# DUAL: Discrete Spoken Unit Adaptive Learning for Textless Spoken Question
Answering

###### Abstract

Spoken Question Answering (SQA) is to find the answer from a spoken document
given a question, which is crucial for personal assistants when replying to
the queries from the users. Existing SQA methods all rely on Automatic Speech
Recognition (ASR) transcripts. Not only does ASR need to be trained with
massive annotated data that are time and cost-prohibitive to collect for low-
resourced languages, but more importantly, very often the answers to the
questions include name entities or out-of-vocabulary words that cannot be
recognized correctly. Also, ASR aims to minimize recognition errors equally
over all words, including many function words irrelevant to the SQA task.
Therefore, SQA without ASR transcripts (textless) is always highly desired,
although known to be very difficult.

This work proposes Discrete Spoken Unit Adaptive Learning (DUAL), leveraging
unlabeled data for pre-training and fine-tuned by the SQA downstream task. The
time intervals of spoken answers can be directly predicted from spoken
documents. We also release a new SQA benchmark corpus, NMSQA, for data with
more realistic scenarios. We empirically showed that DUAL yields results
comparable to those obtained by cascading ASR and text QA model and robust to
real-world data. Our code and model will be open-
sourced111https://github.com/DanielLin94144/DUAL-textless-SQA.

Index Terms: Spoken Question Answering, Textless NLP, Self-Supervised
Representation

##  1 Introduction

Spoken Question Answering (SQA) aims to find the answer from a spoken document
given a question in either text or spoken form. SQA is crucial for personal
assistants when replying to the questions from the user’s spoken queries.
Unlike many spoken language understanding tasks such as speech translation or
intent classification, in which the required understanding of semantics is
primarily on utterance level, the SQA task requires sophisticated
comprehension and reasoning over much longer audio content. In addition to
understanding the question and comprehending the global information in the
audio context, it also needs to catch the fine-grained information to
precisely locate the answer span out of the long audio context. Thus, SQA is
known to be a very challenging task.

Figure 1: The proposed DUAL framework for textless (ASR transcript-free) SQA.
All the passages, questions, and answers are in spoken form. The time
intervals of the spoken answers can be extracted from the spoken passage
without ASR transcripts.

The conventional SQA system consists of the cascade of an Automatic Speech
Recognition (ASR) engine followed by a text-trained QA (TQA) model. However,
speech recognition errors naturally cause catastrophic problems to the TQA.
[1], and several works [2, 3, 4, 5] intended to alleviate such problems by
knowledge distillation, including adapting the TQA model to be more robust
against recognition errors. Some other efforts [6, 7] exploited paired speech
and transcripts to construct a cross-modal speech and text pre-trained model
with aligned semantics fine-tuned end-to-end, in which the speech recognition
error problems can be mitigated to some degree, and SQA performance improved.

However, ASR errors remain to be a major problem for SQA tasks. The correct
answers to the questions often include name entities or out-of-vocabulary
(OOV) words that can never be recognized. The key information is thus
inevitably lost when the audio signals are transformed into transcripts with
errors, and there is no way to recover them in the following TQA stage. Also,
the ASR engine was trained by minimizing the word error rate (WER), which was
evaluated equally overall words including many function words irrelevant to
the SQA task. So the cascade of two stages (ASR and TQA) individually
optimized with two different criteria cannot perform as well as a single-stage
global performance goal. So it is highly desired to capture the information
directly, rather than from the ASR transcripts, and obtain overall performance
not constrained by ASR accuracies.

Also, ASR engines have to be trained with vast quantities of human-annotated
audio-transcript data, which are time-consuming and expensive to collect for
the thousands of low-resourced languages over the world, when low and robust
enough error rates are considered. Furthermore, there exist many languages
without written form worldwide. All the above imply technologies for ASR
transcript-free (textless) SQA are highly desired although challenging.

In this work, we propose the first known textless (i.e., ASR transcript-free)
SQA framework as in Figure 1. Inspired by the concept of Textless NLP [8, 9,
10, 11, 12, 13], which encodes speech signals into discrete units for
modeling, and the pre-trained language models transferability of [14, 15, 16,
17, 18], we propose Discrete Unit Adaptive Learning (DUAL) for textless SQA.
DUAL leverages pre-trained models to obtain quantized, length-condensed speech
representations from audio signals and further adapts the pre-trained language
model to achieve competitive SQA results without any ASR transcripts. The time
span of the answer can be directly located from the audio context and played
to the user, so the extracted answers do not suffer from speech recognition
errors or out-of-vocabulary (OOV) problems because NO ASR is performed.

Furthermore, despite the increasing efforts to build SQA benchmark corpora [1,
19, 20, 21, 22, 23, 24], there is still a lack of natural and large-scale SQA
datasets featuring real-world scenarios. For this purpose, we release a novel
benchmark corpus, Natural Multi-speaker Spoken Question Answering (NMSQA). In
this corpus, the test set was produced by human speakers, and the training and
validation set were synthesized from Amazon Polly TTS service with industrial-
grade quality. We also assign two different speakers to read the pairs of
passage and question, examining whether our textless SQA system is speaker-
independent.

The contributions of this paper are summarized below:

  * •

We propose DUAL as the first known framework for textless SQA, not utilizing
ASR transcripts and not suffering from ASR errors.

  * •

We open-source the NMSQA dataset for SQA in real-world scenarios.

  * •

DUAL achieved performance competitive to those obtained by cascading ASR and
TQA, and significantly better when the word error rate exceeded 30 %.

  * •

DUAL is more robust and retains the performance for the real-speaker testing
set, which was not easily achievable for the cascade approach.

##  2 Method

###  2.1 Problem Formulation

The form of SQA dataset $D$ is $\\{\mathbf{q,p},a\\}$, corresponding to the
question $\mathbf{q}$, passage $\mathbf{p}$, and answer $a$, all in spoken
form in this work. Our goal is to extract the starting and ending time
$(t_{s},t_{e})$, denoted as the answer span $a$, from the spoken passage
$\mathbf{p}$ given the spoken question $\mathbf{q}$.

###  2.2 DUAL framework

The DUAL framework consists of the Speech Content Encoder (SCE) and Pre-
trained Language Model (PLM) as in Figure 2 and introduced below.

####  2.2.1 Speech Content Encoder (SCE)

The SCE transforms the question-passage audio waveform $(\mathbf{q,p})$ pair
to sequences of discrete units $(\mathbf{z_{q},z_{p}})$.
Self-supervised Speech Representation: A self-supervised speech pre-trained
model can extract informative feature representations. We adopted the state-
of-the-art self-supervised speech pre-trained model HuBERT [25] for feature
extraction222We used the open-source S3PRL [26] toolkit to extract the
representations of the HuBERT-Large model.. HuBERT was trained by masked
prediction objectives similar to BERT [27]. The prediction target was the
K-means clustered index for speech signal processing features, e.g., Mel-
frequency cepstral coefficients (MFCC) initially, and then learned latent
representations after clustering in subsequent iterations. We utilized the
HuBERT-Large model containing 24 transformer encoder layers pre-trained on
LibriLight 60k hour dataset. HuBERT encoded the raw waveform into frame-level
1024 dimension features. Each frame was equivalent to 20 ms.
Quantization: The goal of quantization is to discretize the speech features so
they can be fed into the pre-trained language model. K-means clustering was
performed over the layer-wise representations of HuBERT-Large. We used
LibriSpeech [28] 100-hour subset to train the K-means clustering model, and
the number of clusters $K$ is 64, 128, and 512. After clustering, the discrete
units are represented by the clustering indices. The repetitive discrete units
are merged to shorten the sequence length and remove the duration information,
forming the dense discrete unit sequence of the question and passage
$(\mathbf{z_{q}},\mathbf{z_{p}})$. We recorded the duration of duplication
number of repetitions as $\mathbf{c_{q}}$ and $\mathbf{c_{p}}$ for
$\mathbf{z_{q}}$ and $\mathbf{z_{p}}$, so we can recover the frame-level
indices to convert the answer span back to time interval at the inference
stage.

Figure 2: The overview of the DUAL framework.

####  2.2.2 Pre-trained Language Model (PLM)

The learning model is a BERT-like transformer encoder model. The input was the
discrete unit sequences of the spoken questions and passages
$(\mathbf{z_{q}},\mathbf{z_{p}})$. Because SQA is a very challenging task to
train from scratch, we leveraged the cross-disciplinary transferability of PLM
[14, 15, 16, 17, 18] to help the SQA downstream task. Specifically, we used
the weights of text PLM for network initialization, and randomly assigned the
text pre-trained input embeddings for discrete units. The different random
embedding assignments did not significantly affect the final performance. The
input of PLM was the concatenated discrete unit sequences of the question and
passage pair $(\mathbf{z_{q}},\mathbf{z_{p}})$, and the target was the start
and end time $(y_{s},y_{e})$ after the repetitions were reproduced.

Because the length of a discrete unit sequence is much longer than the
corresponding text, and the duration of a spoken passage itself is long, the
standard maximal length of PLM (typically 512) is not enough in our case. As a
result, we leveraged Longformer [29] to model the long
$(\mathbf{z_{q}},\mathbf{z_{p}})$, which is a BERT-like model for long
documents, pre-trained on the unlabeled long text documents and optimized for
training efficiency by sparse attention mechanism, such as local and global
attention, to support up to $4096$ tokens.

