2$-colex, separation between twists $\ell \ge 2$.}\\
\algorithmicensure{ Trivalent lattice with $k$ pairs of color permuting twists.}
\end{flushleft}
\begin{algorithmic}[1]
\State Choose $2k$ vertices of color $c$ in the dual lattice such that any pair of them is at least a distance $\ell$ apart.
Introduce a path between disjoint pairs of vertices so that no two paths overlap.
The $c$-colored vertices on a given path $\pi_i$ satisfy conditions C1) and C2).
Let $\{ e_j^{(i)} \}_{j = 1}^{\ell}$ be the collection of such edges corresponding to the $i^{th}$ path $\pi_i$ in the original lattice, see Fig.~\ref{fig:tc0}.
\For{ $i = 1$ to $k$}
\State Remove the vertices incident on the edge $e_1^{(i)}$ (see Fig.~\ref{fig:tc1}) and connect the closest two-valent vertices, see Figs.~\ref{fig:tc2},~\ref{fig:tc3}. \Comment {Twist Creation}
\State Faces with color $c$ continue to have the same color. Merged face is assigned one of the colors $c^\prime$ or $c^{\prime \prime}$. \Comment Faces that shared the deleted edge as common are merged.
\While{$e_{\ell}^{(i)}$ not reached} \Comment Twist Movement
\State Repeat steps $3$ and $4$ choosing the next edge along the path $\pi_i$, see Figs.~\ref{fig:tm1},~\ref{fig:tm2}.
\EndWhile
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{remark}
If a third $c$-colored vertex from a given vertex $u$ is at is at a distance two, then multiple edges are introduced during the execution of Algorithm~\ref{alg:color-permuting-twists}.
If multiple edges are introduced, then removed the vertices incident on multiple edges and connect the resulting two-valent vertices.
\end{remark}
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[height = 3cm, width = 4cm]{fig-const0}
\subcaption{The path chosen for twist creation is shown in orange solid line and is of length two.}
\label{fig:tc0}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[height = 3cm, width= 4cm]{fig-const1.pdf}
\subcaption{To create twists, remove the vertices that lie along the first edge of the path.}
\label{fig:tc1}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[height = 3cm, width = 4cm]{fig-const2.pdf}
\subcaption{Two-valent vertices created in the lattice as a result of removing vertices.}
\label{fig:tc2}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[height = 3cm, width = 4cm]{fig-const3.pdf}
\subcaption{Connect the nearest two valent vertices. Color of the new faces is as shown. }
\label{fig:tc3}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[height = 3cm, width = 4cm]{fig-move0.pdf}
\subcaption{To move twist, choose one outgoing edge of the same color as the twist.}
\label{fig:tm1}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[height = 3cm, width = 4cm]{fig-move1.pdf}
\subcaption{Repeating the process of twist creation, the twist is moved to the next red face on the path.}
\label{fig:tm2}
\end{subfigure}
\caption{Illustration of twist creation and movement in the square octagon lattice.}
\label{fig:tm}
\end{figure}
\subsection{Stabilizers}
We need not modify a majority of stabilizers for color codes with color permuting twists.
Every face $f$ which is not a twist has an even number of edges.
Hence, two stabilizers are defined on such faces:
\begin{equation}
B_f^X = \prod_{v \in V(f)} X_v, \text{ and } B_f^Z = \prod_{v \in V(f)} Z_v
\end{equation}
However, on a twist face $\tau$, we cannot define two stabilizers as they will anti-commute.
A $Y$-type stabilizer is defined on the twist face~\cite{Kesselring2018}:
\begin{equation}
B_{\tau} = \prod_{v \in V(\tau)} Y_v.
\end{equation}
\noindent \emph{Consistency of stabilizer assignment.} If two faces are adjacent, then they share two common vertices.
To check commutation between adjacent face stabilizers, we have to check commutation on these vertices.
The stabilizers are all either $X$ type or $Z$ type.
If the Pauli operator on the common vertices is same, commutation follows.
On the other hand, if the Pauli operators on the common vertices are different (for instance, $X$ or $Z$ on one of the faces and $Y$ on the other face), then there are two anticommutations and the stabilizers commute.
Therefore, all stabilizers commute.
\noindent \emph{Stabilizer dependencies.}
Giving stabilizer dependency for the general case is difficult.
We restrict to lattices where all twist faces are red.
The constraints are
\begin{subequations}
\begin{eqnarray}
\prod_{f \in \mathsf{F}_b} B_{f}^Z \prod_{f \in \mathsf{F}_g} B_{f}^Z &=& I, \\
\prod_{f \in \mathsf{F}_b} B_{f}^X \prod_{f \in \mathsf{F}_g} B_{f}^X &=& I.
\end{eqnarray}
\label{eqn:color-perm-stab-constraints}
\end{subequations}
where the product over blue faces includes blue modified faces and the outer unbounded blue face.
These constraints indicate the presence of two dependent stabilizers.
We take the dependent stabilizers to be the ones defined on the unbounded blue face.
\begin{proposition}
Algorithm~\ref{alg:color-permuting-twists} creates color permuting twists.
\end{proposition}
\begin{proof}
To prove that Algorithm~\ref{alg:color-permuting-twists} creates color permuting twists, it suffices to show that the algorithm introduces an edge between faces of different color while preserving trivalency of vertices.
When an anyon is moved along this edge, its color is permuted.
Let $u $ and $v$ be the vertices marked for deletion in the algorithm.
Let $N(u) = \{x_1, x_2\}$ and $N(v) = \{ y_1, y_2\}$.
Note that $e_1 = (x_1, x_2)$ and $e_2 = (y_1, y_2)$ do not exist before twist creation.
Due to the property of $2$-colex that the set of faces sharing an edge with any given face can be colored with two colors, the edges $e_1$ and $e_2$ connect faces of different color.
Algorithm~\ref{alg:color-permuting-twists} removes vertices $u$ and $v$ and introduces edges $e_1$ and $e_2$ thereby creating an edge between faces of different color.
Stabilizers on the unmodified faces are not changed.
Also, the stabilizer on the twist face is of $Y$ type.
Therefore, any operator $P_{x_1} P_{x_2}$, where $P \in \{X,Y,Z \} $ will commute with the twist stabilizer while moving the syndrome from face of one color to another.
Apart from these, the modified face in between twist has edges of different colors incident on it.
As a result, whenever an anyon enters and exits this faces through edges of different color, it color label is permuted.
Stabilizers defined on modified faces are of $Z$ and $X$ type.
Therefore, the error operators that move the syndrome across modified faces commute with the stabilizers defined on them.
\end{proof}
\begin{lemma}[Parity of color permuting twists]
Color permuting twists are created in pairs.
\label{lm:parity-color}
\end{lemma}
\begin{proof}
Let $\Gamma$ be the lattice with color permuting twists and let $\Gamma^\ast$ be its dual.
Note that color permuting twists are odd cycles and hence vertices corresponding to them have odd degree in the dual lattice.
Since the number of vertices with odd degree should be even in a graph \cite{Diestel} (and hence its embedding on a surface), color permuting twists are created in pairs.
\end{proof}
\noindent{\em $T$-lines and domain walls.}
When color permuting twists are created in the lattice, local three colorability around twist faces is destroyed.
As a consequence, we can find a sequence of edges, having the same color as the twist face, along which faces of the same color meet. On the (red) shrunk lattice these edges form a
path terminating on the vertices corresponding to the twist faces.
We refer to this sequence of edges as a $T$-line following \cite{Bombin2011}.
For instance, if the twist face is red, then this path is formed by red edges and along this path blue and/or green adjacent faces meet.
An example of trivalent lattice with twists along with $T$-line is shown in Fig.~\ref{fig:hex-lattice}.
Existence of $T$-lines in the presence of color permuting twists is proved in Appendix~\ref{sec:t-line}.
Note that for this specific coloring of the lattice domain walls run parallel to the $T$-line.
A different coloring might be possible where the $T$-line crosses the domain wall as for instance happens
in case of surface codes with twists \cite{GowdaSarvepalli2020}.
\begin{figure}[htb]
\centering
\includegraphics[height = 5.5cm,width = 6.5cm ]{fig-hexagon-twists.pdf}
\caption{Two pairs of color permuting twists in hexagonal lattice. Due to the presence of twists, there is a path, starting from one of the twists and ending on another, along which faces of same color (in this case, blue faces) meet. This path is shown as a solid line. Also, the domain wall is shown as dashed line.}
\label{fig:hex-lattice}
\end{figure}
\noindent \emph{Coding theoretic view of color permuting twist creation.} The process of color permuting twist creation can be understood in terms of classical code puncturing \cite{Huffman2003}.
In classical codes, code puncturing is done by deleting one or more (classical) bits and readjusting the parity check and generator matrices if necessary.
While creating twists in color codes, two qubits are removed from the system, see Fig.~\ref{fig:tc2}.
Suppose that stabilizers are written in symplectic form.
Let $f_1$ and $f_2$ be the top and bottom red faces in the figure and let $f_3$ and $f_4$ be the green and blue faces respectively.
Due to qubit removal, one qubit is removed from the stabilizers on $f_1$.
This leads to anti commutation among the stabilizers on $f_1$.
The anti commutation is rectified by dropping a stabilizer (either $Z$ or $X$ type).
The same holds true for stabilizers on $f_2$.
Now, the stabilizers on $f_3$ and $f_4$ anti commute with stabilizer on $f_1$.
The resolution of this anticommutation comes from merging the corresponding $Z$ and $X$ stabilizers on $f_3$ and $f_4$.
In this process, we have deleted two columns, one row each of stabilizers of $f_1$ and $f_2$ and merged the rows of $f_3$ and $f_4$.
This process is exactly similar to code puncturing in classical codes.
Therefore, color permuting twist creation is the analogue of code puncturing in quantum codes.
For a general 2-colex into which color permuting twists are introduced the following result holds.
\begin{lemma}[Encoded qubits for color permuting twists]
A 2-colex with $t$ color permuting twists encodes $(t - 2)$ qubits where $t$ is even.
\label{lm:encoded-qubits-cp}
\end{lemma}
\begin{proof}
Note that in lattice containing color permuting twists, all vertices are trivalent.
Hence, we have $2e = 3v$, where $e$ and $v = n$ are the number of edges and vertices of the lattice respectively.
Also, $v + f - e = 2$, where $f$ is the number of faces in the lattice.
Therefore, we get, $f = \frac{v}{2} + 2 = \frac{n}{2} + 2$.
We count two stabilizers for every face, including twists.
Later the excess stabilizers, which are $t$ in number, are subtracted.
This gives the number of stabilizers to be $s = 2f - t = n - t + 4$.
Ignoring the outer unbounded face stabilizers (see Equations~\eqref{eqn:color-perm-stab-constraints}), we get, $s = n - 2\left(\frac{t}{2} - 1\right)$.
Hence, the number of encoded qubits is $2(\frac{t}{2} - 1)$.
\end{proof}
\begin{remark}
The number of encoded qubits here is twice that of surface codes with twists~\cite{GowdaSarvepalli2020}.
The reason is, with the introduction of a twist pair in surface codes, one stabilizer was removed and hence one new encoded qubit.
However, in color codes with every twist pair, two stabilizers are removed and hence we get twice the number of encoded qubits.
\end{remark}
\subsection{Pauli operators as strings on a 2-colex with color permuting twists}
In this subsection, we give the string representation of Pauli operators on a 2-colex in the presence of color permuting twists.
Consider Fig.~\ref{fig:Qb-cc-pauli-string-10} where Pauli operator $Z$ is applied on the vertices of the edge.
The red face on the top is twist, green and blue faces on the right and left of twist are unmodified faces and the bottom face with blue color is the modified face.
The Pauli operators create two $ry$ syndromes on the twist face, $bx$ and $gx$ on the normal blue and green faces respectively and $gx$, $bx$ on the modified face
(the part of the modified face to the right of the domain wall supports anyons with green color label).
Anyons on the twist face annihilate each other and hence the string is continuous in the twist face.
The Pauli operator commutes with the stabilizers on the modified face and therefore string is continuous in modified face too, see Fig.~\ref{fig:Qb-cc-pauli-string-11}.
String representation of the Pauli operator in Fig.~\ref{fig:Qb-cc-pauli-string-10} is given in Fig.~\ref{fig:Qb-cc-pauli-string-12}.
Note that the string changes color as it crosses the domain wall.
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-10.pdf}
\subcaption{Syndromes created as a result of applying $Z$. Upper red face is twist and the lower face with mixed color is modified face.}
\label{fig:Qb-cc-pauli-string-10}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-11.pdf}
\subcaption{Syndromes on twist vanish. On the modified face if we move the
syndrome $gx$
across the domain wall it becomes $bx$ and anhilates the other $bx$
syndrome giving a zero syndrome.}
\label{fig:Qb-cc-pauli-string-11}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-cc-pauli-string-12.pdf}
\subcaption{The string corresponding to the operator in Fig.~\ref{fig:Qb-cc-pauli-string-10} is of mixed color. The color change happens at the domain wall.}
\label{fig:Qb-cc-pauli-string-12}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-color-perm-modified-string-X.pdf}
\subcaption{$X$ type stabilizer on modified face.}
\label{fig:Qb-color-perm-modified-string-X}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-Qb-color-perm-modified-string-Z.pdf}
\subcaption{$Z$ type stabilizer on modified face.}
\label{fig:Qb-color-perm-modified-string-Z}
\end{subfigure}
\caption{Representing Pauli operators as strings in the presence of color permuting twists.}
\label{fig:Qb-color-perm-modified-string}
\end{figure}
Stabilizers on modified faces cannot be represented as strings with one color, see Fig~\ref{fig:Qb-color-perm-modified-string-X} and Fig.~\ref{fig:Qb-color-perm-modified-string-Z}.
For the same string, both $X$ and $Z$ stabilizers are defined.
Note the strings changing color as they crosses the domain wall.
As in the case with charge permuting twists, logical operators are closed strings encircling an even number ($ \ge 2$) of twists.
(A string encircling an odd number of twists will be an open string.)
