# To Answer or Not to Answer? Improving Machine Reading Comprehension Model
with Span-based Contrastive Learning

Yunjie Ji, Liangyu Chen, Chenxiao Dou, Baochang Ma, Xiangang Li
Beike Inc., Beijing, China
{jiyunjie001,chenliangyu003,douchenxiao001,
<EMAIL_ADDRESS>
###### Abstract

Machine Reading Comprehension with Unanswerable Questions is a difficult NLP
task, challenged by the questions which can not be answered from passages. It
is observed that subtle literal changes often make an answerable question
unanswerable, however, most MRC models fail to recognize such changes. To
address this problem, in this paper, we propose a span-based method of
Contrastive Learning (spanCL) which explicitly contrast answerable questions
with their answerable and unanswerable counterparts at the answer span level.
With spanCL, MRC models are forced to perceive crucial semantic changes from
slight literal differences. Experiments on SQuAD 2.0 dataset show that spanCL
can improve baselines significantly, yielding 0.86~2.14 absolute EM
improvements. Additional experiments also show that spanCL is an effective way
to utilize generated questions.

##  1 Introduction

Machine Reading Comprehension (MRC) is an important task in Natural Language
Understanding (NLU), aiming to answer specific questions through scanning a
given passageHermann et al. (2015); Cui et al. (2016); Rajpurkar et al.
(2018). As a fundamental NLU task, MRC also plays an essential role in many
applications such as question answering and dialogue tasks Chen et al. (2017);
Gupta et al. (2020); Reddy et al. (2019). With the rapid development of pre-
trained language models (PLMs), there is also a paradigm shift Schick and
Schütze (2020); Dai et al. (2020); Sun et al. (2021) reformulating other NLP
tasks (e.g. information extraction) into MRC format, especially for open-
domain scenarios Li et al. (2019); Yan et al. (2021a).

In most of the application scenarios, there exists a hypothesis that only
answerable questions can be asked, which is somehow unrealistic and
unreasonable. Thus, the model that is capable of distinguishing unanswerable
questions is more welcomed than the model that can only give plausible answers
Rajpurkar et al. (2018). However, the challenge, that a slight literal change
may transfer answerable questions into unanswerable ones, makes MRC models
hard to gain such capabilityRajpurkar et al. (2018). For example, in Figure 1,
the original answerable question becomes unanswerable by only replacing
Twilight with Australian, but the small literal modification towards
paraphrasing does not change the answer. Recent MRC models which predict
answers using context-learning techniques and type-matching heuristics are not
easy to perceive such subtle but crucial literal changesWeissenborn et al.
(2017); Jia and Liang (2017). If different questions share many words in
common, these models are most likely to give them the same answer, i.e., 2005
may be answered for all the three questions in Figure 1.

Figure 1: Question Distortion and Question Paraphrase are derived by slightly
changing Original Question.

To address the aforementioned challenge, we propose a span-based method of
Contrastive Learning (spanCL) in this paper. By explicitly contrasting
answerable questions with their paraphrases and their distortions, MRC models
are forced to recognize the subtle but crucial literal changes. Using pre-
trained language model (PLM) as encoder, most contrastive learning methods
adopt [CLS] as the sentence representation Luo et al. (2020); Wu et al.
(2020); Gao et al. (2021); Yan et al. (2021b); Wang et al. (2021). However, in
this problem, as the differences between contrastive questions are very
subtle, [CLS] is inadequate to capture such small changes. To solve the
challenge, we propose a novel learning method, which incorporates the
comparative knowledge between answerable and unanswerable questions, and
exploits the semantic information of answer spans to improve the sentence
representation. Overall, our contributions are summarized as two folds:

  * •

To improve MRC model’s capability of distinguishing unanswerable questions, we
propose a simple yet effective method called spanCL, which teaches the model
to recognize crucial semantic changes from slight literal differences.

  * •

Comprehensive experiments show that spanCL can yield substantial performance
improvements of baselines. We also show that spanCL is an effective way to
utilize generated questions.

##  2 Related Work

Models for MRC. With the help of various large-scale reading comprehension
datasets Hermann et al. (2015); Hill et al. (2015); Trischler et al. (2016);
Rajpurkar et al. (2016); Lai et al. (2017); Rajpurkar et al. (2018), neural
networks have achieved a great success on MRC in recent years. At first, these
models are typically designed with a LSTM Hochreiter and Schmidhuber (1997) or
CNN LeCun et al. (1998) backbone, based on word embeddings Mikolov et al.
(2013); Pennington et al. (2014), leveraging various attention mechanisms to
build interdependent representations of passage and question Kadlec et al.
(2016); Dhingra et al. (2016); Cui et al. (2016); Seo et al. (2016). Recently,
pre-trained language models (PLMs) made a profound impact on NLP tasks Radford
et al. (2018); Devlin et al. (2018); Yang et al. (2019a); Liu et al. (2019);
Lan et al. (2019); Clark et al. (2020); Brown et al. (2020); Fedus et al.
(2021). With millions, billions even trillions of parameters, PLMs show a
great capacity of capturing contextualized representations, and significantly
boost the performance of MRC models.

MRC with Unanswerable Questions. Knowing what you do not know is a crucial
aspect of model intelligence Rajpurkar et al. (2018). In the field of MRC, a
model should abstain from answering when no answer is available to the
question. To deal with unanswerable questions, previous researchers mostly
focused on designing a powerful answer verification module Clark and Gardner
(2017); Liu et al. (2018); Kundu and Ng (2018); Hu et al. (2019). Recently, a
double checking strategy is proposed, in which an extra verifier is adopted to
rectify the predicted answer Hu et al. (2019); Back et al. (2019); Zhang et
al. (2020a, b). Besides the idea of designing verification modules, some other
studies try to solve the problem through data augmentation, namely to
synthesize more QA pairs Yang et al. (2019b); Alberti et al. (2019); Zhu et
al. (2019b); Liu et al. (2020).

Contrastive Learning. To obtain rich representations of texts for down-stream
NLP tasks, there have been numerous investigations of using contrastive
objectives on strengthening supervised learning Khosla et al. (2020); Gunel et
al. (2020) and unsupervised learning Chen et al. (2020); Gao et al. (2021) in
various domains He et al. (2020); Lin et al. (2020); Iter et al. (2020); Kipf
et al. (2019). The main idea of contrastive learning (CL) is to learn textual
representations by contrasting positive and negative examples, through
concentrating the positives and alienating the negatives. In NLP tasks, CL is
usually devoted to learning rich sentence representations Luo et al. (2020);
Wu et al. (2020), and the main difference between these methods is the
approach to find positive and negative examples. Wang et al. (2021) argued
that using hard negative examples in CL is helpful to improve the semantic
robustness and sensitivity of pre-trained language models. Enlightened by the
promising effects of CL, Kant et al. (2021) proposed to use CL in visual
question answering. He focused on playing CL on MRC by comparing multiple
answer candidates, but neglected the fact that not all questions can be
answered through a given paragraph.

Figure 2: (a) The baseline model for MRC-U. (b) Span-based contrastive
learning on answer-span representation.

##  3 Approach

In this section, we first introduce the task of Machine Reading Comprehension
with Unanswerable Questions (MRC-U). Then, a baseline MRC model based on PLM
is described. At last, we propose a span-based contrastive learning method for
MRC-U, named as spanCL. In this paper, question paraphrase and positive
question, question distortion and negative question are used interchangeably.

###  3.1 Task Description

In this paper, we focus on studying extractive MRC, in which the expected
answer of a question is a word span of a given passage. Thus, given a textual
question $Q$ and a textual passage $P$, our goal is to find the answer span
$(y_{s},y_{e})$ to $Q$ in $P$, where $y_{s}$ is the answer start position in
$P$ and $y_{e}$ is the answer end position in $P$.

###  3.2 Basic MRC Model

We use the model same as Devlin et al. (2018) as the basic model for MRC-U
task. When a question and a passage are input, if the question is answerable,
the model is expected to give a legal answer span $(y_{s},y_{e})$ in the
passage; if the question is unanswerable, the model is expected to output the
[CLS] span $(0,0)$, which indicates no related answer can be found in the
passage. The overall structure of the network is presented in Figure 2.

