# DeToxy: A Large-Scale Multimodal Dataset for Toxicity Classification in
Spoken Utterances

###### Abstract

Toxic speech, also known as hate speech, is regarded as one of the crucial
issues plaguing online social media today. Most recent work on toxic speech
detection is constrained to the modality of text and written conversations
with very limited work on toxicity detection from spoken utterances or using
the modality of speech. In this paper, we introduce a new dataset DeToxy, the
first publicly available toxicity annotated dataset for the English language.
DeToxy is sourced from various openly available speech databases and consists
of over 2 million utterances. We believe that our dataset would act as a
benchmark for the relatively new and un-explored Spoken Language Processing
task of detecting toxicity from spoken utterances and boost further research
in this space. Finally, we also provide strong unimodal baselines for our
dataset and compare traditional two-step and E2E approaches. Our experiments
show that in the case of spoken utterances, text-based approaches are largely
dependent on gold human-annotated transcripts for their performance and also
suffer from the problem of keyword bias. However, the presence of speech files
in DeToxy helps facilitates the development of E2E speech models which
alleviate both the above-stated problems by better capturing speech clues.

Index Terms: Speech Toxicity Analysis, End-to-End, 2-step, multimodal

##  1 Introduction

Social network platforms are generally meant to share positive, constructive,
and insightful content. However, in recent times, people often get exposed to
objectionable content like threats, identity attacks, hate speech, insults,
obscene texts, offensive remarks, or bullying. With the rise of different
forms of content available online beyond just written text, i.e., audio and
video, it is crucial that we device efficient content moderation systems for
these forms of shared media. However, most prior work in literature and
available datasets focus primarily on the modality of conversational text,
with other modalities of conversation ignored at large. Thus, to alleviate
this problem, in this paper, we propose a new dataset DeToxy, for the
relatively new and unexplored Spoken Language Processing (SLP) task of
toxicity classification in spoken utterances, which remains a crucial problem
to solve for interactive intelligent systems, with broad applications in the
field of content moderation in online audio/video content, gaming, customer
service, etc. DeToxy is a large-scale multimodal dataset with both speech and
text cues, manually annotated for toxicity detection in spoken utterances,
with potential applications in building unimodal or multimodal models for
conversational utterance-level Speech Toxicity Classification (STC) and End-
to-End (E2E) speech systems which better capture semantics of the utterance.

The key challenge in toxicity classification in spoken speech is to learn good
representations that capture toxicity signals from raw speech signals which is
inherent in a rich set of acoustic and linguistic content, including semantic
meaning, speaker characteristics, emotion, tone, and possibly even sentiment
information while remaining invariant under different speakers, acoustic
conditions, and other natural speech variations. However, different from
Speech Emotion Recognition (SER) and similar to Speech Sentiment Recognition
(SSER), toxicity classification from spoken utterances involve the system
capturing semantic and syntactic properties of the utterance beyond just
acoustic and prosodic cues which makes this task all the more difficult.

Traditional approaches for various speech downstream tasks involving spoken
utterance classification employ low-level acoustic features, such as band-
energies, filter banks, and MFCC features [1], or raw waveform [2] . We
acknowledge the fact that models trained on these low-level features can
easily overfit to noise or signals irrelevant to the task. One way to remove
variations in speech is to transcribe the audio into text and use text
features to predict toxicity as done with SSER by [3]. Nonetheless, toxicity
signals in the speech, like tone and emotion, can be lost in the
transcription. Thus, it is essential for any system to learn speech
representations that make high-level information from speech signals available
to solve downstream SLP tasks. Inspired by this, we provide both 2-step
cascade and E2E baselines for STC on DeToxy and analyze the advantages and
disadvantages in both, concluding why the modality of speech is crucial for
the task of STC and makes DeToxy an important contribution to the speech
community. Additionally, we show how prosodies and linguistic cues in natural
speech aid our E2E model, which outperforms the 2-step methodology using less
than 10% of the total amount of labeled data available to our 2-step system.

##  2 Related Work

Hate speech or toxic speech detection is a challenging task extensively
studied in the literature, with techniques including dictionary-based search,
distributional semantics, and recent literature exploring the power of neural
network architectures for the same. However, most of the work done on hate
speech detection is constrained to just text, in English and some other
foreign languages [4, 5, 6] with the only existing work done by [7] for audio.

On the other hand, the most commonly explored SLP tasks include Automatic
Speech Recognition (ASR), SER , speaker verification, speaker identification,
speech separation, speech enhancement, Named Entity Recognition (NER), phoneme
recognition, etc, and the most recently explored speech sentiment
classification which is very related to our task. All of these tasks are well
studied with a lot of open-source datasets available online. With recent
advancements in Natural Language Processing (NLP) and SLP, a lot of the
systems achieving state-of-the-art in these tasks leverage E2E multi-layer
neural networks like CNNs or Transformers, including self-supervised and semi-
supervised techniques which have shown to help models in the low-resource data
paradigm and proven to learn powerful speech representations [8, 9, 10].

With downstream SLP tasks like speech sentiment classification and NER
requiring an understanding of the contents and semantics of the spoken
utterance, people have employed both 2-step [11] and E2E methodologies [12,
13]. Some of these self-supervised methodologies, originally invented for
pushing the performance of state-of-the-art in ASR systems, have also shown
success in other downstream tasks [10].