The above mentioned logical operators are not generated by stabilizers.
The reason being one can find another string encircling a pair of twists that anticommutes with the given string, see Fig.~\ref{fig:hexagon-lattice-LO-X} and Fig.~\ref{fig:hexagon-lattice-LO-Z}.
The logical operators for one of the encoded qubits is shown in Fig.~\ref{fig:hexagon-lattice-LO}.
Logical $X$ operator is the string which encircles twists not created together, see Fig.~\ref{fig:hexagon-lattice-LO-X}.
Logical $Z$ operator is the string that encircles twists created together, see Fig.~\ref{fig:hexagon-lattice-LO-Z}.
Note that with every pair of twists introduced, the number of encoded qubits increases by two.
\begin{figure}[htb]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[height = 5.5cm,width = 6.5cm]{fig-hexagon-lattice-LO-X.pdf}
\subcaption{Logical $X$ operator depicted on lattice. Blue circles and green circles with black boundary represent Pauli operator $Z$.}
\label{fig:hexagon-lattice-LO-X}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[height = 5.5cm,width = 6.5cm]{fig-hexagon-lattice-LO-Z.pdf}
\subcaption{Logical $Z$ operator depicted on lattice. Blue circles represent Pauli operator $X$.}
\label{fig:hexagon-lattice-LO-Z}
\end{subfigure}
\caption{Logical operators for an encoded qubit depicted on lattice.}
\label{fig:hexagon-lattice-LO}
\end{figure}
Henceforth, we consider all twists to be of the same color.
The reason for this is that when twist pairs are of different color, logical $X$ operator has the form shown in Fig.~\ref{fig:diff-color-twists}.
To avoid such self-intersecting logical operators, we choose all twists to be of the same color.
Logical operators for encoded qubits in color codes with color permuting twists is given in Fig.~\ref{fig:lo_color}.
The string encircling twists $t_1$ and $t_2$ is the logical $Z$ operator and the one enclosing twists $t_1$ and $t_3$ is the logical $X$ operator.
When two strings of same color intersect they overlap in even number of locations, therefore
they commute.
Only when strings of different color intersect they overlap in one location, therefore they anticommute.
From this we conclude that the operators in Fig.~\ref{fig:lo_color} anticommute.
Even when the support of both logical operators intersect where the colors are same, the size of the common support is always an even number.
We choose to use $\lfloor t/3 \rfloor$ encoding for the same reason as that for charge permuting twists.
The canonical form of logical operators for $\lfloor t/3 \rfloor$ encoding is shown in Fig.~\ref{fig:lo_color_canonical}.
\begin{figure}[htb]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[height = 3cm, width = 3cm]{fig-diff-color-twists.pdf}
\subcaption{Logical $X$ operators when twist pairs are of different color.}
\label{fig:diff-color-twists}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[height = 3cm, width = 6cm]{fig-LO-color.pdf}
\subcaption{Logical operators for encoded qubits in lattice having color permuting twists. Operators $\Bar{Z}_1$ and $\Bar{Z}_2$ have $Z$ and $X$ on their support respectively whereas operators $\Bar{X}_1$ and $\Bar{X}_2 $ have $X$ and $Z$ on their support respectively.}
\label{fig:lo_color}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[height = 3cm, width = 4.5cm]{fig-LO-color-canonical.pdf}
\subcaption{Canonical logical operators for encoded qubits in lattice having color permuting twists for $\lfloor t/3 \rfloor $.}
\label{fig:lo_color_canonical}
\end{subfigure}
\caption{Logical operators for encoded qubits in lattices with color permuting twists.}
\label{fig:color_LO}
\end{figure}
\begin{remark}
With $\lfloor t/3 \rfloor $ encoding, we get $\left(t - 2\right) - \left\lfloor \frac{t}{3} \right\rfloor$ gauge qubits.
\end{remark}
\section{Clifford gates using charge permuting twists}
\label{sec:gates-charge-permuting}
In this section, we show that encoded Clifford gates can be realized with charge permuting twists by braiding alone.
We use $\left \lfloor t / 3 \right \rfloor$ encoding while realizing gates, see Fig~\ref{fig:lo_charge_canonical}.
Let $t_1$, $t_2$ and $t_3$ be twists as shown in Fig.~\ref{fig:braid-single-qubit}.
The list of logical operators and corresponding strings is given in Table~\ref{tab:string_LO}.
In some case a logical operator has an equivalent representation obtained by adding a gauge operator, see Fig.~\ref{fig:equivalent_LO}. Howver strings corresponding to logical $\bar{Y}_1 (\bar{Y}_2) $ operator do not have equivalent strings of different color of the form $\mathcal{W}_{2,3}^c (\mathcal{W}_{4,5}^c)$.
\begin{table}[htb]
\centering
\begin{tabular}{l|c|c}
\hline
\hline
String & Equivalent string & Logical Operator\\
\hline
$\mathcal{W}_{1,2}^{b}$ & $\mathcal{W}_{1,2}^{r}$ & $\bar{Z}_1$ \\
$\mathcal{W}_{1,3}^{g}$ & $\mathcal{W}_{1,3}^{r}$ & $\bar{X}_1$ \\
$\mathcal{W}_{2,3}^{r}$ & $-$ & $\bar{Y}_1$ \\
$\mathcal{W}_{5,6}^{b}$ & $\mathcal{W}_{5,6}^{r}$& $\bar{Z}_2$ \\
$\mathcal{W}_{4,6}^{g}$ & $\mathcal{W}_{4,6}^{r}$ & $\bar{X}_2$ \\
$\mathcal{W}_{4,5}^{r}$ & $-$ & $\bar{Y}_2$ \\
\hline
\end{tabular}
\caption{Strings and the corresponding logical operator associated with them. Note that the strings corresponding to logical $Y$ operators are dependent on those of logical $X$ and $Z$ operators. }
\label{tab:string_LO}
\end{table}
\begin{remark}
The string $\mathcal{W}_{2,3}^r$ corresponding to $\bar{Y}_1$ is not equivalent to $\mathcal{W}_{2,3}^g$, $\mathcal{W}_{2,3}^b$
unlike $\bar{X}_1 $ and $\bar{Z}_1 $.
It turns out that $\mathcal{W}_{2,3}^g$ and $\mathcal{W}_{2,3}^b$ correspond to operators $\bar{X}_1 \bar{Z}_2$ and $\bar{Z}_1 \bar{X}_2$ respectively.
\end{remark}
\begin{theorem}
[Single qubit Clifford gate]
Let $t_1$, $t_2$ and $t_3$ be twists that encode a logical qubit and let $\mathcal{W}_{1,2}^{b}$, $\mathcal{W}_{1,3}^{g}$ and $\mathcal{W}_{2,3}^{r}$ be the strings corresponding to logical operators $\Bar{Z}$, $\Bar{X}$ and $\Bar{Y}$ respectively.
Then,
\begin{compactenum}[i)]
\item Braiding $t_1$ and $t_2$ realizes phase gate ($Z$ rotation by $\pi /2$).
\item Braiding $t_1$ and $t_3$ realizes $X$ rotation by $\pi /2$.
\end{compactenum}
\label{thm:single-qb-Clifford}
\end{theorem}
\begin{proof}
The encoding used is as shown in Fig.~\ref{fig:lo_charge_canonical}.
The logical $Y$ operator (up to gauge operators) is shown in Fig.~\ref{fig:Qb-cc-logY-4}.
After braiding twists $t_1$ and $t_2$ we have, $t_1 \leftrightarrow t_2$, $t_3 \rightarrow t_3$ and $t_4 \rightarrow t_4$. Using this we get the transformation given below.
\begin{eqnarray*}
\begin{array}{ccc}
\mathcal{W}^{b}_{1,2} \mapsto \mathcal{W}_{1,2}^{b} & \vert & \bar{Z} \mapsto \bar{Z}\\
\mathcal{W}^{g}_{1,3} \mapsto \mathcal{W}_{2,3}^{g} & \vert & \bar{X} \mapsto \bar{Y}\\
\mathcal{W}_{2,3}^{r} \mapsto \mathcal{W}_{1,3}^{r} & \vert & \bar{Y} \mapsto \bar{X} \\
\end{array}
\end{eqnarray*}
The string $\mathcal{W}_{1,2}^{b}$ is unchanged since the twists encircled by it are braided.
Transformation of $\mathcal{W}_{1,3}^{g}$ is shown in Appendix~\ref{sec:charge-braiding} and transformation of $\mathcal{W}_{2,3}^{r}$ to $\mathcal{W}_{1,3}^{r}$ is straightforward.
Similarly, braiding $t_1$ and $t_3$, we get the following transformation.
\begin{eqnarray*}
\begin{array}{ccc}
\mathcal{W}_{1,3}^{g} \mapsto \mathcal{W}_{1,3}^{g} & \vert & \bar{X} \mapsto \bar{X},\\
\mathcal{W}_{1,2}^{b} \mapsto \mathcal{W}_{2,3}^{b} & \vert & \bar{Z} \mapsto \bar{Y} \\
\mathcal{W}_{2,3}^{r} \mapsto \mathcal{W}_{1,2}^{r} & \vert & \bar{Y} \mapsto \bar{Z} \\
\end{array}
\end{eqnarray*}
Hence, the theorem follows.
\end{proof}
Hadamard gate can be realized by $X$ rotation and phase gates.
Hence $Z$ and $X$ rotations by $\pi / 2 $ suffice to realize single qubit Clifford gates by braiding.
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-braid-single-qubit-rz.pdf}
\subcaption{To implement a Pauli $Z$ rotation, we have to braid the twists that are encircled by the logical $Z$ operator viz $t_1$ and $t_2$.}
\label{fig:braid-single-qubit-rz}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = 1]{fig-braid-single-qubit-rx.pdf}
\subcaption{Pauli $X$ rotation is implemented by braiding twists $t_1$ and $t_3$ which are encircled by the string corresponding to logical $X$ operator. }
\label{fig:braid-single-qubit-rx}
\end{subfigure}
\caption{Braiding twists to implement single qubit Clifford gates (a) Phase gate (b) Pauli $X$ rotation by $\pi / 2$.
}
\label{fig:braid-single-qubit}
\end{figure}
To complete the Clifford group, we need an entangling gate.
We prove below that braiding the twist pair shared by two logical qubits accomplishes this.
Let $t_{1}$, $t_{2}$ and $t_{3}$ be twists that encode the logical qubit $a$ and let $t_{4}$, $t_{5}$ and $t_{6}$ be twists that encode the logical qubit $b$ in the same block as shown in Fig.~\ref{fig:braid-entangling-adjacent}.
The operator corresponding to string $\mathcal{W}_{3,4}^{b}$ encircling twists $t_{3}$ and $t_{4}$ is the product of operators encircling twists $t_{1}$, $t_{2}$ and $t_{5}$, $t_{6}$ i.e. $\mathcal{W}_{3,4}^{b} = \mathcal{W}_{1,2}^{b} \mathcal{W}_{5,6}^{b}$.
This can be seen easily as the logical operators in the form of closed strings can be opened up and running between boundaries.
This is possible as the logical operators encircling twists created in pairs have support on blue strings which can be deformed to end on the unbounded blue face.
Then, by adding stabilizers, the open strings can be made to encircle twists $t_{3}$ and $t_{4}$.
Similarly, for qubits not in the same block, see Fig.~\ref{fig:braid-entangling-nonadjacent}, the string encircling twists $t_3$, $t_4$ and $t_9$, $t_{10}$ can be deformed in similar way to that in surface codes~\cite{GowdaSarvepalli2020}.
The twists encoding qubits is given in Table~\ref{tab:twists_encoding}.
\begin{table}[htb]
\centering
\begin{tabular}{c|c}
\hline
\hline
Encoded qubit & Twists used \\
\hline
$a$ & $t_1$, $t_2$, $t_3$ \\
\hline
$b$ & $t_7$, $t_8$, $t_9$\\
\hline
$c$ & $t_4$, $t_5$, $t_6$\\
\hline
$d$ & $t_{10}$, $t_{11}$, $t_{12}$\\
\hline
\end{tabular}
\caption{Encoded qubits and the twists used for encoding.}
\label{tab:twists_encoding}
\end{table}
\begin{figure}[htb]
\centering
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = .85]{fig-braid-entangling-adjacent.pdf}
\subcaption{To realize entangling gate between two encoded qubits in the same block, braid the twist pair as shown counterclockwise.}
\label{fig:braid-entangling-adjacent}
\end{subfigure}
~
\begin{subfigure}{.225\textwidth}
\centering
\includegraphics[scale = .85]{fig-braid-entangling-adjacent-log-X.pdf}
\subcaption{After braiding $t_3$ and $t_4$, $\Bar{X}_1$ is modified as shown. This string can be expressed as a combination of string encircling twists $t_3$ and $t_4$ and $\Bar{X}_1$ before braiding.}
\label{fig:braid-entangling-adjacent-log-X}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .9]{fig-braid-entangling-nonadjacent.pdf}
\subcaption{An entangling gate between two encoded qubits in different blocks is realized by braiding twists as shown.}
\label{fig:braid-entangling-nonadjacent}
\end{subfigure}
\caption{Realizing entangling gate between qubits in the same block and different block}
\label{fig:braid-entangling}
\end{figure}
\begin{theorem}[Entangling gate]
Controlled-Z gate is realized up to phase gate on control qubit $a$ and target qubit $b$ by the following braiding.
\begin{compactenum}[i)]
\item Braid $t_3$ and $t_4$ for qubits in the same block.
\item Braid $t_3$ and $t_9$ for qubits in different block.
\end{compactenum}
\label{thm:multi-qb-entangling}
\end{theorem}
\begin{proof}
We prove for the qubits in the same block.
Extension to qubits from nonadjacent block is straightforward.
Encoding used is as shown in Fig.~\ref{fig:lo_charge_canonical}.
After braiding $t_{3}$ and $t_{4}$, we have $t_{3} \leftrightarrow t_{4}$, $t_{m} \rightarrow t_{m}$, $m = 1,2,5,6$.