For illustration, we denote the output of PLM’s last layer as the sequence
representation, $\bm{H}\in\mathbb{R}^{l\times d}$, where $l$ is the sequence
length and $d$ is the dimension. Accordingly, the hidden representation of the
$i$-th token in the sequence is denoted as $\bm{h}_{i}\in\bm{H}$. To find the
start position of an answer, a start weight vector
$\bm{w}_{s}\in\mathbb{R}^{d}$ is introduced to calculate the beginning
possibility of each position. Formally, the probability that the answer starts
at the $i$-th token is defined as

$p^{s}_{i}=\cfrac{\mathrm{exp}(\bm{h}_{i}\cdot\bm{w}_{s})}{\sum\limits_{j\leq
l}\mathrm{exp}(\bm{h}_{j}\cdot\bm{w}_{s})}.$ (1)

Similarly, with a end weight vector $\bm{w}_{e}\in\mathbb{R}^{d}$, the
probability that the answer ends at the $i$-th token is defined as

$p^{e}_{i}=\cfrac{\mathrm{exp}(\bm{h}_{i}\cdot\bm{w}_{e})}{\sum\limits_{j\leq
l}\mathrm{exp}(\bm{h}_{j}\cdot\bm{w}_{e})}.$ (2)

For learning, the cross-entropy loss on identifying the answer start and end
positions is taken as the training objective as

$\mathcal{L}_{span}=-\mathrm{log}(p^{s}_{y_{s}})-\mathrm{log}(p^{e}_{y_{e}}),$
(3)

where $y_{s}$ and $y_{e}$ are the start and end positions of the true answer
span. With the learnt model, the output answer span
$(y_{s}^{\prime},y_{e}^{\prime})$ is predicted according to

$(y_{s}^{\prime},y_{e}^{\prime})=\arg\max\nolimits_{(i,j)|i\leq
j}\bm{h}_{i}\cdot\bm{w}_{s}+\bm{h}_{j}\cdot\bm{w}_{e}.$ (4)

Strategy | Example
---|---
Negation |  Original question: What was Beyonce’s role in Destiny’s Child? Negative question: What wasn’t Beyonce’s role in Destiny’s Child?
Entity replacement |  Original question: What native people lived in the San Diego area before the Europeans arrived? Negative question: What native people lived in the San Diego area before the Mexicans arrived?
Antonym |  Original question: What part of Gothic buildings are often found terminated with enormous pinnacles? Negative question: What other part of Gothic buildings are often found terminated with small pinnacles?
Table 1: Strategies used to generate negative questions.

###  3.3 Span-based Contrastive Learning

In this section, spanCL is introduced from two aspects. First, considering the
contrastive idea of CL, we give the details about how the positive and
negative examples are generated. Second, the training objective of spanCL is
presented.

Positive Examples. In our method, we define the positive examples as the
questions which have slight literal differences but the same answers with
their original questions. Back Translation is an effective data augmentation
method Xie et al. (2019); Zhang et al. (2017); Zhu et al. (2019a), in which a
text is first translated to a target language (e.g. France) from its source
language (e.g. English), and then back translated to the source language. The
final back-translated text is taken as the example of augmentation. Thanks to
Back Translation, the produced examples are lexically different but
semantically same with the original example. Specifically, for each question,
we first produce three question paraphrases by Back Translation using three
languages. Then we select the question that has the most literal differences
with the original question as the positive question.

Negative Examples. In our method, we define the negative examples as the
questions which have slight literal differences and not the same answers with
their original questions. Three simple strategies are adopted to produce
negative examples as the following.

  * •

Negation. A negation word is inserted or removed from the original question.

  * •

Antonym. First, spaCy 111https://github.com/explosion/spaCy is utilized to
conduct segmentation and POS for the original question. Then, one of the words
(verbs, nouns, adjectives, or adverbs) are randomly replaced with its antonym.

  * •

Entity Replacement. With an answerable question, one of its entity words is
randomly placed with another entity word, which has the same entity type but
does not appear in any questions.

Table 1 shows several negative examples derived by these strategies. Note that
question generation is not the main topic of this paper.

Span-based Contrastive Learning. Using PLM as the encoder, [CLS] usually serve
as the sentence representation in CL Gao et al. (2021); Wang et al. (2021);
Yan et al. (2021b). When the difference between the original question and its
paraphrase or distortion is very subtle, a single [CLS] token is not adequate
to capture the difference, making the model hard to answer such question.
Therefore, we propose to improve MRC models by contrasting these questions
according to their answer-span representations. Specifically, given a question
$Q_{org}$ and its answer span $(y_{s},y_{e})$, through the augmentation
methods mentioned previously, we synthesize one positive question $Q_{pos}$
and one negative question $Q_{neg}$. Based on the definition of positive
examples and negative examples, $(y_{s},y_{e})$ is the answer span to both
$Q_{pos}$ and $Q_{org}$ but not to $Q_{neg}$. Denote
$\bm{h}^{Q_{org}}_{y_{s}}$ and $\bm{h}^{Q_{org}}_{y_{e}}$ as the
representation vectors of the $y_{s}$-th token and $y_{e}$-th token in the
input passage $P$ for the question $Q_{org}$, $\bm{h}^{Q_{pos}}_{y_{s}}$ and
$\bm{h}^{Q_{pos}}_{y_{e}}$ as those for $Q_{pos}$, and
$\bm{h}^{Q_{neg}}_{y_{s}}$ and $\bm{h}^{Q_{neg}}_{y_{e}}$ as those for
$Q_{neg}$. The concatenation of ${h}^{Q_{org}}_{y_{s}}$ and
$\bm{h}^{Q_{org}}_{y_{e}}$ is used as the answer-span representation to
$Q_{org}$ and denoted as $\bm{z}^{Q_{org}}$. Similarly, the answer-span
representation to $Q_{pos}$ and $Q_{neg}$ are denoted as $\bm{z}^{Q_{pos}}$
and $\bm{z}^{Q_{neg}}$ respectively. Then, our span-based contrastive loss is
calculated as

$\displaystyle\mathcal{L}_{spanCL}=-\mathrm{log}\cfrac{\mathrm{exp}(\Phi(\bm{z}^{Q_{org}},\bm{z}^{Q_{pos}})/\tau)}{\begin{multlined}\mathrm{exp}(\Phi(\bm{z}^{Q_{org}},\bm{z}^{Q_{pos}})/\tau))\\\
+\mathrm{exp}(\Phi(\bm{z}^{Q_{org}},\bm{z}^{Q_{neg}})/\tau))\end{multlined}\mathrm{exp}(\Phi(\bm{z}^{Q_{org}},\bm{z}^{Q_{pos}})/\tau))\\\
+\mathrm{exp}(\Phi(\bm{z}^{Q_{org}},\bm{z}^{Q_{neg}})/\tau))}$ (5)

where
$\Phi(\bm{u},\bm{v})=\bm{u}^{\top}\bm{v}/\left\|\bm{u}\right\|\left\|\bm{v}\right\|$
computes similarity between $\bm{u}$ and $\bm{v}$ and $\tau>0$ is a scalar
temperature parameter. With the definition, the final objective loss of our
method is presented as the following:

$\mathcal{L}=\lambda_{1}\mathcal{L}_{span}+\lambda_{2}\mathcal{L}_{spanCL}.$
(6)

##  4 Experiments

###  4.1 Datasets and Metrics

We evaluate our method on the well-known dataset SQuAD 2.0 Rajpurkar et al.
(2018), which covers the questions of SQuAD1.1 Rajpurkar et al. (2016) with
new unanswerable questions written adversarially by crowdworkers to imitate
the answerable ones. Moreover, for each unanswerable question, a plausible
answer span is annotated, which indicates the incorrect answer obtained by
type-matching heuristics. The training dataset contains 87k answerable and 43k
unanswerable questions, and half of the examples in the development set are
unanswerable.

Two official metrics are used to evaluate the model performance on SQuAD 2.0:
Exact Match (EM) and F1. EM is used to compute the percentage of predictions
that match ground truth answers exactly. F1 is a softer metric, which measures
the average overlap between the prediction and ground truth answer at token
level.

###  4.2 Experimental Setup

MRC Model. We adopt the model introduced in 3.2 with various PLM encoders for
the MRC-U task. Bert Devlin et al. (2018), RoBERTa Liu et al. (2019), ALBERT
Lan et al. (2019) are selected in our experiments. We download the pre-trained
weights from Hugging Face222https://huggingface.co/bert-base-uncased.

Training Data Construction. For each original answerable question, we use Back
Translation to generate its paraphrase. In SQuAD 2.0, we can find the negative
questions for 18,541 answerable questions in the original dataset. For the
rest 68,280 answerable questions, we use our augmentation strategies to
generate negative questions.

During our training, the span loss is calculated based on $Q_{org}$ and
$Q_{neg}$. In section 4.3, we will explain why $Q_{pos}$ is discarded for
calculating span loss.