##  3 DeToxy and DeToxy-B

In this paper, we present DeToxy, the first annotated dataset for toxic speech
detection in the English language. Our dataset is a subset of various open-
source datasets, detailed statistics about which can be found in Table 1. We
also present DeToxy-B, a balanced version of the dataset, curated from the
original larger version taking into consideration auxiliary factors like
trigger terms and utterance sentiment labels.

###  3.1 Dataset Collection and Annotation

For annotation, we define toxicity as _rude, disrespectful, or otherwise
likely to make someone leave a discussion_. For the initial version of our
dataset, we primarily focus on openly available speech databases with or w/o
text transcripts available. For obtaining transcripts of datasets for which
transcripts were not available, we use pre-trained wav2vec-2.0 to obtain the
transcriptions from a similar domain. In the process of consolidation of our
dataset, primarily, after an empirical analysis of the transcripts of a
majority of the open-source datasets available online, we found that most of
them did not contain toxic utterances. Thus, we follow a 2-step approach to
annotate our dataset. First, we train a textual toxicity classifier using
_BERT BASE_ and use that to filter out datasets that had at least 10% of their
total number of utterances flagged as toxic by the model. Some datasets which
did not fit this criterion were LibriSpeech [14], TIMIT [15], TED-LIUM [16]
and MASS [17]. Next, all these utterances from all datasets filtered through
step 1 were manually annotated by 3 professional annotators using audino [18]
taking both the textual transcript and speech files into account. Finally, we
do a simple majority voting among the 3 annotations to determine the final
toxicity class of each utterance. The Cohen’s Kappa Score for inter-annotator
agreement is 0.76. Table 1 and Table 2 show detailed descriptions of DeToxy
and DeToxy-B datasets respectively.

###  3.2 Sampling DeToxy-B and Test-Trigger Datasets

For DeToxy-B, we keep all the toxic utterances from DeToxy and sample thrice
the number of toxic utterances as our non-toxic utterances. We acknowledge the
fact that sentiment is a strong controlling factor for various speech cues,
and over-sampling any one of the sentiments might lead our E2E model to
overfit the same. Thus, we don’t sample at random but use sentiment labels to
sample non-toxic utterances so that the final DeToxy-B dataset has an equal
distribution of sentiments for its non-toxic utterances. These sentiment
labels were annotated by a single annotator only for the purpose of
constructing DeToxy-B and thus we don’t include these labels in our final
release of the dataset. In this paper, we evaluate all our proposed baselines
on only DeToxy-B, with a Training, Development and Test stratified split of
70:15:15.

Finally, we also collate an explicit test set with just non-toxic utterances
where each utterance consisted of at least one trigger term
111https://hatebase.org/. We do this to evaluate how our baselines perform
against not getting biased to trigger terms, a long-standing problem in
toxicity classification [19, 20].

Table 1: DeToxy Statistics

Dataset | # Utterances | # Toxic | # Non-Toxic | # Sp | Toxic | Emo | Sent | TL (hh:mm:ss)
---|---|---|---|---|---|---|---|---
CMU-MOSEI [21] | 44,977 | 217 | 44,760 | 1,000 | ✓ | ✓ | ✓ | 65:53:36
CMU-MOSI [22] | 2,199 | 67 | 2,132 | 98 | ✓ | ✗ | ✓ | 02:36:17
Common Voice [23] | 1,584,219 | 2,888 | 1,581,331 | 66,173 | ✓ | ✗ | ✗ | $\approx 2181:00:00~{}~{}~{}$
IEMOCAP [24] | 10,087 | 274 | 9,813 | 10 | ✓ | ✓ | ✗ | 11:28:12
LJ Speech [25] | 13,100 | 40 | 13,060 | 1 | ✓ | ✗ | ✗ | 23:55:17
MELD [26] | 13,708 | 142 | 13,566 | 304 | ✓ | ✓ | ✓ | 12:02:44
MSP-Improv [27] | 8,348 | 129 | 8,219 | 13 | ✓ | ✓ | ✗ | 8:25:41
MSP-Podcast [28] | 73,042 | 692 | 72,350 | $\approx 1,273~{}~{}~{}$ | ✓ | ✓ | ✗ | 113:41:00
Social-IQ [29] | 12,024 | 122 | 11,902 | - | ✓ | ✗ | ✗ | 20:39:00
SwitchBoard [30] | 259,890 | 456 | 259,434 | 400 | ✓ | ✗ | ✗ | $\approx 260:00:00~{}~{}~{}$
VCTK [31] | 44,583 | 50 | 44,533 | 110 | ✓ | ✗ | ✗ | 70:22:28