####  2.2.3 Training Objective

Table 1: Word Error Rates on different datasets for the two open-sourced ASR models used in the baseline (cascade). ASR | LS test-clean |  NMSQA dev | NMSQA test
---|---|---|---
SB | 3.1 |  15.6 | 61.7
W2v2 | 1.9 |  10.5 | 11.3

The training objective is similar to the canonical QA fine-tuning in TQA. A
randomly initialized linear layer is added on the top to predict the start and
end time index. As the gradient flow shown in Figure 2, $\theta$ represents
the trainable weights of the model,
$\mathbf{c_{p}}=[\mathbf{c_{p}}_{1},\mathbf{c_{p}}_{2},...,\mathbf{c_{p}}_{n}]$
is the repetition of every discrete units $\mathbf{z_{p}}_{i}$ in
$\mathbf{z_{p}}=[\mathbf{z_{p}}_{1},\mathbf{z_{p}}_{2},...,\mathbf{z_{p}}_{n}]$.
$(t_{s},t_{e})$ is the ground truth start and end time in second, and
$(y_{s},y_{e})$ is the converted the version on the index level. The overall
training objective was to minimize the loss $L(\theta)$ as the sum of the
negative log probabilities of the true start and end indices on all the
examples,

$\displaystyle-\sum\mathop{log}P(y_{s}|\mathbf{z_{q}},\mathbf{z_{p}};\theta)+\mathop{log}P(y_{e}|\mathbf{z_{q}},\mathbf{z_{p}};\theta).$

At the inference stage, we converted the predicted start and end indices
$(\hat{y}_{s},\hat{y}_{e})$ to the frame level by $\mathbf{c_{p}}$ above, and
finally transformed them to the time level $(\hat{t}_{s},\hat{t}_{e})$.

##  3 Experiments

###  3.1 Corpus Description

We developed and released here a new listening comprehension task named
Natural Multi-speaker Spoken Question Answering (NMSQA). The train and dev set
are the spoken version of the SQuAD v1.1 dataset, one of the largest QA
datasets from Wikipedia paragraphs and human-written questions. We randomly
split the SQuAD dev set into disjoint SQuAD-dev-1 and SQuAD-dev-2 for the
NMSQA dev set and test set. The Amazon Polly Text-to-Speech
service333https://aws.amazon.com/tw/polly/ was used for generating natural
speech. We randomly assigned 12 TTS speakers and ensured that different
speakers were used in producing each spoken document-question pair. Overall,
there are 297.18 / 37.61 hours of audio for the train/dev set. Moreover, in
order to have a realistic test set, 60 human speakers (30 male / 30 female)
were requested to produce the SQuAD-dev-2 naturally. This test set included
2.67 hours of audio. The answer intervals were annotated by Montreal Force
Aligner [30].

###  3.2 Evaluation

Since the output target here is the temporal span of the spoken answer,
following the evaluation metrics previously proposed by [1, 6], we adopted the
Frame-level F1 score (FF1) and Audio Overlapping Score (AOS) to evaluate the
performance based on the predicted time intervals. Higher FF1 and AOS scores
imply more overlap between the predicted and ground truth spans.

Table 2: The performance of the proposed (DUAL) and baseline (cascade) approaches on the NMSQA dev and test sets. “Longformer†” indicates the Longformer model fine-tuned on clean text SQuAD-v1.1, while the normal “Longformer” was only pre-trained by unlabeled text data. The number after HuBERT (64, 128, 512) are numbers of clusters. “synth” and “human” represent the synthesized and human speech respectively. Input |  Model | dev (synth) | test (human)
---|---|---|---
FF1 |  AOS | FF1 |  AOS
Baseline - Cascade (with ASR transcripts)
SB |  Longformer† | 56.7 |  49.7 | 17.3 |  15.3
W2v2 |  Longformer† | 65.7 |  58.3 | 64.2 |  57.4
Proposed - DUAL (without ASR transcripts)
HuBERT-64 |  Longformer | 47.8 |  42.2 | 39.0 |  33.0
HuBERT-128 |  Longformer | 54.2 |  48.5 | 55.9 |  49.1
HuBERT-512 |  Longformer | 55.0 |  49.6 | 17.3 |  12.5

###  3.3 Baseline - Cascade (ASR plus TQA)

The conventional SQA approach cascading an ASR model and a TQA model was taken
as the baseline to be compared here. Two open-sourced pre-trained ASR models
were used in the first stage, one from Speechbrain
[31]444https://huggingface.co/speechbrain/asr-crdnn-rnnlm-librispeech,
referred to as SB, the other the Wave2vec 2.0-large with self-training fine-
tuning [32]555https://huggingface.co/facebook/wav2vec2-large-960h-lv60-self,
referred to as W2v2. The Word Error Rates of them on different speech datasets
are listed in Table 1. Both SB and W2v2 utilized LibriSpeech [28] 960-hour
dataset as the supervised training data; however, we see in Table 1 W2v2 was
much more robust than SB on the NMSQA test set, obviously because it leveraged
60k hrs of unlabeled data and the self-training procedure.

The TQA model, or the second stage of the baseline, is a Longformer-based
model fine-tuned on SQuAD v1.1, denoted as Longformer† below. We used the
online available model checkpoint666
https://huggingface.co/valhalla/longformer-base-4096-finetuned-squadv1 for TQA
inference. The Longformer† obtained 91.5 F1 score and 85.1 EM (Exact Match)
score on the text SQuAD v1.1 dataset. For the evaluation metrics used here, we
adopted force alignment [30] to obtain the time intervals of the spoken
answers in seconds.

###  3.4 Implementation Details of DUAL

We use the official Longformer-base
model777https://huggingface.co/allenai/longformer-base-4096 as the PLM. The
learning rate is searched in [3e-5, 5e-5, 7e-5, 1e-4], and we select models
with the best performance on the validation set. The learning rate warmup step
is 500, growing up linearly to the peak value and then linearly decaying to 0.
All the DUAL experiments use 4 Tesla V100s with an overall 128 batch size for
up to 5000 training steps. If the length of discrete units
$(\mathbf{z_{q}},\mathbf{z_{p}})$ input exceeds 4096, we truncate the passage
$\mathbf{z_{p}}$.

##  4 Results

Encouraging results are reported here. Noting that the proposed (DUAL)
approach achieves performance comparable to baselines (cascade), which require
large ASR parallel data.

###  4.1 “dev” set for synthesized speech

The experimental results are shown in Table 2. We first consider results for
synthesized speech on the “dev” set, whose style is very similar to the
training set, in the first column.

The top section of Table 2 is for the baseline (cascade) approach with ASR
transcripts. We see ASR with W2v2 offered much better performance than that
with SB (65.7 vs. 56.7 for FF1), obviously due to its lower recognition error
in Table 1. On the other hand, for the proposed (DUAL) approach without ASR
transcripts in the lower section of Table 2, we see DUAL achieved good FF1
scores (55.0, 54.2, 47.8 respectively for 512, 128, 64 clusters). Here,
“HuBERT-$K$” denotes the input of DUAL is the clustering results from HuBERT-
Large $22th$ layer representations with $K$ clusters. These numbers are
competitive to those in the top section for the baseline, verifying DUAL can
learn semantics in the audio signals and find the answers almost as good as
the case where ASR transcripts were available, though transcripts were not
available at all. The relatively weak performance for 64 clusters suggested
that the too small codebook size may lose important fine-grained content
information. The situation can be improved significantly using larger codebook
sizes (128 or 512 clusters).

###  4.2 “test” set for human speakers

The experimental results on a more realistic scenario of human speech (“test”)
are shown in the right column of Table 2. We observe that for the baseline
(cascade) approach using SB for ASR on the top section, the performance
dropped sharply due to the very high WER (61.7 in Table 2) on the human
speech, while using the W2v2 for ASR model offered more robust results similar
to the “dev” set. The result indicated that the performance of the baseline
(cascade) approach relied heavily on the accuracy and robustness of ASR.

On the other hand, in the lower section of Table 2, we see the proposed (DUAL)
approach could retain outstanding performance when K = 128, showing remarkable
robustness of the approach for realistic human speech (55.9 of FF1 for K =
128). However, the performance dropped drastically for K = 512 (17.3 of FF1).
The observation suggested that the cluster number played a crucial role in
performance. We surmise that 128 clusters of quantization provide a smooth
transformation from the machine-synthesized speech used in training to the
human-speech test set. In contrast, 512 clusters may retain too many details
that differentiate synthesized and human speech. This finding inspires more
research to understand what makes textless SQA/NLP work or not.