Hyper-parameters. We use the default hyper-parameter settings for the SQuAD
2.0 task. Specifically, we set maximum sequence length, doc stride, maximum
query length and maximum answer length to 512, 128, 64 and 30. For fine-tuning
our model, we set the learning rate, batch size, training epoch and warm-up
rate to 2e-5, 12, 2 and 0.1. The temperature in spanCL is set to 0.05. The
weights of span loss and spanCL loss are $\lambda_{1}=\lambda_{2}=0.5$. For
each time, we fix the random seed, ensuring our results are reproducible. We
run our experiments on two Tesla A100 40G GPUs with 5 GPU hours to train a
model.

###  4.3 Main Results

Model | Dev | $\Delta$
---|---|---
EM | F1 | EM | F1
$BERT_{base}$ | 73.37 | 76.34 | - | -
\+ spanCL | 75.51 | 78.34 | +2.14 | +2.00
$BERT_{large}$ | 78.88 | 81.85 | - | -
\+ spanCL | 79.76 | 82.61 | +0.88 | +0.76
$RoBERTa_{base}$ | 78.85 | 81.42 | - | -
\+ spanCL | 80.18 | 82.84 | +1.33 | +1.42
$RoBERTa_{large}$ | 86.12 | 88.88 | - | -
\+ spanCL | 86.98 | 89.70 | +0.86 | +0.82
$ALBERT_{base}$ | 77.84 | 81.27 | - | -
\+ spanCL | 79.52 | 82.97 | +1.68 | +1.7
$ALBERT_{large}$ | 79.99 | 83.27 | - | -
\+ spanCL | 81.51 | 84.67 | +1.52 | +1.4
Table 2: Results (%) on the dev set of SQuAD 2.0. Model | EM | Dev
---|---|---
HasAns | NoAns | EM | F1
$BERT_{base}$ | 70.31 | 74.76 | 73.37 | 76.34
\+ pos | 72.57 | 68.93 | 72.26 | 75.22
\+ neg | 67.05 | 87.74 | 74.02 | 76.38
\+ pos&neg | 66.16 | 78.48 | 72.59 | 75.37
\+ spanCL | 72.52 | 75.91 | 75.51 | 78.34
Table 3: Training with spanCL vs Training with expanded datasets.

From Table 2, we notice that spanCL improves the performance of each baseline
model, yielding 0.86~2.14 absolute EM improvement and 0.76~2.0 absolute F1
improvement, demonstrating spanCL is model-agnostic and effective.

As additional training data (i.e. the extra positive and negative questions)
is used, it is necessary to analyze if the improvements are merely brought by
this additional data. We conduct experiments by training with different
datasets and display the results in Table 3. $\mathrm{BERT}_{base}$ means
training $\mathrm{BERT}_{base}$ with original SQuAD 2.0 training set.
“$\mathrm{+pos}$” and “$\mathrm{+neg}$” mean expanding the original training
set with generated positive questions and generated negative questions
respectively. Surprisingly, Simply expanding the training set can not
guarantee the performance improvement. We find that adding positive examples
into the training set does not improve the performance of MRC model. One
possible reason is that the positive questions make the model over insensitive
and ignore slight literal changes, which is inappropriate for MRC-U task. By
comparing $\mathrm{BERT}_{base}$ with “$\mathrm{+neg}$”, we find that training
with more negative examples, the model tends to predict more NoAns and achieve
a high performance on NoAns, while the performance on the HasAns drops a lot
and the overall improvement of EM is much less than “$\mathrm{+spanCL}$”. From
the results in Table 3, we can conclude that spanCL is effective to utilize
the generated questions.

###  4.4 Influence of Negative Examples

The unanswerable questions generated by our strategies are rather plain. We
believe spanCL can further boost the performance by high-quality unanswerable
questions. Liu et al. (2020) proposed a context-relevant generation method
called CRQDA, which generates delicate negative questions
333https://github.com/dayihengliu/CRQDA. In table 4, “+CRQDA” denotes training
the baseline model with the dataset including the delicate negative questions
generated by CRQDA. “+spanCL with simple negatives“ denotes applying spanCL
with negative questions generated by our three strategies. “+spanCL with
CRQDA” denotes applying spanCL with negative questions generated by CRQDA.
Comparing “+spanCL with simple negatives“ with “+spanCL with CRQDA“, we find
that spanCL can further boost the performance by delicate negative questions.

Model | EM | F1
---|---|---
$BERT_{base}$ | 73.37 | 76.34
\+ CRQDA Liu et al. (2020) | 75.80 | 78.70
\+ spanCL with simple negatives | 75.51 | 78.34
\+ spanCL with CRQDA | 76.12 | 79.09
Table 4: Performance of spanCL with different synthetic negative questions.

###  4.5 Influence of Temperature

Figure 3: The influence of different temperatures in spanCL. The best
performance is achieved when the temperature is set to 0.05.
$\mathrm{BERT}_{base}$ is adopted as the base model.

The temperature $\tau$ in spanCL loss (Equation 5) is used to control the
smoothness of the distribution normalized by the softmax operation. A large
temperature smoothes the distribution while a small temperature sharpens the
distribution. As shown in the Figure 3, spanCL is sensitive to the temperature
value. In general, small temperature results in better performance. A
practical temperature can be obtained within a small range (from about 0.02 to
0.1). We select 0.05 as the temperature in our experiments.

Figure 4: Qualitative Examples.

###  4.6 Selection of Question Representations

Model | EM | F1
---|---|---
$BERT_{base}$ | 73.37 | 76.34
\+ spanCL | 75.51 | 78.34
\+ CL with [CLS] reps | 74.18 | 77.05
\+ CL with span and [CLS] reps | 73.82 | 76.86
Table 5: Results (%) with different question representations used in the contrastive learning. Base Model | Training schemes | EM | F1
---|---|---|---
$BERT_{base}$ | Joint | 75.51 | 78.34
$BERT_{base}$ | Alternate | 74.67 | 77.08
$BERT_{base}$ | pre-train then finetune | 72.19 | 74.87
$BERT_{large}$ | Joint | 79.76 | 82.61
$BERT_{large}$ | Alternate | 79.50 | 82.55
$BERT_{large}$ | pre-train then finetune | 77.77 | 80.30
Table 6: Results (%) with different training schemes.

In this paper, we argue that the answer span representation is better than
[CLS]. We conduct experiments with different question representations in this
section. When applying CL with [CLS] representations, we add a classification
layer on the top of [CLS] to determine if a question is answerable or not
Zhang et al. (2020b), making the representation of [CLS] acquire the
information of the question’s answerability. We also play CL with both [CLS]
and answer-span representations, in which two CL losses are calculated
together. From Table 5, we can see that CL with [CLS] reps improves the model
performance but the improvement is small than that from spanCL, and the
combination of the two CL losses can confuse the model and result in a little
improvement.

###  4.7 Comparison between Different Training Schemes

There are three training schemes to combine the span loss and spanCL loss: 1)
joint training, in which these two losses are used together in each training
step; 2) alternate training, in which the model is updated with spanCL loss
after every $M$ updates with span loss; 3) pre-train and fine-tune, in which
we first pre-train the model with spanCL loss and then fine-tune it with span
loss. For alternate training, we select $M$ from $\\{1,2,3\\}$ and find $M=2$
gives the best results. From Table 6, we conclude that joint training gives
the best performance and alternate training performs a little worse.
Surprisingly, with the pre-train and fine-tune training scheme, the model
performs worse than the baseline model. We guess this is because without the
supervision of answer-span knowledge, it is hard to learn useful question
representations.

###  4.8 Qualitative Analysis

We qualitatively analyze two representative unanswerable questions in Figure
4. It can be seen that the baseline model predicts a plausible answer for each
question while the baseline model trained with spanCL abstain from answering.

To correctly answer the first question, the model is asked to learn the
question’s semantics in sentence level. To correctly answer the second
question, the model is asked to recognize the literal change in word level.
SpanCL can help the model perceive such crucial differences between the
question and passage from both semantic and lexical aspects, and thus enable
the baseline model to abstain from answering for these two questions.

##  5 Conclusion

In this paper, we propose a span-based method of Contrastive Learning (spanCL)
to solve the MRC task with Unanswerable Questions. SpanCL is devised based on
the fact that an answerable question can become unanswerable with slight
literal changes. By explicitly contrasting an answerable question with its
paraphrase and distortion at the answer span level, MRC models can be taught
to perceive subtle but crucial literal changes. Experimental results
demonstrate that spanCL is model-agnostic and can improve MRC models
significantly. Additional experiments show that spanCL is more effective to
utilize the generated questions than other methods. In addition, it should be
noticed that how to generate high-quality question examples is not fully
investigated in this paper, which may introduce a performance bottleneck to
spanCL. Therefore, a study on question generation compatible with spanCL is
encouraged in the future.