Table 2: DeToxy-B Statistics

Dataset | # Utterances | # Toxic | # Non-Toxic | # Sp | # Positive | # Neutral | # Negative | TL (hh:mm:ss)
---|---|---|---|---|---|---|---|---
CMU-MOSEI | 860 | 217 | 643 | - | 233 | 151 | 476 | 1:44:25
CMU-MOSI | 260 | 67 | 193 | - | 77 | 49 | 134 | 0:18:03
Common Voice | 11,551 | 2,888 | 8,663 | 6,350 | 3,241 | 2,199 | 6,111 | 12:38:17
IEMOCAP | 1,090 | 274 | 816 | - | 313 | 195 | 582 | 1:19:26
LJ Speech | 148 | 40 | 108 | 1 | 30 | 27 | 91 | 0:14:57
MELD | 565 | 142 | 423 | $\approx 67~{}~{}~{}$ | 145 | 136 | 284 | 0:31:05
MSP-Improv | 523 | 129 | 394 | 12 | 115 | 91 | 317 | 0:36:32
MSP-Podcast | 2,772 | 692 | 2,080 | $\approx 415~{}~{}~{}$ | 842 | 516 | 1,414 | 4:01:57
Social-IQ | 479 | 122 | 357 | - | 147 | 93 | 239 | 0:36:40
Switchboard | 1,824 | 456 | 1,368 | - | 500 | 345 | 979 | 2:28:57
VCTK | 199 | 50 | 149 | 81 | 58 | 33 | 108 | 0:08:51
Total | 20,271 | 5,077 | 15,194 | - | 5,701 | 3,835 | 10,735 | 24:39:10

##  4 Baselines

###  4.1 Problem Formulation

The task of toxicity detection from spoken utterances involves assigning a
probability score, denoting the toxicity, to each utterance fed to the model.
Formally put, if X = {$x_{1}$, $\cdots$, $x_{i}$, $\cdots$, $x_{N}$} are the
set of input utterances from the dataset $D$ with $N$ utterances, then Y =
{$y_{1}$, $\cdots$, $y_{i}$, $\cdots$, $y_{N}$} are their corresponding labels
where _y i_ $\in\\{0,1\\}$.

In the next 2 sections we describe in detail our proposed 2-step and End-to-
End baselines for DeToxy, the different components involved, and the training
procedure involved in each.

###  4.2 2-step Baselines

Our 2-step approach consists of 2 primary components, an ASR component that
produces transcriptions from spoken utterances, and a Sequence Classification
Model for classifying the toxicity of the output transcript. For our ASR
model, we resort to using wav2vec-2.0 [8]. Wav2vec-2.0 is built on the
transformer architecture and has a Convolutional Neural Network (CNN) feature
encoder layer, which encodes raw audio into speech frames. Wav2vec-2.0 follows
the pre-training and fine-tuning paradigm wherein we first pre-train the model
using a large unlabeled audio corpus in a self-supervised fashion solving a
contrastive task as follows:

$\mathcal{L}_{m}=-\log\frac{\exp\left(\operatorname{sim}\left(\mathbf{c}_{t},\mathbf{q}_{t}\right)/\kappa\right)}{\sum_{\tilde{\mathbf{q}}\sim\mathbf{Q}_{t}}\exp\left(\operatorname{sim}\left(\mathbf{c}_{t},\tilde{\mathbf{q}}\right)/\kappa\right)}$
(1)

where $\mathcal{L}_{m}$ is the InfoNCE loss that wav2vec-2.0 tries to
minimize, $\mathbf{c}_{t}$ is the context representation of a masked frame
obtained from the last transformer layer, $\mathbf{q}_{t}$ is the quantized
representation of obtained through a product quantization based quantization
module and $\tilde{\mathbf{q}}$. Precisely, for every masked input frame which
is input from the feature encoder module to the transformer-based context
encoder, wav2vec-2.0 tries to identify the true quantized representation among
a set of distractors. For more implementation-specific details about
wav2vec-2.0, we refer our readers to [8]. For our 2-step approach, we use the
wav2vec-2.0LARGE architecture to push performance on ASR.

Post SSL pre-training, the model is then fine-tuned by minimizing the CTC
objective function on labeled speech data. We do not use a Language Model in
our final setup to decode our utterances during inference, as we did not find
significant differences in Word Error Rate (WER) with it in our experiments,
and also to keep the setup simple.

Both self-supervised and supervised training paradigms for ASR tend to get
biased to the domain from where the data originates [32] and therefore we make
careful data considerations for pre-training and fine-tuning our model. Most
works in literature report results on a single benchmark dataset and so we
resort to multiple dataset pre-training but single dataset fine-tuning, We
choose to pre-train our model using a combination of Libri-Light, Common Voice
(CV) [23], SwitchBoard (SWBD) [30], and Fisher [33]. These 4 datasets cover a
diverse set of domains and suit well to our downstream ASR fine-tuning needs
without creating a huge domain-shift [32]. For downstream ASR fine-tuning, we
resort to 3 different dataset setups as follows: 1) To fit our model to the
conversational domain, we fine-tune it on 300 hours of SWBD . 2) For read
speech we resort to 960hrs of LibriSpeech fine-tuning and finally 3) We also
fine-tune on CV which comprises of a majority of DeToxy.