##  5 Analysis and Discussion

Ablation study for embedding assignment: Table 3 shows the ablation study
regarding how the discrete units should be assigned to the pre-trained
embeddings, as shown in the middle of Figure 2. In this table on the top two
rows (“Most frequent” and “Least frequent”), we randomly assigned the K (128)
discrete units to the pre-trained embedding of the top-$K$ and the least-$K$
frequent vocabularies, where the vocabulary frequency was determined by Byte-
Pair Encoding (BPE) on unlabeled text data. “Random” refers to randomly
selecting pre-trained input embedding regardless of the frequency. “Re-init”
denotes re-initializing the input embedding by a normal distribution.
”Scratch” means the Longformer model was not pre-trained on the unlabeled text
data. The results in Table 3 indicate that randomly assigning the pre-trained
input embeddings for discrete units did not result in very different
performance, although the “Most frequent” initialization offered the best
results, which are those listed in Table 2.
Performance for Poor ASR Accuracy: We compared the performance of the baseline
cascade approach (SB for ASR which gave poor accuracy) and the proposed DUAL
with HuBERT-128 for different levels of WER. Specifically, we bucketize the
NMSQA dev set into subsets based on the WER (from 0% to 70%) obtained with ASR
(SB). In Figure 3, we observe that for the baseline (cascade) approach, the
FF1 score dropped significantly and continuously as the WER increased. This is
the typical phenomenon of recognition error propagation. In contrast, the
proposed (DUAL) attained very similar FF1 scores for different levels of WER,
even when WER went up to 70 %. Because there is no ASR in DUAL and no ASR
transcripts were used, there was actually no correlation between WER and the
FF1 score. The cascade approach outperformed DUAL when the WER was below 30%,;
but DUAL became much higher when WER exceeded 30%. Since the content of SQuAD
(and thus NMSQA) is based on Wikipedia, it includes many words that are name
entities, abbreviations, and OOV, which led to recognition errors. Many of
these words are part of the answers. DUAL handle such scenario much better
than cascade approaches.

Table 3: Ablation study on embedding assignment. All experiments used the HuBERT-128 setting. Performance was measured on the NMSQA dev set. Embedding Assignment | FF1 | AOS
---|---|---
Most frequent | 54.2 | 48.5
Least frequent | 46.9 | 41.7
Random | 51.7 | 46.2
Re-init | 8.9 | 7.2
Scratch (baseline) | 6.1 | 4.9
Figure 3: Frame-level F1 (FF1) scores for DUAL and cascade approach (SB),
evaluated on the small groups of full NMSQA dev set at different levels of ASR
(SB) WER.

##  6 Conclusion

We propose DUAL, the first textless (i.e., ASR transcript-free) SQA framework
in this work. This framework only utilizes unlabeled speech and text data for
pre-training and fine-tuning by the spoken questions, passages, and answer
time intervals. DUAL directly predicts the answer time span without text
supervision or acoustic word boundaries. Furthermore, we release NMSQA, a new
natural, multi-speaker SQA benchmark corpus, which contains human speakers for
the test set and large-scaled synthesized data for the training and
development sets. The experiments showed that DUAL yields competitive results
with the conventional cascade approach using ASR transcripts and is robust to
real-world scenarios on NMSQA.

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# Localization effects in disordered quantum batteries

Mohammad B. Arjmandi<EMAIL_ADDRESS>Faculty of Physics, University
of Isfahan, P.O. Box 81746-7344, Isfahan, Iran.  Quantum Optics Research
Group, University of Isfahan, Isfahan, Iran  Hamidreza Mohammadi
0000-0001-7046-3818<EMAIL_ADDRESS>Faculty of Physics, University
of Isfahan, P.O. Box 81746-7344, Isfahan, Iran.  Quantum Optics Research
Group, University of Isfahan, Isfahan, Iran  Andreia Saguia
0000-0003-0403-4358<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Gragoatá, 24210-346
Niterói, Rio de Janeiro, Brazil  Marcelo S. Sarandy  0000-0003-0910-4407
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal Fluminense, Av.
Gal. Milton Tavares de Souza s/n, Gragoatá, 24210-346 Niterói, Rio de Janeiro,
Brazil  Alan C. Santos  0000-0002-6989-7958<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de São Carlos, Rodovia Washington
Luís, km 235 - SP-310, 13565-905 São Carlos, SP, Brazil

###### Abstract

We investigate the effect of localization on the local charging of quantum
batteries (QBs) modeled by disordered spin systems. Two distinct schemes based
on the transverse-field random Ising model are considered, with Ising
couplings defined on a Chimera graph and on a linear chain with up to next-to-
nearest neighbor interactions. By adopting a low-energy demanding charging
process driven by local fields only, we obtain that the maximum extractable
energy by unitary processes (ergotropy) is highly enhanced in the ergodic
phase in comparison with the many-body localization (MBL) scenario. As we turn
off the next-to-nearest neighbor interactions in the Ising chain, we have the
onset of the Anderson localization phase. We then show that the Anderson phase
exhibits a hybrid behavior, interpolating between large and small ergotropy as
the disorder strength is increased. We also consider the splitting of total
ergotropy into its coherent and incoherent contributions. This incoherent part
implies in a residual ergotropy that is fully robust against dephasing, which
is a typical process leading to the self-discharging of the battery in a real
setup. Our results are experimentally feasible in scalable systems, such as in
superconducting integrated circuits.

##  I Introduction

Few-body devices able to exploit quantum features, such as entanglement and
coherence, have been referred to as quantum devices, leading to the second
revolution for quantum technologies. They can potentially provide striking
advantages, such as more sensitivity Degen et al. (2017); Zhang and Zhuang
(2021); Crawford et al. (2021); Feng (2019) and high efficient metrology
Poulsen et al. (2022); Giovannetti et al. (2011, 2006). They can also be used
for communication tasks Bennett et al. (1993); Yin et al. (2012); Ma et al.
(2012); Ren et al. (2017), heat and spin current devices like transistors
Geppert (2000); Marchukov et al. (2016); Joulain et al. (2016); Loft et al.
(2018); de Ponte and Santos (2019) and diodes McRae et al. (2019); Shukla and
Eliasson (2008); Shen et al. (2014), among others Devoret and Glattli (1998);
Arute et al. (2019); Bacon and Flammia (2009); Santos et al. (2016); Gottesman
and Chuang (1999); Briegel et al. (1998); DiVincenzo and Loss (1999).
Naturally, each of these devices needs energy to operate and this energy, in
turn, can be supplied by another quantum storage device, i.e., a quantum
battery (QB) Campaioli et al. (2018); Bhattacharjee and Dutta (2021). By
following the seminal work by Alicki and Fannes Alicki and Fannes (2013), QBs
are designed as a set of interacting quantum cells, each of them usually
represented by a two-level quantum system (qubit). These cells can be charged
to store energy (with respect to a reference Hamiltonian) by means of external
agents, such as fields, interactions, or any other stimulus able to inject
energy into the system. Although QBs are typically composed by qubits,
proposals of QB with qutrits Santos et al. (2019) and bosonic systems can also
be found Andolina et al. (2019a); Shaghaghi et al. (2022); Downing and Sturges
(2022).

QBs may exhibit supercharging behavior, with efficiency of energy extraction
growing with the number of cells Alicki and Fannes (2013). Such advantage with
respect to classical batteries has been attributed to entanglement and quantum
correlations Andolina et al. (2019b); Farina et al. (2019); Arjmandi et al.
(2022a); Campaioli et al. (2017); Binder et al. (2015); Rossini et al. (2020);
Crescente et al. (2020); Ferraro et al. (2018). However, both constructive and
destructive scenarios have been found for the interplay between entanglement
and available energy in QBs Andolina et al. (2019b); Quach and Munro (2020);
Kamin et al. (2020); Ghosh et al. (2021). On the other hand, coherence seems
to play an important role for the QB performance Monsel et al. (2020); Kamin
et al. (2020); Santos (2021); Çakmak (2020); Shi et al. (2022); Tirone et al.
(2023). As a consensus, collective effects in the dynamics are behind a robust
route for designing efficient QBs, when compared with the parallel (non-
interacting) charging of the system Gyhm et al. (2022). From the experimental
point of view, the energy storing properties of a diversity of quantum systems
have been exploited, with physical realizations in a number of architectures,
such as nuclear magnetic resonance Joshi and Mahesh (2022), superconducting
integrated circuits Hu et al. (2022); Gemme et al. (2022), metal complex Cruz
et al. (2022), and an all-optical setup Huang et al. (2023).

Here, we aim at investigating quantum phases of matter that may be suitable
for qubit-based QBs. Concretely, we will focus on the ergotropy and charging
power of quantum devices subject to disordered couplings and external fields.
As sketched in Fig. 1, two distinct schemes for QBs based on the transverse-
field random Ising model will be considered, with Ising couplings defined on a
Chimera graph (for a physical realization, see e.g. Ref. da C Filho et al.
(2022)) and on a linear chain with up to next-to-nearest neighbor interactions
Kjäll et al. (2014). Analyses of disorder effects in the performance of QBs
have been previously considered in the literature, such as in Refs. Ghosh et
al. (2020); Rossini et al. (2019). However, differently from previous works,
where time-dependent interactions drive the charging process, we adopt
permanent interactions, so that the QB charging is performed through external
local fields only. Our approach is motivated by the fact that switching on/off
interactions typically costs a significant amount of energy in comparison with
the ergotropy provided by the QB Cruz et al. (2022). As a result, we obtain
that the ergodic phase favors the QB performance, with disorder being
detrimental to ergotropy due to the memory effects in the many-body
localization (MBL) phase. Remarkably, when Anderson localization is available,
we have a hybrid scenario between ergodic and MBL behaviors depending on the
disorder strength. We also address the splitting of ergotropy into coherent
and incoherent contributions, showing that a residual extractable work robust
against dephasing can be stored as incoherent ergotropy Shi et al. (2022).

Figure 1: Two distinct schemes of QBs based on the transverse-field random
Ising model are considered: (i) Ising couplings on a linear chain with up to
next-to-nearest neighbor interactions (left) and (ii) Ising couplings defined
on a Chimera graph (right). For both models, the charging process is locally
driven by an external field. The set of intracell interactions defines the
kind of model adopted, such that different phases can be obtained as a
function of the disorder strength.

##  II Preliminaries

We review now the concept of ergotropy and introduce the quantum spin systems
adopted as models of QB in our work.