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# Encoding quantum bits in bound electronic states of a graphene nanotorus

J. Furtado  0000-0002-1273-519X<EMAIL_ADDRESS>Centro de Ciências e
Tecnologia, Universidade Federal do Cariri, 63048-080, Juazeiro do Norte,
Ceará, Brazil  A. C. A. Ramos<EMAIL_ADDRESS>Centro de Ciências e
Tecnologia, Universidade Federal do Cariri, 63048-080, Juazeiro do Norte,
Ceará, Brazil  J. E. G. Silva<EMAIL_ADDRESS>Centro de Ciências e
Tecnologia, Universidade Federal do Cariri, 63048-080, Juazeiro do Norte,
Ceará, Brazil  R. Bachelard  0000-0002-6026-509X<EMAIL_ADDRESS>Departamento
de Física, Universidade Federal de São Carlos, Rodovia Washington Luís, km 235
- SP-310, 13565-905 São Carlos, SP, Brazil  Université Côte d’Azur, CNRS,
Institut de Physique de Nice, 06560 Valbonne, France  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  Department of Physics, Stockholm University,
AlbaNova University Center 106 91 Stockholm, Sweden

(August 28, 2024)

###### Abstract

We propose to use the quantum states of an electron trapped on the inner
surface of a graphene nanotorus to realize as a new kind of physical quantum
bit, which can be used to encode quantum information. Fundamental tasks for
quantum information processing, such as the qubit initialization and the
implementation of arbitrary single qubit gates, can be performed using
external magnetic and electric fields. We also analyze the robustness of the
device again systematic errors, which can be suppressed by a suitable choice
of the external control fields. These findings open new prospects for the
development an alternative platform for quantum computing, for which the
scalability remains to be determined.

##  I Introduction

Quantum bits (qubits) constitute the most fundamental unit of information for
quantum computation, exploiting the properties of two-level systems to be in a
superposition of quantum states. Considering the potential advantage of
quantum computers in solving hard problems through quantum algorithms, such as
Shor’s Shor (1994, 1997) and Simon’s Simon (1994, 1997) algorithms, efforts
have been dedicated to identify suitable physical platforms to encode qubits.
Qubits have been successfully encoded in degrees of freedom of light Knill et
al. (2001); Kok et al. (2007), spins in nuclear magnetic resonance of atoms
Oliveira et al. (2011), spin in quantum dots Loss and DiVincenzo (1998),
semiconductor spins Vandersypen and Eriksson (2019), and trapped ions Cirac
and Zoller (1995) among others Nielsen and Chuang (2011). Recently, impressive
progresses have also been achieved with superconducting qubits Arute et al.
(2019); Wu et al. (2021), driven by the interest of the community to proposing
different kinds of physical qubits for quantum computation .

On the other hand, the outstanding properties of two-dimensional materials
sparked a revolution in physical science. In particular, the features of
curved graphene has been exploited to devise new electronic devices, such as
helical strips with chiral effects, also known as chiraltronics Dandoloff and
Truong (2004); Atanasov et al. (2009); Atanasov and Saxena (2015), and Möbius
strips with topological insulator properties Guo et al. (2009). Bridges
connecting layers of graphene have also been proposed using a catenoid surface
Dandoloff (2009); Dandoloff et al. (2010); Silva et al. (2020a). These curved
materials are described by an effective Dirac Hamiltonian for a massless and
gapless electron in the single layer of graphene Katsnelson (2012), and the
surface curvature is associated with a geometric potential, the so-called da
Costa potential da Costa (1981, 1982). Among the curved geometries of
interest, the torus can be built by gluing the two edges of a nanotube, and
presents a rich phenomenology Ahlskog et al. (1999); Wang et al. (2001);
Martel et al. (1999); Haddon (1997). In particular, it exhibits a number of
bound states, whose spectrum and geometry can be manipulated with external
electric and magnetic fields Silva et al. (2020b).

In this paper we propose to encode a qubit in the quantum states of an
electron confined on the surface of a graphene nanotorus, as sketched in Fig.
1. The curvature-induced trapping potential leads to a discrete set of quantum
states, which are well characterized by two quantum numbers, namely, the total
angular momentum $l$ and its component along the quantization axis $m$. We
also show how the qubit can be initialized, and how gates can be implemented.
Furthermore, the robustness of such information processing against systematic
errors is characterized, showing how they can be mitigated by external control
fields. Considering the number of techniques to produce and probe such a
nanotorus (and nanorings, a similar geometry) Ahlskog et al. (1999); Wang et
al. (2001); Martel et al. (1999); Haddon (1997); Omachi et al. (2013); Chen et
al. (2005); Thorner et al. (2014); Kharissova et al. (2019); Lehr et al.
(2014); Omachi et al. (2011), our proposal paves the way for the experimental
realization of a new kind of qubit, under well-controlled conditions.

Figure 1: (a) Schematic representation of the nanotorus submitted to a static
magnetic field (used to prepare the qubit) and to an time-dependent electric
field (used to control the qubit and implement gates). The trapping sector in
which the electron is confined is shown in green, for the orbital angular
momentum $m=0$ states. (b) In absence of an external magnetic field, there
exist a single bound state, with $\ket{l=0,m=0}$, and a pair of degenerate
states $\ket{l=1,m=\pm 1}$. (c) When applying an external magnetic field, the
geometrical effects induce a Zeeman-like breaking of degeneracy of the states
with $l=1$, while a new bound state $\ket{l=1,m=0}$, used to realize our two-
level system, arises.

##  II The system and first quantization

We here consider an electron confined on the surface of a nanotorus made of
graphene, and it is characterized by the minor radius $r$ and the major radius
$R$, see Fig. 1a. Thanks to the axial symmetry, the wavefunctions of the time-
independent Schrödinger equation reduces to
$\psi_{n,l,m}(\theta,\phi)=\chi_{n,l}(\theta)e^{im\phi}$, where $\phi$ is the
angle around the torus axis of revolution, and $\theta$ the one around the
tube. As we shall see, $m$ is the quantum number associated with the component
of the angular momentum in direction $\hat{Z}$, and the angular component of
the wavefunction $\chi_{n,l}(\theta)$ takes into account the quantum numbers
related to the electronic energy level $n$ and the total angular momentum of
the system $l$. It satisfies the following time-independent Schrödinger
equation Silva et al. (2020b):

$\displaystyle-\frac{\hbar^{2}}{2m^{*}r^{2}}\frac{d^{2}\chi(\theta)}{d\theta^{2}}+V_{\text{bare}}(\theta)\chi(\theta)=\epsilon\chi(\theta),$
(1)

where the potential $V_{\text{bare}}$ reads

$\displaystyle V_{\text{bare}}(\theta)$
$\displaystyle=\frac{\hbar^{2}}{2m^{*}r^{2}(R+r\cos\theta)^{2}}\left[-\frac{R^{2}}{4}+r^{2}m^{2}+\frac{r^{2}\sin^{2}\theta}{4}\right.$
$\displaystyle\left.+\frac{r(R\cos\theta+r)}{2}\right],$ (2)

with $m^{*}=0.3m_{0}$ the effective mass of the electron on the torus surface,
and $m_{0}$ is the rest mass of the electron Haddon (1997). Under a suitable
choice of the parameters $R$ and $r$, a single confined state exists, for
which the probability density function reveals a “trapping sector” in the
inner region of the nanotorus ($\theta=\pi$). As highlighted in Fig. 1a, its
width at half height is $\delta\theta\approx\pi$, so the electronic
wavefunction forms a ring located around $\theta=\pi$ Silva et al. (2020b).

However, in presence of external fields, whether electric or magnetic, the
potential changes and $V_{\text{bare}}$ in Eq. (1) must be substited by the
effective potential
$V(\theta)=V_{\text{bare}}(\theta)+V_{E}(\theta)+V_{B}(\theta)$, in which
$V_{E}(\theta)$ and $V_{B}(\theta)$ account for electric and magnetic field
contributions, respectively:

$\displaystyle V_{E}(\theta)$ $\displaystyle=-eEr\sin\theta,$ (3)
$\displaystyle V_{B}(\theta)$
$\displaystyle=\frac{e^{2}B^{2}}{8m^{*}}(R+r\cos\theta)^{2}-\frac{me\hbar
B}{2m^{*}}.$ (4)

The Hamiltonian of the system in presence of electric and magnetic static
fields then reads

$\displaystyle
H(\theta)=-\frac{\hbar^{2}}{2m^{*}r^{2}}\frac{d^{2}}{d\theta^{2}}+V(\theta).$
(5)

These external fields can drastically alter the confinement potential of the
electron on the surface of the nanotorus, resulting in new confined electronic
near-ground states. In what follows, we focus on the set of states associated
to the total angular momentum $l=\\{0,1\\}$, since they are already suitable
candidates to encode a quantum bit of information.

