# HDC-MiniROCKET: Explicit Time Encoding in Time Series Classification with
Hyperdimensional Computing ††thanks: ∗all authors are with Chemnitz University
of Technology, Germany _firstname_._lastname_ @etit.tu-chemnitz.de

Kenny Schlegel, Peer Neubert and Peter Protzel∗

###### Abstract

Classification of time series data is an important task for many application
domains. One of the best existing methods for this task, in terms of accuracy
and computation time, is MiniROCKET. In this work, we extend this approach to
provide better global temporal encodings using hyperdimensional computing
(HDC) mechanisms. HDC (also known as Vector Symbolic Architectures, VSA) is a
general method to explicitly represent and process information in high-
dimensional vectors. It has previously been used successfully in combination
with deep neural networks and other signal processing algorithms. We argue
that the internal high-dimensional representation of MiniROCKET is well suited
to be complemented by the algebra of HDC. This leads to a more general
formulation, HDC-MiniROCKET, where the original algorithm is only a special
case. We will discuss and demonstrate that HDC-MiniROCKET can systematically
overcome catastrophic failures of MiniROCKET on simple synthetic datasets.
These results are confirmed by experiments on the 128 datasets from the UCR
time series classification benchmark. The extension with HDC can achieve
considerably better results on datasets with high temporal dependence without
increasing the computational effort for inference.

##  I Introduction

Time series classification has a wide range of applications in robotics,
autonomous driving, medical diagnostic, in the financial sector, and so on. As
elaborated in [1], classification of time series differs from traditional
classification problems because the attributes are ordered. Hence, it is
crucial to create discriminative and meaningful features with respect to the
specific order in time. Over the past years, various methods for
classification of univariate and multivariate time series have been proposed
(for instance, [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]). Often, a high accuracy of a
method comes at the cost of a high computational effort. A very noticeable
exception is MiniROCKET [9] which superseded the earlier ROCKET [8] and
achieves state-of-the-art accuracy at very low computational complexity.
Similar to a convolutional neural network (CNN) layer, MiniROCKET applies a
set of parallel convolutions to the input signal. To achieve a low runtime,
two important design decisions of MiniROCKET are (1) the usage of convolution
filters of small size and (2) accumulation of filter responses over time based
on the Proportion of Positive Values (PPV), which is a special kind of
averaging. However, the combination of these design decisions can hamper the
encoding of temporal variation of signals on a larger scale than the size of
the convolution filters. To address this, the authors of MiniROCKET propose to
use dilated convolutions. A dilated convolution virtually increases a filter
kernel by adding sequences of zeros in between the values of the original
filter kernel [12] (e.g. [-1 2 1] becomes [-1 0 2 0 1] or [-1 0 0 2 0 0 1] and
so on).

Figure 1: MiniROCKET is a fast state-of-the-art approach for time series
classification. However, it is easy to create simple datasets where its
performance is similar to random guessing. The proposed HDC-MiniROCKET uses
explicit time encoding to prevent this failure at the same computational
costs.

The first contribution of this paper is to demonstrate that although the
dilatated convolutions of MiniROCKET perform well on a series of standard
benchmark datasets like UCR [13], it is easy to create datasets where
classification based on MiniROCKET is not much better than random guessing. An
example is illustrated in Fig. 1. There, the task is to distinguish two
different classes of time series signals. Each consists of Gaussian noise and
a single sharp peak either in the first half of the signal (for the first
class) or in the second half of the signal (for each sample from the second
class). Since this is a 2-class problem, random guessing of the class of a
query signal can achieve 50% accuracy. Surprisingly, the MiniROCKET
implementation achieves an only slightly better accuracy of 57% after training
(detail on this experiment will be given in Sec. V-A). This is due to a
combination of two problems: (1) The location of the sharp peaks cannot be
well captured by dilated convolutions with high dilation values due to their
large gaps. (2) Although responses of filters with small or no dilation can
represent the peak, the averaging implemented by PPV removes the temporal
information on the global scale.

Sec. IV will present a novel approach, HDC-MiniROCKET, that addressed this
second problem. It is based on the observation that MiniROCKET’s Proportion of
Positive Values (the second design decision from above) is a special case of a
broader class of accumulation operations known as bundling in the context of
Hyperdimensional Computing (HDC) and Vector Symbolic Architectures (VSA) [14,
15, 16, 17]. This novel perspective encourages a straight-forward and
computational efficient usage of a second HDC operator, binding, in order to
explicitly and flexibly encode temporal information during the accumulation
process in MiniROCKET. The original MiniROCKET then becomes a special case of
the proposed more general HDC-MiniROCKET. As is illustrated in Fig. 1 and
elaborated in Sec. V-A, this explicit time encoding can significantly improve
the performance on the above classification problem to 94% accuracy. Sec. V-B
will demonstrate that the performance also improves on a large collection of
standard time series classification benchmark datasets. The proposed extension
can be implemented efficiently as will be discussed in Sec. IV-C and evaluated
in Sec. V-D.

##  II Related Work

###  II-A Time Series Classification

The related work of time series classification can basically be divided into
two different domains: univariate and multivariate time series. For the first
case, there is a large number of algorithms in the literature, while the
second case is a more general problem formulation with multiple input channels
and only recently received increased attention. There exists are two major
survey paper for these problems: [1] for univariate and [18] for multivariate
time series classification. The survey papers used dataset collections for
benchmarking: 1) the University of California, Riverside (UCR) time series
classification and clustering repository [13] for univariate time series, and
2) the University of East Anglia (UEA) repository [19] for multivariate time
series.

Bagnall et. all [1] compared a large variety of algorithms from different
feature encoding categories like shapelets, dictionary-based or combinations
of these. According to this survey, the most accurate algorithms for
univariate time series classification are BOSS (Bag-of-SFA-Symbols) [2] (with
the more accurate but higher memory consumption extension WEASEL [3]),
Shapelet Transform [4] and COTE (superseded by HIVE-COTE [10]), which is an
ensemble and uses multiple other classifiers. BOSS uses SFA (symbolic fourier
approximation) [20], which is an accurate Bag of Patterns (BoP) method and is
based on the Fourier transformation of the signal. WEASEL extended the BOSS
approach by an additional word extraction from SFA with histogram calculation.

Besides the survey paper by [1], there are other recent accurate
classification methods such as ensemble classifiers like TS-CHIEF [5] and
HIVE-COTE 2.0 [6], the model-based neural network InceptionTime [7] (based on
the Inseption architecture), and ROCKET [8] with its updated version
MiniROCKET [9] as a simple convolution-based method. MiniROCKET is
characterized by a very good trade-off between classification accuracy and
model complexity and is currently state of the art for univariate time series.

[18] provides an overview of approaches for multivariate time series
classification. CIF [21] is an improvement of HIVE-COTE [10] and shows very
good accuracies on the mutlivariate benchmark, but both are extremely slow and
have high memory consumption (according to [18]: CIF is accurate but takes 6
days for all 30 UEA datasets and HIVE-COTE would take over a year). WEASEL
plus MUSE [11] is an extension of WEASEL [3] for the multivariate domain and
provides good results but also lags on computation efficiency. ROCKET was
extended to multivariate classification in [18] and nominated as the clear
winner of the multivariate classification benchmark - it achieves high
accuracy and is by far the fastest classifier. Its superseded version
MiniROCKET [9] has a recent implementation for multivariate time series and
has almost the same accuracy as ROCKET by even lower computational costs.

Therefore, MiniROCKET with its overall good performance is a good starting
point for further improvements in both univariate and multivariate time series
classification.

###  II-B ROCKET and MiniROCKET

As said before, MiniROCKET [9] is a variant of the earlier ROCKET [8]
algorithm. Based on the great success of convolutional neural networks, both
variants build upon convolutions with multiple kernels. However, learning the
kernels is difficult if the dataset is too small, so [8] uses a fixed set of
predefined kernels. While ROCKET uses randomly selected kernels with a high
variety on length, weights, biases and dilations, MiniROCKET is more
deterministic. It uses predefined kernels based on empirical observations of
the behavior of ROCKET. Furthermore, instead of using two global pooling
operation in ROCKET (max-pooling and proportion of positive values, PPV),
MiniROCKET uses just one pooling value – the PPV. This leads to vectors that
are only half as large (about 10,000 instead of 20,000 dimensions). MiniROCKET
is up to 75 times faster than ROCKET on large datasets. To classify feature
vectors, both ROCKET and MiniROCKET use a simple ridge regression. Although
MiniROCKET uses less feature dimensions, the accuracy of both approaches is
almost identical.

###  II-C Hyperdimensional Computing (HDC)

Hyperdimensional computing (also known as Vector Symbolic Architectures, VSA)
is an established approach to solve computational problems using large
numerical vectors (hypervectors) and well-defined mathematical operations.
Basic literature with theoretical background and details on implementations of
HDC are [14, 15, 16, 17]; further general comparisons and overviews can be
found in [22, 23, 24, 25]. HDC has been applied in various fields including
addressing catastrophic forgetting in deep neural networks [26], image feature
aggregation [27], semantic image retrieval [28], medical diagnosis [29],
robotics [30], fault detection [31], analogy mapping [32], reinforcement
learning [33], long-short term memory [34], text classification [35], and
synthesis of finite state automata [36]. Moreover, hypervectors are also
intermediate representations in most artificial neural networks. Therefore, a
combination with HDC can be straightforward. Related to time series, for
example, [30] used HDC in combination with deep-learned descriptors for
temporal sequence encoding for image-based localization. A combination of HDC
and neural networks for multivariate time series classification of driving
styles was demonstrated in [37]. There, the HDC approach was used to first
encode the sensory values and to then combine the temporal and spatial context
in a single representation. This led to faster learning and better
classification accuracy compared to standard LSTM neural networks.

##  III A Primer: The algorithmic steps of MiniROCKET [9]

Input is a time series signal $x\in\mathbb{R}^{T}$ where $T$ is the length of
the time series. MiniROCKET can compute a $D=9,996$ dimensional output vector
$y$ that describes this signal. For application to time series classification,
given training data of time signals with known class, a classifier (e.g. a
simple ridge regression or a neural network) can be trained on these
descriptor vectors and then be used to classify descriptors of new time series
signals.

The first step in MiniROCKET is a dilated convolution of the input signal $x$
with kernels $W_{k,d}$:

$c_{k,d}=x*W_{k,d}$ (1)

$d$ is the dilation parameter, $k\in\\{1,...,84\\}$ refers to 84 pre-defined
kernels $W_{k}$. The length and weights of these kernels are based on insights
from the first ROCKET method [8]: the MiniROCKET kernels have a length of 9,
the weights are restricted to one of two values $\\{-1,2\\}$, and there are
exactly three weights with value 2. To extend the receptive field of the
different kernels, each kernel is applied with one or more dilations $d$
selected from an exponential distribution to ensure that exponentially more
features are computed with smaller dilations. For details on the kernels and
dilations, please refer to [9].

In a second step, each convolution result $c_{k,d}$ is element-wise compared
to one or multiple bias values $B_{b}$:

$c_{k,d,b}=c_{k,d}>B_{b}$ (2)

This is an approximation of the cumulative distribution function of the filter
responses of this particular combination of kernel $k$ and dilation $d$.
MiniROCKET computes the bias values based on the quantiles of the convolution
responses of a randomly chosen training sample. The number of bias values is
such that there are exactly 119 different combinations of dilations $d$ and
bias values $B_{b}$ for each of the 84 kernels $W_{i}$, resulting in $119\cdot
84=9,996$ different binary vectors $c_{k,d,b}\in\\{0,1\\}^{T}$ ($T$ is the
length of the input time series $x$). Again, for the details, please refer to
[9].

The final step of MiniROCKET is to compute each of the 9,996 dimensions of the
output descriptor $y^{PPV}$ as the mean value of one of the $c_{k,d,b}$
vectors (referred to as PPV in [9]). This averaging is where potentially
important temporal information is lost and this final step will be different
in HDC-MiniROCKET.