###  II.1 Ergotropy

The energy available to be extracted in the form of work in a QB depends on
both the state $\rho$ of the quantum device and the reference Hamiltonian
$H_{\mathrm{ref}}$ that sets the energy scale of the system. Given $\rho$ and
$H_{\mathrm{ref}}$, the internal energy is provided by the expectation value
$E(\rho,H_{\mathrm{ref}})=\tr(\rho H_{\mathrm{ref}})$. By keeping a constant
entropy, work can be performed by/on the QB by unitary operations. In this
scenario, we define ergotropy as the maximum energy extractable by realizing
unitary operations on the system Allahverdyan et al. (2004). Therefore,
ergotropy reads

$\mathcal{E}(\rho,H_{\mathrm{ref}})=\max_{U}\left[E(\rho,H_{\mathrm{ref}})-E(U\rho
U^{\dagger},H_{\mathrm{ref}})\right].$ (1)

The maximization over $U$ is achieved by driving the QB towards the passive
state $\bar{\rho}\equiv U\rho U^{\dagger}$ associated with $\rho$, i.e., a
state that cannot provide further energy by unitary operations. We can compute
$\mathcal{E}(\rho,H_{\mathrm{ref}})$ by looking at the spectral decompositions
of $\rho=\sum_{i=1}^{d}{p_{i}\ket{p_{i}}\bra{p_{i}}}$ and
$H_{\mathrm{ref}}=\sum_{i=1}^{d}{\epsilon_{i}\ket{\epsilon_{i}}\bra{\epsilon_{i}}}$,
where we assume the ordering $p_{1}\geq p_{2}\geq...\geq p_{d}$ and
$\epsilon_{1}\leq\epsilon_{2}\leq...\leq\epsilon_{d}$, with $d$ being the
dimension of the Hilbert space. Thus, ergotropy can be rewritten as
Allahverdyan et al. (2004) (see also Refs Moraes et al. (2021); Arjmandi et
al. (2022b))

$\displaystyle\mathcal{E}(\rho,H_{\mathrm{ref}})=E(\rho,H_{\mathrm{ref}})-\sum_{i=1}^{d}{\epsilon_{i}p_{i}}.$
(2)

Notice that both the internal energy and the ergotropy depend on the reference
Hamiltonian $H_{\mathrm{ref}}$. In the charging process of a QB, we also add a
charging Hamiltonian $H_{\mathrm{ch}}(t)$, yielding a total driving
Hamiltonian $H_{\mathrm{d}}(t)=H_{\mathrm{ref}}+H_{\mathrm{ch}}(t)$. Let us
take $H_{\mathrm{ref}}=H_{0}+H_{\mathrm{int}}$, where $H_{0}$ is a local
Hamiltonian and $H_{\mathrm{int}}$ contains the native system interactions.
Then, any additional cost to charge the battery comes from the charging
Hamiltonian $H_{\mathrm{ch}}(t)$. More precisely, the cost to turn-on and
turn-off the term $H_{\mathrm{ch}}(t)$ should be considered in the energy
balance of the device, which may lead to non-efficient quantum batteries Cruz
et al. (2022). Here, to avoid the cost of engineering non-native interactions,
we focus on a charging process driven only by external local fields.

###  II.2 QB Models

As sketched in Fig. 1, we consider two distinct models of QB, with many-body
localization regimes properly characterized. First, let us consider an
$N$-particle random spin-1/2 Ising linear chain with on-site magnetic static
field $h_{k}$. The native interactions are provided by nearest neighbor
($J^{(1)}_{k}$) and next-to-nearest neighbor ($J^{(2)}_{k}$) couplings. The
reference Hamiltonian of the system reads

$\displaystyle
H_{\mathrm{ref}}^{\mathrm{Isi}}=\sum_{k=1}^{N}{h_{k}\sigma_{k}^{z}}-\sum_{k=1}^{N-1}{J^{(1)}_{k}\sigma_{k}^{x}\sigma_{k+1}^{x}}+\sum_{k=1}^{N-2}J^{(2)}_{k}{\sigma_{k}^{x}\sigma_{k+2}^{x}},$
(3)

where $\sigma^{\alpha}$ $(\alpha\\!=\\!x,y,z)$ are the standard Pauli
matrices. The QB is charged according to a transverse local field through a
charging Hamiltonian

$\displaystyle H_{\mathrm{ch}}=\sum_{k}^{N}{\Omega_{k}\sigma_{k}^{x}}.$ (4)

This driving field can be implemented, for instance, through an oscillating
small radio-frequency magnetic field of intensity
$|B_{\mathrm{rf}}|\propto\Omega_{k}$, as in a nuclear magnetic resonance setup
Oliveira et al. (2011). We assume local fields uniformly distributed over the
chain, with $h_{k}=h$ and $\Omega_{k}=\Omega$ constant couplings. In this
model, the random nature of the system relies on the interactions between
nearest neighbors given by $J^{(1)}_{k}$, as selected from a uniform
distribution in the interval $[J_{0}(1-\delta),J_{0}(1+\delta)]$, where
$J_{0}$ sets the energy scale of the system and $\delta$ is a dimensionless
parameter that determines the degree of disorder. Concerning the next-to-
nearest neighbor coupling $J^{(2)}_{k}$, it will also be taken as a constant
chosen as a function of $J_{0}$. For the charging process we assume
$J_{0}=\Omega$.

The localization properties of random spin-1/2 Ising chains have been widely
studied Kjäll et al. (2014). In particular, as the next-to-nearest neighbor
interaction is turned off ($J^{(2)}_{k}=0$), the model reduces to the well
known nearest neighbor transverse field Ising chain, such that an arbitrary
amount of disorder ($\delta>0$) brings the system into an Anderson
localization regime Žnidarič et al. (2008). However, when $J^{(2)}_{k}\neq 0$,
the degree of disorder needs to be strong enough to drive the system to many-
body localization (MBL), otherwise the system remains in an ergodic phase
Žnidarič et al. (2008); Pal and Huse (2010); Luitz et al. (2015). Unless
specified otherwise, we refer to the various phases of this model by adopting
the specific values for the couplings: (i) Anderson localized: $J^{(2)}_{k}=0$
and $\delta=1$; (ii) Ergodic phase: $J^{(2)}_{k}=h/2=0.3J_{0}$ and $\delta=1$,
and (iii) MBL phase: $J^{(2)}_{k}=h/2=0.3J_{0}$ and $\delta=5$.

It is worth mentioning that this model was previously considered to
investigate localization effects on QB Rossini et al. (2019). However, in Ref.
Rossini et al. (2019), the authors assume the QB is charged through turned-on
interactions between the quantum cells of the battery during charging time.
Then, all figures of merit are evaluated with respect to the free part of the
Hamiltonian, i.e., the on-site applied field. As previously mentioned, the
energy cost of switching on/off the interactions might be detrimental to the
performance of the QB, with the energy cost to charge the battery much larger
than the ergotropy itself Cruz et al. (2022); Hovhannisyan et al. (2020).
Here, we analyze the energy cost of the charging process in Appendix A,
showing that the charging through a local field only is an energy friendly
choice.

As our second QB model, we consider random Ising interactions and local fields
defined on the 8-spin Chimera unit cell, as sketched in Fig. 1. The reference
Hamiltonian reads

$\displaystyle
H_{\mathrm{ref}}^{\mathrm{Chi}}=\sum_{k}{h_{k}\sigma_{k}^{z}}+\sum_{\langle
k,j\rangle\in\mathcal{C}_{\mathrm{Chi}}}{J_{kj}\sigma_{k}^{z}\sigma_{j}^{z}},$
(5)

where the $\langle k,j\rangle\in\mathcal{C}_{\mathrm{Chi}}$ refers to a sum
over the connections of the Chimera graph. Concerning the charging Hamiltonian
$H_{\mathrm{ch}}$, it will be defined through transverse local fields, as in
Eq. (4), with constant couplings $\Omega_{k}=\Omega$. For this system, we
consider random values for both $h_{k}$ and $J_{kj}$ from a uniform
distribution $[-\delta J_{0},\delta J_{0}]$, with $\delta$ being the disorder
amplitude, such that $\delta J_{0}$ is the maximum admissible value for the
coupling strength. Taking into account the characterization of the MBL phase
transition as in Ref. da C Filho et al. (2022), with $J_{0}=\Omega$, we will
adopt the specific values for the disorder strength: (i) Ergodic phase:
$\delta=2$, such that $\delta J_{0}=2\Omega$ and (ii) MBL phase: $\delta=6$,
such that $\delta J_{0}=6\Omega$. It is worth mentioning that, while the
Anderson localized phase is well defined for the random Ising chain in the
absence of next-to-nearest-neighbor interactions (i.e., when $J_{2}=0$)
111since by using a Jordan-Wigner transformation Sachdev (1999) the model can
be mapped into a non-interacting fermionic model, for which any arbitrary
degree of disorder is able to localize the system Kjäll et al. (2014), we do
not expect that the chimera topology will exhibit Anderson localization for
the regime of couplings considered here.