For our text-based sequence-classification model, which is the second step of
our 2-step approach, we employ the _BERT BASE_ architecture from the
transformer family. Since its inception, transformers have achieved SOTA on
sequence classification tasks including toxicity or hate-speech classification
[34, 35]. Formally put, we tokenize each word in the sentence and feed it as
input through the transformer architecture. Next, we utilize the hidden state
embedding _e_ corresponding to the _[CLS]_ token, where _e_
$\in\mathbb{R}^{768}$, as the aggregate representation of words in the
transcript of the utterance. This embedding _e_ is now fed through a final
classification layer which learns a parameterized function $f(.)$ and outputs
$f(e)=h_{x}\in\mathbb{R}^{n}$ where $n=2$. Finally, we pass the logits
obtained through a softmax activation function to get the probability
distribution $p$ of toxicity for each sentence. For training our model we
minimize the Cross Entropy (C.E.) loss as follows:

$\text{C.E.}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{M}y_{i_{j}}\log\left(p_{i_{j}}\right)$
(2)

where $y_{i_{j}}$ is corresponds to the $j^{th}$ gold annotated class from the
$i^{th}$ training instance in our dataset and $p_{i_{j}}$ is the probability
score given by the model that the $i^{th}$ instance belongs to the $j^{th}$
class. During inference, we simply perform the $argmax(p_{i})$ to find the
final class for each utterance.

We fine-tune _BERT BASE_ on a publicly available large-scale hate-speech
classification dataset [36] or on the relatively smaller gold toxicity
annotations from the DeToxy-B dataset.

###  4.3 End-to-End Baselines

For our End-to-End (E2E) Baselines, we choose to experiment with different
feature extractors, where we build baselines employing either low-level
feature extractors (eg. Filter-Banks) or high-level features from contextual
feature encoder models learned through SSL on unlabelled speech data. In the
next 2 sub-sections, we briefly describe both our approaches.

####  4.3.1 F-Bank

For our Filter-Bank (F-Bank) based E2E approach, given an audio input sequence
$x$, we extract log-compressed mel-scaled F-Banks with a window size of 25 ms
and a hop size of 10 ms to obtain _T_ frames. Post this, we employ
$GlobalAveragePooling(.)$ over all the _T_ frames [_h 1_, $\cdots$,_h T_],
where $h_{t}$ corresponds to the frame at time-step $t$, to obtain a single
embedding _e_ $\in\mathbb{R}^{80}$ for each utterance. This embedding is then
fed into a toxicity classification decoder, consisting of a single fully
connected layer post which we employ the _softmax_ activation function to the
output of this layer to obtain the probability of toxicity for each utterance.

Figure 1: Proposed Baselines: 2-step approach (left) and E2E approach (right)

####  4.3.2 wav2vec-2.0

For a fair comparison with our 2-step approach, we employ the same wav2vec-2.0
architecture employed in the 2-step procedure for our E2E system and
experiment with only the _BASE_ architecture. We follow the same pre-training
methodology as our 2-step baseline ASR system, but this time we fine-tune the
parameters of our model on DeToxy-B, solving a downstream toxicity
classification task, and train in an E2E fashion. During downstream training,
we either freeze or keep all the parameters of wav2vec-2.0 trainable. In both
cases, the CNN feature extractor remains frozen by default.

To fine-tune our model on the downstream task, we employ the same pooling
strategy and _prediction-head_ as our F-Bank approach, except now we take the
embeddings from the last layer of our ASR encoder and obtain a single
embedding _e_ $\in\mathbb{R}^{768}$ for each utterance by passing it through a
$GlobalAveragePooling(.)$ layer which pools over all the time-steps.

Both these models are trained using Cross-Entropy Loss (2) and similar to our
2-step procedure, during inference, we take the $argmax(.)$ of the output of
the $softmax(.)$ function as the final class for each utterance.

##  5 Experimental Setup

We use the PyTorch Framework for building and evaluating all our Deep Learning
models along with the Transformer implementations, pre-trained models, and
specific tokenizers are taken from the HuggingFace library. We train all our
models with Adam optimizer in batched mode with a batch size of 8 and a
learning rate of $1e^{-4}$ for a maximum of 100 epochs with an early-stopping
of 5 epochs. We make all our code and data publicly available on
GitHub222https://github.com/Sreyan88/Toxicity-Detection-in-Spoken-Utterances

##  6 Results and Result Analysis

Table 3, provides the _macro-averaged F 1_ scores obtained for variations of
both our 2-step and E2E baselines on the Development, Test and Test-T splits
of DeToxy-B post training on the Train split. The category section of 2-step
system is denoted as “Transcript Source - Sequence Classifier Data Source”,
where Gold is gold transcripts and LS, CV, and SWBD stand for fine-tuning on
LibriSpeech, Common Voice, and SwitchBoard respectively. CC [36] and DB stand
for _BERT BASE_ fine-tuned on Civil Comments or DeToxy-B gold text transcripts
respectively. Beyond freezed and un-freezed wav2vec-2.0 E2E setups, based on
findings by [37], we also test our E2E model by taking embeddings from the
$9^{th}$ layer which contains maximum semantic information.

