# BitCoin: Bidirectional Tagging and Supervised Contrastive Learning based
Joint Relational Triple Extraction Framework

Luyao He1, Zhongbao Zhang1, Sen Su1, Yuxin Chen1

###### Abstract

Relation triple extraction (RTE) is an essential task in information
extraction and knowledge graph construction. Despite recent advancements,
existing methods still exhibit certain limitations. They just employ
generalized pre-trained models and do not consider the specificity of RTE
tasks. Moreover, existing tagging-based approaches typically decompose the RTE
task into two subtasks, initially identifying subjects and subsequently
identifying objects and relations. They solely focus on extracting relational
triples from subject$\rightarrow$object, neglecting that once the extraction
of a subject fails, it fails in extracting all triples associated with that
subject. To address these issues, we propose BitCoin, an innovative
Bidirectional tagging and supervised Contrastive learning based joint
relational triple extraction framework. Specifically, we design a supervised
contrastive learning method that considers multiple positives per anchor
rather than restricting it to just one positive. Furthermore, a penalty term
is introduced to prevent excessive similarity between the subject and object.
Our framework implements taggers in two directions, enabling triples
extraction from subject$\rightarrow$object and object$\rightarrow$subject.
Experimental results show that BitCoin achieves state-of-the-art results on
the benchmark datasets and significantly improves the F1 score on Normal, SEO,
EPO, and multiple relation extraction tasks.

## Introduction

Relation triple extraction plays a crucial role in information extraction and
knowledge graph construction, and it aims to extract entity pairs and their
corresponding relations in the form of (subject, relation, object) from
unstructured text. Early RTE approaches mainly relied on pipelined methods,
which involve a two-step paradigm: extracting entities through named entity
recognition (NER) and then identifying relationships through relationship
classification (Zelenko, Aone, and Richardella 2003; Zhou et al. 2005; Chan
and Roth 2011). Despite its flexibility, this paradigm falls short in
overlooking the correlation and interaction between subtasks and potential
error propagation. To address these challenges, researchers have proposed end-
to-end approaches that simultaneously model entity recognition and relation
classification, which have achieved improved RTE performance (Yu and Lam 2010;
Wei et al. 2019; Eberts and Ulges 2019; Wang et al. 2020).

Overlapping triples present a challenge where a single entity or pair of
entities participating in multiple relational triples in the same sentence
(Zeng et al. 2018). Among these existing approaches, tagging-based joint
extraction methods have shown superior performance and capability in handling
overlapping triples (Zheng et al. 2017; Wei et al. 2019; Yu et al. 2019; Ren
et al. 2022; Zheng et al. 2021).

Despite the promising results achieved by existing joint extraction methods,
they still suffer from two major shortcomings. Firstly, existing joint
extraction methods just employ generalized pre-trained models and do not
consider the specificity of RTE tasks and the importance of designing
additional tasks, resulting in unreliable extraction results. Secondly,
existing tagging-based approaches typically decompose RTE tasks into two
subtasks, which identify subjects initially and subsequently identify objects
and relations. They solely focus on extracting relational triples from
subject$\rightarrow$object, neglecting that once the extraction of a subject
fails, it fails in extracting all triples associated with that subject. These
two issues significantly hamper the performance of RTE.

To overcome the limitations of existing approaches and achieve improved
results, we propose BitCoin, an end-to-end framework based on Bidirectional
tagging and supervised Contrastive learning. The core idea is first to employ
contrastive learning method that considers multiple positives per anchor and
designs a penalty term to prevent excessive similarity between subject and
object. Then, we employ taggers in two directions, enabling the extraction of
triples from subject$\rightarrow$object and object$\rightarrow$subject. There
are two main challenges here. Firstly, we aim to design a task to get better
features by making related entities closer and unrelated entities farther
apart, and one way to achieve this is through contrastive learning. However,
existing contrastive learning methods are not applicable in this scenario. The
conventional InfoNCE loss is unsuitable for cases where multiple positives may
exist, as one subject can have relationships with multiple objects. Also, the
subject and object features may become excessively similar, making it
difficult to distinguish them accurately, which results in incorrect triple
extractions. Secondly, bidirectional extraction is not just about designing
extractions in two directions but also about the interaction of information
during the extraction in these two directions.