##  IV Algorithmic approach: HDC-MiniROCKET

The proposed HDC-MiniROCKET is a variant of MiniROCKET. Both use the same set
of dilated convolutions to encode the input signal. The important difference
is that HDC-MiniROCKET uses a Hyperdimensional Computing (HDC) binding
operator to bind the convolution results to timestamps before creation of the
final output. For a general introduction to HDC and explanations, why these
operators work in this application, please refer to one of [14, 15, 16, 17].
Here, we will only very briefly introduce some of the concepts and explain
their usage for encodings in HDC-MiniROCKET. Sec. IV-A introduces HDC bundling
and formulates MiniROCKET in terms of this operation. Sec. IV-B introduces the
second operator, binding, and uses it in combination with time encodings based
on fractional binding in the proposed HDC-MiniROCKET. Finally, Sec. IV-C will
present an efficient implementation.

###  IV-A A HDC perspective on MiniROCKET

The general concept of HDC is to systematically process very high-dimensional
vectors with carefully designed vector operations. One of these operations is
bundling $\oplus$. Input to the bundling operation are two or more vectors
from a vector space $\mathbb{V}$ and the output is a vector of the same size
that is similar to each input vector. Dependent on the underlying vector space
$\mathbb{V}$, there are different Vector Symbolic Architectures (VSAs) that
implement this operation differently. For example, the multiply-add-permute
(MAP) architecture [16] can operate on bipolar vectors (with values
$\\{-1,1\\}$) and implements bundling with a simple element-wise summation –
in high-dimensional vector spaces, the sum of vectors is similar to each
summand [15].

The valuable connection between HDC and MiniROCKET is that the 9,996 different
vectors $c_{k,d,b}\in\\{0,1\\}^{T}$ from eq. 2 in MiniROCKET also constitute a
9,996-dimensional feature vector $F_{t}$ for each timestep $t\in\\{1,...,T\\}$
(think of each vector $c_{k,d,b}$ being a row in a matrix, then these feature
vectors $F_{t}$ are the columns). The averaging in MiniROCKET’s PPV (i.e. the
row-wise mean in the above matrix) is then equivalent to bundling of these
vectors $F_{t}$. More specifically, if we convert $F_{t}$ to bipolar by
$F^{BP}_{t}=1-2F_{t}$ and use the MAP architecture’s bundling operation (i.e.
elementwise addition) then the result is proportional to the output of
MiniROCKET:

$\bigoplus_{t=1}^{T}F^{BP}_{t}=\sum_{t=1}^{T}(1-2F_{t})\propto y_{PPV}$ (3)

This is just another way to compute output vectors that point in the same
directions as the MiniROCKET output – which preserves cosine similarities
between vectors. But much more importantly, it encourages the integration of a
second HDC operation, binding $\otimes$, as described in the following
section.

###  IV-B From MiniROCKET to HDC-MiniROCKET

Figure 2: Different graded similarities for timestamp encodings.

HDC-MiniROCKET extends MiniROCKET by using the HDC binding $\otimes$ operation
to also encode the timestamp $t$ of each feature $F^{BP}_{t}$ in the output
vector (without increasing the vector size).

In HDC, the input to the binding $\otimes$ operation are two vectors from the
same vector space and the output is a single vector, again of the same size,
that is non-similar to each of the input vectors. We will use the property
that binding is similarity preserving, i.e., $\forall
a,b_{1},b_{2}\in\mathbb{V}:sim(a\otimes b_{1},a\otimes b_{2})\approx
sim(b_{1},b_{2})$. A typical usage of binding is to encode key-value pairs. In
the MAP architecture, binding is implemented by simple element-wise
multiplication [16].

The output of HDC-MiniROCKET $y^{HDC}$ is computed by:

$y^{HDC}=\bigoplus_{t=1}^{T}(F^{BP}_{t}\otimes
P_{t})=\sum_{t=1}^{T}(1-2F_{t})\odot P_{t}$ (4)

where $P_{t}$ is a systematic encoding of the timestamp $t$ in a vector of the
same length as $F^{BP}_{t}$, i.e. 9,996-dimensional. Before we provide details
on the creation of $P_{t}$, we want to emphasize that all vectors, including
$y^{HDC}$, are of the same length and that all operations are simple and
efficient local operations. In the MAP architecture, $\oplus$ is simple
element-wise addition, $\otimes$ is element-wise multiplication.

The vectors $P_{t}$ are intended to encode temporal position by making early
events dissimilar to later ones. For creating such hypervectors, we use
fractional binding [38]. It uses a predefined real valued hypervector
$\mathbf{v}$ of dimensionality $D$ and transfers a scalar value $p$ to a
hypervector by applying the following equation:

$\mathbf{v}^{p}:=IDFT\left(\left(DFT_{j}\left(\mathbf{v}\right)^{p}\right)_{j=0}^{D-1}\right).$
(5)

DFT and IDFT are the Discrete Fourier Transform and the Inverse Discrete
Fourier Transform. The resulting vector $\mathbf{v}^{p}$ represents a
systematically encoded hypervector of the scalar $p$, which means that the
procedure preserved the similarity of scalar values (euclidean distance)
within the hyperdimensional vector space (cosine similarity).

To be able of adjust the temporal similarity of feature vectors, we introduce
a parameter $s$, which influences the graded similarity between consecutive
timestamps. In HDC-MiniROCKET the scalar values $p_{t}$ at time $t$ in a time
series with length $T$ is given by $p_{t}=\frac{t\cdot s}{T}$. The scale
factor $s$ adjusts how fast the similarity of timestamp encodings decreases
with increasing difference in $t$. This is visualized in Fig. 2. If $s$ is
equal to 0, all timestamp encodings are the same. $s=1$ means the similarity
decreases for increasing difference of timestamps and only reaches zero for
the difference between the first and the last timestamp of the whole time
series. $s=2$ means that the similarity went to zero after the half of the
length, and so on. $s$ weigths the importance of temporal position – the
higher its value, the more dissimilar become timestamps.

This systematic encoding of timestamps is exploited in HDC-MiniROCKET in Eq.
4. Although binding $\otimes$ and bundling $\oplus$ can be implemented as
simple as element-wise multiplication and addition, these operations have
partially surprising properties in very high-dimensional vector spaces such as
the 9,996-dimensional MiniROCKET features. In HDC-MiniROCKET, the underlying
mechanism is the similarity preserving property of the HDC-binding operation
$\otimes$ from the beginning of this section: the similarity of two feature
vectors, each bound to the encoding of their timestamp, gradually decreases
with increasing difference of the timestamps. This information is maintained
when bundling ($\oplus$) feature vectors to create the output $y^{HDC}$, since
the output of bundling is similar to each input. For more details on these
underlying mechanisms of hyperdimensional computing, please refer to one of
[25, 30, 22].

As described earlier, the proposed temporal encoding creates a more general
variant of MiniROCKET – if parameter $s$ is set to 0, the cosine similarities
based on HDC-MiniROCKET are identical to those of MiniROCKET. The higher $s$,
the more distinctive is the temporal context inside HDC-MiniROCKET’s feature
calculation. The selection of the value of $s$ will be discussed and evaluated
in Sec. V-C.

###  IV-C Efficient implementation

A low computational complexity is an important feature of MiniROCKET. HDC-
MiniROCKET can be implemented with the same (in fact, even slightly lower)
computational complexity during inference.

The efficient implementation of HDC-MiniROCKET builds on two simple
observations: First, $P_{t}$ can be pre-computed for all $t$ since it only
depends on the fixed seed vector $v$ and the scaling parameter $s$, i.e. it is
independent of the signal $x$. Second, we can omit all additions and
multiplications when computing $F_{t}^{BP}\oplus P_{t}=(1-2F_{t})\odot P_{t}$
in Eq. 4. The term $(1-2F_{t})$ in Eq. 4 can be directly computed by replacing
the binarization in MiniROCKET from Eq. 2:by:

$c^{\prime}_{k,d,b}=\begin{cases}1&if\ c_{k,d}>B_{b}\\\ -1&if\
c_{k,d}<B_{b}\end{cases}$ (6)

This can also directly integrate the mutiplication with $P_{t}$:

$c^{\prime\prime}_{k,d,b}=\begin{cases}P_{t,i}&if\ c_{k,d}>B_{b}\\\
-P_{t,i}&if\ c_{k,d}<B_{b}\end{cases}$ (7)

Here, $P_{t,i}$ is the i-th dimension of the vector time encoding of time step
$t$. Finally, the i-th dimension of the output of HDC-MiniROCKET is then
simply:

$y^{HDC}_{i}=\sum_{t=1}^{T}c^{\prime\prime}_{k,d,b}$ (8)

In summary, the computational steps of HDC-MiniROCKET are:

  1. 1.

Compute the filter responses $c_{k,d}$ for each kernel-dilation combination
(Eq. 1).

  2. 2.

Compare each of them to a set of bias values $B_{b}$ to select either a
positive or negative precomputed time encoding $P_{t,i}$ (Eq. 7).

  3. 3.

Compute the sum over all time steps (Eq. 8).

Other than the precomputation of time encodings $P_{t}$, there are no
additional computations compared to MiniROCKET. The PPV computation of the
original MiniROCKET even requires an additional division per output dimension;
however, if cosine similarities are used, then this division could also be
omitted in the original MiniROCKET.

##  V Experimental Evaluation

We evaluate the performance of the proposed HDC-MiniROCKET in two different
setups: (1) the simple synthetic dataset classification task from the
introduction and Fig. 1, and (2) the 128 datasets from the UCR benchmark for
univariate time series classification. Finally, we will analyze the
computational efficiency of our implementation.

We use a Python implementation based on the code of MiniROCKET from the Python
library sktime111sktime, GitHub, https://github.com/alan-turing-
institute/sktime, which contains the original code222minirocket, GitHub,
https://github.com/angus924/minirocket of [9]. For classification, we also use
the same ridge regression classifier as the original MiniROCKET [9].

###  V-A Synthetic Dataset

As described before, the global pooling by PPV in MiniROCKET can neglect
potentially important temporal information if it is not well captured by the
dilated convolutions. To demonstrate possible problems of such behavior, we
constructed a simple synthetic dataset, which consists of two classes with
high temporal dependency. Example signals for both classes are shown in Fig.
1. Each signal consists of a single sharp peak either in the first (class 1)
or second (class 2) half of the signal and additional Gaussian noise ($\mu=0$,
$\sigma=1$). The peak is created using an approximate delta function, which is
based on the normal distribution function:

$\delta(t)=\frac{1}{\sqrt{2\pi a}}\cdot e^{-\frac{t^{2}}{2a}}$ (9)

The parameter $a$ influences the shape of the peak. We use $a=0.03$ which
creates a high and sharp peak that resembles a Dirac impulse.

The length of the time series is 500 time steps. For each time step, we create
one sample in the dataset with the peak centered at this position.
Accordingly, the dataset $D$ consists of 500 samples, 250 samples of each
class.

Classification with a random split of 80% training and 20% test data leads to
the accuracies in table I second column (“standard case“). It can be seen that
MiniROCKET can correctly classify only 65% of the test samples, while the HDC
version with $s=1$ can achieve 97%. An even more drastic result is seen if we
select a particularly challenging subset from the samples; i.e., those class-1
and class-2 sample pairs of the original MiniROCKET-transformed signals that
have high similarity and can be easily confused. The result is a dataset with
250 highly similar data samples that leads to the classification accuracies in
the rightmost column of table I (again using a 80%-20% split). In this
challenging scenario, MiniROCKET without global temporal encoding is not
significantly better than random guessing. However, when HDC-based temporal
encoding is added to MiniROCKET, it can solve the classification problem with
94.1% accuracy.