Table 3: Experimental Results of our 2-step and E2E baselines on DeToxy-B System | Category | Dev | Test | Test-T
---|---|---|---|---
2-step | Gold-CC | 0.787 | 0.801 | 0.258
| Gold-DB | 0.921 | 0.934 | 0.492
| LS-CC | 0.672 | 0.682 | 0.281
| LS-DB | 0.742 | 0.744 | 0.484
| CV-CC | 0.659 | 0.674 | 0.270
| CV-DB | 0.505 | 0.724 | 0.474
| SWBD-CC | 0.672 | 0.682 | 0.270
| SWBD-DB | 0.726 | 0.737 | 0.478
F-Bank | - | 0.610 | 0.620 | 0.491
wav2vec-2.0 | Freezed | 0.448 | 0.457 | 0.497
| Un-freezed | 0.877 | 0.869 | 0.510
wav2vec-2.0 (9) | Un-freezed | 0.897 | 0.877 | 0.540

As we see in Table 3, our E2E system clearly outperforms our 2-step system
setup when gold transcripts are not available to the text sequence classifier.
We hypothesize the true reason to be that ASR models don’t generalize across
domains and generally fail to perform well when trained on one domain and
inferred on another. This is also very evident in our case where we get an
average WER of 33%, 43% and 26.9% on LS, CV and SWBD respectively. 2-step
systems trained on CC consistently under-performs, which reveals that text
sequence classifiers models trained to classify toxicity are not robust to
changes in domain.

Our un-frozen wav2vec-2.0 E2E setup with representations taken from the
$9^{th}$ layer outperforms all our true baselines. This proves that semantic
information of the utterance is crucial to toxicity classification from spoken
speech. This setup also outperforms other baselines on Test-T, in which very
evidently all our baselines struggled on, especially our 2-step baselines
trained on CC.

##  7 Conclusion

In this paper, we introduce the first publicly available human-annotated
dataset DeToxy for the task of toxic speech classification and propose two
strong baselines for this task. Future work includes expanding the dataset at
least 10 fold using more naturally spoken utterances, exploring different
modalities, and developing better neural architectures to solve this problem.

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# Optimal Dynamical Decoupling Sequence for Ohmic Spectrum

Yu Pan1,2 Zai-Rong Xi1<EMAIL_ADDRESS>Wei Cui1,2 1Key Laboratory of Systems
and Control, Institute of Systems Science, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of
China  2Graduate University of Chinese Academy of Sciences, Beijing 100039,
People’s Republic of China

###### Abstract

We investigate the optimal dynamical decoupling sequence for a qubit coupled
to an ohmic environment. By analytically computing the derivatives of the
decoherence function, the optimal pulse locations are found to satisfy a set
of non-linear equations which can be easily solved. These equations
incorporates the environment information such as high-energy (UV) cutoff
frequency $\omega_{c}$, giving a complete description of the decoupling
process. The solutions explain previous experimental and theoretical results
of locally optimized dynamical decoupling (LODD) sequence in high-frequency-
dominated environment, which were obtained by purely numerical computation and
experimental feedback. As shown in numerical comparison, these solutions
outperform the Uhrig dynamical decoupling (UDD) sequence by one or more orders
of magnitude in the ohmic case.

##  I INTRODUCTION

Suppressing decoherence is one of the fundamental issues in the field of
quantum information processing. Decoherence, which has been caused by the
environmental noise, plagues almost all the implementations of quantum bit. To
eliminate the unwanted coupling between a qubit and its environment, several
schemes have been proposed and tested. Among them a promising one is dynamical
decoupling Viola and Lloyd (1998); Ban (1998); Viola et al. (1999); Gordon et
al. (2008), which restores the qubit coherence by applying delicately designed
sequence of control pulses.

For a qubit that can be modeled by a spin-1/2 particle, the oldest dynamical
decoupling sequence is periodic dynamical decoupling (PDD). Originated from
pulse sequences widely used in nuclear magnetic resonance (NMR) Ernst et al.
(1991), the PDD sequence consists of periodic and equidistant $\pi$ pulses. To
achieve better performance, there has been an extensive study in how to
optimize the pulse locations Viola and Knill (2005); Kern and Alber (2005);
Khodjasteh and Lidar (2005, 2007); Uhrig (2007); Yang and Liu (2008); Pryadko
and Quiroz (2008); Pasini and Uhrig (2009); Biercuk et al. (2009a); Uys et al.
(2009). One important progress is the powerful Uhrig DD (UDD) Uhrig (2007),
which employs $n$ pulses located at $t_{j}$ according to the simple rules

$\delta_{j}=\sin^{2}(j\pi/2(n+1)),$

where $\delta_{j}=t_{j}/T$ and $T$ is the total evolution time. UDD is first
derived on spin-boson model and further proved to be universal in the sense
that it can remove the qubit-bath coupling to $n^{th}$ order in generic
environment Yang and Liu (2008).

Beyond UDD, another locally optimized dynamical decoupling (LODD) sequence has
drawn great attention Biercuk et al. (2009a). LODD, along with its simplified
version optimized noise-filtration dynamic decoupling (OFDD) Uys et al.
(2009), generates the decoupling sequence by directly optimizing the
decoherence function using numeric methods as well as experimental feedback.
It has been shown to be able to suppress decoherence effect by orders of
magnitude over UDD for certain noise spectrum, especially for the one with a
high frequency part and sharp high-energy (UV) cutoff.