By disregarding decoherence effects, the dynamics of the QB is governed by the
unitary evolution operator $U(t)$ as $\rho(t)=U(t)\rho(0)U^{\dagger}(t)$. We
assume the device starts from its fully empty state $\rho(0)=\ket{G}\bra{G}$,
where $\ket{G}$ is the ground state of the reference Hamiltonian for each
model (empty energy state of the battery). Given the randomness of local
fields and interaction strengths, we will have a distinct reference
Hamiltonian $H_{n,\mathrm{ref}}$ for each disorder realization. Therefore, it
is expected that the ergotropy will change for different choices of the random
parameters. For this reason, to define a fair figure of merit to evaluate the
QB performance, we will consider the average ergotropy over $N_{\mathrm{r}}$
realizations of disorder Arjmandi et al. (2022b)

$\displaystyle\bar{\mathcal{E}}(t)=\frac{1}{N_{\mathrm{r}}}\sum_{n=1}^{N_{\mathrm{r}}}\frac{\mathcal{E}(\rho_{n}(t),H_{n,\mathrm{ref}})}{\mathcal{E}_{n}^{\mathrm{max}}},$
(6)

where $\mathcal{E}(\rho_{n}(t),H_{n,\mathrm{ref}})$ is the time-dependent
ergotropy associated with the evolved state $\rho_{n}(t)$ for the $n$-th
realization. We also define
$\mathcal{E}_{n}^{\mathrm{max}}=\tr(\rho_{n,\mathrm{fc}}H_{n,\mathrm{ref}})-\tr(\rho_{n,G}H_{n,\mathrm{ref}})$
as the maximum capability of the battery, i.e., the ergotropy stored in its
fully charged state $\rho_{n,\mathrm{fc}}$, with $\rho_{n,G}$ denoting the
density operator for the ground state of $H_{n,\mathrm{ref}}$. Because we
assume pure states (unitary evolution), $\rho_{n,\mathrm{fc}}$ is the
eigenstate of $H_{n,\mathrm{ref}}$ with the highest eigenvalue Allahverdyan et
al. (2004); Santos et al. (2020a). In this way, we can measure the charge in
the QB with the reference Hamiltonian $H_{n,\mathrm{ref}}$ for a given
realization $n$, with respect to the maximally charged state
$\rho_{n,\mathrm{fc}}$.

##  III Results and discussion

###  III.1 Ergotropy and charging power

Now, we present our main results. First of all, we evaluate the ergotropy by
charging the battery in the different phases aforementioned. In Fig. 2, we
show the dynamics of the average ergotropy for both models considered in our
work, where we conclude that the MBL phase turns out to be a poor choice for a
QB in the sense of energy storage. This can be explained by the emergence of
memory effects attributed to the MBL phase with respect to the initial state
Nandkishore and Huse (2015); da C Filho et al. (2022), which in turn prevents
the QB from a significant evolution with respect to its ground state. If the
dynamics starts from a different state, an enhancement of the charging
performance is expected, but it requires the initialization step by using a
different reference Hamiltonian. For example, in Ref. Rossini et al. (2019),
the authors considered the initial state as the lowest energy level of the
non-interacting quantum cells, which will demand an extra cost as we turn on
the interactions throughout the charging process.

Figure 2: The dynamics of average ergotropy for (a) the $N=8$ random Ising
chain with next-to-nearest neighbors and (b) the random Ising model on the
8-spin Chimera graph. The number of realizations is taken as
$N_{\mathrm{r}}=100$. The Hamiltonian parameters for each phase are set as
described in Sec. II.2.

Differently from the MBL phase, where the “frozen” dynamics handicaps the
energy charging, the battery presents a charging enhancement in either the
ergodic or Anderson phases. In the case of the ergodic phase, such a result
can be explained by its delocalization behavior, which makes the ergodic phase
more suitable for ergotropy deposition. Notice that the Anderson phase follows
a similar pattern as obtained in the ergodic phase for the chosen disorder
strength, also showing a remarkable difference with respect to the MBL phase
in terms of ergotropy. Due to the absence of strong memory effects typical of
the MBL phase, the dynamics of QB in the Anderson phase is not frozen so that
it shows a suitable performance. Moreover, for a wide time interval, which is
around the range $[10^{0},10^{2}]$, the Anderson phase presents a performance
even better than the ergodic case.

Let us now analyze the time-averaged charging power of the QB as a function of
time. Notice that, as shown in from Fig. 2, the maximum (local) instantaneous
amount of ergotropy is achieved for different instants of time for each phase
considered. Then, it is meaningful to consider the time-averaged charging
power in the parameterized time interval $\Omega\Delta t=\Omega(t-t_{0})$
(assuming here $t_{0}=0$). In this context, by using the notation introduced
in Eq. (6), we can introduce the dimensionless power parameter

$\displaystyle\bar{P}_{0}(\Delta
t)=\frac{\bar{\mathcal{E}}(t)-\bar{\mathcal{E}}(0)}{\Omega\Delta t}.$ (7)

Notice that $\bar{P}_{0}(\Delta t)$ is directly related to the average power
$P(\Delta t)$. Indeed, $P(\Delta t)$ can be written in terms of the average of
the maximum ergotropy capability of the battery
$\bar{\mathcal{E}}^{\mathrm{max}}=\sum_{n=1}^{N_{\mathrm{r}}}\mathcal{E}_{n}^{\mathrm{max}}/N_{\mathrm{r}}$,
the charging field amplitude $\Omega$, and the parameter $\bar{P}_{0}(\Delta
t)$ as $P(\Delta t)=\Omega\bar{\mathcal{E}}^{\mathrm{max}}\bar{P}_{0}(\Delta
t)$. For both models, as it can be seen from Fig. 3, the QB in the ergodic
phase is more powerful than the other phases for the range of time considered.
Then, while the Anderson phase may allow for storing more energy than the
ergodic phase for a specific time range, its charging power is not as
efficient as the ergodic QB.

Figure 3: The dynamics of averaged charging power for (a) the $N=8$ random
Ising chain with next-to-nearest neighbors and (b) the random Ising model on
the 8-spin Chimera graph. The number of realizations is taken as
$N_{\mathrm{r}}=100$. The Hamiltonian parameters for each phase are set as
described in Sec. II.2.

So far, our results suggest that the disorder strength seems to be the main
determinant for ergotropy and power rather than the quantum (ergodic or
Anderson) phase the system actually is. In fact, the charging of the QB in the
Anderson localized and ergodic phases are performed by using an equal degree
of disorder ($\delta=1$), while the MBL phase requires stronger disorder. The
role played by disorder can be confirmed by considering the Anderson phase for
different degrees of randomness. This is shown in Fig. 4, where we have
plotted ergotropy and dimensionless power in the Anderson phase for the Ising
linear chain for different values of $\delta$. Notice that the Anderson phase
then interpolates between the ergodic and MBL behaviors as we increase the
strength of disorder.

###  III.2 Incoherent and coherent ergotropy

Let us now split ergotropy into its coherent and incoherent contributions Shi
et al. (2022). The generation of quantum superpositions (coherence) in the
energy basis during the charging leads to a non-zero amount of coherent
ergotropy. On ther other hand, part of the total ergotropy is stored as
incoherent ergotropy. In particular, the incoherent part of the ergotropy is
relevant in our discussion because it takes into account the “residual”
ergotropy of the dephased state $\rho^{D}$ in energy eigenbasis of the
reference Hamiltonian, with
$\rho^{D}=\sum_{i}^{d}{\bra{\epsilon_{i}}\rho\ket{\epsilon_{i}}\ket{\epsilon_{i}}\bra{\epsilon_{i}}}$.
Physically speaking, it quantifies the total amount of ergotropy that remains
in the system, even in absence of an external charging field, after a long
time of action of a dephasing process. Therefore, we can identify this amount
of remaining ergotropy as related to the robustness of a QB against its self-
discharging process Santos (2021).

Figure 4: The dynamics of (a) average ergotropy and (b) power for the $N=8$
random Ising chain in the Anderson phase for different degrees of disorders.
The number of realizations is taken as $N_{\mathrm{r}}=100$. The Hamiltonian
parameters for each phase are set as described in Sec. II.2.

Inspired by Ref. Shi et al. (2022), we introduce the average normalized
incoherent and coherent ergotropy, respectively, as

$\displaystyle\bar{\mathcal{E}}_{\mathrm{Inco}}(t)$
$\displaystyle=\frac{1}{N_{\mathrm{r}}}\sum_{n=1}^{N_{\mathrm{r}}}\frac{\mathcal{E}(\rho_{n}^{D}(t),H_{n,\mathrm{ref}})}{\mathcal{E}_{n}^{\mathrm{max}}},$
(8) $\displaystyle\bar{\mathcal{E}}_{\mathrm{Cohe}}(t)$
$\displaystyle=\frac{1}{N_{\mathrm{r}}}\sum_{n=1}^{N_{\mathrm{r}}}\frac{\mathcal{E}(\rho_{n}(t),H_{n,\mathrm{ref}})-\mathcal{E}(\rho_{n}^{D}(t),H_{n,\mathrm{ref}})}{\mathcal{E}_{n}^{\mathrm{max}}}.$
(9)

Fig. 5 shows the time evolution of $\bar{\mathcal{E}}_{\mathrm{Inco}}(t)$ and
$\bar{\mathcal{E}}_{\mathrm{Cohe}}(t)$ for different phases of the models
considered in our work. Given the good qualitative agreement between the
behavior of the total extractable energy and the its coherent and incoherent
parts, we can safely state that the performance of the QB is mainly related to
the coherence generated during its charging, since the extractable work is
mainly stored as coherent ergotropy. However, by comparing Fig. 2 and Fig. 5,
it can be inferred that, even though the total ergotropy is mostly consisted
of coherent part, there remains a nonzero amount of incoherent ergotropy
during charging process. This can be useful in a real setup since, as
discussed above, the incoherent ergotropy is robust against dephasing
channels, since its corresponding state is already dephased in the basis of
eigenenergies.