###  II.1 Electronic near-ground states

In Fig. 2, we present the change in energy levels induced by the external
magnetic field (from $B=0T$ to $B=0.45T$), with the lowest-lying states
associated with $l=\\{0,1\\}$ represented. As sketched in Fig. 2a, by applying
an external magnetic field the degenerate subspace $\\{\ket{l=1,m=\pm 1}\\}$
becomes non-degenerate. In the absence of magnetic field, the bound states
correspond to $\ket{l=0,m=0}$ and $\ket{l=1,m=\pm 1}$, whereas the state
$\ket{l=1,m=0}$ is not bound. According to Eq. (2), the term $r^{2}m^{2}$
leads to a degenerate subspace for the states $\\{\ket{l=1,m=\pm 1}\\}$,
illustrated in Fig. 2b. However, applying a strong magnetic field allows one
to modify substantially the potential, so the state $\ket{l=1,m=0}$ becomes
bound and the last term in Eq. (4) lifts the degeneracy of the subspace
$\\{\ket{l=1,m=\pm 1}\\}$, see Fig. 2c.

Figure 2: (a) Schematic representation of the Zeeman-like splitting. (b)
Energy diagram of the three lowest-lying bound states of the system in absence
of fields ($B\\!=\\!0T$), when the electron is subject only to the potential
induced by curvature effects. (c) By turning on the magnetic (here we set
$B\\!=\\!0.45\leavevmode\nobreak\ T$), new bound states appear and a Zeeman-
like splitting coours, here stemming from both magnetic field and geometrical
effects. This allows one to encode a bit of quantum information in the energy
subspace spanned by the states $\ket{l=\\{0,1\\},m=0}$.

We point out that when under the action of an electric field, the number of
confined states also increases, and become spatially displaced over the
nanotorus surface. Nevertheless, these states are doubly degenerate to
$m=\\{\pm 1,\pm 2,\cdots\\}$, a situation which is undesirable for the
initialization of the qubit state. Indeed, degenerate states are not adequate
for quantum information encoding, since multiple states may become populated.
This motivates the choice of a static magnetic field over the nanotorus to
build the qubit. In this context, we emphasize that the breaking of degeneracy
by the magnetic field is different in nature from the Zeeman effect, in the
sense that the field drastically modifies the geometric potential, in addition
to shifting the energy of the associated levels.

##  III Electronic qubit encoding

Let us adopt DiVincenzo’s approach to characterize our system as a qubit
suitable to encode quantum information DiVincenzo (2000), focusing on the two
following criteria: A physical system (1) with well characterized qubit (i.e.,
two-level system), and (2) which can be coherently controlled, is a candidate
for quantum information processing. The additional DiVicenzo’s criteria
concern the scalability, decoherence effects and the readability of the
quantum information encoded in the qubits, which we discuss only briefly in
the conclusions.

We first consider the case when an external magnetic field is applied over the
system and the orbital angular momentum $m$ is taken to be zero. The
$m\\!=\\!0$ states are often considered, for example, in quantum computing
with trapped ions Leibfried et al. (2003); Barreiro et al. (2011); Hu et al.
(2018, 2020). Returning to Eq. (5) and doing an expansion around the trapping
region ($\theta\\!\approx\\!\pi$), one obtains

$\displaystyle
V_{\text{bare}}(\theta_{\pi})+V_{B}(\theta_{\pi})=\frac{\beta^{2}}{2m^{*}}\theta_{\pi}^{2}+\delta\theta_{\pi}^{4}+\epsilon,$
(6)

where we have introduced the variable $\theta_{\pi}\\!=\\!\theta-\pi$. The
coefficients $\beta$, $\delta$ and $\epsilon$ are given by

$\displaystyle\delta$
$\displaystyle=-\frac{1}{96m^{*}}\left[\frac{\hbar^{2}r(R+8r)}{(R-r)^{4}}+e^{2}B^{2}r(R-4r)\right],$
(7a) $\displaystyle\beta^{2}$
$\displaystyle=\frac{r}{4}\left[\frac{\hbar^{2}}{(R-r)^{3}}+e^{2}B^{2}(R-r)\right],$
(7b) $\displaystyle\epsilon$
$\displaystyle=\frac{1}{8m^{*}(R-r)^{2}}\left[e^{2}B^{2}(R-r)^{4}-\frac{\hbar^{2}\left(R^{2}-2Rr+2r^{2}\right)}{r^{2}}\right].$
(7c)

The first quantization is realized by identifying the canonical variables of
position, $\theta_{\pi}\rightarrow\hat{\theta}_{\pi}$, and momentum
$\hat{p}_{\theta}\\!=\\!(i/r)d(\bullet)/d\theta_{\pi}$, as operators. In this
first-quantization picture, the Hamiltonian of the system now reads

$\displaystyle\hat{H}_{0}=-\frac{\hbar^{2}}{2m^{*}r^{2}}\frac{d^{2}}{d\theta_{\pi}^{2}}+\epsilon+\frac{\beta^{2}}{2m^{*}}\hat{\theta}_{\pi}^{2}+\delta\hat{\theta}_{\pi}^{4}.$
(8)

This Hamiltonian describes a modified harmonic potential with annharmonicity
$\delta$, which is reminiscent of the one obtained for superconducting qubits
Blais et al. (2004); Koch et al. (2007); Krantz et al. (2019); Rasmussen et
al. (2021). In particular, the change of variables
$\hat{\theta}_{\pi}\rightarrow\hat{q}/r$ leads to

$\displaystyle\hat{H}_{0}=\frac{\hat{p}^{2}}{2m^{\ast}}+\frac{m^{\ast}\omega^{2}}{2}\hat{q}^{2}+\frac{\delta}{r^{4}}\hat{q}^{4}+\epsilon\mathbbm{1}.$
(9)

The connection with artificial superconducting atoms is made more
straightforward by defining the creation and annihilation operators,
$\hat{a}^{\dagger}=\left(\hat{q}-id/d\hat{q}\right)/\sqrt{2}$ and
$\hat{a}=\left(\hat{q}+id/d\hat{q}\right)/\sqrt{2}$ respectively, so we obtain
the anharmonic oscillator Hamiltonian:

$\displaystyle\hat{H}_{0}=\mathbbm{1}E_{0}+\hbar\omega\hat{a}^{\dagger}\hat{a}+\delta\left(\frac{\hbar}{2m^{\ast}\omega
r^{2}}\right)^{2}\left(\hat{a}+\hat{a}^{\dagger}\right)^{4},$ (10)

with $E_{0}\\!=\\!\epsilon+\hbar\omega/2$ and $\omega=|\beta|/m^{*}r$. The
first two terms correspond to the quantum harmonic oscillator with ground-
state energy $E_{0}$ and a spectrum of energies separated by quanta of energy
$\hbar\omega$. The third term of the above equation stands for the
anharmonicity parameter. We hereafter refer to this Hamiltonian as our bare
Hamiltonian, which defines the typical properties of our system, such as the
qubit energy scale and qubit transition frequency.