TABLE I: Results on the synthetic dataset of random guessing and MiniROCKET without and with temporal encoding Method | Accuracy [%] | Accuracy [%]
---|---|---
| standard case | challenging case
random guess | 50.0 | 50.0
MiniROCKET | 65.0 | 56.8
HDC-MiniROCKET (s=1) | 97.0 | 94.1

Figure 3: Similarity matrices on the synthetic dataset. (left) Original
MiniROCKET (right) HDC-MiniROCKET with explicit temporal encoding. Bright
pixels indicate high and dark low similarity. On top and left to the matrices
are overlayed examples of the synthetic signals colored in red for class 1 and
blue for class 2.

Figure 4: Relative accuracy improvements for selecting the optimal $s$ and
automatically calculated $s$ while training. Only datasets with a change
unequal zero are shown.

To better illustrate the effect of the time encoding, Fig. 3 shows the two
similarity matrices of the original encoded MiniROCKET descriptors (left) and
the HDC-MiniROCKET descriptors (right). The i-th row as well as the i-th
column both correspond to the signal from the dataset with the peak at time
step i. The matrix entries are the cosine similarities. The blue and red
signals exemplify the shape of samples of classes 1 and 2. The first half of
the rows and columns of the similarity matrices refer to class 1 and the
second half to class 2. In the left similarity matrix, it can be seen that for
the original MiniROCKET without explicit temporal coding, class 1 signals
(red) sometimes resemble class 2 signals (blue) and vice versa. In contrast,
the right similarity matrix separates the two different classes by explicit
temporal coding - class 1 signals are less likely to be similar to class 2
signals.

###  V-B UCR benchmark

To evaluate the practical benefit of the proposed HDC-MiniROCKET, we show
results on the 128 datasets from the UCR benchmark [13]. This standard
benchmark has previously been used to compare different time series
classification algorithms and contains datasets from a broad range of domains
[13]. Since MiniROCKET showed excellent performance (especially computational
efficiency) on this dataset collection compared to other state-of-the-art
approaches like [5, 6], we only compare against MiniROCKET in our experiments.

Figure 5: Classification accuracies on the 128 UCR datasets when using
MiniROCKET or HDC-MiniROCKET. There are two points for each dataset. Blue
color is with an oracle that selects the best scale parameter, orange color is
with the simple cross validation for parameter selection. For points in the
top-left triangle, HDC-MiniROCKET is better.

Fig. 5 shows the classification accuracies on all UCR datasets for MiniROCKET
in comparison to HDC-MiniROCKET. The blue colored dots show the performance of
HDC-MiniROCKET with an oracle that selects the optimal scale value $s$ for
each dataset from the set $\\{0,1,2,...,6\\}$. It is important to keep in mind
that dependent on the underlying task and dataset, explicit time encoding can
be intended and helpful or also unintended and harmful. This oracle and a
simple data-driven approach to select the scale $s$ are discussed in the next
Sec. V-C. The evaluation with oracle is intended to demonstrate the potential
of the explicit time encoding in HDC-MiniROCKET. Fig. 4 shows the relative
change in accuracy for the individual URC datasets. For 81 out of 128
datasets, there is an improvement by the explicit time encoding. Although the
improvement is sometimes rather small (the average improvement across these 81
datasets is 3.1%), it also reaches more than 27% for one dataset. For those
datasets without improvement, the performance is the same as for MiniROCKET
(which is the special case of $s=0$) – if we are able to select the optimal
scale.

###  V-C Selecting the scale $s$

TABLE II: Results for different similarity parameters $s$ on UCR benchmark. Param. $s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | optimal s | automatic selected s
---|---|---|---|---|---|---|---|---|---
Mean | 0.8540 | 0.8537 | 0.8500 | 0.8450 | 0.8407 | 0.8366 | 0.8336 | 0.8681 | 0.8569
Worst Case | 0.2906 | 0.3080 | 0.3054 | 0.2788 | 0.1827 | 0.0865 | 0.0769 | 0.3080 | 0.2991
Best Case | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0

The evaluation from the previous section used an oracle to select the scale
$s$. Since this time scale parameter is physically grounded, we assume that in
a practical application, there will be expert knowledge that guides the
selection of this parameter. When chosing the scale $s$, it is important to
keep two properties in mind:

(1) There is no single value of the scale parameter that works for all
datasets. This is supported by table II that shows mean, worst-case and best-
case performances across all 128 UCR datasets for each scale parameter. No
scale choice is considerably better than all others. These average statistics
show that the average benefit of scale selection across all datasets is rather
small – however, as illustrated in Fig. 4, the benefit for individual datasets
can be quite high.

(2) The influence of $s$ is quite smooth. Fig. 6 shows the influence of
different values of $s$ on the performance on individual datasets. Small
changes of $s$ create rather small changes of the performance. Typically,
there is a single local maximum (which can be one of the extrem values of the
evaluated range of $s$).

Although we think that expert knowledge about the task at hand is very
valuable to decide whether explicit time encoding is helpful and for selection
of $s$, we also propose a purely data-driven approach to select $s$: Since the
classifier is trained in a supervised fashion, we can assume a set of labeled
training samples. For automatic scale selection, we can simple apply a cross-
validation by further splitting the training set into training and validation
splits. We use a 10-fold cross-validation to select $s$ from the same possible
values as in the oracle. For each fold, we train the classifier on the train
split and evaluate on the corresponding validation split (which is, of course,
also taken from the UCR train split). We select the value $s$ that performs
best in the highest number of splits. Ties are broken in favour of smaller
values.

Fig. 5 and 4 also provide results for this simple automatic scale selection.
Although the performance is considerably below that of the oracle, the results
are again better than those of the original MiniROCKET. In particular, since
MiniROCKET is a special case with $s=0$, HDC-MiniROCKET with oracle cannot be
worse than MiniROCKET. Very importantly, even without any knowledge about the
tasks, the automatic scale selection is in worst-case only 0.5% below the
performance of MiniROCKET.

Figure 6: Differences in absolute accuracies between HDC-MiniROCKET and
MiniROCKET for different values of $s$.

Figure 7: Pairwise comparison of inference time of MiniROCKET and HDC-
MiniROCKET.

###  V-D Computational effort

As discussed before, apart from the precomputation of time encodings (which
only has to be done once), the computational effort for MiniROCKET and HDC-
MiniROCKET is almost identical. Fig. 7 shows the pairwise comparison of the
inference times of both approaches (computed on a standard desktop computer
with Intel i7-5775C CPU). It can be seen that HDC-MiniROCKET is occasionally
slower for some datasets and is consistently marginally faster for fast data
sets. Presumably this is due to the difference in the calculation of PPV in
MiniROCKET that averages values, while HDC-MiniROCKET only accumulates them.
On average, both algorithms require about 1.57s per dataset.

##  VI Conclusion

We proposed to extend the MiniROCKET algorithm for time series classification
by incorporating an explicit temporal context through hyperdimensional
computing. Very importantly, the explicit time encoding with HDC does not
affect computational costs during inference. HDC-MiniROCKET is a
generalization of MiniROCKET where a parameter $s$ can be used to adjust the
importance of time encoding in the descriptor of the time series. MiniROCKET
becomes the special variant with $s=0$.

Experiments on a classification task on a synthetic dataset with high temporal
dependences demonstrated that such explicit temporal coding can prevent
MiniROCKET from failure. An evaluation on the 128 UCR datasets demonstrated
the potential of the proposed approach on a standard time series
classification benchmark. It is very important to keep in mind that not all
tasks and datasets benefit from explicit temporal encoding (e.g.. we can
similarly construct datasets where temporal encoding is harmful). In HDC-
MiniROCKET, this is reflected with the scale parameter $s$. The experiments
demonstrated that the proposed approach with an oracle that selects the best
parameter $s$ can achieve classification improvements on about 2/3 of the
datasets and up to 27%. In practice, selecting $s$ should incorporate
knowledge about the particular task. We demonstrated that selecting $s$ using
a simple purely data-driven approach increased performance for 17 of the 128
data sets. However, the presented automatic selection of $s$ is only a simple,
prelimary approach – the very promising results using the oracle demonstrate
the general potential of the proposed time encodings and that the automatic
scale selection is a promising direction for future research.