However, in spite of the great experimental success, analytical results about
the LODD sequence is insufficient. Until recently S. Pasini and G. S. Uhrig
has made an analytical progress in optimizing the decoherence function for
power law spectrum (PLODD) Pasini and Uhrig (2009). The power law spectrum
$\omega^{\alpha}$ for $\alpha<1$ without UV cutoff is considered. They
minimize the decoherence function through expanding the function and
separating, canceling divergences from the relevant terms and solving
variation problems. Inspired by Pasini’s work, we try to analyze the LODD
problem with respect to the ohmic spectrum $S(\omega)\sim\omega$ and a sharp
UV cutoff. Ohmic noise is the major decoherence source often found in a
qubit’s environment, for example, the semiconducting quantum dot Leggett et
al. (1987) and superconducting qubit Weiss (2008). Optimal performance pulse
sequence is found analytically which entirely differs from the UDD sequence in
such environment. We call this kind of optimal sequence HLODD (LODD for ohmic
spectrum) for short.

We organize this paper as follows. In the second section we propose the
optimization problem of the decoherence function. In section III, we derive
the analytical equations for the optimal pulse sequence. In the following
section, we run a simulation to verify our results. Conclusions are put in
section V.

##  II OPTIMIZATION OF THE DECOHERENCE FUNCTION

Given a two-level quantum system, when the environmental noise behaves
quantum-mechanically, we use the long-established spin-boson model with pure
dephasing

$H=\sum_{i}\omega_{i}b_{i}^{\dagger}b_{i}+\frac{1}{2}\sigma_{z}\sum_{i}\lambda_{i}(b_{i}^{\dagger}+b_{i}).$
$None$

Here we ignore the qubit free evolution hamiltonian. On the other hand, when
the qubit is subjected to classical noise, the system is modeled as Cywinski
et al. (2008); Kuopanportti et al. (2008)

$H=\frac{1}{2}[\Omega+\beta(t)]\sigma_{z},$ $None$

where the $\Omega$ is the qubit energy splitting and $\beta(t)$ the classical
random noise. Let $t$ be the total evolution time, and $n$ pulses are applied
at $t_{1}<t_{2}<...<t_{n}$ in sequence with negligible pulse durations . We
use the notation $\delta_{j}=\frac{t_{j}}{t}$. This naturally leads to the
definition of $t_{0}=0$ and $t_{n+1}=1$. In either ($1$) or ($2$), the decay
of coherence under the dynamical decoupling sequence can be described by the
decoherence function Uhrig (2007); Cywinski et al. (2008); Uhrig (2008);
Biercuk et al. (2009b) $e^{-2\chi(t)}$ with

$\chi(t)=\int_{0}^{\infty}\frac{S(\omega)}{\omega^{2}}{|y_{n}(\omega{t})|}^{2}d\omega,$
$None$

where $S(\omega)$ is environmental noise spectrum. The filter function
$y_{n}(t)$ is given by

$y_{n}(t)=1+(-1)^{n+1}e^{\mbox{i}\omega{t}}+2\sum_{j=1}^{n}(-1)^{j}e^{\mbox{i}\omega{t}\delta_{j}}.$
$None$

Thus minimization of $\chi(t)$ with respect to $\delta_{j}$ gives the optimal
decoupling sequence.

We now consider the case when the noise spectrum is ohmic with a sharp cutoff
at $\omega_{c}$, i.e. $S(\omega)=S_{0}\omega\Theta(\omega_{c}-\omega)$.
$S_{0}$ is an irrelevant constant factor and $\Theta$ is unit step function.
Then minimization of ($3$) turns to minimization of $I_{n}$ with

$I_{n}=\int_{0}^{z_{c}}\frac{{|y_{n}(z)|}^{2}}{z}dz,$ $None$

where $z_{c}=\omega_{c}{t}$. Since $y_{n}(0)=0$, the IR convergence insures
the integral converges to a finite value Pasini and Uhrig (2009).

##  III DERIVATION OF OPTIMAL PULSE SEQUENCE

We follow the approach of Pasini and Uhrig Pasini and Uhrig (2009) to treat
the integral ($5$). Here we use notation

$q_{j}=\left\\{\begin{array}[]{ll}0&\mbox{if $j=0,n+1$,}\\\ 1&\mbox{if
$j\in\\{1,2,...,n\\}$,}\end{array}\right.$

and

$\Delta_{ij}=\mbox{i}(\delta_{i}-\delta_{j}),$

from which we get

$|y_{n}(z)|^{2}=\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}e^{z\Delta_{ij}}.$

Then the integral $I_{n}$ can be expressed as

$I_{n}=\lim_{x\to 0^{+}}I_{n}(x),$
$I_{n}(x)=\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}I_{ij}(x),$ $None$

where the integrals $I_{ij}(x)$ are

$\displaystyle I_{ij}(x)$ $\displaystyle=$
$\displaystyle\int_{x}^{z_{c}}\frac{e^{\Delta_{ij}z}}{z}dz$ (7)
$\displaystyle=$
$\displaystyle\int_{-\Delta_{ij}x}^{-\Delta_{ij}z_{c}}\frac{e^{-z}}{z}dz.$

The limit $x\to 0^{+}$ is carried out because $I_{ij}(0)$ does not exist for
arbitrary $i,j$. Making use of the series representation of exponential
function Abramowitz and Stegun (1964)