We have the following transformation.
\begin{eqnarray*}
\begin{array}{ccc}
\mathcal{W}_{3,4}^{b} \mapsto \mathcal{W}_{3,4}^{b} & \vert & \bar{Z}_1\bar{Z}_2 \mapsto \bar{Z}_1\bar{Z}_2,\\
\mathcal{W}_{1,2}^{b} \mapsto \mathcal{W}_{1,2}^{b} & \vert & \bar{Z}_1 \mapsto \bar{Z}_1 \\
\mathcal{W}_{1,3}^{g} \mapsto \mathcal{W}_{1,4}^{g} & \vert & \bar{X}_1 \mapsto \bar{X}_1 \bar{Z}_1 \bar{Z}_2 = \bar{Y}_1 \bar{Z}_2\\
\mathcal{W}_{5,6}^{b} \mapsto \mathcal{W}_{5,6}^{b} & \vert & \bar{Z}_2 \mapsto \bar{Z}_2 \\
\mathcal{W}_{4,6}^{g} \mapsto \mathcal{W}_{3,6}^{g} & \vert & \bar{X}_2 \mapsto \bar{X}_2 \bar{Z}_1 \bar{Z}_2 = \bar{Z}_1 \bar{Y}_2\\
\end{array}
\end{eqnarray*}
In the third and the last transformations, the deformed strings can be expressed as $\mathcal{W}_{1,3}^r \mathcal{W}_{3,4}^b$ and $\mathcal{W}_{3,4}^b \mathcal{W}_{4,6}^r$ respectively, see Fig.~\ref{fig:Qb-cc-logY} in Appendix~\ref{sec:charge-braiding}.
This transformation is similar to that of combining $\Bar{Z}$ and $\Bar{X}$ to give $\Bar{Y}$.
By noting the logical operator transformation on the right side, we can conclude that braiding twists $t_3$ and $t_4$ results in controlled-$Z$ gate up to a phase gate on control and target qubits.
\end{proof}
To obtain Controlled-Z gate after performing the braids as indicated in Theorem~\ref{thm:multi-qb-entangling}, perform phase gate on control and target qubits.
Phase gate is performed by braiding the twist pair encircled by the logical $Z$ operator.
The procedure to realize the entangling gate stated in Theorem~\ref{thm:multi-qb-entangling} between the other encoded qubit pairs is given in Table~\ref{tab:entangling_braid_nonadjacent}.
\begin{table}[htb]
\centering
\begin{tabular}{c|c}
\hline
\hline
Encoded qubits & Braid to be performed \\
\hline
$(a,d)$ & $(t_{3},t_{10})$ \\
\hline
$(b,c)$ & $(t_{4},t_{9})$\\
\hline
$(b,d)$ & $(t_{4},t_{10})$\\
\hline
\end{tabular}
\caption{The entangling gate between encoded qubits is realized by performing the braiding indicated. The first qubit in the pair $(a,d)$, viz. $a$, is the control qubit and the other is the target qubit.}
\label{tab:entangling_braid_nonadjacent}
\end{table}
\begin{theorem}
In a color code with charge permuting twists, Clifford gates can be realized by braiding alone.
\end{theorem}
\begin{proof}
Follows from Theorem~\ref{thm:single-qb-Clifford} and Theorem~\ref{thm:multi-qb-entangling}.
\end{proof}
The braiding protocol for realizing encoded Clifford gates is given in Table~\ref{tab:gate_braiding}.
\begin{table}[htb]
\centering
\begin{tabular}{c|c}
\hline
\hline
Gate & Twists to be braided \\
\hline
$R_Z(\pi / 2)$ & $t_1$ and $t_2$ (Theorem~\ref{thm:single-qb-Clifford}) \\
\hline
$R_X(\pi / 2)$ & $t_1$ and $t_3$ (Theorem~\ref{thm:single-qb-Clifford}) \\
\hline
Adjacent CZ & $t_3$ and $t_4$, $t_1$ and $t_2$, $t_5$ and $t_6$ (Theorem~\ref{thm:multi-qb-entangling})\\
\hline
Nonadjacent CZ & $t_3$ and $t_9$, $t_1$ and $t_2$, $t_7$ and $t_8$ (Theorem~\ref{thm:multi-qb-entangling})\\
\hline
\end{tabular}
\caption{Braiding protocol for Clifford gates.}
\label{tab:gate_braiding}
\end{table}
\section{Clifford Gates using color permuting twists}
\label{sec:gates-color-permuting}
In this section, we describe the implementation of Clifford gates using color permuting twists.
We use four color permuting twists to encode a logical qubit, see Fig.~\ref{fig:Qb-cc-four-twist-encoding-color}.
One of the benefits of using quadruple twist encoding is that ancillas can be added and removed from the lattice as needed by lattice surgery.
For realizing phase and Hadamard gates, we use Pauli frame update~\cite{Hastings2015}.
Pauli frame update is done classically.
In this procedure, one has to keep track of the Pauli frame for each encoded qubit i.e. one has to maintain the information whether the canonical logical operator labels are exchanged.
Phase gate is realized by interchanging the labels of $X$ and $Y$ logical operators and for realizing Hadamard gate, the labels of $X$ and $Z$ logical operators are interchanged.
If a $Z$ measurement is to be done after Hadamard gate, then after Pauli frame update, we take into account the Pauli frame update and measure the canonical $X$ logical operator.
We implement CNOT gate by making use of an additional ancilla qubit and holes~\cite{Terhal2015, Brown2017}.
This is done by joint $X$ parity measurement and joint $Z$ parity measurement with an ancilla qubit.
The protocol is given in Table~\ref{tab:cc_cnot}.
\begin{table}[htb]
\centering
\begin{tabular}{ll}
\hline
& Protocol for implementing CNOT gate~\cite{Terhal2015} \\
\hline
(1) & Prepare ancilla in the state $|0\rangle$. \\
(2) & Perform joint $X$ parity measurement on ancilla and target qubit. \\
(3) & Perform joint $Z$ parity measurement on ancilla and control qubit.\\
(4) & Perform Hadamard on ancilla qubit and measure it in $X$ basis.\\
(5) & Measure ancillas and apply correction as given in Equation~\ref{eqn:CNOT}.\\
\hline
\end{tabular}
\caption{Protocol for implementing CNOT gate by joint measurements with an ancilla.}
\label{tab:cc_cnot}
\end{table}
Suppose that $|c\rangle$ and $|t\rangle$ are the states of control and target qubits with $c = 0,1$ and $t = 0,1$.
Let the measurement outcome of joint $X$ parity measurement be $m_{xx}$ and those of joint $Z$ parity measurement and ancilla measurement be $m_{zz}$ and $m_{x}$ respectively.
Then it can be shown that at the end of the protocol the initial state is modified as~\cite{Terhal2015}
\begin{equation}
|c\rangle |t\rangle |0\rangle \rightarrow |c\rangle Z^{m_{xx}} X^{m_{zz}}|c + t\rangle Z^{m_{x}} |+\rangle.
\label{eqn:CNOT}
\end{equation}
Ancilla qubit is disentangled by measuring in the $X$ basis.
\begin{figure}[htb]
\centering
\begin{subfigure}{.15\textwidth}
\centering
\includegraphics[scale = .75]{fig-Qb-cc-four-twist-encoding-color.pdf}
\subcaption{Four twist encoding used for realizing encoded gates with color permuting twists.}
\label{fig:Qb-cc-four-twist-encoding-color}
\end{subfigure}
~
\begin{subfigure}{.15\textwidth}
\centering
\includegraphics[scale = .75]{fig-primal-qubit-hole.pdf}
\subcaption{Three holes are used to encode a logical (primal) qubit.}
\label{fig:primal-qubit-hole}
\end{subfigure}
~
\begin{subfigure}{.15\textwidth}
\centering
\includegraphics[scale = .75]{fig-hole-encoding.pdf}
\subcaption{Three holes are used to encode a logical (dual) qubit.}
\label{fig:hole-encoding}
\end{subfigure}
\caption{Four twist encoding that used red color permuting twists. Also shown is the dual logical qubit encoded with a triple of holes.}
\label{fig:twist-hole-encoding}
\end{figure}
\noindent \emph{Motivation for using holes.} The encoded CNOT gate protocol given in Table~\ref{tab:cc_cnot} involves joint parity measurement of logical $Z$ and logical $X$ operators.
These operators are high weight owing to the separation between the twists.
(We keep the twist separation large to make the distance large.)
Therefore, measuring the the joint operators by doing gates between physical qubits in the support of logical operators and ancilla qubit fault-tolerant will be quite complex.
However, an alternate scheme exists for doing fault-tolerant measurement of these operators.
This alternate scheme in the context of surface codes involves creating holes, braiding them around twists and measuring them~\cite{Brown2017}.
We adapt this technique to the color permuting twists.
\noindent \emph{Holes in color code lattices.}
Here we briefly review holes in color code lattices.
For a more detailed treatment of holes in color codes, we refer the reader to Ref.~\cite{Fowler2011}.
A hole is a region in the lattice where stabilizers are not measured.
Suppose that we have to introduce a red hole in the lattice.
We choose a red face and do not measure the stabilizers defined on it.
To expand the hole, we take the vertices on one of the edges incident on the face.
Then $ZZ$ and $XX$ measurements are performed on the qubits on the vertices of the chosen edge.
Using this procedure, a hole can be made arbitrarily big.
We use triple hole encoding~\cite{Fowler2011}.
Two more twists, one each of green and blue colors, are created by the same procedure.
A logical qubit encoded in a hole is shown in Fig.~\ref{fig:primal-qubit-hole} and Fig.~\ref{fig:hole-encoding}.
Logical operators are the strings encircling hole and the string net (in the shape of T).
Depending on whether we choose $Z$ or $X$ logical operator to encircle the hole, we get primal and dual qubits respectively.
The encoding shown in Fig.~\ref{fig:primal-qubit-hole} is that of a primal qubit whereas the encoding in Fig.~\ref{fig:hole-encoding} is that of a dual qubit.
\subsection{Hole-twist braiding}
We now show how logical operators are modified by braiding a hole around twists.
Three hole encoding~\cite{Fowler2011} is used, see Fig.~\ref{fig:primal-qubit-hole} and Fig.~\ref{fig:hole-encoding}.
In this scheme, a logical qubit is encoded using three holes (of different color).
For primal qubit, the $Z$ logical operator is the string encircling a hole and the logical $X$ operator is the string having tree like structure connecting the three holes.
For the dual qubit, the $X$ logical operator is the string encircling the hole and $Z$ logical operator is the string having tree like structure.
The idea of braiding holes around twists to implement CNOT gate was proposed in Ref.~\cite{Brown2017} in the context of surface codes.
In this paper, we explore the same idea in the context of color permuting twists.
\begin{figure*}
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-twist-hole-braid-1.pdf}
\subcaption{After crossing the $T$-line, the hole has changed its color from blue to green.
Note that $\bar{X}$ does not change color because red twists do not permute the color of red strings.}
\label{fig:twist-hole-braid-1}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-twist-hole-braid-2.pdf}
\subcaption{Hole regains its color after crossing the second domain wall. Logical operators of the dual qubit encoded using holes are not transformed.}
\label{fig:twist-hole-braid-2}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-twist-hole-braid-3.pdf}
\subcaption{The hole is returned to its position after braiding. Note that the deformed logical $Z$ operator of dual qubit now encircles twists $t_2$ and $t_3$ and is equivalent to $\bar{Z}_H \bar{X}_T$.}
\label{fig:twist-hole-braid-3}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-twist-hole-braid-4.pdf}
\subcaption{The equivalence of the deformed logical $Z$ to $\bar{Z}_H \bar{X}_T$ can be seen by adding a stabilizer as shown.}
\label{fig:twist-hole-braid-4}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-twist-hole-braid-5.pdf}
\subcaption{Logical $Z$ of the qubit encoded using twists, $\Bar{Z}_T$, is deformed by hole movement as blue string cannot have terminal points on green hole.}
\label{fig:twist-hole-braid-5}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-twist-hole-braid-6.pdf}
\subcaption{$\Bar{Z}_T$ and hole change color as they crosses the domain wall of twists $t_1$ and $t_2$. Color of hole and $\Bar{Z}_T$ are different and hence $\Bar{Z}_T$ is further deformed.}
\label{fig:twist-hole-braid-6}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-twist-hole-braid-7.pdf}
\subcaption{The deformed logical $Z$ operator of twist qubit is equivalent to $\Bar{Z}_T \Bar{X}_H$.}
\label{fig:twist-hole-braid-7}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-twist-hole-braid-8.pdf}
\subcaption{By adding the stabilizer as shown, one can see the equivalence between deformed logical $Z$ operator to $\Bar{Z}_T \Bar{X}_H$.}
\label{fig:twist-hole-braid-8}
\end{subfigure}
\caption{Braiding hole around twists $t_2$ and $t_4$ to implement the entangling gate given in Equation~\eqref{eqn:controlled-X}. The effect on the logical operators of the dual qubit is shown from Fig.~\ref{fig:twist-hole-braid-1} to Fig.~\ref{fig:twist-hole-braid-4} and that of qubit encoded using twists is shown from Fig.~\ref{fig:twist-hole-braid-5} to Fig.~\ref{fig:twist-hole-braid-8}.}
\label{fig:twist-hole-braid}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-hole-twist-braid-1.pdf}
\subcaption{Logical $Z$ operator of the dual qubit is stretched as hole is braided around twists.}
\label{fig:hole-twist-braid-1}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-hole-twist-braid-2.pdf}
\subcaption{After braiding $\bar{Z}_H$ is deformed around twists $t_3$ and $t_4$ and encircles them.}
\label{fig:hole-twist-braid-2}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-hole-twist-braid-3.pdf}
\subcaption{The deformed logical $Z$ operator of the primal qubit is equivalent to $\bar{Z}_T \bar{Z}_H$}
\label{fig:hole-twist-braid-3}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-hole-twist-braid-4.pdf}
\subcaption{The equivalence of the deformed $\bar{Z}_H$ to $\bar{Z}_T \bar{Z}_H$ can be seen by adding the stabilizer as shown.}
\label{fig:hole-twist-braid-4}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-hole-twist-braid-5.pdf}
\subcaption{The operator $\bar{X}_T$ deforms as hole is moved. This is because green string cannot have terminal points in a blue hole.}
\label{fig:hole-twist-braid-5}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-hole-twist-braid-6.pdf}
\subcaption{The operator $\bar{X}_T$ is dragged along with the hole and at the end of braiding, $\bar{X}_T$ encircles the hole.}
\label{fig:hole-twist-braid-6}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-hole-twist-braid-7.pdf}
\subcaption{The deformed $\bar{X}_T$ is equivalent to $\bar{X}_T \bar{X}_H$.}
\label{fig:hole-twist-braid-7}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-hole-twist-braid-8.pdf}
\subcaption{The equivalence stated in Fig.~\ref{fig:hole-twist-braid-7} can be seen by adding the stabilizer as shown.}
\label{fig:hole-twist-braid-8}
\end{subfigure}
\caption{Braiding hole around twists $t_3$ and $t_4$ to implement CNOT gate with qubit encoded using twists as control and qubit encoded using holes as target. The effect on the logical operators of the primal qubit is shown in Fig.~\ref{fig:hole-twist-braid-1} to Fig.~\ref{fig:hole-twist-braid-4} and the same for qubit encoded using twists is shown from Fig.~\ref{fig:hole-twist-braid-5} to Fig.~\ref{fig:hole-twist-braid-8}.}
\label{fig:hole-twist-braid}
\end{figure*}
Consider the braiding as shown in Fig~\ref{fig:twist-hole-braid}.