## References

  * [1] A. Bagnall, J. Lines, A. Bostrom, J. Large, and E. Keogh, “The great time series classification bake off: a review and experimental evaluation of recent algorithmic advances,” _Data Mining and Knowledge Discovery_ , vol. 31, no. 3, pp. 606–660, 2017.
  * [2] P. Schäfer, “The BOSS is concerned with time series classification in the presence of noise,” _Data Mining and Knowledge Discovery_ , vol. 29, no. 6, pp. 1505–1530, 2015. [Online]. Available: http://dx.doi.org/10.1007/s10618-014-0377-7
  * [3] P. Schäfer and U. Leser, “Fast and Accurate Time Series Classification with WEASEL,” in _Proceedings of the 2017 ACM on Conference on Information and Knowledge Management_ , no. 8.5.2017. New York, NY, USA: ACM, nov 2017, pp. 637–646. [Online]. Available: https://dl.acm.org/doi/10.1145/3132847.3132980
  * [4] A. Bostrom and A. Bagnall, “Binary shapelet transform for multiclass time series classification,” in _Big Data Analytics and Knowledge Discovery_ , S. Madria and T. Hara, Eds. Cham: Springer International Publishing, 2015, pp. 257–269.
  * [5] A. Shifaz, C. Pelletier, F. Petitjean, and G. I. Webb, _TS-CHIEF: a scalable and accurate forest algorithm for time series classification_. Springer US, 2020, vol. 34, no. 3. [Online]. Available: https://doi.org/10.1007/s10618-020-00679-8
  * [6] M. Middlehurst, J. Large, M. Flynn, J. Lines, A. Bostrom, and A. Bagnall, _HIVE-COTE 2.0: a new meta ensemble for time series classification_. Springer US, 2021, vol. 110, no. 11-12. [Online]. Available: https://doi.org/10.1007/s10994-021-06057-9
  * [7] H. Ismail Fawaz, B. Lucas, G. Forestier, C. Pelletier, D. F. Schmidt, J. Weber, G. I. Webb, L. Idoumghar, P. A. Muller, and F. Petitjean, “InceptionTime: Finding AlexNet for time series classification,” _Data Mining and Knowledge Discovery_ , vol. 34, no. 6, pp. 1936–1962, 2020\.
  * [8] A. Dempster, F. Petitjean, and G. I. Webb, _ROCKET: exceptionally fast and accurate time series classification using random convolutional kernels_. Springer US, 2020, vol. 34, no. 5. [Online]. Available: https://doi.org/10.1007/s10618-020-00701-z
  * [9] A. Dempster, D. F. Schmidt, and G. I. Webb, “MiniRocket: A Very Fast (Almost) Deterministic Transform for Time Series Classification,” _Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , pp. 248–257, 2021.
  * [10] J. Lines, S. Taylor, and A. Bagnall, “Time series classification with HIVE-COTE: The hierarchical vote collective of transformation-based ensembles,” _ACM Transactions on Knowledge Discovery from Data_ , vol. 12, no. 5, 2018.
  * [11] P. Schäfer and U. Leser, “Multivariate Time Series Classification with WEASEL+MUSE,” _arXiv preprint_ , 2017.
  * [12] F. Yu and V. Koltun, “Multi-scale context aggregation by dilated convolutions,” in _International Conference on Learning Representations (ICLR)_ , May 2016.
  * [13] H. A. Dau, A. Bagnall, K. Kamgar, C. C. M. Yeh, Y. Zhu, S. Gharghabi, C. A. Ratanamahatana, and E. Keogh, “The UCR time series archive,” _IEEE/CAA Journal of Automatica Sinica_ , vol. 6, no. 6, pp. 1293–1305, 2019\.
  * [14] P. Kanerva, “Hyperdimensional Computing: An Introduction to Computing in Distributed Representation with High-Dimensional Random Vectors.” _Cognitive Computation_ , vol. 1, no. 2, pp. 139–159, 2009.
  * [15] T. A. Plate, “Distributed representations and nested compositional structure,” Ph.D. dissertation, University of Toronto, Toronto, Ont., Canada, Canada, 1994.
  * [16] R. W. Gayler, “Vector Symbolic Architectures answer Jackendoff’s challenges for cognitive neuroscience,” in _Int. Conf. on Cognitive Science_ , 2003\.
  * [17] C. Eliasmith, “How to build a brain: from function to implementation.” _Synthese_ , vol. 159, no. 3, pp. 373–388, 2007.
  * [18] A. P. Ruiz, M. Flynn, J. Large, M. Middlehurst, and A. Bagnall, _The great multivariate time series classification bake off: a review and experimental evaluation of recent algorithmic advances_. Springer US, 2021, vol. 35, no. 2. [Online]. Available: https://doi.org/10.1007/s10618-020-00727-3
  * [19] A. Bagnall, H. A. Dau, J. Lines, M. Flynn, J. Large, A. Bostrom, P. Southam, and E. Keogh, “The UEA multivariate time series classification archive,” pp. 1–36, oct 2018. [Online]. Available: http://arxiv.org/abs/1811.00075
  * [20] P. Schäfer and M. Högqvist, “SFA: A symbolic fourier approximation and index for similarity search in high dimensional datasets,” _ACM International Conference Proceeding Series_ , pp. 516–527, 2012.
  * [21] M. Middlehurst, J. Large, and A. Bagnall, “The Canonical Interval Forest (CIF) Classifier for Time Series Classification,” _Proceedings - 2020 IEEE International Conference on Big Data, Big Data 2020_ , pp. 188–195, 2020\.
  * [22] D. Kleyko, D. A. Rachkovskij, E. Osipov, and A. Rahimi, “A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations,” pp. 1–27, 2021. [Online]. Available: http://arxiv.org/abs/2111.06077
  * [23] ——, “A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part II: Applications, Cognitive Models, and Challenges,” pp. 1–36, 2021. [Online]. Available: http://arxiv.org/abs/2112.15424
  * [24] E. P. Frady, D. Kleyko, and F. T. Sommer, “A theory of sequence indexing and working memory in recurrent neural networks.” _Neural Comput._ , vol. 30, no. 6, 2018.
  * [25] K. Schlegel, P. Neubert, and P. Protzel, “A comparison of vector symbolic architectures,” _Artificial Intelligence Review_ , dec 2021. [Online]. Available: https://link.springer.com/10.1007/s10462-021-10110-3
  * [26] B. Cheung, A. Terekhov, Y. Chen, P. Agrawal, and B. A. Olshausen, “Superposition of many models into one.” in _NeurIPS_ , 2019.
  * [27] P. Neubert and S. Schubert, “Hyperdimensional computing as a framework for systematic aggregation of image descriptors,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 16 938–16 947.
  * [28] P. Neubert, S. Schubert, K. Schlegel, and P. Protzel, “Vector semantic representations as descriptors for visual place recognition,” in _Proc. of Robotics: Science and Systems (RSS)_ , 2021.
  * [29] D. Widdows and T. Cohen, “Reasoning with Vectors: A Continuous Model for Fast Robust Inference,” _Logic journal of the IGPL / Interest Group in Pure and Applied Logics_ , no. 2, p. 141–173, 2015.
  * [30] P. Neubert, S. Schubert, and P. Protzel, “An introduction to hyperdimensional computing for robotics.” _Künstliche Intell._ , vol. 33, no. 4, pp. 319–330, 2019.
  * [31] D. Kleyko, E. Osipov, N. Papakonstantinou, V. Vyatkin, and A. Mousavi, “Fault detection in the hyperspace: Towards intelligent automation systems,” in _International Conference on Industrial Informatics (INDIN)_ , 2015.
  * [32] D. A. Rachkovskij and S. V. Slipchenko, “Similarity-based retrieval with structure-sensitive sparse binary distributed representations,” _Computational Intelligence_ , vol. 28, no. 1, pp. 106–129, 2012.
  * [33] D. Kleyko, E. Osipov, R. W. Gayler, A. I. Khan, and A. G. Dyer, “Imitation of honey bees’ concept learning processes using Vector Symbolic Architectures,” _Biologically Inspired Cognitive Architectures_ , vol. 14, pp. 57 – 72, 2015.
  * [34] I. Danihelka, G. Wayne, B. Uria, N. Kalchbrenner, and A. Graves, “Associative long short-term memory,” _CoRR_ , vol. abs/1602.03032, 2016. [Online]. Available: http://arxiv.org/abs/1602.03032
  * [35] D. Kleyko, A. Rahimi, D. A. Rachkovskij, E. Osipov, and J. M. Rabaey, “Classification and Recall With Binary Hyperdimensional Computing: Tradeoffs in Choice of Density and Mapping Characteristics,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 29, no. 12, pp. 5880–5898, 2018.
  * [36] E. Osipov, D. Kleyko, and A. Legalov, “Associative synthesis of finite state automata model of a controlled object with hyperdimensional computing,” in _Conference of the IEEE Industrial Electronics Society (IECON)_ , 2017.
  * [37] K. Schlegel, F. Mirus, P. Neubert, and P. Protzel, “Multivariate Time Series Analysis for Driving Style Classification using Neural Networks and Hyperdimensional Computing,” in _Intelligent Vehicles Symposium (IV)_ , no. Iv, 2021.
  * [38] B. Komer, T. C. Stewart, A. R. Voelker, and C. Eliasmith, “A neural representation of continuous space using fractional binding,” in _41st Annual Meeting of the Cognitive Science Society_ , 2019.


# Thermal drag effect in quantum Hall circuits

Edvin G. Idrisov  Department of Physics and Materials Science, University of
Luxembourg, Luxembourg  Ivan P. Levkivskyi  Dropbox Ireland, One Park Place,
Hatch Street Upper, Dublin, Ireland  Eugene V. Sukhorukov  Département de
Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland

###### Abstract

We study the thermal drag between two mesoscopic quantum Hall (QH) circuits.
Each circuit consists of Ohmic contact perfectly coupled to quantum Hall edge
states. The drag is caused by strong capacitive coupling between Ohmic
contacts. The non-equilibirum conditions and the electron-electron interaction
are taken into account by using the non-equilibrium bosonization technique.
The thermal drag current in the passive circuit, the noise power of the
corresponding heat current, and the Fano factor are calculated and analyzed.

The mutual electron drag effect is the phenomenon that arises in a system of
two isolated (in the absence of the transfer of charge) quantum circuits
coupled via the long-range Coulomb interaction [1]. Typically, one is
interested in the so-called Coulomb drag effect, where the charge current in
the “active” part of the circuit causes the charge drag current in the
“passive” part of the circuit [2], and normally, this effect is perturbative
and weak. The Coulomb drag effect in various systems was thoroughly studied
both experimentally and theoretically [1]. It was further shown that the
charge drag current can also be mediated by the combined electron-phonon,
electron-photon, and electron-ion interactions [3, 4, 5]. All these effect
have to be differentiated with the thermal drag effect, which can also be
cause by the long-range Coulomb interaction [6, 7, 8, 9]. In this case the
flow of heat current in the active circuit results in the heat current in the
passive circuit. Of particular interest is the thermal drag effect due to the
temperature difference between active and passive circuits. The effect of
thermal drag was studied in the context of quantum refrigerators and engines,
heat diodes and heat pumps [10, 11, 12, 13, 14, 15].

Figure 1:  Schematic of the possible experimental setup. The systems consists
of two identical quantum circuits: upper (active) and lower (passive) parts.
Each part includes an Ohmic contact (metallic granula with capacitance $C$ and
negligible level spacing) which is perfectly coupled to chiral integer QH
channel (the channel enters an Ohmic contact and leaves it without electron
backscattering). The incoming channels are kept at different temperatures
$T_{\rm in,u}-T_{\rm in,d}>0$. The capacitive coupling between Ohmic contacts,
$Q_{\rm u}Q_{\rm d}/C_{\rm ud}$, results in thermal drag effect: The upper
mesoscopic circuit induces extra heat current in the lower circuit in the
outgoing channel.

The recent progress in the fabrication of hybrid systems that are based on
chiral QH edge states made it possible to do accurate mesoscopic experiments
with strongly interacting electron systems [16]. A particular interesting
example of such system is a QH edge state perfectly coupled to an Ohmic
contact, a mesoscale metallic granula with negligible level spacing and finite
charging energy comparable to the temperature [17]. It is worth mentioning
that such an Ohmic contact immediately became a key element of new
experimental and theoretical studies [16], the remarkable examples being the
suppression of charge quantization caused by quantum and thermal fluctuations
[18, 19], the heat Coulomb blockade effect [20, 21, 22, 23], the interaction
induced recovery of the phase coherence [24, 25, 26], the charge Kondo effect
[27, 28], the quantization of the anyonic heat flow [29], and the observation
of the half-integer thermal Hall conductance [30, 31].

In this Letter, we propose a mesoscopic QH system with embedded Ohmic
contacts, building up strong Coulomb interactions, for studying thermal drag
effect (see Fig. 1). The system consists of the active (upper) and passive
(lower) quantum circuits. Each circuit includes an Ohmic contact perfectly
coupled to a chiral QH edge state: The edge state enters an Ohmic contact and
leaves it without electron backscattering. The thermal drag effect is caused
by the capacitive coupling between Ohmic contacts of the active and passive
circuits. The temperature difference $T_{\rm in,u}-T_{\rm in,d}>0$ is applied
between circuits which causes the temperature imbalance between contacts and
results in the thermal heat flow from the upper to the lower part. We assume
the full thermalization of the electron systems inside Ohmic contacts. In our
study we focus on the thermal drag current, its zero-frequency noise power,
and the associated Fano factor.

Model and theoretical approach. We use the low-energy effective field theory
[32, 33] to describe QH edge states strongly coupled to the Ohmic contacts.
The Ohmic contact is modeled by extending an edge state inside the metallic
granula and splitting it in two uncorrelated channels [20, 21]. The
Hamiltonian of each quantum circuit ($\alpha=\rm u,d$) contains two terms (see
Fig. 2)

$\begin{split}&H_{\alpha}=\frac{v_{F}}{4\pi}\sum_{\sigma=\pm}\int
dx(\partial_{x}\phi_{\alpha\sigma})^{2}+Q^{2}_{\alpha}/2C,\\\
&Q_{\alpha}=\int_{-\infty}^{0}dxe^{\epsilon
x/v_{F}}\left[\rho_{\alpha+}(x)+\rho_{\alpha-}(x)\right],\end{split}$ (1)

where $v_{F}$ is Fermi velocity, $C$ is capacitance of each Ohmic contact,
$Q_{\alpha}$ is an operator of the total charge accumulated at each Ohmic
contact, and $\epsilon$ is a small regularization parameter. Here the first
term accounts for the dynamics of incoming and outgoing edge channels, and the
second term describes the charging energy of an Ohmic contact of a finite
size. The set of scalar fields $\phi_{\alpha\sigma}(x,t)$, where $\alpha=\rm
u,d$ and $\sigma=\pm$, are introduced in Eq. (1) to describe the low energy
physics. The operators of edge charge densities and currents for incoming,
$\sigma=-$, and outgoing, $\sigma=+$, states are given by
$\rho_{\alpha\sigma}=(1/2\pi)\partial_{x}\phi_{\alpha\sigma}$ and
$j_{\alpha\sigma}=-(1/2\pi)\partial_{t}\phi_{\alpha\sigma}$. The bosonic
fields satisfy the standard canonical commutation relations

$[\partial_{x}\phi_{\alpha\sigma}(x,t),\phi_{\alpha^{\prime}\sigma^{\prime}}(x^{\prime},t)]=2\pi
i\sigma\delta_{\alpha\alpha^{\prime}}\delta_{\sigma\sigma^{\prime}}\delta(x-x^{\prime}),$
(2)

where $\delta_{\alpha\alpha^{\prime}}$ is the Kronecker delta and $\delta(x)$
is the Dirac delta function. It is worth mentioning that the presented above
model of an Ohmic contact was successfully used to explained the experiments
on the charge quantization, heat Coulomb blockade and heat quantization of
anyonic flow [18, 22, 23, 29, 30, 31].