Figure 5: The dynamics of coherent and incoherent ergotropy for (a), (c) the
$N=8$ random Ising chain with next-to-nearest neighbors and (b), (d) the
random Ising model on the 8-spin Chimera graph. The number of realizations is
taken as $N_{\mathrm{r}}=100$. The Hamiltonian parameters for each phase are
set as described in Sec. II.2.

We can now analyze the average normalized coherence
${\overline{\mathcal{Q}\mathcal{C}}}(t)$ generated for each dynamics as
quantified by the $l_{1}$-norm, which is defined as

$\displaystyle{\overline{\mathcal{Q}\mathcal{C}}}(t)=\frac{1}{N_{\mathrm{r}}}\sum\nolimits_{n=1}^{N_{\mathrm{r}}}\left[\frac{1}{C_{max}}\sum\nolimits_{i,j\neq
i}|\rho_{ij}|\right],$ (10)

In Eq. (10), the term in “brackets” is the $l_{1}$-norm coherence of the
density matrix for each realization $n$, where $C_{max}$ denotes the coherence
of the quantum state $(\ket{e}+\ket{g})/\sqrt{2}$, which is the maximally
coherent state in the local basis $\\{\ket{e},\ket{g}\\}$. The results for
${\overline{\mathcal{Q}\mathcal{C}}}(t)$ are shown in Fig. 6 for both the
Ising chain and the Chimera graph. Comparing Figs. 2b and 6b, for the Chimera
graph, quantum coherence can be regarded as a suitable figure of merit of the
stored ergotropy, at least qualitatively. As time passes by, both quantities
sharply increase when the QB is in the ergodic phase but they are small for
the MBL phase. This is not the case for the Ising chain. The model contains
some amount of initial quantum coherence but localized phases reach more
coherence in comparison with the ergodic phase. The relation between
generation of ergotropy and coherence is then model-dependent ,i.e., different
patterns of interactions may imply in different qualitative relations.

##  IV Conclusions

We have investigated quantum phases of disordered spin systems as a model of
qubit-based QBs. Differently from previous investigations, we have focused on
the local ergotropy injection through external fields only. This is due to the
low energy demanding property of this scheme in comparison with the time-
dependent switchable interacting systems. For the models considered in our
work, ergodicity has shown to improve the QB performance, with disorder being
detrimental to ergotropy due to the memory effects of the discharged state in
the MBL phase. In the case of the linear Ising chain, where Anderson
localization is induced for any amount of disorder as the next-to-nearest
interaction is turned off, we have shown the presence of a hybrid scenario
interpolating between ergodic and MBL behaviors depending on the disorder
strength. It is worth highlighting here that, while the MBL phase is
attractive for time-dependent interacting QBs Rossini et al. (2019), it
develops a poor performance for the local charging scheme. Concerning quantum
coherence, we identified that most of ergotropy is stored as coherent
ergotropy, which provides the quantum character of the energy storing device.
On the other hand, we also obtained a nonvanishing amount of incoherent
ergotropy, which indicates robustness of the storage capability against the
self-discharging of QBs due to dephasing (see, e.g., Refs. Santos (2021) and
Shi et al. (2022)). Further investigations of lattice topologies and quantum
phases in disordered systems as well as experimental implementations are left
for a future analysis.

Figure 6: Average quantum coherence for (a) the $N=8$ random Ising chain with
next-to-nearest neighbors and (b) the random Ising model on the 8-spin Chimera
graph. The number of realizations is taken as $N_{\mathrm{r}}=100$. The
Hamiltonian parameters for each phase are set as described in Sec. II.2.

##  Appendix A Energy cost of switching on/off interactions

In this section, we analyze the energy cost of turning on/off the intracell
interactions for both models of QB considered in our work. For the spin-$1/2$
Ising chain with nearest and next-to-nearest neighbor interactions we have
$H_{\mathrm{ref}}^{\mathrm{Isi}}=H_{0}^{\mathrm{Isi}}+H_{\mathrm{int}}^{\mathrm{Isi}}$,
where

$\displaystyle H_{0}^{\mathrm{Isi}}$
$\displaystyle=h\sum_{k=1}^{N}{\sigma_{k}^{z}},\leavevmode\nobreak\
\leavevmode\nobreak\
H_{\mathrm{int}}^{\mathrm{Isi}}=-\sum_{k=1}^{N-1}{J_{k}\sigma_{k}^{x}\sigma_{k+1}^{x}}+J_{2}\sum_{k=1}^{N-2}{\sigma_{k}^{x}\sigma_{k+2}^{x}}.$
(11)

For the Ising couplings defined on the Chimera graph, we also have
$H_{\mathrm{ref}}^{\mathrm{Chi}}=H_{0}^{\mathrm{Chi}}+H_{\mathrm{int}}^{\mathrm{Chi}}$
where

$\displaystyle
H_{0}^{\mathrm{Chi}}=\sum_{k}{h_{k}\sigma_{k}^{z}},\leavevmode\nobreak\
\leavevmode\nobreak\
H_{\mathrm{int}}^{\mathrm{Chi}}=\sum_{kj}{J_{kj}\sigma_{k}^{z}\sigma_{j}^{z}}.$
(12)

We introduce the energy cost $\mathcal{C}_{\mathrm{int}}$ to implement the
interactions as the difference between the norm of the reference Hamiltonian
including interactions ($H_{\mathrm{ref}}$) and the norm of the reference
Hamiltonian without interactions ($H_{0}$). Mathematically, we have

$\displaystyle\mathcal{C}_{\mathrm{int}}=\mathcal{N}(H_{\mathrm{ref}})-\mathcal{N}(H_{0}),$
(13)

where $\mathcal{N}(A)=\sqrt{Tr(A^{\dagger}A)}$ is the Hilbert-Schmidt norm of
a given (finite-dimensional) matrix $A$. Our definition based on the Hilbert-
Schmidt norm is motivated by previous works showing that such a quantity is a
good measure of energy cost for quantum control in different contexts Santos
and Sarandy (2015); Coulamy et al. (2016); Campbell and Deffner (2017); Santos
and Sarandy (2018), including experiments Hu et al. (2018); Santos et al.
(2020b). More precisely, it is related to a measure of the thermodynamic cost
of an arbitrary unitary quantum evolution Deffner (2021). In this way, one can
show that, for the Ising chain, the cost reads

$\displaystyle\mathcal{C}^{\mathrm{Isi}}_{\mathrm{int}}=16\left(-2\sqrt{4h^{2}}+\sqrt{8h^{2}+6J_{2}^{2}+\sum_{k}{J_{k}^{2}}}\right),$
(14)

while, for the Chimera graph, we have

$\displaystyle\mathcal{C}^{\mathrm{Chi}}_{\mathrm{int}}=16\left(-\sqrt{\sum_{k}{h_{k}^{2}}}+\sqrt{\sum_{k}{h_{k}^{2}+\sum_{kj}{J_{kj}}^{2}}}\right).$
(15)

In our work, interactions are always present and we only turn-on and turn-off
the charging field. Then, we define the energy cost
$\mathcal{C}_{\mathrm{ch}}$ of charging with local chargers as the difference
between the norm of the driving Hamiltonian (including reference Hamiltonian
and any local charging fields) and the norm of the reference Hamiltonian.
Mathematically, we write

$\displaystyle\mathcal{C}_{\mathrm{ch}}=\mathcal{N}(H)-\mathcal{N}(H_{\mathrm{ref}}),$
(16)

where $H=H_{\mathrm{ref}}+H_{\mathrm{ch}}$ is the total driving Hamiltonian.
Again, one can simply obtain the explicit form of this cost for the two models

$\displaystyle\mathcal{C}^{\mathrm{Isi}}_{\mathrm{ch}}$
$\displaystyle=16\left(\sqrt{\Xi^{2}+8\Omega^{2}}-|\Xi|\right),$ (17)
$\displaystyle\mathcal{C}^{\mathrm{Chi}}_{\mathrm{ch}}$
$\displaystyle=16\left(\sqrt{\Lambda^{2}+8\Omega^{2}}-|\Lambda|\right).$ (18)

with $\Xi^{2}=8h^{2}+6J_{2}^{2}+\sum_{k}{J_{k}^{2}}$, and
$\Lambda^{2}=\sum_{k}{h_{k}^{2}}+\sum_{kj}{J_{kj}}^{2}$.

Figure 7: Cost of turning on/off (a) intracell interactions, and (b) local
charging fields as function of the degree of disorder for the $N=8$ random
Ising chain with next-to-nearest neighbors and the random Ising model on the
8-spin Chimera graph. The number of realizations is taken as
$N_{\mathrm{r}}=100$. The Hamiltonian parameters for each phase are set as
described in Sec. II.2.