Let $\Bar{Z}_{T} $ and $\Bar{X}_{T}$ be the $Z$ and $X$ logical operators of logical qubit encoded by twists and let $\Bar{Z}_{H} $ and $\Bar{X}_{H}$ be the $Z$ and $X$ logical operators of logical qubit encoded by holes.
The hole is braided around twists $t_2$ and $t_4$.
An interesting phenomenon occurs as the hole crosses the domain wall.
Suppose that the hole in question has blue color.
To move the hole across the domain wall, it has to pass through the same.
In doing so, we merge the hole with one of the blue modified faces.
Note that the outgoing edges from the modified faces are incident on green faces, see Fig.~\ref{fig:hex-lattice}.
When the hole is to be further moved from the domain wall, we now have to merge the modified face with one of the green faces.
When the hole is completely deformed away from the domain wall, its color will have permuted.
As the hole the crosses the first domain wall, the color changes from blue to green as shown in Fig.~\ref{fig:twist-hole-braid-1}.
Again when the hole crosses the domain wall second time, its color is changed back to blue as shown in Fig.~\ref{fig:twist-hole-braid-2}.
The hole is returned to its original position as shown in Fig.~\ref{fig:twist-hole-braid-3}.
The operator around twists $t_2$ and $t_4$ is $\bar{X}_T \bar{Z}_H$ which can be seen by adding a stabilizer as shown in Fig.~\ref{fig:twist-hole-braid-4} which decomposes the operator into $\bar{X}_T$ and $\bar{Z}_H$.
The transformation of the logical operators of the qubit encoded using twists is shown from Fig.~\ref{fig:twist-hole-braid-5} to Fig.~\ref{fig:twist-hole-braid-8}.
The logical operator transformation effected by this braiding is given in Table~\ref{tab:hole_twist_horizontal}.
From the transformations, it is clear that braiding hole around twists realizes an entangling gate.
\begin{table}[htb]
\centering
\begin{tabular}{ccc}
$\Bar{Z}_{T}$ & $\longrightarrow$ & $\Bar{Z}_{T}\Bar{X}_{H}$\\
$\Bar{X}_{T}$ & $\longrightarrow$ & $\Bar{X}_{T}$ \\
$\Bar{Z}_{H}$ & $\longrightarrow$ & $\Bar{X}_{T}\Bar{Z}_{H}$\\
$\Bar{X}_{H}$ & $\longrightarrow$ & $\Bar{X}_{H}$
\end{tabular}
\caption{Transformation of logical operators after braiding hole around twists as shown in Fig.~\ref{fig:twist-hole-braid}.}
\label{tab:hole_twist_horizontal}
\end{table}
One can see that this transformation is equivalent to that obtained by applying Hadamard gate on the control qubit before and after CNOT gate.
This gate is used with the target qubit initialized in $|0\rangle$ to measure $X$ operator~\cite{Nielsen2010}.
\begin{lemma}[Twist-hole braiding-1]
Braiding hole encoding the dual qubit around twists $t_3$ and $t_4$ as shown in Fig.~\ref{fig:twist-hole-braid} realizes the gate
\begin{equation}
(H \otimes I) CNOT (H \otimes I) = |+\rangle \langle +| \otimes I + |-\rangle \langle -| \otimes X
\label{eqn:controlled-X}
\end{equation}
with twist qubit as the control and dual qubit as the target.
\end{lemma}
\begin{remark}
The string encircling twists $t_2$ and $t_4$ in Fig.~\ref{fig:Qb-cc-four-twist-encoding-color} is equivalent to logical $X$ operator.
\end{remark}
One can verify that by using primal qubit, i.e. the operator encircling the hole is $\Bar{Z}$ and the tree like operator is $\Bar{X}$, and braiding around $t_2$ and $t_4$ the following transformation is realized:
$\Bar{Z}_T \rightarrow \Bar{Z}_T \Bar{Z}_H $, $ \Bar{X}_T \rightarrow \Bar{X}_T$, $ \Bar{Z}_H \rightarrow \Bar{Z}_H$, $\Bar{X}_H \rightarrow \Bar{X}_T\Bar{X}_H$.
One can see that this is CNOT gate between primal qubit and qubit encoded with twists with the primal qubit as control.
To realize CNOT gate between qubits encoded using twists and holes, we use primal qubit.
We do Pauli frame update on the hole so that the logical $X$ and $Z$ are interchanged.
If the primal qubit encoded using holes is deformed around twists $t_3$ and $t_4$, then the transformation given in Table~\ref{tab:hole_twist_vertical} is realized.
The transformation of the logical operators of the primal qubit and the qubit encoded by twists is shown in Fig.~\ref{fig:hole-twist-braid}.
This is CNOT gate between qubit encoded by twist and primal hole with twist qubit as the control.
\begin{table}[htb]
\centering
\begin{tabular}{ccc}
$\Bar{Z}_{T}$ & $\longrightarrow$ & $\Bar{Z}_{T}$\\
$\Bar{X}_{T}$ & $\longrightarrow$ & $\Bar{X}_{T}\Bar{X}_H$ \\
$\Bar{Z}_{H}$ & $\longrightarrow$ & $\Bar{Z}_T\Bar{Z}_{H}$\\
$\Bar{X}_{H}$ & $\longrightarrow$ & $\Bar{X}_{H}$
\end{tabular}
\caption{Transformation of logical operators after braiding hole around twists as shown in Fig.~\ref{fig:hole-twist-braid}.}
\label{tab:hole_twist_vertical}
\end{table}
\begin{lemma}[Twist-hole braiding-2]
Performing Pauli update $\bar{Z} \leftrightarrow \bar{X}$ on the primal qubit and braiding hole encoding primal qubit around twists $t_3$ and $t_4$ as shown in Fig.~\ref{fig:hole-twist-braid} realizes CNOT gate with qubit encoded using twists as control and qubit encoded using holes as target.
\end{lemma}
\begin{remark}
The string encircling twists $t_3$ and $t_4$ in Fig.~\ref{fig:Qb-cc-four-twist-encoding-color} is equivalent to logical $Z$ operator.
\end{remark}
Similarly, by using primal qubit encoded using holes and braiding around $t_3$ and $t_4$, one can realize controlled phase gate between twist and hole with twist as the control.
\begin{figure*}
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-color-perm-CNOT-1.pdf}
\subcaption{This protocol requires an ancilla to implement CNOT gate. Twists used for encoding control (C), target (T) and ancilla (A) qubits are shown. Logical $Z$ and $X$ operators are in the canonical form.}
\label{fig:color-perm-CNOT-1}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .85]{fig-color-perm-CNOT-2.pdf}
\subcaption{ Braiding hole along the operator $\Bar{X}_T \Bar{X}_A$ and measuring it in the end to implement joint $X$ measurement on target and ancilla qubits.}
\label{fig:color-perm-CNOT-2}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .8]{fig-color-perm-CNOT-3.pdf}
\subcaption{Joint $Z$ measurement on control and ancilla qubits is performed by braiding hole along the string $\Bar{Z}_C \Bar{Z}_A$ and measuring it.}
\label{fig:color-perm-CNOT-3}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[scale = .8]{fig-color-perm-CNOT-4.pdf}
\subcaption{Ancilla qubit is measured by braiding hole along $\Bar{X}_A$ which is the string encircling twists $t_{11}$ and $t_{12}$ after Pauli frame update measuring it.}
\label{fig:color-perm-CNOT-4}
\end{subfigure}
\caption{Protocol for implementing CNOT gate using twists and holes.}
\label{fig:color-perm-CNOT}
\end{figure*}
\subsection{CNOT Gate}
We now present the encoded version of the CNOT gate protocol given in Table~\ref{tab:cc_cnot}.
This protocol makes use of the encoded gates realized using twists and holes as discussed in the previous subsection.
\begin{table}[htb]
\centering
\begin{tabular}{ll}
\hline
& Protocol for implementing Encoded CNOT gate \\
\hline
(1) & Prepare ancilla encoded using twists in the state $|\Bar{0}\rangle$. \\
(2) & Perform $\Bar{X}_T \Bar{X}_A$ parity measurement on ancilla and target qubit \\
& by braiding hole around twists encircled by the operator $\Bar{X}_T \Bar{X}_A$.\\
(3) & Perform $\Bar{Z}_C \Bar{Z}_A$ parity measurement on ancilla and control qubit\\
& by braiding hole around twists encircled by the operator $\Bar{Z}_C \Bar{Z}_A$.\\
(4) & Perform Hadamard on ancilla qubit and measure it in $X$ basis.\\
(5) & Measure the ancillas encoded using holes and apply\\
& correction as given in Equation~\ref{eqn:CNOT}.\\
\hline
\end{tabular}
\caption{Protocol for implementing encoded CNOT gate by joint measurements with an ancilla.}
\label{tab:cc_cnot_encoded}
\end{table}
\begin{remark}
Fresh ancillas are prepared for steps $2$ to $4$ in the protocol given in Table~\ref{tab:cc_cnot_encoded} for encoded CNOT gate.
\end{remark}
The procedure for implementing the encoded CNOT gate using color permuting twists is shown in Fig.~\ref{fig:color-perm-CNOT}.
In Fig.~\ref{fig:color-perm-CNOT-1}, control, target and ancilla qubits encoded using a quadruple of twists is shown.
We abbreviate control, target and ancilla qubits as C, T and A respectively.
The ancilla is initialized in the $|\Bar{0}\rangle$ state.
Joint $\Bar{X}_T \Bar{X}_A$ parity measurement by using dual qubit encoded using holes is shown in Fig.~\ref{fig:color-perm-CNOT-2}.
This parity measurement is equivalent to doing the gate indicated in Equation~\eqref{eqn:controlled-X} with target and ancilla $A$ qubits as control and the hole as target.
Recall that braiding a dual qubit around twists encircled by logical $X$ operator realizes the gate in Equation~\eqref{eqn:controlled-X} with twist qubit being control and dual qubit being target.
Braiding hole as shown in Fig.~\ref{fig:color-perm-CNOT-2} realizes the gate in Equation~\eqref{eqn:controlled-X} between target qubit and ancilla qubit encoded using twists as control and qubit encoded using holes as target.
The dual qubit is initialized in the $|\bar{0}\rangle$ state, braided around twists as shown and measured in the end.
The measurement outcome reveals the parity of the operator $\Bar{X}_T \Bar{X}_A$.
Similarly, to carry out $\Bar{Z}_C \Bar{Z}_A$ parity measurement, a primal qubit encoded by holes is created and its logical $X$ and $Z$ operators are interchanged by Pauli frame update.
The primal qubit is initialized in the $+1$ eigenstate of $\Bar{X}_H$ so that after Pauli frame update, the state of the primal qubit is $|\bar{0}\rangle$, deformed around twists and measured in the end, see Fig.~\ref{fig:color-perm-CNOT-3}.
This braiding realizes CNOT gate between control and ancilla qubits encoded using twists as control and the dual qubit encoded with holes as target.
Finally, Hadamard on ancilla is performed by Pauli frame update and measurement in the $X$ basis is done as shown in Fig.~\ref{fig:color-perm-CNOT-4}.
\subsection{Non-Clifford gate}
In this section we discuss implementing a non-Clifford gate.
We use magic state distillation to implement the non-Clifford gate.
The protocol for implementing the non-Clifford gate is given in Table~\ref{tab:non-clifford}.
This protocol works for both charge permuting and color permuting twists.
\begin{table}[htb]
\begin{tabular}{ll}
\hline
& Protocol for $\Lambda\left( e^{i \theta} \right)$ gate\\
\hline
(0) & Prepare the magic state $|\phi\rangle = 2^{-1/2}(|0\rangle + e^{i \theta}|1\rangle)$. \\
(1) & Perform CNOT gate between data qubit and magic state\\
& with the former as control and the latter as target.\\
(2) & Measure the magic state qubit. \\
\hline
\end{tabular}
\caption{Protocol for implementing non-Clifford gate.}
\label{tab:non-clifford}
\end{table}
We have dropped the joint $Z$ measurement in the protocol given in Ref.~\cite{Bravyi2006} so as to suit fault-tolerant topological quantum computation.
Larger the code distance, more the number of qubits in the support of logical operator and hence more CNOT gate to be done between physical qubits in the lattice and ancilla.