As an alternative to genuine two-level systems, such as spin and photon
polarization qubits, one can exploit the large anharmonicity to induce energy
shifts in the upper-excited levels, so the system operates effectively as a
two-level system. Indeed, when driving the lowest excited state, the
probability to gain a second quantum of energy is negligible. This scheme is
often used in artificial atoms, such as in superconducting qubits Krantz et
al. (2019); Rasmussen et al. (2021). In the case of the nanotorus, increasing
the external magnetic field $B$ allows one to produce an effective two-level
system which spans the subspace
$\mathcal{S}_{\mathrm{qb}}=\\{\ket{l=0,m=0},\ket{l=1,m=0}\\}$, with the main
quantum number $m\\!=\\!0$ and $l=\\{0,1\\}$, see Fig. 2. Now assuming that
the system only explores the two energy levels of $\mathcal{S}_{\mathrm{qb}}$,
the bosonic operators $\hat{a}$ and $\hat{a}^{\dagger}$ can be substituted by
$\hat{\sigma}_{-}$ and $\hat{\sigma}_{+}$, respectively, the Pauli matrices
defined as $\hat{\sigma}_{-}=\ket{l=0,m=0}\bra{l=1,m=0}$ and
$\hat{\sigma}_{+}=(\hat{\sigma}_{-})^{\dagger}$. The Hamiltonian for the qubit
then reads

$\displaystyle\hat{H}_{\mathrm{qb}}=\hbar\omega\hat{\sigma}_{+}\hat{\sigma}_{-}+\alpha\left(\hat{\sigma}_{+}+\hat{\sigma}_{-}\right)^{4}+\mathbbm{1}E_{0},$
(11)

with $\alpha\\!=\\!\delta(\hbar/2m^{\ast}\omega r^{2})^{2}$. Moreover, since
the second term in the above equation can be rewritten as
$\alpha\left(\hat{\sigma}_{+}+\hat{\sigma}_{-}\right)^{4}=\alpha\mathbbm{1}$,
we simplify the above Hamiltonian by redefining the zeroth energy level to get

$\displaystyle\hat{H}_{\mathrm{qb}}=\hbar\omega\hat{\sigma}_{+}\hat{\sigma}_{-}.$
(12)

We stress that this Hamiltonian is valid only when transitions occur only
inside the subspace $\mathcal{S}_{\mathrm{qb}}$, where no change in the main
quantum number is promoted. With a stronger magnetic field, new bound states
with $m=0$ will appear, so the system will turn in an effective multilevel
atom.

##  IV Qubit initialization and control

As mentioned before, when the magnetic field is below a certain value
$B_{\mathrm{ini}}$, hereafter called initialization static magnetic field, the
system presents a single low-lying state for which $m=0$. The others states
have a number $m\neq 0$, as illustrated in Fig. 2. In this case, an electron
trapped in the nanotorus tends to populate this specific state, since it has
the lowest energy. The qubit initialization is then achieved by introducing an
external magnetic field which is strong enough to turn the electronic state
$\ket{l=1,m=0}$ into a bound state, but not too strong so that the subspace
corresponding to $m=0$ contains only two states. Hence, the system behaves as
an effective two-level system. Since the system has previously been
initialized in the state $\ket{l=0,m=0}$, and since the magnetic field
variation does not promote any transition, the system remains in the state
$\ket{\psi(0)}=\ket{l=0,m=0}$, which concludes the initialization of the
qubit.

Figure 3: Energy levels (with $m=0$) of an electron confined on the inner
radius of a nanotorus as function of the external magnetic field $B$ (in
Tesla) for two configurations: (a) $r=350\AA$ and $R=900\AA$, and for (b)
$r=350\AA$ and $R=3600\AA$. In both cases, the first and second curves, from
bottom to top, correspond to the fundamental level $\ket{l=0,m=0}$ and first
excited one $\ket{l=1,m=0}$, respectively. The other curves correspond to
other energy levels, which are undesirable in the present scheme to encode a
qubit.

In Fig. 3, we present the energy of the bound states as a function of the
magnetic field, showing the value of $B_{\mathrm{min}}$ and $B_{\mathrm{max}}$
for two different values of the external radius $R$. Indeed, the precise range
of values for the magnetic field required to initialize the qubit depends on
the specific geometry of the nanotorus, through the minor $r$ and major radius
$R$.

###  IV.1 Qubit control and universal single-qubit quantum gates

As for the majority of quantum emitters, the driving of a specific transition
can be achieved using an external electric field, using a local field to
control the qubit. Introducing the electric field $\vec{E}=E\hat{Z}$
manifests, in Hamiltonian (5), in an additional contribution:
$\hat{H}=\hat{H}_{0}+\hat{H}_{E}$, with

$\displaystyle\hat{H}_{E}=erE\sqrt{\frac{\hbar}{2r|\beta|}}\left[\left(\hat{a}+\hat{a}^{\dagger}\right)-\frac{\hbar}{12r|\beta|}\left(\hat{a}+\hat{a}^{\dagger}\right)^{3}\right].$
(13)

Considering again an external magnetic field such that we have only the low-
lying states $\ket{n}\\!=\\!\ket{l=n,m=0}$, we can replace the operators
$\hat{a}$ and $\hat{a}^{\dagger}$ by $\hat{\sigma}_{-}$ and $\hat{\sigma}_{+}$
in Eq. (13), which leads to

$\displaystyle\hat{H}_{E}=\hbar\Omega(E)(\hat{\sigma}_{+}+\hat{\sigma}_{-}).$
(14)

We have used that
$\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)^{3}=\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)$
and introduced the frequency $\Omega(E)=\mu E/\hbar$, with $\mu$ a parameter
given by

$\displaystyle\mu=er\left[\sqrt{\frac{\hbar}{2r|\beta|}}-\frac{1}{6}\left(\sqrt{\frac{\hbar}{2r|\beta|}}\right)^{3}\right].$
(15)

Since the quantity $\hbar/r|\beta|$ is dimensionless, $\mu$ has dimension of a
‘charge-distance’, i.e., the dimension of an electric dipole moment (see
Appendix A for further details). Furthermore, the amplitude of $\Omega(E)$
here depends on both the electric and magnetic fields, in addition to other
parameters related to the system geometry. The controllability of the system
is thus strongly related to the control of the driving electric field $E$,
once the other parameters have been fixed. In particular, by considering a
time-dependent oscillating radio-frequency (RF) field $E$ given by

$\displaystyle E=E_{0}\cos(\omega_{\mathrm{rf}}t+\phi),$ (16)

with $\phi$ an adjustable phase, the driving Hamiltonian becomes

$\displaystyle
H_{\mathrm{dv}}(t)=\hbar\omega\sigma^{+}\sigma^{-}+\hbar\Omega(E_{0})\cos(\omega_{\mathrm{rf}}t+\phi)(\sigma^{+}+\sigma^{-}).$
(17)

As a final step, let us derive the system effective dynamics, close to
resonance. According to Schrödinger equation, the system evolves as

$\displaystyle i\hbar\ket*{\dot{\psi}(t)}=H_{\mathrm{dv}}(t)\ket{\psi(t)},$
(18)

since no external dissipative channel is taken into account. In order to
explore the two-level nature of the system, we then consider the case of a
resonant driving ($\Omega(E_{0})\ll\omega$). Furthermore, we move to the
rotating frame of the external field, consider the Rotating Wave
Approximation, so the dynamics of the system is dictated by the driving
effective Hamiltonian (see Appendix B)

$\displaystyle
H_{\text{eff}}=\hbar\Delta\sigma_{+}\sigma_{-}+\frac{\hbar\Omega(E_{0})}{2}\left(e^{i\phi}\sigma_{-}+e^{-i\phi}\sigma_{+}\right),$
(19)

where $\Delta\\!=\\!\omega-\omega_{\mathrm{rf}}$ is the detuning between the
qubit transition frequency and the external field frequency. The above
Hamiltonian, where the phase $\phi$ is controlled through the driving field,
can then be used to implement arbitrary single qubit gates Santos (2018), so
one can conclude that the di Vincenzo’s criteria to use a physical qubit for
quantum computation are fulfilled.

###  IV.2 Preparation of an arbitrary input state

In this section we show how to choose adequately the parameters of the
effective Hamiltonian $H_{\text{eff}}$ in order to prepare an arbitrary input
state given by

$\displaystyle\ket{\psi(\theta,\eta)}=\sin\left(\theta/2\right)\ket{0}+e^{i\eta}\cos\left(\theta/2\right)\ket{1},$
(20)

for any $\theta\in(0,\pi]$ and $\eta\in[0,\pi]$. The system is initially
prepared in the state $\ket{0}$ using the procedure discussed previously.
Then, using a resonant RF field $(\omega_{\mathrm{rf}}=\omega)$, the system
evolves toward
$\ket{\Psi(t,\phi)}=e^{-\frac{i}{\hbar}H_{\text{eff}}t}\ket{0}$, which reads

$\displaystyle\ket{\Psi(t,\phi)}$
$\displaystyle=\cos\left(\frac{t\Omega}{2}\right)\ket{0}+e^{-i\left(\phi+\frac{\pi}{2}\right)}\sin\left(\frac{t\Omega}{2}\right)\ket{1}.$
(21)

Figure 4: (a) Trajectories on the Bloch sphere obtained for different choices
of the parameter $\phi$. The total evolution time $\tau=\pi/2\Omega$ brings
the system to one of the state in the equator of the sphere, where any state
in $\mathcal{E}$ can be prepared. The states in the $x$\- and $y$-axis of the
Bloch sphere are defined as $\ket{\pm}_{x}=(\ket{0}\pm\ket{1})/\sqrt{2}$ and
$\ket{\pm}_{y}=(\ket{0}\pm i\ket{1})/\sqrt{2}$. (b) Schematic representation
of each step in the implementation of the quantum circuit shown in (c), where
a magnetic field is used to initialize the system in the ground state
$\ket{0}$, followed by the coherent control using the external electric field
which drives the system to a desired final state. In (c) we show the quantum
gates Hadamard ($H$) and phase gate ($R_{\eta}$), satisfying
$H\ket{n}=(\ket{0}+(-1)^{n}\ket{1})/\sqrt{2}$, with $n\in\\{0,1\\}$, and
$R_{\eta}(a\ket{0}+b\ket{1})=a\ket{0}+be^{i\eta}\ket{1}$.