To compute the energy cost we define here the quantity

$\displaystyle\bar{\mathcal{C}}_{\mathrm{ch}}=\frac{1}{N_{\mathrm{r}}}\sum_{n=1}^{N_{\mathrm{r}}}\frac{\mathcal{C}_{\mathrm{ch},n}}{\mathcal{E}_{n}^{\mathrm{max}}},$
(19)

where $\mathcal{C}_{\mathrm{ch},n}$ denotes the charging cost for each model,
as given by Eqs. (14), (15), (17) and (18), in the $n$-th realization. It
quantifies the average normalized energy cost of the charging (either with
interactions or local fields) as a multiple of the maximum capability
$\mathcal{E}_{n}^{\mathrm{max}}$ of the QB for each model. We apply this
definition to all models considered in this work in order to compare the
energy cost for charging by using time-dependent intracell interactions and
the cost for the case where we charge the battery by using local fields. In
Fig. 7 we present the cost for each model as function of the degree of
disorder. For all models considered here as energy storing quantum devices,
the charging by local fields is energetically more efficient than the time-
dependent interaction engineering, particularly when disorder is strong
enough. In particular, the advantage of the local charging comes from the fact
that, for high disorder, the interaction dominates over the local charging
field as $\mathcal{N}(H)\rightarrow\mathcal{N}(H_{\mathrm{ref}})$, providing
$\mathcal{C}_{\mathrm{ch},n}\ll\mathcal{E}_{n}^{\mathrm{max}}$ for each model.
We stress that while this limit is consistent, it does not imply that we have
zero cost to implement the charging.

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# Group-valued invariant of knots in the full torus

Vassily Olegovich Manturov111Moscow Institute of Physics and technology, Kazan
Federal University, Northeastern University (China),
Igor Mikhailovich Nikonov

AMS MSC2020: 05C10, 05C60, 57K10, 57K12, 57M15

Keywords: slice knot, parity, concordance

##  1 Introduction

Knot concordance plays a crucial role in the low dimensional topology [3].

The better understanding of knot sliceness and the concordance group of
classical knots led to significant results [4, 5].

In the last decades, detecting new ${\mathbb{Z}}^{\infty}$ summands in the
group $\Theta^{3}_{\mathbb{Z}}$ were considered as significant results [1].

To tackle these problems, a very elaborated techniques was developed. Say,
Heegaard-Floer Homology originally used some count of holomorphic discs and
$spin^{c}$-structures on manifolds.

The present paper deals with the sliceness problems for knots in the full
torus222Note that once forever we deal only with homologically trivial knots
(having winding number $0$ with respect to $S^{1}$. $S^{1}\times D^{1}$.

We propose a very elementary techniques which allows one to construct a lot of
sliceness obstructions for such knots.

Our approach deals with group theoretical techniques; it is completely
combinatorial, and the groups are very easy to deal with.

##  2 Definitions

Let $K$ be a knot in the full torus $S^{1}\times{\mathbb{R}}^{2}$. The full
torus can be considered as a thickening of the cylinder
$S^{1}\times{\mathbb{R}}^{1}$. Hence, the knot $K$ is presented by its diagram
$D$ in the cylinder.

Consider the group

$G^{\prime}=\left\\{a,b,b^{\prime},B,B^{\prime}\mid
a^{2}=1,ab=(b^{\prime})^{-1}a,aB=(B^{\prime})^{-1}a,\right.\\\
\left.bB^{-1}=B^{\prime}(b^{\prime})^{-1},b^{-1}B=(B^{\prime})^{-1}b^{\prime}\right\\}.$
(1)

There is a bijection $\phi$ on $G^{\prime}$ given by the formula

$\phi(x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots
x_{n}^{\alpha_{n}})=x_{2}^{-\alpha_{2}}\cdots
x_{n}^{-\alpha_{n}}x_{1}^{-\alpha_{1}}$

where $x_{i}=a,b,b^{\prime},B$ or $B^{\prime}$, and $\alpha_{i}=\pm 1$. Let
$\tilde{G}^{\prime}=G^{\prime}/\phi$.

With an oriented knot $K$ in the full torus $S^{1}\times{\mathbb{R}}^{2}$
having winding number $0$ we associate an element
$f^{\prime}(K)\in\tilde{G}^{\prime}$ as follows.

We fix a reference point $P$ (not a crossing) on the Gauss diagram $G(D)$ of
the knot. We enumerate the chord endpoints in the core circle as they appear
according to the orientation. The endpoints are called to be in odd position
if the number of endpoints containing itself from the starting point of $G(K)$
is odd. Otherwise the endpoint is called to be in even position.

We start walking from the reference point along the core circle of $G(D)$ and
write the letters according to the following rule:

  1. 1.

We associate $a$ with an even chord end;

  2. 2.

We associate $b,b^{\prime},B$ or $B^{\prime}$ to chord ends in odd positions
and we associate $b^{-1},(b^{\prime})^{-1},B^{-1}$ or $(B^{\prime})^{-1}$ to
chord ends in even position;

  3. 3.

We choose $b,b^{-1},B$ or $B^{-1}$ if the chord considered is linked with
evenly many even chords and
$b^{\prime},(b^{\prime})^{-1},B^{\prime},(B^{\prime})^{-1}$, otherwise;

  4. 4.

We associate $b,b^{-1},b^{\prime}$ or $(b^{\prime})^{-1}$ to undercrossings,
and $B,B^{-1},B^{\prime},(B^{\prime})^{-1}$ to overcrossings.

The product of the letters is a word $f^{\prime}(K)$ considered as an element
of $\tilde{G}^{\prime}$.

######  Theorem 1.

The element $f^{\prime}(K)$ is an invariant of oriented knots $K$ in the full
torus.

The theorem is proved by standard checking Reidemeister moves. The bijection
$\phi$ appears whenever we change the reference point.

It turns out that the invariant sees much more than just knot invariance:

######  Theorem 2.

Let $K$ be a knot in the full torus. If
$f^{\prime}(K)=\tilde{1}\in\tilde{G}^{\prime}$ then K is not slice.

The theorem follows from a more general statement.

Let $G$ be a group and $\Phi$ be a family of bijections $\phi\colon G\to G$
such that $\phi(1)=1$. For example,

$\Phi=\\{Ad_{g}\mid\ g\in G\\}\mbox{ where }Ad_{g}(h)=ghg^{-1},h\in G.$

Let $\tilde{G}=G/\Phi$ be the set of equivalence classes of elements of $G$
modulo the relations $g\sim\phi(g)$, $g\in G$, $\phi\in\Phi$. We will denote
the equivalence class of an element $g\in G$ by $\tilde{g}$. Note that by
definition $\tilde{1}=\\{1\\}$.

Let $f$ be an invariant of oriented links with ordered components whose value
on a link $L=K_{1}\cup\cdots\cup K_{n}$ is an element
$f(L)=(f_{i}(L))_{i=1}^{n}\in\prod_{i=1}^{n}\tilde{G}=\tilde{G}^{n}$. Assume
$f$ obeys the following conditions:

  1. 1.

For any permutation $\sigma\in\Sigma_{n}$ $\sigma\circ f(L)=f\circ\sigma(L)$.
The permutation interchanges the components of a link $L$ and interchanges the
components of the value $f(L)$;

  2. 2.

If the component $K_{i}$ is trivial then $f_{i}(L)=\tilde{1}$;

  3. 3.

Let $L=K_{1}\cup\cdots\cup K_{n}$, $L^{\prime}=K_{1}\cup\cdots\cup
K_{n-2}\cup(K_{n-1}\\#K_{n})$ and $f(L)=(f_{i}(L))_{i=1}^{n}\in\tilde{G}^{n}$.
Then $f(L^{\prime})=(f_{i}(L^{\prime}))_{i=1}^{n-1}\in\tilde{G}^{n-1}$ where
$f_{i}(L^{\prime})=f_{i}(L)$ for $1\leq i\leq n-2$ and
$f(L^{\prime})_{n-1}=\widetilde{g_{n-1}g_{n}}$ for some representatives
$g_{n-1},g_{n}\in G$ of the classes $f_{n-1}(L)$ and $f_{n}(L)$ respectively.

######  Theorem 3.

The map $f$ is a concordance invariant of links.

######  Corollary 1.

If $K$ is a slice knot then $f(K)=\tilde{1}$.

###  2.1 Concordance of links in the thickened torus

Let $L=L_{1}\cup L_{2}\cup\cdots\cup L_{n}$ be a link in $S^{3}$ such that
$L_{1}\cup L_{2}$ forms the Hopf link. Then $S^{3}\setminus(L_{1}\cup L_{2})$
is diffeomorphic to the thickening $M=T^{2}\times(0,1)$ of the torus. Hence,
the link $\bar{L}=L_{3}\cup\cdots\cup L_{n}$ can be considered as a link in
the thickened torus $M$.

######  Theorem 4.

Two links $L=L_{1}\cup L_{2}\cup\cdots\cup L_{n}$ and
$L^{\prime}=L^{\prime}_{1}\cup L^{\prime}_{2}\cup\cdots\cup L^{\prime}_{n}$ in
$S^{3}$ such that $L_{1}\cup L_{2}$ and $L^{\prime}_{1}\cup L^{\prime}_{2}$
are Hopf links, are concordant if and only if the links $\bar{L}$ and
$\bar{L}^{\prime}$ are concordant in $M$, i.e. there exists a smooth embedding
$\bar{W}\colon\sqcup_{i=3}^{n}S^{1}\times[0,1]\to M\times[0,1]$ such that
$\bar{W}\cap M\times\\{0\\}=\partial_{0}\bar{W}=\bar{L}\times\\{0\\}$ and
$\bar{W}\cap
M\times\\{1\\}=\partial_{1}\bar{W}=\bar{L}^{\prime}\times\\{1\\}$.

###### Proof.

After some isotopy of $L^{\prime}$ we can assume that $L_{1}=L^{\prime}_{1}$
and $L_{2}=L^{\prime}_{2}$.