Similarities and differences with previous protocols is given in Table~\ref{tab:comp_protocol}.
\begin{table}[htb]
\centering
\begin{tabular}{p{3cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}}
\hline\hline
Step & Ref.~\cite{Bravyi2005} & Ref.~\cite{Bravyi2006}& Proposed protocol\\
\hline
Joint $Z$ parity measurement & present & present & absent\\
\hline
CNOT gate & present & present & present\\
\hline
Ancilla measurement & absent & present & present\\
\hline
\end{tabular}
\caption{Similarities and differences of our protocol with those in literature.}
\label{tab:comp_protocol}
\end{table}
Let $|\psi \rangle = a|0\rangle + b|1\rangle$ be the state of the qubit on which the non-Clifford gate has to be performed.
The state of ancillary qubit prepared in magic state is $|\phi \rangle = 2^{-1/2}(|0\rangle + e^{i \theta}|1\rangle)$.
The joint state of the two qubits is $|\alpha \rangle = |\psi\rangle |\phi\rangle = 2^{-1/2}(a|00\rangle + ae^{i \theta}|01\rangle + b|10\rangle + be^{i \theta}|11\rangle)$.
After implementing CNOT gate, the joint state is $|\beta \rangle = CNOT|\alpha \rangle$,
\begin{eqnarray*}
|\beta \rangle &=& \frac{1}{\sqrt{2}}\left(a|00\rangle + a e^{i \theta}|01\rangle + b|11\rangle + b e^{i \theta}|10\rangle \right) \\
&=& \frac{1}{\sqrt{2}}\left(\left( a|0\rangle + be^{i \theta}|1\rangle \right) |0\rangle + e^{i \theta}\left( a|0\rangle + be^{-i \theta}|1\rangle \right) |1\rangle \right) \\
&=& \frac{1}{\sqrt{2}}\left(\Lambda\left( e^{i \theta} \right)|\psi \rangle |0\rangle + e^{i \theta} \Lambda\left( e^{-i \theta} \right)|\psi \rangle |1\rangle \right).
\end{eqnarray*}
Upon measuring ancillary qubit, either the desired gate or its conjugate is implemented on the data qubit with equal probability.
If $\theta = \pi / 4$, then, if the measurement outcome is $1$, then the desired transformation on data qubit can be obtained by performing phase gate.
In color codes with twists, non-Clifford gate can be implemented by initializing an encoded qubit in the magic state and implementing the protocol given above.
Alternately, one can use lattice surgery~\cite{Fowler2011} to merge the lattice containing encoded magic state with the lattice having encoded qubit.
Then, the non-Clifford gate is implemented by performing the encoded versions of operations described in the protocol.
After measuring the ancilla qubit i.e. qubit initialized in the magic state, the (sub) lattice containing the same can be detached.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we have presented a systematic way to introduce charge permuting and color permuting twists in $2$-colexes.
We have also discussed the coding theoretic aspects of the same.
We have shown that Clifford gates can be implemented using charge permuting twists by braiding alone.
Encoded gates in the case of color permuting twists are implemented by a combination of Pauli frame update for single qubit gates and joint parity measurements with an ancilla for CNOT gate.
Joint measurements are carried out by braiding holes around twists.
To implement non-Clifford gate, we use magic state injection.
A direction for future research could be to use techniques other than magic state distillation.
A non-Clifford gate was implemented in surface codes by making use of multiple locally interacting copies~\cite{Brown2020}.
Similar techniques, as presented in Ref.~\cite{Brown2020}, could be used in the case of color codes with twists.
Decoding color codes with twists is another fruitful area for further exploration.
\section*{Appendix}
|
{
"timestamp": "2021-04-09T02:14:46",
"yymm": "2104",
"arxiv_id": "2104.03669",
"language": "en",
"url": "https://arxiv.org/abs/2104.03669"
}
|
\section*{Appendix}
\subsection{Samples}
\label{AppendixSample}
The sample was made in the same batch as the one used in \cite{Iftikhar2016}, with additional fabrication steps.
It consists of a Ga(Al)As two-dimensional electron gas buried $105$\,nm below the surface, of density $2.5 \times 10^{11}\,\mathrm{cm}^{-2}$ and of mobility $10^6\,\mathrm{cm}^2\mathrm{V}^{-1}\mathrm{s}^{-1}$.
Its nanostructuration is performed by standard e-beam lithography, dry etching and metallic deposition.
The central metallic island (nickel (30\,nm), gold (120\,nm) and germanium (60\,nm)) was thermally annealed (440\,$^\circ$C for 50\,s) to achieve a good ohmic contact with the 2DEG.\\
The contact quality between the metallic island and the 2DEG is fully characterized, through the individual determination of the electron reflection probability at the interface for each connected quantum Hall channel, with the same experimental procedure previously detailed in Methods of \cite{Iftikhar2015}.
We find a reflection probability below $\lesssim0.001\%$ (the statistical uncertainty) for all the used channels.\\
The typical electronic level spacing in the metallic island is estimated to be negligibly small ($\delta\approx k_\mathrm{B}\times0.2\,\mu$K), based on the electronic density of states of gold ($\nu_\mathrm{F}\approx1.14\times 10^{47}$\,J$^{-1}$m$^{-3}$) and the metallic island volume ($\approx3\,\mu$m$^3$).\\
Due to technical problems, the initial electrostatic gates (shown on Fig.~\ref{fig-sample}) were etched out, redefined and redeposited with $40$\,nm of aluminum.\\
An important device parameter is the charging energy $E_\mathrm{C}\equiv e^2/2C$ of the island.
For this specific device, one of the channel remains open and standard Coulomb diamond determination of $E_\mathrm{C}$ could not be performed.
Instead, the value $E_\mathrm{C}\simeq k_\mathrm{B}\times 0.37$\,K is obtained by fitting the overall $G(V)$ tunnel data at a uniform temperature, at all $T\simeq T_\mathrm{node}\simeq T_\mathrm{env}$ (from $8$ to $90$\,mK) and for all the series resistances $R_\mathrm{K}/N$ with $N\in\{2,3,4\}$.
Note that this value is slightly higher ($\sim+20\%$) than the one found in \cite{Iftikhar2016}.
The reduced geometrical capacitance $C$ might come from imperfect new gates.
However, note also that $E_\mathrm{C}$ is sufficiently large to have a relatively small impact, notably in the data-theory comparison as a function of a temperature bias.
This can be seen in Fig.~\ref{fig-DCBtunnelT} from the small difference between black continuous lines and red dashed lines at the largest $T_\mathrm{node}$.\\
The resistance in series with the studied, non-ballistic channels is simply taken as the dc electrical resistance $R_\mathrm{K}/N$ of the constitutive $N$ quantum Hall channels in parallel.
In principle, deviations from this value could occur at high frequencies.
However, these deviations are limited by the high-frequency cutoff introduced by the capacitance $C$ of the island.
For instance, such deviations could result from a non-zero conductivity across the 2D bulk at the frequency $\nu$ when the energy $h\nu$ becomes comparable with the quantum Hall gap.
Yet, in our sample the quantum Hall gap is about two orders of magnitude higher than $h/RC=N\times E_\mathrm{C}$.
Also, the transit time $t_\mathrm{transit}$ along the micron-scale distance between the back-scattering location in the non-ballistic channel and the island could lead to an inductive correction to $R$.
Yet, given the typical velocity of $\sim 10^5\,$m/s for the propagation of charge along the quantum Hall edge, the associated energy scale $h/t_\mathrm{transit}$ is about one order of magnitude higher than $h/RC$.
In practice, we check the validity of our $RC$ circuit description by comparing the conductance data with the DCB theory in the well-established regimes of a uniform temperature (see Figs.~2, 9, 10, and full symbols in Figs.~3 and 4).
\subsection{$P(E)$ theory of dynamical Coulomb Blockade for a tunnel junction}
\label{DCB}
Here we focus on the predictions for the DCB renormalization of the transmission probability $\tau$ across an electronic channel in the tunnel regime ($\tau,\tau_\infty\ll1$), when it is embedded in an $RC$ circuit.
These results can be applied to a high-resistance tunnel junction including many such channels replacing $\tau/R_\mathrm{K}$ and $\tau_\infty/R_\mathrm{K}$ by, respectively, the junction renormalized and intrinsic differential conductance.
\subsubsection{Numerically efficient formulation at arbitrary voltage and uniform temperature $T=T_\mathrm{node}=T_\mathrm{env}$ with an $RC$ environment}
\label{DCBtunnel}
Numerical calculations of the conductance of a coherent conductor in the tunnel limit in presence of environmental back-action were made with the efficient formulation of the DCB theory for small tunnel junctions given in \cite{Joyez1997}.
In this section, we recapitulate the expressions used in the case of uniform temperatures and finite bias voltages.
The transmission probability $\tau\equiv R_\mathrm{K}\mathrm{d}I/\mathrm{d}V$ across a short electronic channel in the tunnel regime, embedded in an electromagnetic environment described by the series impedance $Z(\omega)$, at a uniform temperature $T=T_\mathrm{node}=T_\mathrm{env}$, and for a voltage $V$ applied here across the channel (corresponding to $V-V_\mathrm{node}$ in Fig.~\ref{fig-dcbTunnelV}, as shown in Fig.~\ref{fig-sample}(a)) reads \cite{Joyez1998}:
\begin{equation}
\begin{split}
\tau (&V,T)/\tau_\infty-1 = \\
&\int_0^{+\infty}\!dt\,\frac{2 \pi t}{\sinh^{2} \frac{\pi t k_\mathrm{B} T}{\hbar}}\left(\frac{k_\mathrm{B} T}{\hbar}\right)^2 \mathrm{Im}\left[e^{J(t)}\right] \cos \frac{e V t}{\hbar},
\label{PhilEquilibrium}
\end{split}
\end{equation}
with the channel intrinsic resistance $R_\mathrm{K}/\tau_\infty$ assumed to be very large compared to the environmental impedance $\mathrm{Re}[Z(\omega)] \ll R_\mathrm{K}/\tau_\infty$.
For the simplified RC model of the electromagnetic environment shown in article Fig.~\ref{fig-sample}(b) ($Z(\omega)=R/(1+iRC\omega)$), $J(t)$ reads:
\begin{equation}
\begin{split}
J(t)=\frac{\pi R}{R_\mathrm{K}}\Bigg(& \left(1-e^{- \mid t\mid/RC} \right) \left( \mathrm{cot}\frac{\hbar}{2RCk_\mathrm{B}T}-i \right)\\
&-\frac{2 k_\mathrm{B} T \mid t \mid}{\hbar}
+ 2 \sum_{n=1}^{+\infty}\frac{ 1-e^{-\omega_n \mid t\mid}}{ \pi n \left[ (RC\omega_n)^2-1\right]} \Bigg),
\end{split}
\label{eqJ(t)}
\end{equation}
where $\omega_n=2\pi n k_\mathrm{B} T/\hbar$ are Matsubara's frequencies and
\begin{equation}
\begin{split}
2 \sum_{n=1}^{+\infty}\frac{ 1-e^{-\omega_n t}}{ \pi n \left[ (RC\omega_n)^2-1\right]} = - \frac{1}{\pi}\big[2\gamma +\Psi(-x)+\Psi(x)\\
+2\ln(1-y) + \frac{y}{1+x}~_2 F_1(1,1+x,2+x,y)
\\+ \frac{y}{1-x} ~_2F_1(1,1-x,2-x,y) \big],
\end{split}
\end{equation}
where $\gamma\simeq0.5772$ is Euler's constant, $\Psi$ is the logarithmic derivative of the Gamma function, $_2F_1$ is the hypergeometric
function, $y=\mathrm{exp}(\frac{-2 \pi t k\mathrm{_B} T}{\hbar})$, and $x=E_\mathrm{C} R_\mathrm{K}/(2\pi^2R k\mathrm{_B} T)$ with $E_\mathrm{C}=e^2/(2C)$ the charging energy.
\subsubsection{Analytical asymptotic expressions versus $V$ at $T=T_\mathrm{node}=T_\mathrm{env}=0$, and versus $T=T_\mathrm{node}=T_\mathrm{env}$ at $V=0$}
\label{DCBtunnelTV0}
We detail here the derivation of analytical expressions at a uniform temperature $T=T_\mathrm{node}=T_\mathrm{env}$ for the asymptotic limits $k_\mathrm{B} T\ll eV\ll \hbar/RC$ (abbreviated in equations as $V\rightarrow 0$, $T=0$), plotted on Fig.~\ref{fig-dcbTunnelV}, and $eV\ll k_\mathrm{B} T \ll \hbar/RC$ (abbreviated as $T\rightarrow 0$, $V=0$), plotted on Fig.~\ref{fig-DCBtunnelT}.
Although the expression given by Eq.~\eqref{PhilEquilibrium} is convenient for performing numerical evaluations in most practical situations, the singularity of the integrand in Eq.~\eqref{PhilEquilibrium} is troublesome when attempting to obtain analytical results.
To express asymptotic limits in the case of a $RC$ environment, other formulations are required.
(\textit{i}) At $T=T_\mathrm{node}\ll eV/k_\mathrm{B}$, in \cite{Devoret1990}, the conductance at low voltages with respect to the capacitive cutoff ($eV\ll \hbar/RC$) is obtained from a $P(\epsilon)$ formulation such as Eq.~\eqref{eqTunnelP(E)}.
The transmission probability then reads:
\begin{equation}
\begin{split}
\frac{\tau(T=0,V\rightarrow 0)}{\tau_\infty}\simeq \frac{1+2R/R_\mathrm{K}}{\Gamma(2+2R/R_\mathrm{K})}
\left(\frac{\pi R\,eV}{e^\gamma R_\mathrm{K} E_\mathrm{C}}\right)^{2R/R_\mathrm{K}}.
\end{split}
\label{TunnelVtoZero}
\end{equation}
(\textit{ii}) For the linear conductance at $eV\ll k_\mathrm{B}T$, we have developed an equivalent formulation of the tunnel conductance by extending the analysis to the complex plane in $t$, managing the poles and shifting the integral contour.
We then get:
\begin{equation}
\frac{\tau(V=0,T)}{\tau_\infty} = \int_0^{+\infty}\!dt\, \frac{\pi k_\mathrm{B} T /\hbar}{\cosh^{2} \frac{\pi t k_\mathrm{B} T}{\hbar}} \times e^{J^\dagger (t,T)}.