By comparing the above equation with Eq. (20), it becomes clear the total
evolution time and the phase $\phi$ can be chosen to achieve any state of the
form (20). As an illustration, we show in Fig. 4a the trajectory of the
quantum state on the Bloch sphere, driven by Hamiltonian (19). After
initializing the system in the state $\ket{0}$, we set the total evolution
time as $\tau=\pi/2\Omega$ such that the final state is one of the set of
quantum states $\mathcal{E}$ at the equator of the Bloch sphere, which is a
set of states with maximum quantum coherence in basis $\\{\ket{0},\ket{1}\\}$
Streltsov et al. (2017). That is, the set $\mathcal{E}$ of all states of the
form $\ket{\xi(\eta)}\\!=\\!(\ket{0}+e^{i\eta}\ket{1})/\sqrt{2}$. Hence, by
tuning the parameter $\phi$, we can prepare any state in $\mathcal{E}$. In
Fig. 4b, we show sequential steps of the implementation for each evolution
considered in Fig. 4a. In Fig. 4c the equivalent quantum circuit where a
Hadamard and phase gate are implemented, two genuine quantum gates which do
not have an equivalent in classical computation Nielsen and Chuang (2011).

##  V Systematic errors and gate fidelity

When the external fields used to qubit initialization and control present
fluctuations around a reference value, it directly affects the qubit control
in different ways. In this section we study the performance of the gate
implementation in presence of systematic errors. In our system, according to
Eqs. (7), (15) and (19), fluctuations in the magnetic field have an impact on
the qubit energy (and thus on its frequency) and on the effective electric
dipole moment. On the other hand, any error associated to the electric field
amplitude only affects the control of the qubit.

Figure 5: (a) Average infidelity of the Hadamard gate under systematic errors
in magnetic field over $N=10,000$ random input states, for different values of
$B_{0}$. We set $E_{0}=100$ V/m. (b) By choosing the electric field amplitude
in an adequate way, gate error mitigation is possible. In our example we use
$B_{0}=0.45$ T. (c) Average infidelity for errors in the electric field for
different values of $E_{0}$, where we choose $B_{0}=0.45$ T. (d) By changing
the magnetic field inside the limit imposed in Fig. 3a, no enhancement is
observed. Here we set $E_{0}=1,000$ V/m. In all graphs we used the parameters
considered in Fig. 3a.

We consider an analysis of the systematic error based on the deviation between
a desired quantum gate $G$ (target unitary operation) and the real operation
implemented through the perturbed Hamiltonian. We call the unperturbed and
perturbed Hamiltonians $H_{\mathrm{eff}}$ and
$H_{\mathrm{per}}=H_{\mathrm{per}}(\delta B,\delta E)$, respectively, where
$\delta B$ and $\delta E$ encode the fluctuations of the magnetic and electric
fields, respectively. We then let the system evolves under the action of
$H_{\mathrm{per}}$ and compare the outcome state $\ket{\psi_{\mathrm{eff}}}$
with the state $\ket*{\psi_{\mathrm{per}}}$ expected when we drive the system
with $H_{\text{eff}}$ during the time interval $\Delta t$ necessary to
implement the gate. Given the evolution operators

$\displaystyle U_{\mathrm{eff}/\mathrm{per}}(\Delta
t)=\exp\left(\frac{1}{i\hbar}H_{\mathrm{eff}/\mathrm{per}}\Delta t\right),$
(22)

where the unperturbed dynamics gives $U_{\mathrm{eff}}(\Delta t)=G$, we
compute the infidelity as given by the Bures metric for pure states Nielsen et
al. (2006)

$\displaystyle\mathcal{F}(\ket{\psi_{\mathrm{eff}}},\ket*{\psi_{\mathrm{per}}})=1-|\langle\psi_{\mathrm{eff}}|\psi_{\mathrm{per}}\rangle|^{2},$
(23)

with
$\ket*{\psi_{\mathrm{eff}/\mathrm{per}}}=U_{\mathrm{eff}/\mathrm{per}}(\Delta
t)\ket*{\psi_{\mathrm{input}}}$ and $\ket*{\psi_{\mathrm{input}}}$ an
arbitrary input state. Since we need to implement single qubit gates to
arbitrary input states, the error analysis is realized using the average over
many input states of a given operation. We implement each dynamics and take
the average of
$\mathcal{F}(\ket{\psi_{\mathrm{eff}}},\ket*{\psi_{\mathrm{per}}})$ over $N$
random input states $\ket*{\psi_{\mathrm{input}}}$.

Let us here consider the average infidelity to implement the Hadamard gate as
a function of the deviation amplitude of the magnetic field $\delta B/B_{0}$,
where $B_{0}$ is a reference value. From the maximum deviation we estimated
the fluctuations in the final state from the relation
$\delta\mathcal{F}=\max_{\\{\psi_{\mathrm{inp}}\\}}\mathcal{F}(\ket{\psi_{\mathrm{eff}}},\ket*{\psi_{\mathrm{per}}})$.
In Fig. 5a we show the behaviour of the infidelity with respect to $\delta
B/B_{0}$ for two different values of magnetic field $B_{0}$. In the cases
considered here, small deviations in $B_{0}$ (less than $1\%$) are enough to
promote a significant in the gate implementation. Note that errors of this
magnitude are within reach of several experiments Fei et al. (1997); Goerbig
(2011), especially in state-of-the-art setups Gross et al. (2016); Miller et
al. (2017); Thiele et al. (2018), where the control over the magnetic field is
of order of $\mu$T. However, as shown in Fig. 5b, the negative effects induced
by the fluctuations of the magnetic field can be mitigated by a suitable
choice of the electric field, which sets the time of the gate implementation.
In the case where the fluctuations are present in the electric field, with a
relative error amplitude of $\delta E/E_{0}$ around the reference value
$E_{0}$, the qubit control is much more robust to errors than for a
fluctuating $B_{0}$ see Figs. 5c and 5d. However, our results suggest the
magnetic field, within the values presented in 3b, does not allow to reduce
the effects of the fluctuating electric field.

##  VI Conclusions and prospects

In this paper we have introduced a new kind of physical two-level system which
can be used to process quantum information. In particular, we showed that the
curvature of a graphene surface induces a confining potential for an electron.
The electronic energy level structure is suitable to the encoding of quantum
information as a superposition of the two lowest energy states
$\ket{l=\\{0,1\\},m=0}=\ket{\\{0,1\\}}$, and arbitrary single-qubit states in
the form $\ket{\psi}=a\ket{0}+b\ket{1}$ can be prepared and manipulated
through external magnetic and electric fields. Our system obeys the
DiVincenzo’s criteria that states that any new proposal of qubit must allow
for a perfect encoding of information and a coherent control of information.
As a first analysis of errors in such a system, we quantified how undesired
fluctuations in magnetic and electric field may affect the gate fidelity. As a
proposal to partially suppress such effects, we showed an optimization scheme
that explores an adequate choice in amplitude of the electric field used to
implement single-qubit gates.

It is timely to say that any proposal of quantum computer begins with a
proposal of highly controllable single qubits. However, the scalability and
selective controllability of multi-qubit gates are other fundamental
requirements in DiVincenzo’s view DiVincenzo (2000). In particular, a system
able to implement arbitrary single-qubit gates and any entangling two-qubit
gate (such as the CNOT, or controlled flip phase gate) is eligible as a device
able to perform universal quantum computation Barenco et al. (1995). In our
system, building a two-qubit system is, in principle, possible due to the
recent progress in the production and control of multiple organic nanorings
Chen et al. (2005); Thorner et al. (2014); Kharissova et al. (2019), along
with progress on their realization on a larger scale Lehr et al. (2014).
Moreover, understanding the influence of the environment over the system, and
thus deccoherence effects Breuer and Petruccione (2007), is a fundamental
issue which remains to be addressed. Due to the coupling of electrons with the
phonon modes in graphene Castro Neto et al. (2009); Mañes (2007); Venezuela et
al. (2011); Park et al. (2014); Woods and Mahan (2000); Sohier (2015), in
addition to the hot-electron relaxation Hwang and Das Sarma (2013) and
curvature effects Pretko and Radzihovsky (2018); Sedrakyan et al. (2021), the
environment is bound to exert an influence over the computational performance
of our system. In this sense, our results should not be understood as the
whole story, but rather as the initial step of a long and challenging path
towards a new potential platform for quantum computation.