To address the first challenge, we propose a novel supervised contrastive
learning method. Firstly, we consider multiple positives per anchor rather
than limiting ourselves to just one positive. These positives are selected
from samples of the same class, in contrast to the data augmentation approach
commonly used in self-supervised learning. We designate a subject as an
anchor, consider all relevant objects as positives, and then designate all
non-relevant entities as negatives. To augment the number of positives and
negatives, we employ dropout as a minimal form of data augmentation. Secondly,
we design a penalty term to prevent excessive similarity between subject and
object. When the similarity between subject and object reaches a threshold,
this penalty term prevents its similarity from expanding further and
stabilizes the similarity around the threshold.

To address the second challenge, we design taggers in two directions, enabling
the extraction of triples from subject$\rightarrow$object and
object$\rightarrow$subject. During extraction, information from both
directions can interact with each other. For example, when extracting objects
in the s2o direction, we can draw on the information of objects in the o2s
direction for more accurate extraction. The extracted triples can be cross-
validated by the results of two-way taggers, which means extractions from two
directions complement each other. Also, we consider the fundamental properties
of a triple, which are the interdependency and indivisibility of its entity
pairs and relations. We may get unreliable results if we fail to utilize the
relational information when extracting relevant objects fully but rely solely
on the subject information to extract objects. To adequately consider the
information conveyed by the relation, we design a relationship prediction
module to get relational features and combine them with sentence and entity
features to get reliable triples. This comprehensive input integration enables
us to fully exploit the relational information within the sentence,
facilitating easier and more accurate triple extraction.

Experimental results show that BitCoin achieves state-of-the-art results on
the benchmark datasets and significantly improves the F1 score on Normal, SEO,
EPO, and multiple relation extraction tasks.

The main contributions of this work are as follows:

  * •

This is the first attempt to employ supervised contrastive learning with a
penalty term for RTE tasks.

  * •

We propose a novel end-to-end bidirectional tagging and supervised contrastive
learning based framework BitCoin. It significantly alleviates the problems of
one-direction subject extraction failure and neglecting relationship
information.

  * •

We evaluate our model on four public datasets, and the results indicate that
our method outperforms all the state-of-the-art baselines, especially in
complex scenarios.

## Related Works

Researchers have proposed two kinds of methods for RTE: pipeline and joint
learning. Traditional pipelined approaches initially perform named entity
recognition to extract entities, followed by identifying relationships through
relationship classification (Zelenko, Aone, and Richardella 2003; Zhou et al.
2005; Chan and Roth 2011). Although flexible, these methods neglect the
interdependence between the two subtasks and are prone to error propagation,
consequently undermining overall performance.

To tackle this problem, several joint models that extract entities and
relations jointly have been proposed. Among them, feature-based joint models
require a process of constructing features manually. Neural network-based
models have shown considerable improvement in both performance and efficacy in
RTE tasks by replacing manual feature construction, which is a complicated
process of feature-based joint models. Zheng et al. (2017) proposed a novel
tagging schema that unified the role of the entity and the relation between
entities and converted the task of RTE to an end-to-end sequence tagging
problem. However, they ignored the problem of overlapping triples.

To address this problem, various neural network-based joint extraction methods
are proposed. Zeng et al. (2018) proposed a sequence-to-sequence model with
copy mechanism to address the overlapping triples problem, although it
struggles to generate multi-word entities. Fu, Li, and Ma (2019) also delved
into the issue and devised a novel method based on Graph Convolutional
Networks (GCNs). As an improvement, Nayak and Ng (2020) adopted an encoder-
decoder model, where the decoder incrementally extracts words, just like
machine translation models. Ye et al. (2021) propose a contrastive triple
extraction method with a generative transformer. Wang et al. (2020) treated
entity recognition and relation classification as a table-filling problem,
where each entry represents the interaction between two words. Shang, Huang,
and Mao (2022) proposed to frame the joint extraction task as a fine-grained
triple classification problem, which can extract triples from sentences in a
one-module one-step manner.