The total Hamiltonian of the system includes three terms (see Fig. 1)

$H=H_{\rm u}+H_{\rm d}+H_{\rm ud},$ (3)

where the Hamiltonians of the active and passive circuit, $H_{\alpha}$, are
given in Eq. (1), while capacitive coupling between Ohmic contacts has the
form

$H_{\rm ud}=Q_{\rm u}Q_{\rm d}/C_{\rm ud},$ (4)

where $1/C_{\rm ud}$ is mutual capacitive coupling constant.

Using commutation relations from Eq. (2), the Hamiltonian (1) and (3), and the
definition of the current operators

Figure 2:  The model of an Ohmic contact with the capacitive interaction is
illustrated, see Eq. (1). The charge current $j_{\rm in,\alpha}(t)$ arriving
at the Ohmic contact enters it at the interface at $x=0$ and continues as a
neutral current $j^{\rm c}_{\rm out,\alpha}(t)$. The neutral current $j^{\rm
c}_{\alpha}(t)$ arriving at the interface from inside the Ohmic contact
continues as a charge current $j_{\rm out,\alpha}(t)$. In order to account for
the negligible level spacing in the Ohmic contact, we extend edge states
inside it to minus infinity and introduce the small regularization parameter
$\epsilon$ in Eq. (1).

one can present the equations of motion for the currents and the charges in a
form of Langevin equations [21, 19], namely

$\begin{split}&\partial_{t}Q_{\alpha}(t)=j_{\rm in,\alpha}(t)-j_{\rm
out,\alpha}(t),\\\ &j_{\rm out,\alpha}(t)=[Q_{\alpha}(t)+\lambda
Q_{-\alpha}(t)]/\tau_{\rm c}+j_{\rm c,\alpha}(t),\\\ &j_{\rm
out,\alpha}(t)+j^{\rm c}_{\rm out,\alpha}(t)=j_{\rm in,\alpha}(t)+j_{\rm
c,\alpha}(t),\end{split}$ (5)

where $\lambda=C/C_{\rm ud}\leq 1$ is the dimensionless coupling constant,
$\tau_{\rm c}=R_{\rm q}C$ is the charge relaxation time, $R_{\rm
q}=2\pi\hbar/e^{2}$ is the resistance quantum, and $-\alpha=\rm d,u$ simply
denotes swapping the indexes $\alpha=\rm u,d$. Here, the first equation
expresses the conservation of charge in the circuit $\alpha$. The second line
is the Langevin equation, which has the simple meaning: The outgoing current
acquires one contributions $[Q_{\alpha}(t)+\lambda Q_{-\alpha}(t)]/\tau_{\rm
c}$ from the time dependent potential of an Ohmic contact $\alpha$, and the
second contribution $j_{\rm c,\alpha}$ that can be viewed as a Langevin
source. The third equation expresses the conservation of incoming and outgoing
currents at each Ohmic contact.

The Eqs. (5) can be easily solved, and the results may be presented as

$\mathbf{J}_{\rm out}(\omega)=\mathcal{\hat{U}}(\omega)\mathbf{J}_{\rm
in}(\omega),$ (6)

where the four component vectors of outgoing and incoming currents are given
by $\mathbf{J}_{\rm out}=(j_{\rm out,u},j_{\rm out,d},j^{\rm c}_{\rm
out,u},j^{\rm c}_{\rm out,d})$, and $\mathbf{J}_{\rm in}=(j_{\rm in,u},j_{\rm
in,d},j_{\rm c,u},j_{\rm c,d})$ The scattering matrix in Eq. (6) has the form

$\mathcal{\hat{U}}(\omega)=\begin{bmatrix}\mathcal{A}&\mathcal{B}&\mathcal{C}&-\mathcal{B}\\\
\mathcal{B}&\mathcal{A}&-\mathcal{B}&\mathcal{C}\\\
\mathcal{C}&-\mathcal{B}&\mathcal{A}&\mathcal{B}\\\
-\mathcal{B}&\mathcal{C}&\mathcal{B}&\mathcal{A}\end{bmatrix},$ (7)

where $\mathcal{A}(\omega)=[i\omega\tau_{\rm
c}+\lambda^{2}-1]/[(\omega\tau_{\rm c}+i)^{2}+\lambda^{2}]$,
$\mathcal{B}(\omega)=i\lambda\omega\tau_{\rm c}/[(\omega\tau_{\rm
c}+i)^{2}+\lambda^{2}]$, and $\mathcal{C}(\omega)=\omega\tau_{\rm
c}[\omega\tau_{\rm c}+i]/[(\omega\tau_{\rm c}+i)^{2}+\lambda^{2}]$. One can
easily check the unitary of the scattering matrix:
$\mathcal{\hat{U}}^{\dagger}(\omega)=\mathcal{\hat{U}}(\omega)^{-1}$, which
reflects the conservative character of the equations of motion, before
Langevin sources are traced out. It is also worth mentioning that at
$\lambda=0$ or $1$ one recovers the results for scattering matrix obtained in
Refs. [21, 19, 25].

Next, we assume that currents originating from Ohmic contacts are equilibrium.
Thus, the two-point correlation functions of the incoming currents as well as
the Langevin sources are given by the equilibrium spectral function [34]

$\langle\delta j_{l,\alpha}(\omega)\delta
j_{k,\beta}(\omega^{\prime})\rangle=2\pi\delta_{lk}\delta_{\alpha\beta}\delta(\omega+\omega^{\prime})S_{l,\alpha}(\omega),$
(8)

where $S_{l,\alpha}(\omega)=R^{-1}_{\rm q}\omega/(1-e^{-\omega/T_{l,\alpha}})$
and $l,k={\rm in,c}$, and $\alpha,\beta=\rm u,d$. These correlation functions
are used for further calculations. Note, however, the temperatures of Ohmic
contacts $T_{\rm c,u/d}$ are yet unknown. They will be found below.

Heating effect and thermal drag current. In a chiral QH channel, the energy
current operator is equal to the energy density operator multiplied by group
velocity ${\cal
J}_{l,\alpha}(t)=(v^{2}/4\pi)[\partial_{x}\phi_{l,\alpha}(x,t)]^{2}$. Using
the equation of motion for the bosonic field, $\phi_{l,\alpha}(x,t)$, one can
rewrite the energy current as ${\cal J}_{l,\alpha}(t)=(R_{\rm
q}/2)j^{2}_{l,\alpha}(t),$ where $j_{l,\alpha}(t)$ are given in Eqs. (5) and
(6). The heat current can be obtained by subtracting the vacuum (zero
temperature) contribution:

$J_{l,\alpha}=(R_{\rm q}/2)[\langle j^{2}_{l,\alpha}\rangle-\langle
j^{2}_{l,\alpha}\rangle_{\rm vac}].$ (9)

For a ballistic equilibrium channel at the filling factor $\nu=1$ and
temperature $T$ one uses the noise spectral function Eq. (8) to arrive at the
known result [21, 35]: $J=\pi k^{2}_{B}T^{2}/12\hbar\equiv J_{\rm Q}$, where
$J_{Q}$ is called the heat flux quantum.

Next, in order to find temperatures $T_{\rm c,u}$ and $T_{\rm c,d}$ in Eq.
(8), we solve self-consistently the energy balance equations, i.e., the
conservation of incoming and outgoing heat currents at the Ohmic contacts
(coupling to phonons, neglected in the present paper, can also be taken into
account as in Ref. [22, 23, 36]). These equations read $J_{\rm c,u}=J^{\rm
c}_{\rm out,u}$, $J_{\rm c,d}=J^{\rm c}_{\rm out,d}$, where the heat currents
are defined in Eq. (9). Using the relations between incoming and outgoing
currents Eq. (6) and spectral noise functions from the Eq. (8), one arrives at
the system of coupled nonlinear equations for temperatures of Ohmic contacts:

$\begin{split}\frac{\pi T^{2}_{\rm c,u}}{12}=\frac{\pi T^{2}_{\rm
in,u}}{12}+J^{\mathcal{B}}_{\begin{subarray}{c}\rm in,d\\\ \rm
in,u\end{subarray}}(\lambda)+J^{\mathcal{A}}_{\begin{subarray}{c}\rm c,u\\\
\rm in,u\end{subarray}}(\lambda)+J^{\mathcal{B}}_{\begin{subarray}{c}\rm
c,d\\\ \rm in,u\end{subarray}}(\lambda),\\\ \frac{\pi T^{2}_{\rm
c,d}}{12}=\frac{\pi T^{2}_{\rm
in,d}}{12}+J^{\mathcal{B}}_{\begin{subarray}{c}\rm in,u\\\ \rm
in,d\end{subarray}}(\lambda)+J^{\mathcal{A}}_{\begin{subarray}{c}\rm c,d\\\
\rm in,d\end{subarray}}(\lambda)+J^{\mathcal{B}}_{\begin{subarray}{c}\rm
c,u\\\ \rm in,d\end{subarray}}(\lambda).\end{split}$ (10)

The functions on the right hand side are given by

$\begin{split}&J^{i}_{\begin{subarray}{c}l,\alpha\\\
k,\beta\end{subarray}}(\lambda)=\pi[T^{2}_{l,\alpha}\cdot
I_{i}(\lambda,\tau_{\rm c}T_{l,\alpha})-T^{2}_{k,\beta}\cdot
I_{i}(\lambda,\tau_{\rm c}T_{k,\beta})]/12,\\\ &I_{i}(\lambda,a)=\frac{6}{(\pi
a)^{2}}\int_{0}^{\infty}\\!\frac{dzzg_{i}(z,\lambda)}{\exp(z/a)-1},\quad
i=\mathcal{A},\mathcal{B},\end{split}$ (11)

where
$g_{\mathcal{A}}(z,\lambda)=|iz+\lambda^{2}-1|^{2}/|(z+i)^{2}+\lambda^{2}|^{2}$,
$g_{\mathcal{B}}(z,\lambda)=\lambda^{2}z^{2}/|(z+i)^{2}+\lambda^{2}|^{2}$, and
$z=\omega\tau_{\rm c}$ is the dimensionless integration variable. Note, that
the solution of Eq. (10) must be an even function of $\lambda$, since
$g_{i}(z,-\lambda)=g_{i}(z,\lambda)$.

We first concentrate on the limit of small temperatures, ${\rm
max}\\{\tau_{\rm c}T_{l,\alpha}\\}\ll 1$. In the case of ultimately strong
coupling, $\lambda=1$, the system of equations simplifies, and to the leading
order one obtains equal temperatures:

$T^{2}_{\rm c,u}=T^{2}_{\rm c,d}=\left(T^{2}_{\rm in,u}+T^{2}_{\rm
in,d}\right)/2.$ (12)

This result has a simple physical meaning: at $\lambda=1$ and small
temperatures two Ohmic contacts merge into one, and the total incoming heat
flux $\pi(T^{2}_{\rm in,u}+T^{2}_{\rm in,d})/12$ is equally distributed
between outgoing channels. In the case of weak coupling, $\lambda\ll 1$, the
Eqs. (10) can be expanded to include corrections to the leading order in
$\lambda$:

$\begin{split}&T^{4}_{\rm c,u}=T^{4}_{\rm in,u}+2\lambda^{2}\left(T^{4}_{\rm
in,d}-T^{4}_{\rm in,u}\right),\\\ &T^{4}_{\rm c,d}=T^{4}_{\rm
in,d}+2\lambda^{2}\left(T^{4}_{\rm in,u}-T^{4}_{\rm in,d}\right).\end{split}$
(13)

In the opposite limit of large temperatures, ${\rm min}\\{\tau_{\rm
c}T_{l,\alpha}\\}\gg 1$, one arrives at the result that holds for arbitrary
values of $\lambda$ in the interval from $0$ to $1$ [37],

$\begin{split}&T^{2}_{\rm c,u}=T^{2}_{\rm in,u}+3\lambda^{2}(T_{\rm
in,d}-T_{\rm in,u})/2\pi\tau_{\rm c},\\\ &T^{2}_{\rm c,d}=T^{2}_{\rm
in,d}+3\lambda^{2}(T_{\rm in,u}-T_{\rm in,d})/2\pi\tau_{\rm c}.\end{split}$
(14)

This is because at large temperatures the interactions are effectively weak
for the arbitrary strength of coupling.