###### Acknowledgements.

R.B. and A.C.S. are supported by the São Paulo Research Foundation (FAPESP)
through Grants Nos. 2019/22685-1, 2019/13143-0, 2018/15554-5 and 2021/10224-0.
R.B. received support from the National Council for Scientific and
Technological Development (CNPq) Grant Nos. 313886/2020-2 and 409946/2018-4.

##  Appendix A Dimensional analysis of Eq. (15)

Let us consider

$\displaystyle\mu_{0}=er\left[\sqrt{\frac{\hbar}{2r|\beta|}}-\frac{1}{2}\left(\sqrt{\frac{\hbar}{2r|\beta|}}\right)^{3}\right].$
(24)

where the dimensional analysis of $\beta$ follows

$\displaystyle[\beta]$
$\displaystyle=\sqrt{[r]\left[\frac{[\hbar]^{2}}{[(R-r)]^{3}}+[e]^{2}[B]^{2}[(R-r)]\right]}$
$\displaystyle=\sqrt{m\left[\frac{(J\cdot
s)^{2}}{m^{3}}+C^{2}T^{2}m\right]}=\sqrt{\frac{J^{2}\cdot
s^{2}}{m^{2}}+C^{2}\frac{Kg^{2}}{C^{2}\cdot s^{2}}m^{2}}$
$\displaystyle=\sqrt{\frac{J^{2}\cdot
s^{2}}{m^{2}}+\frac{Kg^{2}}{s^{2}}m^{2}}=\sqrt{\frac{Kg^{2}\cdot
m^{4}}{s^{4}}\frac{s^{2}}{m^{2}}+\frac{Kg^{2}}{s^{2}}m^{2}}$
$\displaystyle=\sqrt{\frac{Kg^{2}\cdot m^{2}}{s^{2}}}=\frac{Kg\cdot m}{s}$
(25)

such that

$\displaystyle\frac{[\hbar]}{[r][|\beta|]}$ $\displaystyle=\frac{J\cdot
s}{m}\frac{s}{Kg\cdot m}=J\frac{s^{2}}{Kg\cdot m^{2}}$
$\displaystyle=\frac{Kg\cdot m^{2}}{s^{2}}\frac{s^{2}}{Kg\cdot m^{2}}=1,$ (26)

therefore we get

$\displaystyle[\mu_{0}]=[e][r]=C\cdot m,$ (27)

the dimension of electric dipole moment.

##  Appendix B Effective dynamics

Consider the total Hamiltonian $H(t)=H_{0}+H_{1}(t)$, in which
$H_{0}=\hbar\omega\sigma_{+}\sigma_{-}$, as

$\displaystyle
H(t)=\hbar\omega\sigma_{+}\sigma_{-}+\hbar\Omega(E_{0})\cos(\omega_{\mathrm{rf}}t+\phi)\left(\sigma_{-}+\sigma_{+}\right).$
(28)

Since in the above equation we have an small highly oscillating term, it is
convenient to identify how to approximate it to an effective Hamiltonian using
the Rotating Wave Approximation (RWA). To this end, we write the Schrödinger
equation in the interaction picture using the transformation
$\ket{\psi_{\text{I}}(t)}=e^{iH_{0}t/\hbar}\ket{\psi(t)}$ we get the dynamics
for $\ket{\psi_{\text{I}}(t)}$ as

$\displaystyle\ket*{\dot{\psi}_{\text{I}}(t)}=i\hbar
V_{\text{I}}(t)\ket{\psi_{\text{I}}(t)},$ (29)

with $V_{\text{I}}(t)=e^{iH_{0}t/\hbar}H_{1}(t)e^{-iH_{0}t/\hbar}$. As a first
remark, note that
$[H_{1}(t_{1}),H_{1}(t_{2})]=[V_{\text{I}}(t_{1}),V_{\text{I}}(t_{2})]=0$ for
any $(t_{1},t_{2})$. It allows to write the solution for the system dynamics
as

$\displaystyle\ket{\psi_{\text{I}}(t)}=U_{\text{I}}(t)\ket{\psi_{\text{I}}(0)},$
(30)

with

$\displaystyle
U_{\text{I}}(t)=\exp\left[\frac{1}{i\hbar}\int_{0}^{t}V_{\text{I}}(\xi)d\xi\right].$
(31)

We use that

$\displaystyle V_{\text{I}}(t)$
$\displaystyle=e^{iH_{0}t/\hbar}H_{1}(t)e^{-iH_{0}t/\hbar}$
$\displaystyle=\frac{\hbar\Omega(E_{0})}{2}\left(e^{i(\omega_{\mathrm{rf}}t+\phi)}+e^{-i(\omega_{\mathrm{rf}}t+\phi)}\right)e^{iH_{0}t/\hbar}(\sigma_{+}+\sigma_{-})e^{-iH_{0}t/\hbar}$
$\displaystyle=\frac{\hbar\Omega(E_{0})}{2}\left[\left(e^{i(\omega_{-}t+\phi)}+e^{-i(\omega_{+}t+\phi)}\right)\sigma_{-}+\mathrm{H.c.}\right],$
(32)

where $\omega_{\pm}=\omega_{\mathrm{rf}}\pm\omega$ and ‘H.c.’ denotes the
Hermitian conjugate. Therefore, by integrating the above expression

$\displaystyle\int_{0}^{t}V_{\text{I}}(\xi)d\xi$
$\displaystyle=\frac{i\hbar\Omega(E_{0})}{2}\left[\frac{e^{-i\phi}\left(e^{-i\omega_{+}t}-1\right)}{\omega_{+}}-\frac{e^{i\phi}\left(e^{i\omega_{-}t}-1\right)}{\omega_{-}}\right]\sigma_{-}$
$\displaystyle+\frac{i\hbar\Omega(E_{0})}{2}\left[\frac{e^{-i\phi}\left(e^{-i\omega_{-}t}-1\right)}{\omega_{-}}-\frac{e^{i\phi}\left(e^{i\omega_{+}t}-1\right)}{\omega_{+}}\right]\sigma_{+}.$
(33)

Thus, for a highly oscillating external field, we can approximate the above
expression to

$\displaystyle\int_{0}^{t}V_{\text{I}}(\xi)d\xi\approx\frac{\hbar\Omega(E_{0})}{2}\left[\frac{-ie^{i\phi}\left(e^{i\omega_{-}t}-1\right)}{\omega_{-}}\sigma_{-}+\mathrm{H.c.}\right].$
(34)

For this reason, in our analysis we can neglect highly oscillating terms. In
this case, we follow by writing the Hamiltonian $H(t)$ in the rotating frame
by using $R(t)=e^{i\omega_{\mathrm{rf}}t\sigma_{+}\sigma_{-}}$, such that we
find

$\displaystyle H_{R}$
$\displaystyle=R(t)H(t)R^{\dagger}(t)+i\hbar\dot{R}(t)R^{\dagger}(t)=\hbar\Delta\sigma_{+}\sigma_{-}+R(t)H_{1}(t)R^{\dagger}(t)$
$\displaystyle=\hbar\Delta\sigma_{+}\sigma_{-}+\frac{\hbar\Omega(E_{0})}{2}\left(e^{i(\omega_{\mathrm{rf}}t+\phi)}+e^{-i(\omega_{\mathrm{rf}}t+\phi)}\right)e^{-i\omega_{\mathrm{rf}}t}\sigma_{-}+\mathrm{H.c.},$
(35)

where we define $\Delta=\omega-\omega_{\mathrm{rf}}$. Now, by neglecting the
highly oscillating terms $e^{\pm 2i\omega_{\mathrm{rf}}t}$, it is possible to
proof that we get

$\displaystyle
H_{R}=\hbar\Delta\sigma_{+}\sigma_{-}+\frac{\hbar\Omega(E_{0})}{2}\left(e^{i\phi}\sigma_{-}+e^{-i\phi}\sigma_{+}\right).$
(36)

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