Among these approaches, tagging-based joint extraction methods have shown
superior performance and capability in handling overlapping triples. Wei et
al. (2019) proposed a novel cascade binary tagging framework that models
relations as functions that map subjects to objects rather than treating
relations as discrete labels on entity pairs, achieving competitive results.
It first identifies all possible subjects and then identifies the relevant
objects under all relations for each subject. Zheng et al. (2021) proposed an
end-to-end framework that decomposed joint extraction into three subtasks:
relationship determination, entity extraction, and subject-object alignment.
Experiments show that tagging-based joint extraction methods achieve
competitive results and have a solid ability to extract triples from sentences
that contain overlapping triples or multiple triples.

Figure 1: The overview structure of BitCoin. In this example, given a
sentence, BitCoin detects two candidate subjects, three candidate objects and
two potential relations. Then for each candidate subject, the relation-
specific object tagger will extract relevant objects under all relations. The
relation-specific subject tagger extracts relevant subjects under all
relations for each candidate object. Finally, we take the union set of two-
direction results.

## Method

The architecture of BitCoin is shown in Fig. 1. It consists of the following
six components: encoder based on BERT and supervised contrastive learning,
subject tagger, object tagger, relationship prediction module, relation-
specific object tagger and relation-specific subject tagger. During training,
we adopt a multi-task learning approach that allows each module to be trained
with ground truth, resulting in a more reliable model. During the inference
stage, BitCoin operates in three stages: 1) The encoder generates
representations for each token. 2) The subjects, objects and potential
relationships are identified through the subject tagger, object tagger, and
relationship prediction module. 3) The relation-specific object and subject
taggers are used to tag the relevant objects and subjects under all
relationships based on extracted subjects, objects and potential relations.

### Contrastive Learning based Encoder

The encoder module is designed to convert sentences into word embeddings.
Here, we first encode the input sentence using a pre-trained BERT model
(Devlin et al. 2018) to generate the representations of each token in the
sentence and then use our contrastive learning method to learn better
features. We use the BERT-BASE-CASED model to ensure a fair comparison with
other models, but it is theoretically possible to use other pre-trained
models, such as Roberta and BART, etc.

In the knowledge graph, the ideal situation is that entities that are related
are close to each other and entities that are not related are far apart. One
way to achieve this is through contrastive learning. However, existing self-
supervised contrastive learning methods are not applicable in this scenario,
as one entity may have relations with multiple entities. Also, if we adopt the
previous idea of comparative learning that pulls in the positives and pushes
out the negatives, the subject and object features may become excessively
similar, making it difficult to distinguish them accurately, resulting in
incorrect triple extractions.

To address these problems and obtain better features from the encoder for the
RTE task, we have designed a supervised contrastive learning method with a
penalty term. After generating the token representations through BERT, we use
this contrastive learning method to train the BERT model further to obtain an
encoder better suited for the RTE task.

Input | Tom was born in New York at 2000.
---|---
London is the capital of England.
Triples | (Tom, birth_place, New York)
(Tom, birth_date, 2000)
(London, capital_of, England)
(London, belong_to, England)
anchor | Tom | London
Positives | New York, 2000 | England
Negatives | London, England | New York, 2000
Table 1: Example of positives and negatives in supervised contrastive
learning.

The idea of our supervised contrastive learning method is easy to understand.
Firstly, we design a new loss function, as the self-supervised contrastive
loss is incapable of handling the case where more than one sample is known to
belong to the same class. Here, we set a subject as an anchor, all relevant
objects as positives, and all non-relevant entities as negatives. Tab. 1 shows
an example of positives and negatives. To increase the negatives and
positives, we use dropout noise as the minimum form of data augmentation (Gao,
Yao, and Chen 2021) by inputting the sentence to the encoder twice and getting
the embeddings of subjects and objects with different dropout masks. Our loss
function is structurally similar to the self-supervised contrastive learning
loss, and the basic idea is both to pull in the positives and push out the
negatives. Differently, we consider many positives per anchor instead of only
one positive, which can handle the situation where one subject may have
multiple relevant objects. The supervised contrastive loss is computed based
on the embeddings of subjects and objects, and the loss function is shown in
Eq. 1.