In order to investigate the thermal drag effect, we assume that the lower
circuit is passive and cold, $T_{\rm in,d}=0$, and concentrate on the most
interesting limit of low temperatures: $\tau_{\rm c}T_{\rm in,u}\ll 1$. In the
case of strong coupling, $\lambda\to 1$, the temperatures of Ohmic contacts
are equal (see Eq. (12)), therefore the total incoming heat flux $J_{\rm
in,u}=J_{\rm Q}$ in the upper incoming channel with the temperature $T_{\rm
in,u}$ splits in two equal outgoing fluxes. Thus the thermal drag current in
the lower circuit takes its maximum value of

$J_{\rm out,d}=J_{\rm in,u}/2.$ (15)

For small $\lambda\ll 1$ the the thermal drag current acquires the following
form [37]

$J_{\rm out,d}=(8\pi^{2}\lambda^{2}/5)(\tau_{\rm c}T_{\rm in,u})^{2}J_{\rm
in,u},$ (16)

i.e., it is suppressed by two small parameters, $\lambda$ and $\tau_{\rm
c}T_{\rm in,u}$. In what follows, we use Eqs. (15) and (16) to calculate the
Fano factor.

Noise of thermal drag current. The spectral density of heat current
fluctuations at zero frequency and the Fano factor of the heat current are
defined as

$\mathcal{S}_{l,\alpha}=\int dt\langle\delta J_{l,\alpha}(t)\delta
J_{l,\alpha}(0)\rangle,\quad F_{l,\alpha}\equiv\mathcal{S}_{l,\alpha}/\langle
J_{l,\alpha}\rangle,$ (17)

where $\delta J_{l,\alpha}(t)=J_{l,\alpha}(t)-\langle J_{l,\alpha}(t)\rangle$.
We first consider an equilibrium ballistic channel at the temperature $T$ as a
reference. By substituting the operator of heat current from Eq. (9) into Eq.
(17) and applying the Wick’s theorem, we arrive at the following result [35,
37]:

$\mathcal{S}=2k_{B}TJ_{Q}\equiv\mathcal{S}_{\rm Q}\quad{\rm and}\quad
F=2k_{B}T\equiv F_{Q}.$ (18)

Thus, the upper incoming channel carries the heat current noise
$\mathcal{S}_{\rm in,u}=\mathcal{S}_{\rm Q}$ with the temperature $T_{\rm
in,u}$, and the Fano factor is equal to $F_{\rm Q}$.

Next, using Eqs. (6) and (7) after some algebra we arrive at the following
expression

$\mathcal{S}_{\rm out,d}=\sum_{\begin{subarray}{c}lk\\\
\alpha\beta\end{subarray}}\int\frac{d\omega}{2\pi}S_{l,\alpha}(-\omega)\mathcal{M}_{l,\alpha;k,\beta}(-\omega,\omega)S_{k,\beta}(\omega),$
(19)

where $S_{l,\alpha}(\omega)$ are given in Eq. (8), and we introduced the
function
$\mathcal{M}_{l,\alpha;k,\beta}(-\omega,\omega)=m_{l,\alpha}(-\omega)m_{k,\beta}(\omega)/2$
with $m_{\rm in,u}(\omega)=|\mathcal{B}(\omega)|^{2}$, $m_{\rm
in,d}(\omega)=|\mathcal{A}(\omega)|^{2}$, $m_{\rm
c,d}(\omega)=|\mathcal{C}(\omega)|^{2}$, and $m_{\rm c,u}(\omega)=m_{\rm
in,u}(\omega)$. This general result can be applied to a number of physical
situations. Below, we concentrate on the noise of the thermal drag current in
the most interesting limit of $T_{\rm in,d}=0$ and $\tau_{\rm c}T_{\rm
in,u}\ll 1$ and in the regimes of strong and weak coupling.

In the limit $\lambda=1$, one can set $m_{\rm in,u}=m_{\rm in,d}=m_{\rm
c,u}=m_{\rm c,d}=1/4$, and the straightforward calculation gives [37]

$\mathcal{S}_{\rm out,d}=(3\mathcal{I}/2\pi^{2})\mathcal{S}_{\rm in,u},$ (20)

where $\mathcal{I}\approx 2.5782$. By using the expression for heat current in
the lower outgoing channel, Eq. (15), we obtain the Fano factor of the thermal
drag current:

$F_{\rm out,d}/F_{\rm in,u}\approx 1.5673.$ (21)

In the limit of weak coupling, $\lambda\ll 1$, one obtains [37]

$\mathcal{S}_{\rm out,d}=(3\lambda^{2}\mathcal{K}/2)(\tau_{\rm c}T_{\rm
in,u})^{2}\mathcal{S}_{\rm in,u},$ (22)

where $\mathcal{K}\approx 10$. By using Eq. (16) for the heat current, one
obtains the Fano factor:

$F_{\rm out,d}/F_{\rm in,u}\approx 0.9498.$ (23)

The above result deserves additional discussion. First, we note that the Fano
factor of the noise of the equilibrium heat current can be estimated as a size
of the typical fluctuation of heat, that is of the order of the average heat
current times the correlation time. In equilibrium, the correlation time of
the heat current noise is of the order of $1/T$ (the only available time
scale). That is why the Fano factor of the equilibrium heat current noise
scales linearly with the temperature (see Eq. (18)). Therefore, it appears
somewhat surprising, from the first glance, that in the weak coupling regime,
$\lambda\ll 1$, the Fano factor of the thermal drag current is close to the
one of the incoming equilibrium channel with the temperature $T_{\rm in,u}$,
despite the fact that the effective temperature of the thermal drag current
scales as $\lambda\tau_{c}T^{2}_{in,u}$, i.e., it is smaller by the
dimensionless factor $\lambda\tau_{c}T_{in,u}\ll 1$. The explanation of this
effect lies in the fact that in the weak coupling regime the thermal drag
effect can be viewed as essentially non-equilibrium rare Poissonian process of
the emission and re-absorption of photons between the upper and lower parts of
the circuit. In this case the Fano factor acquires the values of the order of
the average energy of transmitted photons, that is of the order of the
temperature of the incoming equilibrium channel.

To summarize, we have proposed and theoretically analyzed a strongly
interacting mesoscopic electron system for studying the thermal drag effect.
The system that is based on QH edge states perfectly coupled to Ohmic contacts
is accessible to modern experiments. It consists of an active circuit, where
the heat is generated, and a passive circuit, where the heat flux is induced
by non-local Coulomb interactions. The model of the system is exactly
integrable with the help of non-equilibrium bosonization technique. We have
calculated the thermal drag current, the corresponding zero frequency noise
power, and the Fano factor of the thermal noise. It has been shown that
depending on the interaction strength the Fano factor can be larger or smaller
compared to the Fano factor of the equilibrium ballistic channel.

EGI acknowledges financial support from the National Research Fund Luxembourg
under Grants CORE 13579612. EVS acknowledges the financial support from the
Swiss National Science Foundation.

## References

  * [1] B. N. Narozhny and A. Levchenko, Rev. Mod. Phys. 88, 025003 (2016)
  * [2] A. Levchenko and A. Kamenev, Phys. Rev. Lett. 101, 216806 (2008)
  * [3] O. E. Raichev, G. M. Gusev, F. G. G. Hernandez, A. D. Levin, and A. K. Bakarov Phys. Rev. B 102, 195301 (2020)
  * [4] J. H. Strait, G. Holland, W. Zhu, C. Zhang, B. R. Ilic, A. Agrawal, D. Pacifici, and H. J. Lezec, Phys. Rev. Lett. 123, 053903 (2019)
  * [5] V. L. Gurevich and M. I. Muradov, Journal of Experimental and Theoretical Physics 121, 9981006 (2015)
  * [6] P. Ben-Abdallah, S. Biehs, and K. Joulain, Phys. Rev. Lett. 107, 114301 (2011)
  * [7] B. Bhandari, G. Chiriacò, P. A. Erdman, R. Fazio, and F. Taddei, Phys. Rev. B 98, 035415 (2018)
  * [8] P. Ben-Abdallah, Phys. Rev. B 99, 201406(R) (2019)
  * [9] W. Berdanier, T. Scaffidi, and J. E. Moore, Phys. Rev. Lett. 123, 246603 (2019)
  * [10] D. Venturelli, R. Fazio, and V. Giovannetti, Phys. Rev. Lett. 110, 256801 (2013)
  * [11] Y. Zhang, G. Lin, and J. Chen, Phys. Rev. E 91, 052118 (2015)
  * [12] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin, PNAS 111, 13786 (2014)
  * [13] R. S. Whitney, R. Sánchez, F. Haupt, and J. Splettstoesser, Physica E 75, 257 (2016)
  * [14] T. Ruokola, and T. Ojanen, Phys. Rev. B 83, 241404 (2011).
  * [15] D. Sánchez and M. Moskalets, Entropy 22(9), 977 (2020)
  * [16] Z. Iftikhar, Charge Quantization and Kondo Quantum Criticality in Few-Channel Mesoscopic Circuits, 1st ed. (Springer International Publishing, 2018)
  * [17] S. Jezouin, F. D. Parmentier, A. Anthore, U. Gennser, A. Cavanna, Y. Jin, and F. Pierre, Science 342, 601 (2013)
  * [18] S. Jezouin, Z. Iftikhar, A. Anthore, F. D. Parmentier, U. Gennser, A. Cavanna, A. Ouerghi, I. P. Levkivskyi, E. Idrisov, E. V. Sukhorukov, L. I. Glazman, and F. Pierre, Nature 536, 58 (2016).
  * [19] E. G. Idrisov, I. P. Levkivskyi, and E. V. Sukhorukov, Phys. Rev. B 96, 155408 (2017).
  * [20] A. Furusaki and K. A. Matveev, Phys. Rev. B 52, 16676 (1995)
  * [21] A. O. Slobodeniuk, I. P. Levkivskyi, and E. V. Sukhorukov, Phys. Rev. B 88, 165307 (2013).
  * [22] E. Sivre, A. Anthore, F. D. Parmentier, A. Cavanna, U. Gennser, A. Ouerghi, Y. Jin, and F. Pierre, Nature Physics 14, 145 (2018).
  * [23] C. Lin, M. Hashisaka, T. Akiho, K. Muraki, and T. Fujisawa, Nature Communications 12, 131 (2021).
  * [24] A. A. Clerk, P. W. Brouwer, and V. Ambegaokar, Phys. Rev. Lett. 87, 186801 (2001).
  * [25] E. G. Idrisov, I. P. Levkivskyi, and E. V. Sukhorukov, Phys. Rev. Lett. 121, 026802 (2018).
  * [26] H. Duprez, E. Sivre, A. Anthore, A. Aassime, A. Cavanna, U. Gennser, and F. Pierre, Science 366, 1243 (2019).
  * [27] Z. Iftikhar, S. Jezouin, A. Anthore, U. Gennser, F. D. Parmentier, A. Cavanna, and F. Pierre, Nature 526, 233 (2015).
  * [28] Z. Iftikhar, A. Anthore, A. K. Mitchell, F. D. Parmentier, U. Gennser, A. Ouerghi, A. Cavanna, C. Mora, P. Simon, and F. Pierre, Science 360, 1315 (2018).
  * [29] M. Banerjee, M. Heiblum, A. Rosenblatt, Y. Oreg, D. E. Feldman, A. Stern, and V. Umansky, Nature 545, 75 (2017).
  * [30] M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, Nature 559, 205 (2018).
  * [31] D. F. Mross, Y. Oreg, A. Stern, G. Margalit, and M. Heiblum, Phys. Rev. Lett. 121, 026801 (2018).
  * [32] X. G. Wen, Phys. Rev. B 41, 12838 (1990).
  * [33] T. Giamarchi, Quantum Physics in One Dimension (Claverdon Press Oxford, 2004).
  * [34] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2: (Landau and Lifshits Course of Theoretical Physics, Vol.9) (Butterworth-Heinemann Oxford, 1980).
  * [35] J. P. Pekola and B. Karimi, Rev. Mod. Phys. 93, 041001 (2021)
  * [36] A. Rosenblatt, S. Konyzheva, F. Lafont, N. Schiller, J. Park, K. Snizhko, M. Heiblum, Y. Oreg, and V. Umansky, Phys. Rev. Lett. 125, 256803 (2020).
  * [37] See Supplemental Material at [] for the detailed calculations of temperature, heat current and corresponding noise