$\displaystyle E_{1}(z)$ $\displaystyle=$
$\displaystyle\int_{z}^{\infty}\frac{e^{-t}}{t}dt$ $\displaystyle=$
$\displaystyle-\gamma-\ln{z}+\sum\limits_{k=1}^{\infty}\frac{(-1)^{k+1}}{k!k}z^{k},$

where $\gamma$ is the Euler-Mascheroni constant and the sum converges for all
the complex $z$, $I_{ij}(x)$ can be written as

$\displaystyle I_{ij}(x)$ $\displaystyle=$ $\displaystyle
E_{1}(-\Delta_{ij}x)-E_{1}(-\Delta_{ij}z_{c})$ (8) $\displaystyle=$
$\displaystyle\ln{{(z_{c}}/{x})}+\sum\limits_{k=1}^{\infty}\frac{\Delta_{ij}^{k}}{k!k}(z_{c}^{k}-x^{k}).$

Since we always have $y_{n}(0)=0$ which implies

$|y_{n}(0)|^{2}=\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}=0,$ $None$

we can now proceed by taking the limit $x\to 0+$ in $I_{n}$

$\displaystyle I_{n}$ $\displaystyle=$ $\displaystyle\lim_{x\to
0^{+}}\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}[\ln{{(z_{c}}/{x})}+\sum\limits_{k=1}^{\infty}\frac{\Delta_{ij}^{k}}{k!k}(z_{c}^{k}-x^{k})]$
(10) $\displaystyle=$ $\displaystyle\lim_{x\to
0^{+}}\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}\sum\limits_{k=1}^{\infty}\frac{\Delta_{ij}^{k}}{k!k}(z_{c}^{k}-x^{k})$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{i}+q_{j}}(-1)^{i+j}\frac{\Delta_{ij}^{k}}{k!k}z_{c}^{k}.$

To minimize $I_{n}$, UDD requires the first $n$ derivatives of $y_{n}$ vanish
while OFDD simplifies the optimization process by replacing $S(\omega)$ by a
constant. Here we attempt to minimize $I_{n}$ directly to obtain optimal pulse
sequence. We notice that at the optimal pulse locations $\delta_{j}\
(j=1,2,...,n)$, the gradient of $I_{n}$ vanishes. So we impose the following
conditions $\frac{\partial{I_{n}}}{\partial\delta_{m}}=0,$ for $m$ from $1$ to
$n$. Although ($10$) are complex infinite series, we can still explicitly
compute the derivatives of ($10$) as long as these derivatives converge. For
arbitrary $m$ we have

$\displaystyle\frac{\partial{I_{n}}}{\partial\delta_{m}}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\delta_{m}}\sum_{i,j=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{i}+q_{j}}(-1)^{i+j}\frac{\Delta_{ij}^{k}}{k!k}z_{c}^{k}$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\delta_{m}}\\{\sum_{i=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{i}+q_{m}}(-1)^{i+m}\frac{\Delta_{im}^{k}}{k!k}z_{c}^{k}$
$\displaystyle+\sum_{i=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{m}+q_{i}}(-1)^{m+i}\frac{\Delta_{mi}^{k}}{k!k}z_{c}^{k}\\}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{m}+q_{i}}(-1)^{m+i}\frac{z_{c}^{k}}{k!}\mbox{i}[\Delta_{mi}^{k-1}-(-1)^{k-1}\Delta_{mi}^{k-1}].$

The terms with $k$ odd cancel, so the result can be simplified as

$\displaystyle=$
$\displaystyle\sum_{i=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{m}+q_{i}+1}(-1)^{m+i}\frac{z_{c}^{2k}}{(2k)!}{\mbox{i}}^{2k}{(\delta_{m}-\delta_{i})}^{2k-1}$
$\displaystyle=$
$\displaystyle\sum_{i\neq{m}}^{n+1}\frac{1}{\delta_{m}-\delta_{i}}2^{q_{m}+q_{i}+1}(-1)^{m+i}\sum\limits_{k=1}^{\infty}\frac{z_{c}^{2k}}{(2k)!}{\mbox{i}}^{2k}{(\delta_{m}-\delta_{i})}^{2k}$
$\displaystyle=$
$\displaystyle\sum_{i\neq{m}}^{n+1}\frac{1}{\delta_{m}-\delta_{i}}2^{q_{m}+q_{i}+1}(-1)^{m+i}\\{\cos[(\delta_{m}-\delta_{i})z_{c}]-1\\}.$

Here we have used the expansion
$\cos{(z)}=\sum\limits_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}z^{2k}$ which
converges on the whole complex plane. From ($12$) we know that the derivatives
of $I_{n}$ indeed converge to a finite value. Thus the optimal pulse locations
$\\{\delta_{1},\delta_{2},...\delta_{n}\\}$ shall satisfy the following non-
linear equations

$\sum_{i\neq{m}}^{n+1}\frac{1}{\delta_{m}-\delta_{i}}2^{q_{m}+q_{i}+1}(-1)^{m+i}\\{\cos[(\delta_{m}-\delta_{i})z_{c}]-1\\}=0.$
$None$

Equations ($13$) are main results of this paper. The optimal sequence obtained
from ($13$) is quite different from the UDD sequence obeying

$\sum\limits_{j=1}^{n+1}2^{q_{j}}(-1)^{j}\delta_{j}^{p}=0$

for $p=\\{1,2,...n\\}$. For the ohmic spectrum, our equations incorporate the
UV cutoff frequency $\omega_{c}$, indicating that the solutions are specially
tailored to combat this kind of noise. Although the UDD sequence is universal
in suppressing decoherence, we believe that the HLODD sequence will outperform
the UDD sequence in the ohmic environment. In the next section, we use numeric
methods to illustrate the performance of HLODD sequence.