$\mathcal{L}_{1}^{i}=\frac{-1}{|P(i)|}\log\frac{\sum\nolimits_{p=1}^{P(i)}{e^{sim\left(\bm{h_{i}},\bm{h_{i}^{p}}\right)/\tau}}}{\sum\nolimits_{a=1}^{A(i)}e^{sim\left(\bm{h_{i}},\bm{h_{i}^{a}}\right)/\tau}},$
(1)

where $P(i)$ is the set of all positives, and $|P(i)|$ is its cardinality.
$A(i)$ is the set of all negtives and positives, $h_{i}$ is the representation
of the anchor, $\tau$ is the temperature hyperparameter and
$sim(\bm{h_{1}},\bm{h_{2}})$ is the cosine similarity
$\frac{\bm{h_{1}^{T}}\bm{h_{2}}}{{\|\bm{h_{1}}\|}\cdot{\|\bm{h_{2}}\|}}$.

Secondly, supposing we only use Eq. 1 as our loss function, we will keep
drawing the subject closer to the relevant objects during training, making
distinguishing the subjects and objects hard, resulting in incorrect triples.
To solve this problem, we design a penalty term to prevent excessive
similarity between the subject and object. When the similarity between the
subject and the object reaches a threshold, this penalty term prevents its
similarity from expanding further and stabilizes the similarity around the
threshold. The penalty term is obtained by multiplying the following two
terms, where the first term determines the strength of the penalty and the
second term determines how the penalty is calculated. For the first term, we
compute the difference between the similarity of the subject-object and
threshold: the more the difference, the stronger the penalty. For the second
term, it is structurally similar to the Eq. 1, but the idea is changed to pull
in the non-relevant objects and push out the relevant objects. The penalty
term is shown in Eq. 2, and the total loss function for supervised contrastive
learning is shown in Eq. 3.

$\displaystyle\mathcal{L}_{2}^{i}=\bigg{[}-$
$\displaystyle\Big{(}\frac{1}{|P(i)|}\sum\limits_{p=1}^{P(i)}sim\left(\bm{h_{i}},\bm{h_{i}^{p}}\right)-\beta\Big{)}$
(2)
$\displaystyle\times\frac{1}{|N(i)|}\log\frac{\sum\nolimits_{n=1}^{N(i)}{e^{sim\left(\bm{h_{i}},\bm{h_{i}^{n}}\right)/\tau}}}{\sum\nolimits_{a=1}^{A(i)}e^{sim\left(\bm{h_{i}},\bm{h_{i}^{a}}\right)/\tau}}\bigg{]}_{+},$

$\mathcal{L}_{c}=\sum\limits_{i=1}^{I}\mathcal{L}_{c}^{i}=\sum\limits_{i=1}^{I}\left(\omega_{1}\mathcal{L}_{1}^{i}+\omega_{2}\mathcal{L}_{2}^{i}\right),$
(3)

where $N(i)$ is the set of all positives, $|N(i)|$ is its cardinality and
$\beta$ is the threshold.

### Bidirectional Tagging based Decoder

BitCoin is a bi-directional framework that extracts the relational triples in
two directions: (1) extracting the subject, followed by both the relation and
the object (s2o); and (2) extracting the object, followed by both the relation
and the subject (o2s). The model structure is identical in both directions,
and we will only present direction from subject$\rightarrow$object here.

#### Subject Tagger

The Subject Tagger extracts all subjects by decoding h obtained from the
encoder. Since the subjects, relations and objects in a triple have unique
characteristics, we do not use the same features $h$ like CasRel, TPLinker and
PRGC but use different features for each. Here, after obtaining the
representations of each token from the input sentences, we can get the
subject-specific representations through a fully-connected layer, denoted as
$\bm{h}_{{sub}}^{{i}}$, which is computed with Eq. 4.

$\bm{h}_{{sub}}^{{i}}=\bm{W}_{{sub}}\bm{h}^{{i}}+\bm{b}_{{sub}}$ (4)

where ${\bm{W}_{sub}\in{\mathbb{R}^{{d_{h}}\times{d_{h}}}}}$ is a trainable
matrix, ${\bm{b}_{sub}\in\mathbb{R}^{{d}_{h}}}$ is a bias vector and
${{\bm{d}_{h}}}$ is the dimension of ${{\bm{h}_{i}}}$.