\label{Geq2}
\end{equation}
For the simplified $RC$ model of the electromagnetic environment shown in Fig.~\ref{fig-sample}(b), $J^\dagger$ reads:
\begin{equation}
\begin{split}
J^\dagger (t,T)=&\frac{\pi R}{R_\mathrm{K}}\left(\frac{\cos{\frac{\hbar}{2RCk_\mathrm{B} T}}-e^{-t/RC}}{\sin{ \frac{\hbar}{2RCk_\mathrm{B} T}}}-\frac{2tk_\mathrm{B}T}{\hbar}\right)\\
&+\frac{2R}{R_\mathrm{K}}\sum_{n=1}^{\infty}\frac{1-(-1)^n e^{-\omega_nt}}{n(\omega_n^2R^2C^2-1)}.
\end{split}
\label{eqJdagger}
\end{equation}
The sum over Matsubara's frequencies then becomes:
\begin{equation
\begin{split}
-\sum_{n=1}^{\infty}\frac{1-(-1)^n e^{-\omega_nt}}{n(\omega_n^2R^2C^2-1)} =
\Psi(x)+\frac{1}{2x}+\gamma
+\ln\left(\sqrt{y}+\frac{1}{\sqrt{y}}\right)\\
+\frac{\pi y^x}{2\sin (\pi x )}
-\frac{y}{2(1+x)}~_2 F_1(1,1+x,2+x,-y)\\
- \frac{y}{2(1-x)} ~_2F_1(1,1-x,2-x,-y),
\end{split}
\label{eqJdaggerMatsubara}
\end{equation}
recalling for clarity that $\omega_n=2\pi n k_\mathrm{B} T/\hbar$ are Matsubara's frequencies, $\gamma$ is Euler's constant, $\Psi$ is the logarithmic derivative of the Gamma function, $~_2F_1$ is the hypergeometric function, $y=\mathrm{exp}(-2 \pi t k_\mathrm{B} T/\hbar)$, and $x=E_\mathrm{C} R_\mathrm{K}/(2\pi^2R k_\mathrm{B} T)$.
For the asymptotic limit $k_\mathrm{B}T\ll \hbar/RC$, the low temperature behavior is dominated by large values of $t\gg RC$ in Eq.~\eqref{Geq2}.
We can thus use the long $t$, small $T$ expansion \cite{Ingold1994}: \begin{equation}
J^\dagger(t)\simeq -\frac{2R}{R_\mathrm{K}}\left(\ln\left[2\cosh\left(\frac{\pi t k_\mathrm{B} T}{\hbar}\right)\right]+\ln(2x)+\gamma\right).
\end{equation}
Inserting this expansion in Eq.~\eqref{Geq2}, we find the asymptotic ($eV\ll k_\mathrm{B}T\ll \hbar/RC$) analytical expression of the transmission probability:
\begin{equation}
\begin{split}
\frac{\tau(V=0,T\rightarrow 0)}{\tau_\infty} \simeq& \frac{\sqrt{\pi}}{2}\frac{\Gamma (1+R/R_\mathrm{K})}{\Gamma (1.5+R/R_\mathrm{K})}\left(\frac{\pi^2 R\,k_\mathrm{B} T}{ e^\gamma R_\mathrm{K} E_\mathrm{C} }\right)^{2R/R_\mathrm{K}}.
\end{split}
\label{LowTeqPowerLaw}
\end{equation}
Note that previously, in \cite{Iftikhar2016}, we proposed a slightly different empirical expression extracted from \cite{Odintsov1991}, which differs by a factor $(\pi e^ {-2 \gamma})^{R/R_\mathrm{K}}$ from the exact asymptotic expression Eq.~\eqref{LowTeqPowerLaw} (this factor deviates from 1 by less than $1\%$ for $R/R_\mathrm{K}\leq1$).
\subsubsection{Extension to different bath temperatures $T, T_\mathrm{node}, T_\mathrm{env}$}
\label{Appendix3T}
We now focus on the case where the voltage-biased tunnel contact is embedded between two electrodes $L$ and $R$ at different temperatures $T_\mathrm{node}$ and $T$, and with an electromagnetic environment at the temperature $T_\mathrm{env}$.
Following Joyez \textit{et al.} and their notations \cite{Joyez1997}, the relative conductance reduction in the tunnel regime reads:
\begin{equation}
\begin{split}
\frac{\tau}{\tau_\infty}-1=\int{dE}\int{d\epsilon}\, P_\mathrm{Tenv}(\epsilon )f_\mathrm{Tnode}(E-eV)\\
\frac{\partial}{\partial E}\left[f_\mathrm{T}(E+\epsilon )-f_\mathrm{T}(E-\epsilon )\right],
\end{split}
\end{equation}
with $P_\mathrm{Tenv}(\epsilon )$ the probability distribution to exchange the energy $\epsilon$ with the electromagnetic environment (previously introduced in Eq.~\eqref{eqTunnelP(E)}), and $f_{\mathrm{T}x}$ the Fermi function at temperature $T_x$ with $x\in\{L,R\}$.
Equivalently, in the time-domain, the relative conductance reduction reads:
\begin{equation}
\begin{split}
\frac{\tau(V,T_\mathrm{node},T_\mathrm{env},T)}{\tau_\infty}-1= \int_0^{+\infty}\!dt\, 2\pi t\,\mathrm{Im} \left[e^{J(t,T_\mathrm{env})}\right]\\
\times \frac{ k_\mathrm{B} T/\hbar}{\sinh (\pi t k_\mathrm{B} T/\hbar)} \frac{ k_\mathrm{B} T_\mathrm{node}/\hbar}{\sinh (\pi t k_\mathrm{B} T_\mathrm{node}/\hbar)}\cos \frac{e V t}{\hbar}.
\label{AA3T}
\end{split}
\end{equation}
In the limit $T=0$, one finds:
\begin{equation}
\begin{split}
\frac{\tau(V,T_\mathrm{node},T_\mathrm{env},T=0)}{\tau_\infty}-1= 2 \int_0^{+\infty}\!dt\, \mathrm{Im} \left[e^{J(t,T_\mathrm{env)}}\right]\\
\times \frac{k_\mathrm{B} T_\mathrm{node}/\hbar}{\sinh (\pi t k_\mathrm{B} T_\mathrm{node}/\hbar)}\cos \frac{e V t}{\hbar}.
\label{AAoneTnul}
\end{split}
\end{equation}
A natural approximation for the environment temperature is the average temperature $T_\mathrm{env}=T_\mathrm{node}/2$; if $eV\ll k_\mathrm{B}T_\mathrm{node}$, Eq.~\eqref{AAoneTnul} then simplifies to:
\begin{equation}
\tau(V=0,T_\mathrm{node}=2 T_\mathrm{env},T_\mathrm{env},T=0)/\tau_\infty=e^{J^\dagger (0,T_\mathrm{env})}.
\label{AAoneTnul2}
\end{equation}
Note that Eq.~\eqref{AAoneTnul2} is equivalent in the energy domain to:
\begin{equation}
\begin{split}
&\tau(V=0,T_\mathrm{node}=2 T_\mathrm{env},T_\mathrm{env},T=0)/\tau_\infty=\\
&\int_{-\infty}^\infty d\epsilon\frac{2P_{\mathrm{Tenv}}(\epsilon)}{1+e^{\epsilon/2k_\mathrm{B}T_\mathrm{env}}}=
\int_{-\infty}^\infty d\epsilon P_{\mathrm{Tenv}}(\epsilon)e^{-\epsilon/2k_\mathrm{B}T_\mathrm{env}}.
\end{split}
\end{equation}
When $T_\mathrm{env}=T_\mathrm{node}/2\ll \hbar/k_\mathrm{B}RC$, we have $J^\dagger(0,T_\mathrm{env})\simeq -\frac{2R}{R_\mathrm{K}}\left(\ln(2x)+\gamma \right)$.
The low temperature asymptotic behavior of Eq.~\eqref{AAoneTnul2} then reads :
\begin{equation}
\begin{split}
\tau(V=0,T_\mathrm{node}\ll \hbar/k_\mathrm{B}RC,T_\mathrm{env}=T_\mathrm{node}/2,T=0)/\tau_\infty
\simeq\\
\left(\frac{\pi^2 R\,k_\mathrm{B} T_\mathrm{node} }{2\,e^\gamma R_\mathrm{K} E_\mathrm{C}}\right)^{2R/R_\mathrm{K}}.
\label{LowToneNulPowerLaw}
\end{split}
\end{equation}
Equation~\ref{LowToneNulPowerLaw} with one null temperature and Eq.~\eqref{LowTeqPowerLaw} with uniform temperatures are both temperature power laws with the same exponent $2R/R_\mathrm{K}$.
They correspond to, respectively, the red and black dashed lines plotted on Fig.~\ref{fig-DCBtunnelT}.
The constant ratio between these limits allows us to calculate the temperature rescaling factor $\alpha$ discussed in the main paper:
\begin{equation}
\begin{split}
\frac{\tau(V=0,T_\mathrm{node}\ll \hbar/k_\mathrm{B}RC, T_\mathrm{env}=T_\mathrm{node}/2,T=0)}{\tau(V=0,T_\mathrm{node}=T_\mathrm{env}=T\ll\hbar/k_\mathrm{B}RC)}=\\
\frac{2}{\sqrt{\pi}}\frac{\Gamma(1.5+R/R_\mathrm{K})}{\Gamma(1+R/R_\mathrm{K})}2^{-2R/R_\mathrm{K}}
=\alpha^{2R/R_\mathrm{K}}.
\end{split}
\end{equation}
Consequently, the temperature reduction factor $\alpha$ reads, in the tunnel regime:
\begin{equation}
\alpha = \frac{1}{2}\left[\frac{2}{\sqrt{\pi}}\frac{\Gamma(1.5+R/R_\mathrm{K})}{\Gamma(1+R/R_\mathrm{K})}\right]^{R_\mathrm{K}/2R}.
\label{eq-rescaledTI}
\end{equation}
For the implemented series resistances $R=R_\mathrm{K}/2,$ $R_\mathrm{K}/3$ and $R_\mathrm{K}/4$, we obtain $\alpha\simeq0.637,$ $0.648$ and $0.655$, respectively.
\subsection{Dynamical Coulomb blockade theory in the near-ballistic regime}
\subsubsection{Quantitative predictions at a uniform temperature $T=T_\mathrm{node}=T_\mathrm{env}$}
\label{CMballistic}
The power law exponent in the vicinity of the ballistic regime is known from the duality predicted between strong back-scattering (tunnel) and weak back-scattering (near ballistic) regimes across an impurity in a Luttinger liquid of interaction parameter $K$ and $1/K$, respectively \cite{Kane1992b,Fendley1998,Lesage1999}.
The corresponding prefactor is nonetheless not universal and depends on the microscopic details, such as the high frequency capacitive cutoff.
It has been inferred in the particular case $R=R_\mathrm{K}$ ($K=1/2$) in \cite{Anthore2018}, adapting \cite{Furusaki1995b}.
We extend here such a quantitative prediction to $R=R_\mathrm{K}/N$ with $N\in\mathbb{N}$.
Remarkably, we find that the duality also exactly applies to the prefactor.
Following Ref.~\cite{Furusaki1995b}, we assume an energy-independent back-scattering at the contact in the absence of DCB, and describe the $N$ fully ballistic channels and the weakly reflected one with bosonic variables $\phi_j (x)$ ($j=1,\ldots,N+1$).
The nearly ballistic (weakly reflected) edge channel is denoted by $j=1$.
Each channel has a fictitious right(left)-moving part, corresponding to the edge path before (after) entering the charged island in the region $x>0$.
The island electric charge is thus
\begin{equation}
\hat{Q} = - \frac{e}{\pi} \sum_{j=1}^{N+1} \int_0^{+\infty} d x \partial_x \phi_j (x) = \frac{e}{\pi} \sum_j \phi_j,
\end{equation}
where we set the notation $\phi_j \equiv \phi_j (0)$.
It is in fact easier to work with the (properly normalized) total charge field $\tilde{\phi}_1 = \frac{1}{\sqrt{N+1}} \sum_j \phi_j$. Employing current conservation, the Kubo formula can be written as
\begin{equation}
G = G_{\rm max} \frac{2 \omega_n}{\pi} \, \langle \tilde{\phi}_2 (\omega_n) \tilde{\phi}_2 (-\omega_n) \rangle_{i \omega_n \to 0^ +},
\end{equation}
where the imaginary (Matsubara) frequency $\omega_n = 2 \pi n k_\mathrm{B}T/\hbar$ is analytically continued to the real axis and then sent to zero.
Here we have introduced a second linear combination of the original fields $\tilde{\phi}_2 = \sqrt{N/(N+1)} \left( \phi_1 - \frac{1}{N} \sum_{j \ne 1} \phi_j \right)$, with coefficients orthogonal to those of $\tilde{\phi_1}$. The advantage of performing this orthogonal change of variables is that the Hamiltonian then couples only the two fields $\tilde{\phi}_1$ and $\tilde{\phi}_2$. We can factor out the other linear combinations $\tilde{\phi}_j$ ($j\ge 2$) for the evaluation of the Kubo formula.
With this formulation, the Euclidean action that governs the dynamics of the two relevant bosonic fields is $S = \sum_{j=1,2} \sum_{n=0}^{+\infty} \tilde{\phi}_j (i \omega_n) K_j \tilde{\phi}_j (-i \omega_n) + S_{\rm BS}$ with the inverse Green's functions $\pi K_j = |\omega_n| + \delta_{j,1} (N+1) E_\mathrm{C}/\pi $ and the back-scattering term
\begin{equation}
S_{BS} = \frac{D \sqrt{1 - \tau_\infty}}{\pi} \! \! \int_0^{ \hbar/k_\mathrm{B} T} \! d \tau \cos \left( \frac{2 \tilde{\phi}_1(\tau) +2 \sqrt{N}\tilde{\phi}_2 (\tau) }{\sqrt{N+1}} \right) ,
\end{equation}
with $D$ the edge electrons' energy bandwidth that is necessarily introduced in bosonization.