Figure 1: Comparison between UDD and HLODD sequence for different UV cutoff
frequency $\omega_{c}$. Pulse sequences $\delta_{i}$ for $n=2$ and $n=5$ are
plotted in one figure under the same $\omega_{c}$.

Figure 2: Semi-log plot of $I_{n}$ versus pulse number $n$. UDD and HLODD
sequences are compared.

##  IV NUMERICAL RESULTS

We start our simulation by solving the non-linear equations ($13$) and
evaluate the decoherence function with these solutions. First, we set the
total evolution time $t=1$ without loss of generality. Then $z_{c}=\omega_{c}$
and we can concentrate on analyzing the influence of the cutoff frequency
$\omega_{c}$. Computing solutions to ($13$) for different $\omega_{c}$, we
find that the optimal pulse sequences behave differently. We also evaluate the
UDD sequence for comparison.

As shown in Fig. 1, deviation of the pulse locations $\delta_{i}$ in HLODD
sequence from their UDD counterparts increases with $\omega_{c}$. This agrees
with our intuition since UDD focuses on suppressing decoherence by minimizing
$|y_{n}(z)|$ in the neighborhood of $y_{n}(0)$, weakening its ability to
maintain small $|y_{n}(z)|$ on the other end of the spectrum. For large
$\omega_{c}$, UDD sequence is no longer optimal. In addition, we can see pulse
number $n$ plays an important role. By increasing $n$, UDD can narrow the
difference from HLODD. The difference between the two sequences when $n=2$ is
greatly reduced when we increase $n$ to $5$, see Fig. 1. Especially for the
case $\omega_{c}=1$, the difference is completely removed. However, for larger
$\omega_{c}$ this gap can’t be removed by increasing $n$.

Next, to demonstrate the optimal decoupling ability of HLODD sequence, we
compute $I_{n}$ versus $n$ while $\omega_{c}$ is chosen to be $5$. The results
are depicted in Fig. 2, and again are compared with UDD. The obtained
solutions yield a significant improvement over UDD. For fixed $n$, the HLODD
suppresses decoherence better than UDD by one or two orders of magnitude which
is in agreement with the results in Biercuk et al. (2009a); Uys et al. (2009),
where LODD and OFDD sequences are tested for 9Be+ qubits in a penning ion trap
and various spectrum. The qubit error rates are below $10^{-5}$ when $n>5$,
and we see that HLODD is capable of suppressing the error rates far below the
Fault-Tolerance error threshold Nielsen and Chuang (2000) by increasing $n$.
Furthermore, by inspecting the points on the HLODD curve, we expect the HLODD
sequence suppresses decoherence in power law $n$ as UDD.

At last, we would like to explain the numerical results in another way. If we
fixed UV cutoff frequency $\omega_{c}$ at the beginning, and compare the HLODD
performance for $t=1$, $t=5$, and $t=10$, the numerical results would be the
same since $z_{c}$ did not change. So we can also conclude that for the same
number of pulses $n$, HLODD will beat UDD with increasing total evolution time
$t$.

##  V CONCLUSIONS

In this paper we analytically find the optimal pulse locations to decouple a
qubit in an ohmic environment. By deriving the analytical expressions for the
derivatives of decoherence function, we obtain a set of non-linear equations
which the optimal pulse sequence must obey. These equations are completely
different from UDD and are more accurate, because they incorporate the effect
of UV cutoff frequency $\omega_{c}$.

In our numerical simulation, the analytical results provide an improvement
over UDD sequence by an order or two of magnitude, which is consistent with
previous results in LODD and OFDD obtained by purely numerical minimization
and experimental feedback. We have to mention that the pulse performance is
influenced by the sharp UV cutoff frequency $\omega_{c}$ greatly. The larger
the UV cutoff $\omega_{c}$, the more HLODD deviates from UDD. Early work
Pasini and Uhrig (2009); Biercuk et al. (2009a); Cywinski et al. (2008); Uhrig
(2008) has pointed out that for soft large UV cutoff, UDD performs even worse
and LODD is still a better choice. However, the integral ($3$) for $S(\omega)$
with a soft cutoff is hard to analyze.

In conclusion, our work provides an analytical solution to optimal dynamical
decoupling for ohmic case. Our derivation is based on ohmic spectrum, but we
believe it can be extended to super-ohmic case
$S(\omega)\sim\omega^{\alpha}(\alpha>1)$ via slight modification.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(No. 60774099, No. 60821091), the Chinese Academy of Sciences (KJCX3-SYW-S01),
and by the CAS Special Grant for Postgraduate Research, Innovation and
Practice.

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