The Subject Tagger comprises two binary tagging modules that use a simple
$(0/1)$ token to indicate whether the token is the start or end position of
the subject. For each token, two separate probability formulas are used to
calculate the probability that it is the start or end position of the subject.
If the probability is higher than a threshold, the corresponding token
position is tagged as $1$, indicating that the token is the start position of
the subject. If it is lower, it is tagged as $0$. As there may be more than
one subject in a sentence, we consider the closest starting and ending
positions as one subject. The probabilities of the start and end positions of
the subject are calculated by Eqs. 5 and 6.

$\displaystyle{p}_{{sub\\_start}}^{{i}}$
$\displaystyle={\sigma}\left(\bm{W}_{{sub\\_start}}\bm{h}_{{sub}}^{{i}}+\bm{b}_{{sub\\_start}}\right)$
(5) $\displaystyle{p}_{{sub\\_end}}^{{i}}$
$\displaystyle={\sigma}\left(\bm{W}_{{sub\\_end}}\bm{h}_{{sub}}^{{i}}+\bm{b}_{{sub\\_end}}\right)$
(6)

where ${{{p}_{sub\\_start}^{i}}}$ and ${{{p}_{sub\\_end}^{i}}}$ represent the
probabilities of identifying the $i$-th token in the input sequence as the
start and end position of a subject respectively. ${{\bm{h}_{sub}^{i}}}$ is
the encoded representation of the $i$-th token with subject feature in the
input sequence and ${\sigma}$ is the sigmoid activation function.

#### Relation Prediction Module

To obtain the relation-specific representations, we pass the representation of
each token obtained from the encoder through a fully-connected layer, as shown
in Eq. 7.

$\bm{h}_{{rel}}^{{i}}=\bm{W}_{{rel}}\bm{h}^{{i}}+\bm{b}_{{rel}}$ (7)

where ${\bm{W}_{rel}\in{\mathbb{R}^{{d_{h}}\times{d_{h}}}}}$ is a trainable
matrix, ${\bm{b}_{rel}\in\mathbb{R}^{{d}_{h}}}$ is a bias vector and
${{\bm{d}_{h}}}$ is the dimension of ${{\bm{h}_{i}}}$.

After obtaining the relation-specific representations, the Relation Prediction
Module performs an Avgpool operation on these representations to obtain the
representation of the input sentence. This representation is then used to
extract all possible relations. Precisely, we can calculate the probability of
each relation being present using a probability formula based on the average
pooled representation of the input sentence. The corresponding relation is
tagged as $1$ if the probability is higher than a threshold. The potential
relations in the input sentence are obtained from Eq. 9.

$\displaystyle\bm{h}_{{rel}}^{\operatorname{avg}}$
$\displaystyle={\operatorname{Avgpool}}\left(\bm{h}_{{rel}}\right)\in{\mathbb{R}^{{d_{h}}\times{1}}}$
(8) $\displaystyle{p}_{rel}$
$\displaystyle={\sigma}\left(\bm{W}_{rel}\bm{h}_{\
{rel}}^{avg}+\bm{b}_{rel}\right)$ (9)

where ${p}_{rel}$ represents the probability of identifying the relations in
the input sequence and $\bm{d}_{r}$ is the number of relations.

#### Relation-Specific Object Tagger

The Relation-Specific Object Tagger extracts objects under all relations.
Unlike the Subject Tagger, which uses encoder output directly, Relation-
Specific Object Tagger also considers the subject tagged by the subject
tagger, the relations predicted by the relation prediction module and object-
specific representations from the object tagger in the other direction. This
allows us to extract relations and objects simultaneously, which can solve the
EPO and SEO problems. The module consists of $2\times{num}_{rels}$ binary
tagging modules. Similar to the subject tagger, we calculate the probability
that it is the start or end position of the corresponding object under each
relation using two probability formulas, respectively. If the probability
exceeds a threshold, the position corresponding to that token is tagged as
$1$; otherwise, it is tagged as $0$. The probabilities of the start and end
positions of the object under each relation are calculated by Eqs. 11 and 12.