Thermal drag effect in quantum Hall circuits
Supplemental material

##  I A: Heating effects

In the case of $\lambda=1$, from Eq. (11) of main text, we have
$g_{\mathcal{A}}(z,\lambda=1)=g_{\mathcal{A}}(z,\lambda=1)=1/(z^{2}+4)$ and
the asymptotics of the integral in Eq. (11) of main text have the simple form

$I_{i}(\lambda=1,a)=\frac{6}{(\pi
a)^{2}}\int_{0}^{\infty}\frac{dzz}{e^{z/a}-1}\frac{1}{z^{2}+4}\simeq\begin{cases}1/4-(\pi
a)^{2}/40,\quad a\ll 1,\\\ 3/2\pi a,\quad a\gg 1.\end{cases}$ (24)

Substituting these asymptotics into Eq. (10) from the main text, we further
obtain the final results for the temperatures $T_{\rm c,u}$ and $T_{\rm c,d}$
in Eq. (12) of the main text.

In the case of weak coupling, $\lambda\ll 1$, one can substitute the expansion
of amplitudes of scattering matrix $g_{\mathcal{A}}(z,\lambda\ll
1)\approx[1-4\lambda^{2}z^{2}/(1+z^{2})^{2}]/(1+z^{2})$ and
$g_{\mathcal{B}}(z,\lambda\ll 1)\approx\lambda^{2}z^{2}/(1+z^{2})^{2}$ into
Eq. (11) from main text and get

$I_{\mathcal{A}}(\lambda\ll 1,a)\simeq
I_{\mathcal{A}}(\lambda=0,a)-4\lambda^{2}\cdot\delta
I_{\mathcal{A}}(\lambda=0,a),\quad\text{and}\quad I_{\mathcal{B}}(\lambda\ll
1,a)\simeq\lambda^{2}\cdot\delta I_{\mathcal{B}}(\lambda=0,a),$ (25)

where the asymptotics of integrals on the right hand side are given by

$I_{\mathcal{A}}(\lambda=0,a)=\frac{6}{(\pi
a)^{2}}\int_{0}^{\infty}\frac{dzz}{e^{z/a}-1}\frac{1}{1+z^{2}}\simeq\begin{cases}1-2(\pi
a)^{2}/5,\quad a\ll 1,\\\ 3/\pi a,\quad a\gg 1,\end{cases}$ (26) $\delta
I_{\mathcal{A}}(\lambda=0,a)=\frac{6}{(\pi
a)^{2}}\int_{0}^{\infty}\frac{dzz}{e^{z/a}-1}\frac{z^{2}}{(1+z^{2})^{3}}\simeq\begin{cases}2(\pi
a)^{2}/5,\quad a\ll 1,\\\ 3/2\pi a,\quad a\gg 1,\end{cases}$ (27) $\delta
I_{\mathcal{B}}(\lambda=0,a)=\frac{6}{(\pi
a)^{2}}\int_{0}^{\infty}\frac{dzz}{e^{z/a}-1}\frac{z^{2}}{(1+z^{2})^{2}}\simeq\begin{cases}2(\pi
a)^{2}/5,\quad a\ll 1,\\\ 3/8\pi a,\quad a\gg 1.\end{cases}$ (28)

Substituting the above asymptotics in the case of ${\rm max}\\{\tau_{\rm
c}T_{l,\alpha}\\}\ll 1$ ($a\ll 1$) into Eq. (10) from the main text, one
obtains the following system of equations

$\begin{cases}\begin{split}&\frac{\pi T^{2}_{\rm c,u}(\lambda)}{12}-\frac{\pi
T^{2}_{\rm in,u}}{12}=\lambda^{2}\left[\frac{\pi T^{2}_{\rm
in,d}}{12}\frac{2(\pi\tau_{\rm c}T_{\rm in,d})^{2}}{5}-\frac{\pi T^{2}_{\rm
in,u}}{12}\frac{2(\pi\tau_{\rm c}T_{\rm in,u})^{2}}{5}\right]\\\ &+\frac{\pi
T^{2}_{\rm c,u}(\lambda)}{12}\left[1-\frac{2(\pi\tau_{\rm c}T_{\rm
c,u}(\lambda))^{2}}{5}-4\lambda^{2}\frac{2(\pi\tau_{\rm c}T_{\rm
c,u}(\lambda))^{2}}{5}\right]\\\ &-\frac{\pi T^{2}_{\rm
in,u}}{12}\left[1-\frac{2(\pi\tau_{\rm c}T_{\rm
in,u})^{2}}{5}-4\lambda^{2}\frac{2(\pi\tau_{\rm c}T_{\rm
in,u})^{2}}{5}\right]\\\ &+\lambda^{2}\left[\frac{\pi T^{2}_{\rm
c,d}(\lambda)}{12}\frac{2(\pi\tau_{\rm c}T_{\rm
c,d}(\lambda))^{2}}{5}-\frac{\pi T^{2}_{\rm in,u}}{12}\frac{2(\pi\tau_{\rm
c}T_{\rm in,u})^{2}}{5}\right],\end{split}\\\ &\\\ \begin{split}&\frac{\pi
T^{2}_{\rm c,d}(\lambda)}{12}-\frac{\pi T^{2}_{\rm
in,d}}{12}=\lambda^{2}\left[\frac{\pi T^{2}_{\rm
in,u}}{12}\frac{2(\pi\tau_{\rm c}T_{\rm in,u})^{2}}{5}-\frac{\pi T^{2}_{\rm
in,d}}{12}\frac{2(\pi\tau_{\rm c}T_{\rm in,d})^{2}}{5}\right]\\\ &+\frac{\pi
T^{2}_{\rm c,d}(\lambda)}{12}\left[1-\frac{2(\pi\tau_{\rm c}T_{\rm
c,d}(\lambda))^{2}}{5}-4\lambda^{2}\frac{2(\pi\tau_{\rm c}T_{\rm
c,d}(\lambda))^{2}}{5}\right]\\\ &-\frac{\pi T^{2}_{\rm
in,d}}{12}\left[1-\frac{2(\pi\tau_{\rm c}T_{\rm
in,d})^{2}}{5}-4\lambda^{2}\frac{2(\pi\tau_{\rm c}T_{\rm
in,d})^{2}}{5}\right]\\\ &+\lambda^{2}\left[\frac{\pi T^{2}_{\rm
c,u}(\lambda)}{12}\frac{2(\pi\tau_{\rm c}T_{\rm
c,u}(\lambda))^{2}}{5}-\frac{\pi T^{2}_{\rm in,d}}{12}\frac{2(\pi\tau_{\rm
c}T_{\rm in,d})^{2}}{5}\right],\end{split}\end{cases}$ (29)

The solution of these equations for ${\rm max}\\{\tau_{\rm
c}T_{l,\alpha}\\}\ll 1$ is given by Eq. (13) from the main text.

In the opposite regime, ${\rm min}\\{\tau_{\rm c}T_{l,\alpha}\\}\gg 1$, one
can obtain the result which is non-perturbative in $\lambda$. Namely, by
calculating the integrals in this regime ($a\gg 1$) we get the following
asymptotic results

$\begin{split}&I_{\mathcal{A}}(\lambda,a)=a\int_{0}^{\infty}dz\frac{z/a}{e^{z/a}-1}\frac{z^{2}+(\lambda^{2}-1)^{2}}{z^{4}+2(\lambda^{2}+1)z^{2}+(\lambda^{2}-1)^{2}}=[a\to\infty]\\\
&\simeq
a\int_{0}^{\infty}dz\frac{z^{2}+(\lambda^{2}-1)^{2}}{z^{4}+2(\lambda^{2}+1)z^{2}+(\lambda^{2}-1)^{2}}=a\cdot\mathcal{F}_{\mathcal{A}}(\lambda),\quad\mathcal{F}_{\mathcal{A}}(\lambda)=\frac{\pi}{4}(2-\lambda^{2}),\end{split}$
(30)

and similarly

$\begin{split}&I_{\mathcal{B}}(\lambda,a)=a\int_{0}^{\infty}dz\frac{z/a}{e^{z/a}-1}\frac{z^{2}\lambda^{2}}{z^{4}+2(\lambda^{2}+1)z^{2}+(\lambda^{2}-1)^{2}}=[a\to\infty]\\\
&\simeq
a\int_{0}^{\infty}dz\frac{z^{2}\lambda^{2}}{z^{4}+2(\lambda^{2}+1)z^{2}+(\lambda^{2}-1)^{2}}=a\cdot\mathcal{F}_{\mathcal{B}}(\lambda),\quad\mathcal{F}_{\mathcal{B}}(\lambda)=\frac{\pi\lambda^{2}}{4}.\end{split}$
(31)

After the substitution of Eq. (30) and (31) into Eq. (10) from the main text,
the system of equations to find the temperatures of Ohmic contacts takes the
form

$\begin{cases}\frac{\pi T^{2}_{\rm c,u}(\lambda)}{12}=\frac{\pi T^{2}_{\rm
in,u}}{12}+\left[\frac{\pi T^{2}_{\rm in,d}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm
in,d}}-\frac{\pi T^{2}_{\rm in,u}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm
in,u}}\right]\cdot\mathcal{F}_{\mathcal{B}}(\lambda)\\\ +\left[\frac{\pi
T^{2}_{\rm c,u}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm c,u}}-\frac{\pi T^{2}_{\rm
in,u}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm
in,u}}\right]\cdot\mathcal{F}_{\mathcal{A}}(\lambda)+\left[\frac{\pi
T^{2}_{\rm c,d}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm c,d}}-\frac{\pi T^{2}_{\rm
in,u}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm
in,u}}\right]\cdot\mathcal{F}_{\mathcal{B}}(\lambda),\\\ \\\ \frac{\pi
T^{2}_{\rm c,d}(\lambda)}{12}=\frac{\pi T^{2}_{\rm in,d}}{12}+\left[\frac{\pi
T^{2}_{\rm in,u}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm in,u}}-\frac{\pi
T^{2}_{\rm in,d}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm
in,d}}\right]\cdot\mathcal{F}_{\mathcal{B}}(\lambda)\\\ +\left[\frac{\pi
T^{2}_{\rm c,d}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm c,d}}-\frac{\pi T^{2}_{\rm
in,d}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm
in,d}}\right]\cdot\mathcal{F}_{\mathcal{A}}(\lambda)+\left[\frac{\pi
T^{2}_{\rm c,u}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm c,u}}-\frac{\pi T^{2}_{\rm
in,d}}{12}\frac{6}{\pi\tau_{\rm c}T_{\rm
in,d}}\right]\cdot\mathcal{F}_{\mathcal{B}}(\lambda).\end{cases}$ (32)

As a first approximation, we choose the solution at $\lambda=0$, namely, we
substitute $T_{\rm c,u}=T_{\rm in,u}$ and $T_{\rm c,d}=T_{\rm in,d}$ into the
right hand side. Therefore, only the first term on the right hand side
survives, and we get the final solution at ${\rm min}\\{\tau_{\rm
c}T_{l,\alpha}\\}\gg 1$

$\begin{split}&\frac{\pi T^{2}_{\rm c,u}(\lambda)}{12}=\frac{\pi T^{2}_{\rm
in,u}}{12}+\frac{3\lambda^{2}}{2}\left[\frac{\pi T^{2}_{\rm
in,d}}{12}\frac{1}{\pi\tau_{\rm c}T_{\rm in,d}}-\frac{\pi T^{2}_{\rm
in,u}}{12}\frac{1}{\pi\tau_{\rm c}T_{\rm in,u}}\right],\\\ &\frac{\pi
T^{2}_{\rm c,d}(\lambda)}{12}=\frac{\pi T^{2}_{\rm
in,d}}{12}+\frac{3\lambda^{2}}{2}\left[\frac{\pi T^{2}_{\rm
in,u}}{12}\frac{1}{\pi\tau_{\rm c}T_{\rm in,u}}-\frac{\pi T^{2}_{\rm
in,d}}{12}\frac{1}{\pi\tau_{\rm c}T_{\rm in,d}}\right].\end{split}$ (33)

This result is provided in Eq. (14) of main text.