$D$ acts as a high-energy regularization which cancels out when evaluating the conductance.
Equipped with this action, we follow Appendix~A.1 from Ref.~\cite{Furusaki1995b} and compute the conductance to leading non-vanishing order in the back-scattering amplitude $\sqrt{1 - \tau_\infty} \ll 1$. We find the analytical result
\begin{equation}
\begin{split}
G_\mathrm{max}-G=&
\frac{R_\mathrm{K}\sqrt{\pi}\left( 1-\tau_\mathrm{\infty}\right)}{2\left(R+R_\mathrm{K}\right)^2}
\frac{\Gamma\left[R_\mathrm{K}/\left(R+R_\mathrm{K}\right)\right]}{\Gamma\left[1/2+R_\mathrm{K}/\left(R+R_\mathrm{K}\right)\right]}
\\
&\times \left[\frac{e^\gamma E_\mathrm{C} (1+R_\mathrm{K}/R)}{\pi^2 k_\mathrm{B}T}\right]^{\frac{2 R}{R+R_\mathrm{K}}},
\end{split}
\label{eqGmaxMinusGNearBallistic}
\end{equation}
where we recall that $G_{\rm max} = (R+R_K)^{-1}$.
We obtain the desired temperature scaling $T^{2 K -2}$ also predicted from the duality tunnel-near ballistic with, in addition, an exact prediction for the prefactor in terms of the transmission $\tau_\infty$, the charging energy $E_\mathrm{C}=e^2/2 C$ and the ratio of resistances $R_\mathrm{K}/R$ corresponding to the number of ballistic channels connecting the island.
Remarkably, although this was not expected to our knowledge, we find that the duality between tunnel and near ballistic regimes also applies for the exact value of the multiplicative factor, despite the dependence of this prefactor on the capacitive cutoff.
More precisely, we compare the expressions of $\tau/\tau_\infty$ in the tunnel regime (obtained from Eq.~\eqref{LowTeqPowerLaw}), with $(G_\mathrm{max}-G)/(G_\mathrm{max}-G_\infty)$ in the near ballistic regime (obtained from Eq.~\eqref{eqGmaxMinusGNearBallistic}), where $G_\infty\equiv(R_\mathrm{K}/\tau_\infty+R)^{-1}$ is the device conductance in the absence of DCB renormalization.
It turns out that these two expressions map \textit{exactly} onto one another provided $K=R_\mathrm{K}/(R+R_\mathrm{K})$ is replaced by $1/K=R/R_\mathrm{K}+1$.
This remarkable robustness of the duality also suggests that Eq.~\eqref{eqGmaxMinusGNearBallistic}, which was obtained for $R=R_\mathrm{K}/N$, may apply for arbitrary values of $R$ (a theoretical treatment of arbitrary $R$ is in preparation \cite{Safi_preparation}).
\subsubsection{Comparison quantitative predictions-experiments at a uniform temperature $T=T_\mathrm{node}=T_\mathrm{env}$}
\label{ballisticCMvsData}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{SI_GmaxMinusGofTnodeExpThy.pdf}
\caption{Insets: symbols show illustrative measurements at $\tau_\infty\sim0.98$ of $G-G_\mathrm{max}$ for $R=R_\mathrm{K}/N$ (top and bottom panel: $N=4$ and $3$, resp.) versus equilibrium temperature $T$ at $V=0$, with $G_\mathrm{max}=\frac{1}{R_\mathrm{K}}\frac{N}{N+1}$ (data also shown in Fig.~\ref{fig-tauClose1T});
Continuous lines are quantitative theoretical predictions of Eq.~\eqref{eqGmaxMinusGNearBallistic}, using $\tau_\infty=\tau(V=-58\,\mu$V) without any fit parameter;
Dashed lines are fits using $\tau_\infty$ as a free parameter adjusted by matching the data point at $T=90$\,mK.
Main panel: symbols represent the fitted values of $\tau_\infty$ versus the corresponding large bias voltage measurements $\tau(V=-58\,\mu$V).}
\label{fig-SI-Close1PowerLaw}
\end{figure}
In Fig.~\ref{fig-SI-Close1PowerLaw}, we confront the new predictions of Eq.~\eqref{eqGmaxMinusGNearBallistic} with the experimental conductance measured at equilibrium in the near ballistic regime, for $R=R_\mathrm{K}/3$ and $R_\mathrm{K}/4$.
The insets display a direct comparison of the predicted (continuous lines) and measured (symbols) conductance at a representative channel tuning of $\tau_\infty\sim0.98$.
The quantitative predictions of Eq.~\eqref{eqGmaxMinusGNearBallistic} are calculated without any fit parameter, assuming $\tau_\mathrm{\infty}\simeq\tau(V=-58\,\mu\mathrm{V})$ on the basis that for such large dc bias voltage only a relatively small renormalization due to DCB is expected ($V$ being of the order of the capacitive cutoff $N E_\mathrm{C}/ \pi e=h/2\pi eRC$).
We observe here a relatively small quantitative discrepancy of $\sim 7\%$ and $\sim16\%$ for $R=R_\mathrm{K}/4$ and $R_\mathrm{K}/3$, respectively.
This small discrepancy could result from the experimental uncertainty on $\tau_\infty$, due to a residual DCB renormalization as well as a non-negligible energy dependence of $\tau_\infty$ at large bias voltages.
In the main panel, we perform a quantitative data/theory comparison over a broad span of $\tau_\infty\in[0.96,1]$.
For this purpose, the fitted value $\tau_\mathrm{\infty}^\mathrm{fit}$ is obtained by matching the prediction of Eq.~\eqref{eqGmaxMinusGNearBallistic} with the conductance measured at $T\simeq90$\,mK.
The resulting $\tau_\mathrm{\infty}^\mathrm{fit}$ is plotted as symbols versus the measured transmission probability at high bias voltage $\tau(V=-58\,\mu\mathrm{V})$.
In the ideal case where $\tau_\mathrm{\infty}=\tau(V=-58\,\mu\mathrm{V})$, we would expect the $\tau_\mathrm{\infty}^\mathrm{fit}$ points to fall on the continuous straight line corresponding to $\tau_\mathrm{\infty}^\mathrm{fit}=\tau(V=-58\,\mu\mathrm{V})$.
We observe that the data points are relatively close to this line, and that the distance reduces as $\tau_\infty$ approaches one.
This comparison establishes the quantitative predictions of Eq.~\eqref{eqGmaxMinusGNearBallistic} at a good relative accuracy, which we believe is here limited by experimental discrepancies between $\tau_\infty$ and $\tau(V=-58\,\mu\mathrm{V})$.
\subsubsection{Near ballistic theory with different bath temperatures $T, T_\mathrm{node}, T_\mathrm{env}$}
\label{ISballistic3T}
The QPC is here coupled to two electrodes at temperatures $T_\mathrm{node}$ and $T$.
In that case one cannot use the Euclidian action employed at equilibrium, and an adapted Keldysh approach is required.
Here, we restrict ourselves to a simple resistance $R=R_\mathrm{K}/N$ in series with the QPC.
A full analysis including exactly the parallel capacitance $C$ will be performed separately \cite{Safi_preparation}.
The QPC is modeled by a weak local back-scattering term at $x=0$:
\begin{equation}
H_\mathrm{BS}=\sqrt{1-\tau_{\infty}}\cos(2\phi_1(0))/2\pi t_0,
\end{equation}
with $1-\tau_{\infty}\ll 1$, $t_0$ a short time cutoff of the order of $\hbar/E_\mathrm{C}$, and $\phi_1$ the bosonic field introduced in Appendix~\ref{CMballistic}.
We treat the remaining $N$ channels as a linear resistance $R=R_\mathrm{K}/N$ (note that the present approach applies to arbitrary values of $R$).
The coupling term between the QPC and this environment reads: $e\phi_1(0)(V-\hat{u})/\pi$, with $V$ the voltage applied to the all device (QPC and series resistance), $e\phi_1(0)/{\pi}$ the total charge transferred through the QPC, and $\hat{u}$ the voltage operator across the resistance, whose fluctuations are given by $\partial_{t}^2J(t)$ (see Eq.~\eqref{eqJ(t)}) and are determined by $R$ and $T_\mathrm{env}$.
First, we need to distinguish the right and left going electron fields $\Psi_{R,1}, \Psi_{L,1}$.
$\Psi_{R,1}$ moves away from the island, at a temperature $T_\mathrm{node}$.
$\Psi_{L,1}$ moves toward it, corresponding to electrons injected from the right electrode at temperature $T$ (see article Fig.~\ref{fig-sample}(b)).
Second, we adopt a similar strategy to Ref.~\cite{Safi2004} by integrating out the environment, ending up with an effective Keldysh action for the bosonic field $\phi_1(0)$ \cite{Safi_preparation}.
The mapping of this DCB problem to a one-dimensional Luttinger liquid with an impurity breaks down (also when including $C$, which corresponds to finite-range interactions as discussed in the supplemental material of \cite{Jezouin2013}).
Yet, it is still convenient to use the parameter $K=(1+R/R_\mathrm{K})^{-1}$, which determines the non-diagonal element of the Keldysh matrix Green's function for $\phi_1(0)$:
\begin{eqnarray}\label{eqC_neq}
C^\mathrm{neq}(t;T_\mathrm{node},T_\mathrm{env},T)&=&\frac{K}2\left[C^\mathrm{eq}(t;T_\mathrm{node})+C^\mathrm{eq}(t;T)\right]\nonumber\\&&+(1-K)C^\mathrm{eq}(t;T_\mathrm{env}).
\end{eqnarray}
Here we use the Green's function obtained at a uniform temperature $T$ in a Luttinger liquid with parameter $K$ and at the same cutoff $t_0$: $C^\mathrm{eq}(t;T)= -(K/2)\ln{ \left[\hbar \sinh\left( \pi k_\mathrm{B}T(-t+it_0)/\hbar\right)/\pi k_\mathrm{B}Tt_0)\right]}$.
To lowest order with respect to the back-scattering amplitude $\sqrt{1-\tau_{\infty}}$, the Green's function $C^\mathrm{neq}(t;T_\mathrm{node},T_\mathrm{env},T)$ determines fully the current as a function of the voltage $V$ and temperatures $T_\mathrm{node},T_\mathrm{env},T$.
Here we restrict ourselves to the experimentally measured linear conductance at zero dc voltage, using an extension of Kubo's formula \cite{ines_philippe,ines_PRB_2019}.
We find:
\begin{eqnarray}\label{X_explicit}
G_\mathrm{max}-G(T_\mathrm{node},T_\mathrm{env},T)=&\nonumber\\-iK^2\frac{1-\tau_{\infty}}{\pi t_0^2 R_\mathrm{K}}\int_{-\infty}^{\infty}\!\!dt\; t\; &e^{4C^\mathrm{neq}\!(t;T_\mathrm{node},T_\mathrm{env},T)}.
\end{eqnarray}
We can determine the effective time cutoff $t_0$ by comparison to the prediction for a uniform temperature $T$ given in Eq.~\eqref{eqGmaxMinusGNearBallistic}: $t_0=\hbar\pi Ke^{-\gamma}/E_\mathrm{C}$, with $\gamma$ the Euler's constant.
Note that this prefactor can also be recovered from a complete analysis including $C$, as will be detailed elsewhere \cite{Safi_preparation}.
Injecting Eq.~\eqref{eqC_neq} into Eq.~\eqref{X_explicit} and restricting ourselves to the experimental hypothesis $T_\mathrm{env}=(T+T_{\mathrm{node}})/2$, we finally obtain:
\begin{equation}
\begin{split}
G_\mathrm{max}-G(T_\mathrm{node},\frac{T_\mathrm{node}+T}{2},T)= \frac {2K^2(1-\tau_{\infty})}{\pi R_\mathrm{K}} \sin \pi K \\
\times \left[\frac{2e^{\gamma} E_\mathrm{C}}{\pi^2 K k_\mathrm{B} (T_\mathrm{node}+T)}\right]^{2(1-K)}
\left[\frac{8\;T_\mathrm{node}T}{(T_\mathrm{node}+T)^2}\right]^{K^2} \\
\int_0^{\infty}\! dt\frac{t}{\left(\sinh t\right)^{2K(1-K)}}\left[\cosh 2t-\!\cosh \left(2t\frac{T_\mathrm{node}-T}{T_\mathrm{node}+T}\right)\right]^{-K^2}.
\end{split}
\label{eqDTweakbs}
\end{equation}
We recall that $K=(1+R/R_\mathrm{K})^{-1}$ and $G_{\rm max} = (R+R_\mathrm{K})^{-1}=K/R_\mathrm{K}$.
The integral in Eq.~\eqref{eqDTweakbs} can be readily evaluated numerically.
This allows one to compute the conductance at arbitrary values of $T$, $T_\mathrm{node}$, as long as both remain small with respect to the high energy cutoff ($T,T_\mathrm{node}\ll E_\mathrm{C}/k_\mathrm{B}$) and that the back-scattering remains weak ($1-G/G_\mathrm{max}\ll1$).
The continuous lines in Fig.~\ref{fig-tauClose1T} are the predictions of Eq.~\eqref{eqDTweakbs}.
\subsection{Experimental electronic temperatures}
\label{App-NoiseThermometry}
Having a good knowledge of the different electronic temperatures (base $T$ and node $T_\mathrm{node}$) is crucial for the present experiments.
In this section, we first summarize how these temperatures are separately measured.
Then we detail how $T_\mathrm{node}$ can be calculated based on the heat Coulomb blockade theory previously established, and compare with our measurements.
Finally, we discuss the possible choices for the temperature $T_\mathrm{env}$ of the electromagnetic environment composed of the series $RC$ circuit.
\subsubsection{Measurement of the electrons' temperature $T$ in the large electrodes}
Following \cite{Iftikhar2016}, we have obtained $T$ from shot noise measurements in a device configuration where the metallic island is bypassed (thanks to lateral gates visible in Fig.~\ref{fig-sample}(a), which are operated as short-circuit switches).