$\displaystyle\bm{h}_{rel\\_obj}^{i}$
$\displaystyle=\bm{W}_{obj}\left(\left(\bm{h}_{obj}^{i}+\bm{v}_{sub}^{n}\right)\oplus\bm{p}_{rel}\right)+\bm{b}_{obj}$
(10) $\displaystyle{p}_{obj\\_start}^{i}$
$\displaystyle={\sigma}\left(\bm{W}_{obj\\_start}^{r}\left(\bm{h}_{rel\\_obj}^{i}\right)+\bm{b}_{obj\\_start}^{r}\right)$
(11) $\displaystyle{p}_{obj\\_end}^{i}$
$\displaystyle={\sigma}\left(\bm{W}_{obj\\_end}^{r}\left(\bm{h}_{rel\\_obj}^{i}\right)+\bm{b}_{obj\\_end}^{r}\right)$
(12)

where ${{{p}_{obj\\_start}^{i}}}$ and ${{{p}_{obj\\_end}^{i}}}$ represent the
probabilities of identifying the $i$-th token as the start and end position of
an object respectively. ${{\bm{h}_{obj}^{i}}}$ is the representation of the
$i$-th token with object feature, ${{\bm{v}_{sub}^{n}}}$ is the encoded
representation of the $n$-th subject extracted in the subject tagger and
${\oplus}$ means concatenating two tensors.

Category | Dataset | Details of Test Set
---|---|---
Train | Valid | Test | Relation | Normal | SEO | EPO | N=1 | N=2 | N=3 | N=4 | N$\geq$5 | Triples
NYT* | 56,195 | 4999 | 5000 | 24 | 3,266 | 1,297 | 978 | 3,244 | 1,045 | 312 | 291 | 108 | 8,110
WebNLG* | 5,019 | 500 | 703 | 171 | 245 | 457 | 26 | 266 | 171 | 131 | 90 | 45 | 1,591
NYT | 5,6195 | 5,000 | 5,000 | 24 | 3,222 | 1,273 | 969 | 3,240 | 1,047 | 314 | 290 | 109 | 8,120
WebNLG | 5,019 | 500 | 703 | 216 | 239 | 448 | 6 | 256 | 175 | 138 | 93 | 41 | 1607

Table 2: Statistics of datasets. Note that a sentence can belong to both EPO
class and SEO class.

### Training Strategy

All sub-modules of this framework work in a multi-task learning manner. In
this way, each sub-modules have its loss function. The model is trained
jointly by optimizing the combined objective function during training. The
loss functions for the three modules mentioned above are denoted as
${{\mathcal{L}_{sub\\_head}}}$, ${{\mathcal{L}_{sub\\_tail}}}$,
${{\mathcal{L}_{rel}}}$, ${{\mathcal{L}_{rel\\_obj\\_head}}}$ and
${{\mathcal{L}_{rel\\_obj\\_tail}}}$, and they all utilize the binary cross-
entropy loss function, as shown in Eqs. 13, 14, 15 and 16.

$\displaystyle{\operatorname{BCE}}\left(\hat{y},{y}\right)$
$\displaystyle=-[{y}{log}\left({\hat{y}})+(1-{y}\right)\log\left(1-{\hat{y}}\right)]$
(13) $\displaystyle\mathcal{L}_{{s\\_\left(h,t\right)}}$
$\displaystyle=\frac{1}{{l}}\sum_{{i}=1}^{l}{\operatorname{BCE}}\left({p}_{{s\\_\left(h,t\right)}}^{i},{{{y}}}_{{s\\_\left(h,t\right)}}^{{i}}\right)$
(14) $\displaystyle\mathcal{L}_{rel\\_{o}\\_(h,t)}$
$\displaystyle=\frac{1}{{l}}\sum_{{i}=1}^{l}{\operatorname{BCE}}\left(p_{o\\_{\left(h,t\right)}}^{i},{{y}}_{{o}\\_\left(h,t\right)}^{{i}}\right)$
(15) $\displaystyle\mathcal{L}_{rel}$
$\displaystyle=\frac{1}{r}\sum_{{i}=1}^{r}{\operatorname{BCE}}\left(p_{rel}^{i},{{y}}_{rel}^{{i}}\right)$
(16)

where ${\operatorname{BCE}}\left(\hat{y},{y}\right)$ is a binary cross entropy
loss, $\hat{y}\in\left(0,1\right)$ is the calculated probability and y is the
true label. l is the number of tokens in the input sentence, and r is the
number of relations.