Let $\bar{L}$ and $\bar{L}^{\prime}$ are concordant in $M$, i.e. exists a set
of cylinders $\bar{W}\subset M\times[0,1]=S^{3}\setminus(L_{1}\cup
L_{2})\times[0,1]$. Then $W=\bar{W}\cup L_{1}\times[0,1]\cup L_{2}\times[0,1]$
establishes a concordance between $L$ and $L^{\prime}$ in $S^{3}$.

Let $L$ and $L^{\prime}$ are concordant in $S^{3}$. Then there exists an
embedding $W\colon\sqcup_{i=1}^{n}S^{1}\times[0,1]\to S^{3}\times[0,1]$ so
that $W\cap S^{3}\times\\{0\\}=\partial_{0}W=L\times\\{0\\}$ and $W\cap
S^{3}\times\\{1\\}=\partial_{1}W=L^{\prime}\times\\{1\\}$.

Assume that $L_{i}\times\\{0\\}$ and $L_{i}\times\\{1\\}$ belong to one
component of $W$, $i=1,2$. Assume also that there exists an isotopy of
$S^{3}\times[0,1]$ which is fixed on the boundary and transforms the
components of $W$ containing $L_{i}\times\\{0\\}$, $i=1,2$, to
$L_{i}\times[0,1]$. Denote the sufrace $W$ after the isotopy by $W^{\prime}$.
Then $\bar{W}=W^{\prime}\setminus(L_{1}\cup L_{2})\times[0,1]$ is a
concordance of $\bar{L}$ and $\bar{L}^{\prime}$ in $S^{3}\setminus(L_{1}\cup
L_{2})=M$. ∎

##  3 Proof of Theorem 3

###### Proof.

Consider first the case of knots. Let $K_{0}$ and $K_{1}$ be two concordant
knots. Then there exists an annulus $W\subset{\mathbb{R}}^{3}\times[0,1]$ such
that $W\cap{\mathbb{R}}^{3}\times\\{i\\}=K_{i}$, $i=0,1$.

Let $t$ be the coordinate on the interval $[0,1]$. We can assume that $t$ is a
simple Morse function on $W$. Consider the Reeb graph $T$ of the function $t$
on $W$. That means the graph $T$ is the quotient space $W/\sim$ where $x\sim
y$ iff $x,y$ belong to the same connected component of the level
$t^{-1}(c)\subset W$ for some $c\in[0,1]$. The graph $T$ is a trivalent graph
and its vertices correspond to components of $t^{-1}(c)$ which contains
singular points of the map $t$ except the vertices $v_{0}$, $v_{1}$ which
corresponds to the knots $K_{0}$ and $K_{1}$.

The graph $T$ is a tree because $W$ is an annulus. Then there is a unique path
$P$ in $T$ with ends $v_{0}$ and $v_{1}$.

For any edge $e$ in the graph $T$ choose a point $z\in e$. Let $c=t(z)$,
$L_{c}=t^{-1}(c)\subset W$ and $K_{i}$ be the component of the link $L_{c}$
that corresponds to $z$. Denote the element $f(L_{c})_{i}\in\tilde{G}$ by
$f(z)$. Since $f$ is an invariant, the value $f(z)$ is the same for all
internal points $z\in e$. Hence, there is a well defined map $f\colon
E(T)\to\tilde{G}$ where $E(T)$ is the set of edges of $T$.

We will prove that $f(e)=\tilde{1}$ for any $e\not\in P$ and $f(e)=f(K_{0})$
for any $e\in P$.

Let $e\in E(T)\setminus P$. Let $T(e)$ be the component of the graph
$T\setminus e$ which does not contain the path $P$ and $v(e)$ be the end of
the edge $e$ in $T_{e}$. The height $h(e)$ of the edge $e$ is the number of
the edges in $T(e)$.

Let us prove that $f(e)$ for any $e\in E(T)\setminus P$ by induction on the
height $h(e)$.

Let $h(e)=0$. Then $e$ is a leaf and it corresponds to a trivial component
$K_{i}$. Then $f(e)=\tilde{1}$ by the second property of $f$.

Let $h(e)>0$. Denote the other edges incident to $v(e)$ by $e^{\prime}$ and
$e^{\prime\prime}$. Then $h(e^{\prime})<h(e)$ and $h(e^{\prime\prime})<h(e)$,
hence, $f(e^{\prime})=f(e^{\prime\prime})=\tilde{1}$. With respect to the
function $t$ there are two cases.

1\. Let $e^{\prime}$ and $e^{\prime\prime}$ merge to the edge or the edge $e$
split into $e^{\prime}$ and $e^{\prime\prime}$. Then the edges
$e,e^{\prime},e^{\prime\prime}$ correspond to link components $K,K^{\prime}$
and $K^{\prime\prime}$ such that $K=K^{\prime}\\#K^{\prime\prime}$. By the
third property of the invariant $f$, there exist elements $g\in f(e)$,
$g^{\prime}\in f(e^{\prime})$, $g^{\prime\prime}\in f(e^{\prime\prime})$ such
that $g=g^{\prime}g^{\prime\prime}$. Since $g^{\prime}=g^{\prime\prime}=1$, we
have $g=1$, hence $f(e)=\tilde{1}$.

2\. Let $e$ and $e^{\prime}$ merge to the edge $e^{\prime\prime}$ or the edge
$e^{\prime\prime}$ split into $e$ and $e^{\prime}$. Then there exist $g\in
f(e)$, $g^{\prime}\in f(e^{\prime})$, $g^{\prime\prime}\in
f(e^{\prime\prime})$ such that $g^{\prime\prime}=gg^{\prime}$. Since
$g^{\prime}=g^{\prime\prime}=1$ by the induction, $g=1$ and $f(e)=\tilde{1}$.

Now, let us prove that $f(e)=f(K_{0})$ for any $e\in P$. The path $P$ is a
sequence of edges $e_{0},e_{1},\dots,e_{n}$ where $v_{0}$ is incident to
$e_{0}$ and $v_{1}$ is incident to $e_{n}$. We prove that $f(e_{i})=f(K_{0})$
by induction on $i$. By definition, $f(e_{0})=f(v_{0})=f(K_{0})$.

Assume that $f(e_{i})=f(K_{0})$. There can be two cases. Let the edge $e_{i}$
split into the edge $e_{i+1}$ and some edge $e\in E(T)\setminus P$. By the
third property of the invariant $f$, there exist elements $g_{i}\in f(e_{i})$,
$g_{i+1}\in f(e_{i+1})$, $g\in f(e)$ such that $g_{i}=g_{i+1}g$. Since $e\in
E(T)\setminus P$, $f(e)=\tilde{1}$ and $g=1$. Then $g_{i}=g_{i+1}$ and
$f(e_{i+1})=f(e_{i})=f(K_{0})$.

If the edge $e_{i}$ merge with some edge $e\in E(T)\setminus P$ to the edge
$e_{i+1}$ then $g_{i+1}=g_{i}g$ for some elements $g_{i}\in f(e_{i})$,
$g_{i+1}\in f(e_{i+1})$, $g\in f(e)$. Since $e\in E(T)\setminus P$,
$f(e)=\tilde{1}$ and $g=1$. Then $g_{i}=g_{i+1}$ and
$f(e_{i+1})=f(e_{i})=f(K_{0})$.

Thus, we have $f(K_{1})=f(v_{1})=f(e_{n})=f(K_{0})$. That means, $f$ is a
concordance invariant.

The concordance invariance for links can be proved analogously.

∎

######  Example 1.

Consider the RGB-link from [5] with parameters $(0,1,0,1,2,1)$, see Fig. 1.

Figure 1: The RGB-link

The red and the green components of the link form the Hopf link. Then the
green and the blue components can be considered as a 2-component link in the
thickened cylinder (see Fig. 2 left), and the blue component can be considered
as a knot $K$ in the thickened torus (see Fig. 2 right).

Figure 2: The RGB-link defines a GB-link in the cylinder and a B-knot in the
torus

Let us calculate the invariant $f^{\prime}$ of the knot $K$:

$f^{\prime}(K)=bb^{-1}B^{\prime}(B^{\prime})^{-1}BB^{-1}BB^{-1}B^{\prime}aB^{\prime}B^{-1}b^{\prime}abb^{-1}bab^{\prime}aa(b^{\prime})^{-1}\cdot\\\
\cdot
ab^{-1}aB^{-1}b^{\prime}aaB^{-1}Bb^{-1}ab^{-1}=B^{\prime}B^{-1}B^{\prime}B^{-1}b^{\prime}b^{-1}b^{\prime}b^{-1}.$

Note that the group $G^{\prime}$ contains a subgroup isomorphic to
${\mathbb{Z}}\oplus{\mathbb{Z}}$ which consists of the elements
$(B^{\prime}B^{-1})^{k}(b^{\prime}b^{-1})^{l}$, $k,l\in{\mathbb{Z}}$. The
bijection $\phi$ maps the element
$(B^{\prime}B^{-1})^{k}(b^{\prime}b^{-1})^{l}$ to the element
$(B^{\prime}B^{-1})^{-k}(b^{\prime}b^{-1})^{-l}$. This means that the
invariant $f^{\prime}(K)$ is not trivial in $\tilde{G}^{\prime}$.

Thus, the knot $K$ in the thickened torus is not slice.

## Aknowlegements

The work of V.O.Manturov was funded by the development program of the Regional
Scientific and Educational Mathematical Center of the Volga Federal District,
agreement N 075-02-2020.

## References

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