For temperatures $T\geq40$\,mK, we used the mixing chamber temperature measured by a RuO$_2$ thermometer, which was previously shown to match very closely the electrons' temperature on the same setup \cite{Iftikhar2016}.
\subsubsection{Measurement of the electrons' temperature in the metallic island $T_\mathrm{node}$}
Following \cite{Sivre2019}, the temperature increase of the electrons in the central node is inferred from two independent noise measurements, performed on the electrodes 1 and 3 of Fig.~\ref{fig-sample}(a).
Here, one noise measurement is realized behind the partially transmitted channel (on electrode numbered $3$, schematically connected to an amplifier and resonator) and the other one behind the $N_1$ ballistic channels (on electrode numbered $1$, also schematically connected to an amplifier).
Specifically, we measure the difference with respect to equilibrium in the auto-correlation signals $\Delta S_{11}$ and $\Delta S_{33}$, and in the cross-correlation signal $\Delta S_{13}$.
From current conservation and the negligible charge accumulation in the device at the MHz measurement frequencies, we find following \cite{Sivre2018,Sivre2019} that the thermal noise increase $\Delta S^\mathrm{th}\equiv 2 k_\mathrm{B}(T_\mathrm{node}-T)/R_\mathrm{K}$ is given by the excess (increase in) noise signals:
\begin{equation}
\Delta S^\mathrm{th}=\Delta S_{11}\frac{N_1+N_2}{N_1N_2}-\Delta S_{33}\frac{N_1}{N_2\left(N_1+N_2\right)}
\end{equation}
or alternatively
\begin{equation}
\Delta S^\mathrm{th}=\Delta S_{11}\frac{N_1+N_2}{N_1N_2}+\frac{\Delta S_{13}}{N_2}.
\end{equation}
Note that both expressions allow one to extract $T_\mathrm{node}-T$ independently.
We have checked that they were equivalent.
None of these expressions depend on $\tau$.
In practice, each data point is averaged over about $10$\,min to get a temperature resolution of $\sim 0.1$\,mK.
At this resolution, we are also sensitive at our lowest temperature to a small heating of the central node by the spurious low-frequency noise induced by vibrations.
This noise, which depends on the device configuration, is separately determined to be $\delta V_\mathrm{noise}\sim 0.4\,\mathrm{\mu V}$.
It results in a small heating of the central node of $\lesssim 0.2$\,mK.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{SI-FigTcalcVersusTmeas.pdf}
\caption{Symbols: relative difference between the calculated node temperature $T_\mathrm{node}^\mathrm{calc}$ and the measured one $T_\mathrm{node}^\mathrm{meas}$, shown at base temperature $T\sim 8$\,mK for both $R=R_\mathrm{K}/3$ and $R=R_\mathrm{K}/4$, over the full range of explored $\tau$ values.}
\label{fig-siTcalcVersusTmeas}
\end{figure}
\subsubsection{Calculation of the electrons' temperature in the metallic island $T_\mathrm{node}$}
We also relied on our knowledge of heat flow in the device \cite{Sivre2019} to calculate $T_\mathrm{node}$.
These calculated values were used in the tunnel regime and also for the out-of-equilibrium measurements performed at temperatures $T$ larger than our base temperature $\sim8$\,mK.
The node temperature is determined by balancing the injected Joule power in the metallic node ($P_\mathrm{J}$) with the outgoing heat currents, from electrons to phonons ($J_\mathrm{ph}^Q$) and through the connected electronic channels ($J_\mathrm{el}^Q$).
The electron-phonon heat flow is determined when the device only hosts ballistic channels.
We find:
\begin{equation}
J_\mathrm{ph}^Q\simeq 1.8 \times 10^{-8}\left(T_\mathrm{node}^{5.5}-T^{5.5}\right)\,\mathrm{W}.
\end{equation}
The flow of heat across the electronic channels reads \cite{Sivre2019}:
\begin{equation}
\begin{split}
J_\mathrm{el}^Q=(& N + \tau) \frac{\pi^2k_\mathrm{B}^2}{6h}\left(T_\mathrm{node}^{2}-T^{2}\right)\\
+&(N+\tau)\frac{(N+\tau^2) E_\mathrm{C}^2}{\pi^2 h}\\
&\times\left[I\left(\frac{(N+\tau)E_\mathrm{C}}{\pi k_\mathrm{B} T}\right)-I\left(\frac{(N+\tau)E_\mathrm{C}}{\pi k_\mathrm{B} T_\mathrm{node}}\right)\right],
\end{split}
\end{equation}
with $I(x)=\frac{1}{2}\left[\ln{\left(\frac{x}{2\pi}\right)}-\frac{\pi}{x}-\psi\left(\frac{x}{2\pi}\right)\right]$.
Knowing the injected power $P_\mathrm{J}=\frac{N_1 V_1^2 + N_2 V_2^2}{2R_\mathrm{K}}$, $N=N_1+N_2$, the charging energy $E_\mathrm{C}=k_\mathrm{B}\times370$\,mK and the temperature $T$, we can solve the heat balance equation for each measured point $\tau$ and thereby find the only unknown parameter $T_\mathrm{node}$.
The relative accuracy of the heat Coulomb blockade theory on the present device is tested at base temperature $T\sim8\,$mK in Fig.~\ref{fig-siTcalcVersusTmeas}, where we plot as symbols the relative difference between calculated and measured node temperatures.
The agreement is better than $4\%$ over the full $\tau$ and $T_\mathrm{node}$ ranges.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{SI-DCBTunnel-Tenv.pdf}
\caption{Open symbols: renormalized transmission probability $\tau/\tau_\infty$ of the generic channel in series with a resistance $R_\mathrm{K}/2$, $R_\mathrm{K}/3$ and $R_\mathrm{K}/4$ versus the node temperature at $V=0$ at the base temperatures $T\simeq8$\,mK in a log-log scale.
Black lines: predictions of the full tunnel DCB theory for different temperatures (see Appendix~\ref{Appendix3T}) calculated using $T=8$\,mK, $T_\mathrm{node}$, and $T_\mathrm{env}=(T+T_\mathrm{node})/2$.
$\tau_\infty$ is the only adjustable parameter per value of $R/R_\mathrm{K}$ in the data-theory comparison.
The gray areas correspond to the predicted range of conductance for $T_\mathrm{env}\in [T,T_\mathrm{node}]$, using $T=8$\,mK and $T_\mathrm{node}$.}
\label{fig-SI-DCBTunnel-Tenv}
\end{figure}
\subsubsection{Temperature of the electromagnetic environment $T_\mathrm{env}$}
\label{AppendixTenv}
The environment temperature appears as a separate parameter $T_\mathrm{env}$ in the tunnel DCB theory as well as in the novel theory developed in the near ballistic regime (Appendix~\ref{ISballistic3T}).
In the main paper, we use the average value $T_\mathrm{env}=(T_\mathrm{node}+T)/2$.
Here, we determine the range of $T_\mathrm{env}$ over which the tunnel DCB theory is compatible with the data.
For this purpose, we show in Fig.~\ref{fig-SI-DCBTunnel-Tenv} the same tunnel data points as in Fig.~\ref{fig-DCBtunnelT}, and the black continuous lines also correspond to the tunnel DCB theory predictions with $T_\mathrm{env}=(T_\mathrm{node}+T)/2$.
In addition, the gray areas enclose the tunnel DCB predictions for the full interval $T_\mathrm{env}\in [T,T_\mathrm{node}]$.
In practice the data points are close or slightly above the prediction for $T_\mathrm{env}=(T_\mathrm{node}+T)/2$, suggesting that $T_\mathrm{env}\gtrsim(T_\mathrm{node}+T)/2$.
However, the interval $T_\mathrm{env}\in [\sim(T_\mathrm{node}+T)/2,T_\mathrm{node}]$ remains within our experimental uncertainty (approximately the size of the points).
\subsection{Complementary data}
\subsubsection{DCB under a bias voltage in the tunnel regime at $R_\mathrm{K}/2$ and $R_\mathrm{K}/4$}
\begin{figure}[htb!]
\centering
\includegraphics[width=\columnwidth]{SI-DCBofVRKs2and4.pdf}
\caption{Tunnel DCB theory-data comparison under a bias voltage at $R_\mathrm{K}/2$ and $R_\mathrm{K}/4$.
Symbols: measured renormalized transmission probability across the generic channel in series with a resistance $R_\mathrm{K}/2$ ($R_\mathrm{K}/4$) plotted versus the channel bias voltage $V-V_\mathrm{node}$ at different temperatures $T$ in a log-log scale.
Black lines: full tunnel DCB theory calculated with the parameters $C=3.1$\,fF ($C=2.5$\,fF) and the measured temperature $T$.
Red dashed line: asymptotic power law predictions at zero temperature with no fit parameter.}
\label{fig-SI-DCBTunnel-V}
\end{figure}
To complement Fig.~\ref{fig-dcbTunnelV} focusing on $R=R_\mathrm{K}/3$, we plot in Fig.~\ref{fig-SI-DCBTunnel-V} the measured conductances (symbols) and DCB predictions (lines) in the tunnel regime for the series resistance $R=R_\mathrm{K}/2$ and $R_\mathrm{K}/4$ at different temperatures $T$.
\subsubsection{Full Tomonaga-Luttinger conductance renormalization curve at equilibrium}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.9\columnwidth]{SI-HomoGVersusT.pdf}
\caption{Universal renormalization flow of the conductance at equilibrium.
Colored continuous lines represent the experimental curves (green for $R_\mathrm{K}/4$ is shifted vertically by $0.1$, purple shows $R_\mathrm{K}/3$) obtained by averaging the ensemble of data measured from $T=8$\,mK to $T=90$\,mK for each $\tau_\infty$ configuration (see \cite{Anthore2018} for the detailed procedure).
The exact theoretical predictions $G^\mathrm{eq}_{R/R_\mathrm{K}}(T/T_\mathrm{I})$ derived in \cite{Boulat2020} are shown as black dashed lines.
The full renormalization curve has not been measured for $R=R_\mathrm{K}/2$.}
\label{fig-SI-HomodGsdLnT}
\end{figure}
As in \cite{Anthore2018,Anthore2020}, we show on Fig.~\ref{fig-SI-HomodGsdLnT} the pertinence of the mapping to a TLL by comparing the measured conductance $G(V=0,T)$ versus $T/T_I$ at equilibrium (colored continuous lines) to the predictions $G^\mathrm{eq}_{R/R_\mathrm{K}}(T/T_\mathrm{I})$ from \cite{Boulat2020} (black dashed lines).
\begin{figure*}[htb!]
\centering
\includegraphics[width=\textwidth]{SI-GeneralTAuRKs2RKs4.pdf}
\caption{Conductance of a generic channel in series with $R=R_\mathrm{K}/2$ (panel (a)) and $R_\mathrm{K}/4$ (panel (b)) under a temperature bias ($T_\mathrm{node}\geq T$) at base temperature $T\simeq8$\,mK.
Open symbols: measured sample conductance $G$ for different channel settings of $\tau_\infty$, each shown using a different color and symbol shape.
For a given channel setting, a unique value of the renormalization temperature $T_\mathrm{I}$ is determined by matching the first data point at equilibrium ($T_\mathrm{node}\simeq T$) with the predicted universal conductance curve at equilibrium $G^\mathrm{eq}_{R/R_\mathrm{K}}(T_\mathrm{node}/T_\mathrm{I})$ (gray dash-dotted line).
Black lines: universal conductance curve at equilibrium with the same effective reduction in temperature expected at large $T_\mathrm{node}/T$ from the tunnel DCB theory, namely $G^\mathrm{eq}_{R/R_\mathrm{K}}(\alpha T_\mathrm{node}/T_\mathrm{I})$ with $\alpha$ given in Eq.~\eqref{eq-rescaledTI}.}
\label{fig-SI-FullRKs2RKs4}
\end{figure*}
\subsubsection{DCB of a generic channel under a temperature bias with $R_\mathrm{K}/2$ and $R_\mathrm{K}/4$}
The figure \ref{fig-SI-FullRKs2RKs4} complements the data at arbitrary channel tuning shown in main manuscript Fig.~\ref{fig-GeneralTau} for $R=R_\mathrm{K}/3$, with here the conductance measured at $R_\mathrm{K}/2$ (panel (a)) and $R_\mathrm{K}/4$ (panel (b)).
\end{document}
|
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"timestamp": "2021-04-09T02:19:51",
"yymm": "2104",
"arxiv_id": "2104.03812",
"language": "en",
"url": "https://arxiv.org/abs/2104.03812"
}
|
"\\section{Introduction}\n\\label{sec:introduction}\nAt their very core, distributed systems consist(...TRUNCATED)
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"\n\n\n\n\n\n\n\n\\section{Proposed Method}\n\n\\noindent\\textbf{Overview.} In this section, we int(...TRUNCATED)
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"\\section{Introduction} \n\\label{sect:intro}\n\n4U 1957+11 is one of the few known persis(...TRUNCATED)
| {"timestamp":"2021-04-12T02:11:52","yymm":"2104","arxiv_id":"2104.03740","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\r\nPenning traps have been proven as a versatile tool for fundamental physi(...TRUNCATED)
| {"timestamp":"2021-04-09T02:16:27","yymm":"2104","arxiv_id":"2104.03719","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nThe meanings of words are constantly evolving through a process called s(...TRUNCATED)
| {"timestamp":"2021-04-09T02:18:37","yymm":"2104","arxiv_id":"2104.03776","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nQuantum gravity is one of the main issues in modern physic, and several (...TRUNCATED)
| {"timestamp":"2021-04-09T02:21:26","yymm":"2104","arxiv_id":"2104.03849","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nDeep Learning \\cite{LeCun2015} has revolutionized the domains of Comput(...TRUNCATED)
| {"timestamp":"2021-04-09T02:20:46","yymm":"2104","arxiv_id":"2104.03838","language":"en","url":"http(...TRUNCATED)
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