Similar to the subject-to-object direction, there are four tagger losses in
the object-to-subject direction, which are denoted as
${{\mathcal{L}_{obj\\_head}}}$, ${{\mathcal{L}_{obj\\_tail}}}$,
${{\mathcal{L}_{rel\\_sub\\_head}}}$ and ${{\mathcal{L}_{rel\\_sub\\_tail}}}$,
and are calculated by a formula similar to Eqs. 14, 15 and 16. The total loss
is the sum of these parts, and it can be expressed by the following equation.

$\displaystyle\mathcal{L}_{total}=\mathcal{L}_{\left(s,o\right)\\_\left(h,t\right)}+\mathcal{L}_{rel\\_\left(s,o\right)\\_\left(h,t\right)}+\mathcal{L}_{rel}+\mathcal{L}_{c}$
(17)

## Experiment

### Experiment Settings

#### Datasets

Following previous works (Wei et al. 2019; Wang et al. 2020; Zheng et al.
2021; Shang, Huang, and Mao 2022), we evaluate our framework on datasets NYT
(Riedel, Yao, and McCallum 2010) and WebNLG (Gardent et al. 2017). NYT and
WebNLG have two different versions: one version only annotates the last word
of entities, and the other annotates the whole span of entities. We denote the
datasets based on the first standard as NYT* and WebNLG* and those based on
the second standard as NYT and WebNLG.

According to the different patterns of relational triple overlap, we classify
the sentences into Normal, Entity-Pair-Overlap (EPO) and Single-Entity-overlap
(SEO) classes to further study the capability of the proposed BitCoin in
handling complex scenarios. Detailed statistics of the two datasets are
described in Tab. 2.

Model | Partial Match | Exact Match
---|---|---
NYT* | WebNLG* | NYT | WebNLG
Prec. | Rec. | F1 | Prec. | Rec. | F1 | Prec. | Rec. | F1 | Prec. | Rec. | F1
ETL-Span (Yu et al. 2019) | 84.9 | 72.3 | 78.1 | 84.0 | 91.5 | 87.6 | 85.5 | 71.7 | 78.0 | 84.3 | 82.0 | 83.1
CasRel (Wei et al. 2019) | 89.7 | 89.5 | 89.6 | 93.4 | 90.1 | 91.8 | - | - | - | - | - | -
RIN (Sun et al. 2020) | 87.2 | 87.3 | 87.3 | 87.6 | 87.0 | 87.3 | 83.9 | 85.5 | 84.7 | 77.3 | 76.8 | 77.0
TPLinker (Wang et al. 2020) | 91.3 | 92.5 | 91.9 | 91.8 | 92.0 | 91.9 | 91.4 | 92.6 | 92.0 | 88.9 | 84.5 | 86.7
SPAN (Sui et al. 2023) | 93.3 | 91.7 | 92.5 | 93.1 | 93.6 | 93.4 | 92.5 | 92.2 | 92.3 | - | - | -
CGT (Ye et al. 2021) | 94.7 | 84.2 | 89.1 | 92.9 | 75.6 | 83.4 | - | - | - | - | - | -
CasDE (Ma, Ren, and Zhang 2021) | 90.2 | 90.9 | 90.5 | 90.3 | 91.5 | 90.9 | 89.9 | 91.4 | 90.6 | 88.0 | 88.9 | 88.4
RIFRE (Zhao et al. 2021) | 93.6 | 90.5 | 92.0 | 93.3 | 92.0 | 92.6 | - | - | - | - | - | -
StereoRel (Tian et al. 2021) | 92.0 | 92.3 | 92.2 | 91.6 | 92.6 | 92.1 | 92.0 | 92.3 | 92.2 | - | - | -
PRGC (Zheng et al. 2021) | 93.3 | 91.9 | 92.6 | 94.0 | 92.1 | 93.0 | 93.5 | 91.9 | 92.7 | 89.9 | 87.2 | 88.5
R-BPtrNet (Chen et al. 2021) | 92.7 | 92.5 | 92.6 | 93.7 | 92.8 | 93.3 | - | - | - | - | - | -
BiRTE (Ren et al. 2022) | 92.2 | 93.8 | 93.0 | 93.2 | 94.0 | 93.6 | 91.9 | 93.7 | 92.8 | 89.0 | 89.5 | 89.3
OneRel (Shang, Huang, and Mao 2022) | 92.8 | 92.9 | 92.8 | 94.1 | 94.4 | 94.3 | 93.2 | 92.6 | 92.9 | 91.8 | 90.3 | 91.0
BitCoin (ours) | 92.9 | 93.3 | 93.1 | 94.4 | 94.5 | 94.4 | 93.1 | 92.6 | 92.8 | 91.9 | 90.5 | 91.2

Table 3: Precision(%), Recall (%) and F1-score (%) of our proposed BitCoin and
baselines.