##  II B: Thermal drag current

In this section we provide the calculation of outgoing heat current in down
passive circuit in case of $\lambda\ll 1$ and $T_{\rm in,d}=0$ and $\tau_{\rm
c}T_{\rm in,u}\ll 1$. Using the definition of heat current Eq. (9) in main
text one has

$\begin{split}&J_{\rm d}=\frac{R_{\rm
q}}{2}\int\frac{d\omega}{2\pi}\left[|\mathcal{B(\omega)}|^{2}(S_{\rm
in,u}(\omega)-S_{0}(\omega))+|\mathcal{A(\omega)}|^{2}(S_{\rm
in,d}(\omega)-S_{0}(\omega))\right]\\\ &+\frac{R_{\rm
q}}{2}\int\frac{d\omega}{2\pi}\left[|\mathcal{B(\omega)}|^{2}(S_{\rm
c,u}(\omega)-S_{0}(\omega))+|\mathcal{C(\omega)}|^{2}(S_{\rm
c,d}(\omega)-S_{0}(\omega))\right],\end{split}$ (34)

where $S_{0}(\omega)=R^{-1}_{\rm q}\omega\theta(\omega)$ ground state
contribution, $S_{\rm in,u}(\omega)=R^{-1}_{\rm q}\omega/(1-e^{-\omega/T_{\rm
in,u}})$, $S_{\rm in,d}=S_{0}(\omega)$, $S_{\rm c,u}(\omega)=R^{-1}_{\rm
q}\omega/(1-e^{-\omega/f_{\rm c,u}(\lambda)T_{\rm in,u}})$, $S_{\rm
c,d}(\omega)=R^{-1}_{\rm q}\omega/(1-e^{-\omega/f_{\rm c,d}(\lambda)T_{\rm
in,u}})$ and according to Eq. (13) from main text $f_{\rm
c,u}(\lambda)=\sqrt[4]{1+3\lambda^{2}}$, $f_{\rm
c,d}(\lambda)=\sqrt[4]{2\lambda^{2}}$. Introducing the dimensionless
integration variable $y=\omega/T_{\rm in,u}$, we write the heat current as a
sum of four terms

$J=J_{1}+J_{2}+J_{3}+J_{4},$ (35)

where in the leading order with respect to $\tau_{\rm c}T_{\rm in,u}\ll 1$
they read

$\begin{split}&J_{1}=\frac{T^{2}_{\rm
in,u}}{2}\int\frac{dy}{2\pi}\frac{(\lambda\tau_{\rm c}T_{\rm
in,u})^{2}y^{2}}{\tau^{4}_{\rm c}T^{4}_{\rm in,u}y^{4}+2\tau^{2}_{\rm
c}T^{2}_{\rm
in,u}y^{2}(1+\lambda^{2})+(1-\lambda^{2})^{2}}\left[\frac{y}{1-e^{-y}}-y\theta(y)\right]\\\
&\simeq\frac{T^{2}_{\rm in,u}}{2}(\lambda\tau_{\rm c}T_{\rm
in,u})^{2}\int\frac{dy}{2\pi}y^{3}\left[\frac{1}{1-e^{-y}}-\theta(y)\right],\end{split}$
(36) $J_{2}=0,$ (37) $\begin{split}&J_{3}=\frac{T^{2}_{\rm
in,u}}{2}\int\frac{dy}{2\pi}\frac{(\lambda\tau_{\rm c}T_{\rm
in,u})^{2}y^{2}}{\tau^{4}_{\rm c}T^{4}_{in,u}y^{4}+2\tau^{2}_{\rm c}T^{2}_{\rm
in,u}y^{2}(1+\lambda^{2})+(1-\lambda^{2})^{2}}\left[\frac{y}{1-e^{-y/f_{\rm
c,u}(\lambda)}}-y\theta(y)\right]\\\ &=(f_{\rm c,u}(\lambda)\to
1)\simeq\frac{T^{2}_{\rm in,u}}{2}(\lambda\tau_{\rm c}T_{\rm
in,u})^{2}\int\frac{dy}{2\pi}y^{3}\left[\frac{1}{1-e^{-y}}-\theta(y)\right],\end{split}$
(38) $\begin{split}&J_{4}=\frac{T^{2}_{\rm
in,u}}{2}\int\frac{dy}{2\pi}\frac{\left[1+(\tau_{\rm c}T_{\rm
in,u}y)^{2}\right](\tau_{\rm c}T_{\rm in,u}y)^{2}}{\tau^{4}_{\rm c}T^{4}_{\rm
in,u}y^{4}+2\tau^{2}_{\rm c}T^{2}_{\rm
in,u}y^{2}(1+\lambda^{2})+(1-\lambda^{2})^{2}}\left[\frac{y}{1-e^{-y/f_{\rm
c,d}(\lambda)}}-y\theta(y)\right]\\\ &\simeq\frac{T^{2}_{\rm
in,u}}{2}(\tau_{\rm c}T_{\rm in,u})^{2}\left[f_{\rm
c,d}(\lambda)\right]^{4}\int\frac{dy}{2\pi}y^{3}\left[\frac{1}{1-e^{-y}}-\theta(y)\right]=(\left[f_{\rm
c,d}(\lambda)\right]^{4}\simeq 2\lambda^{2})\\\ &\simeq T^{2}_{\rm
in,u}(\lambda\tau_{\rm c}T_{\rm
in,u})^{2}\int\frac{dy}{2\pi}y^{3}\left[\frac{1}{1-e^{-y}}-\theta(y)\right].\end{split}$
(39)

Thus the sum of four terms gives the result presented in Eq. (16) of the main
text

$J_{\rm d}=\frac{T^{2}_{\rm in,u}}{\pi}(\lambda\tau_{\rm c}T_{\rm
in,u})^{2}\underbrace{\int
dyy^{3}\left[\frac{1}{1-e^{-y}}-\theta(y)\right]}_{2\pi^{4}/15}=\frac{8\lambda^{2}(\pi\tau_{\rm
c}T_{\rm in,u})^{2}}{5}J_{\rm Q},$ (40)

where $J_{\rm Q}=\pi T^{2}_{\rm in,u}/12$.

##  III C: Noise of thermal current for ballistic channel

Using the definition (9) of the heat flux in the main text, we get the
following expression for the noise power of thermal current

$\mathcal{S}_{\rm Q}(\omega)=(R_{\rm q}/2)^{2}\int dte^{i\omega
t}\left[\langle j^{2}(t)j^{2}(0)\rangle-\langle j^{2}(t)\rangle\langle
j^{2}(0)\rangle\right].$ (41)

According to Wick’s theorem $\langle j^{2}(t)j^{2}(0)\rangle=\langle
j^{2}(t)\rangle\langle j^{2}(0)\rangle+2\langle j(t)j(0)\rangle\langle
j(t)j(0)\rangle$, since each current operator is the combination of bosonic
creation and annihilation operators. Subtracting the second term we get

$\mathcal{S}_{\rm Q}(\omega)=(R^{2}_{\rm q}/2)\int dte^{i\omega t}\langle
j(t)j(0)\rangle^{2}.$ (42)

Next, using the FDT in Eq. (8) of the main text, we calculate the above
expression explicitly

$\begin{split}&\mathcal{S}_{\rm Q}(\omega)=(R^{2}_{\rm
q}/2)\prod^{4}_{i=1}\int\frac{d\omega_{i}}{(2\pi)^{4}}\int
dte^{i(\omega-\omega_{1}-\omega_{3})t}\langle
j(\omega_{1})j(\omega_{2})\rangle\langle
j(\omega_{3})j(\omega_{4})\rangle=\frac{1}{2}\int\frac{d\omega_{1}}{2\pi}R_{\rm
q}S(\omega-\omega_{1})\cdot R_{\rm q}S(\omega_{1})\\\
&=\frac{1}{2}\int\frac{d\omega_{1}}{2\pi}\frac{\omega-\omega_{1}}{1-e^{-(\omega-\omega_{1})/T}}\cdot\frac{\omega_{1}}{1-e^{-\omega_{1}/T}}=\frac{T^{3}}{2}\int\frac{dy}{2\pi}\frac{\frac{\omega}{T}-y}{1-e^{-\frac{\omega}{T}+y}}\cdot\frac{y}{1-e^{-y}}.\end{split}$
(43)

By introducing the dimensionless variable $x=y-\omega/2T$ and performing the
integral with respect to $x$ we obtain

$\mathcal{S}_{\rm Q}(\omega)=\frac{T^{3}}{4\pi}\int
dx\frac{\frac{\omega}{2T}-x}{1-e^{x-\frac{\omega}{2T}}}\cdot\frac{x+\frac{\omega}{2T}}{1-e^{-\left(x+\frac{\omega}{2T}\right)}}=\frac{\omega}{48\pi}[(2\pi
T)^{2}+\omega^{2}][1+\coth(\omega/2T)].$ (44)

It is worth mentioning that the non-symmetrized noise is given by
$\mathcal{S}_{\rm Q}(\omega)/2+\mathcal{S}_{\rm Q}(-\omega)/2$. Because of the
quadratic term $\omega^{2}$ in square brackets the finite frequency noise at
zero temperature $T=0$ does not vanish: $\mathcal{S}_{\rm
Q}(\omega)=\omega^{3}\text{sgn}(\omega)/24\pi$. Finally, taking the limit
$\omega\to 0$ in Eq. (44) we arrive to Eq. (18) from the main text.

##  IV D: Noise of thermal drag current

Similar calculations as in Sec. B can be done for the noise of heat current in
the case of $\lambda\to 1$ and $\lambda\ll 1$ at $T_{\rm in,d}=0$ and
$\tau_{\rm c}T_{\rm in,u}\ll 1$. In the case of $\lambda\to 1$ the result is
given by Eq. (20) from the main text, where

$\mathcal{I}=\int\frac{dy}{16}P(y)P(-y)\approx 2.5782,$ (45)

where $P(y)=y/(1-e^{-y})+y\theta(y)+2y/(1-e^{-\sqrt{2}y})$ and $\theta(y)$ is
the Heaviside step function. $\sqrt{2}$ in the integrand of $\mathcal{I}$
originates from Ohmic contact temperatures $T_{\rm c,u}=T_{\rm c,d}=T_{\rm
in,u}/\sqrt{2}$. In the case of $\lambda\ll 1$, the heat currents noise in the
passive circuit (see Eq. (19) in the main text) contains sixteen terms. After
some algebra similar to that of Sec. B, one can show that the leading order
behavior of the noise of heat current in the lower channel for $\lambda\ll 1$
and $\tau_{\rm c}T_{\rm in,d}\ll 1$ is given by Eq. (22) in the main text.