Model | NYT* | WebNLG*
---|---|---
Normal | SEO | EPO | N=1 | N=2 | N=3 | N=4 | N$\geq$5 | Normal | SEO | EPO | N=1 | N=2 | N=3 | N=4 | N$\geq$5
CasRel | 87.3 | 91.4 | 92.0 | 88.2 | 90.3 | 91.9 | 94.2 | 83.7 | 89.4 | 92.2 | 94.7 | 89.3 | 90.8 | 94.2 | 92.4 | 90.9
TPLinker | 90.1 | 93.4 | 94.0 | 90.0 | 92.8 | 93.1 | 96.1 | 90.0 | 87.9 | 92.5 | 95.3 | 88.0 | 90.1 | 94.6 | 93.3 | 91.6
SPN | 90.8 | 94.0 | 94.1 | 90.9 | 93.4 | 94.2 | 95.5 | 90.6 | - | - | - | 89.5 | 91.3 | 96.4 | 94.7 | 93.8
PRGC | 91.0 | 94.0 | 94.5 | 91.1 | 93.0 | 93.5 | 95.5 | 93.0 | 90.4 | 93.6 | 95.9 | 89.9 | 91.6 | 95.0 | 94.8 | 92.8
OneRel | 90.6 | 94.8 | 95.1 | 90.5 | 93.4 | 93.9 | 96.5 | 94.2 | 91.9 | 94.7 | 95.4 | 91.4 | 93.0 | 95.9 | 95.7 | 94.5
BitCoin | 91.1 | 95.0 | 94.9 | 91.2 | 93.6 | 93.8 | 95.9 | 94.4 | 92.0 | 94.9 | 95.7 | 91.1 | 93.4 | 95.7 | 95.7 | 94.7

Table 4: F1-score (%) on sentences with different overlapping patterns and
different triple numbers

#### Evaluation Metrics

The standard micro precision, recall, and F1 score are used to evaluate the
results. There are two match standards for the RTE task: one is Partial Match
that an extracted triple is regarded as correct if the predicted relation and
the head of both the subject and object are correct, and the other is Exact
Match that a triple is regarded as correct only when its entities and
relationships are completely matched with a correct triple. We follow previous
work (Wei et al. 2019; Wang et al. 2020; Zheng et al. 2021; Shang, Huang, and
Mao 2022) and use Partial Match on NYT* and WebNLG*, use Exact Match on NYT
and WebNLG.

#### Implementation Details

In our experiments, all training process is completed on a workstation with an
Intel(R) 4210R 2.40G CPU, 128G memory, a single RTX 3090 GPU, and Ubuntu
20.04. We use a small batch mechanism to train the model, with 4, 6, and 8
batch sizes. The learning rate is a linear warmup, the maximum learning rate
is set to 1e-5, and the warmup step is set to the first quarter of the epoch.
The threshold for judging whether there is a subject, an object, or a relation
is set to 0.5-0.6. The threshold for supervised contrastive learning in Eq. 2
is set to 0.85. The pre-trained BERT model is $[BERT-base,cases]$, which
contains 12 Transformer blocks and 110M parameters, and the hidden size d is
768. All parameters are optimized by Adam algorithm (Kingma and Ba 2014). The
dropout probability is 0.1. For a fair comparison, the maximum length of our
model input sentences is set to 100 words, as suggested in previous works
(Zeng et al. 2018; Fu, Li, and Ma 2019).

We also use an early stopping mechanism to prevent the overfitting of the
model. Specifically, we stop the training process when the performance on the
validation set does not obtain any improvement for at least ten consecutive
cycles. All involved hyperparameters are determined based on the results of
the development set. Other parameters are initialized randomly.

#### Baselines

We compare BitCoin with 13 strong baseline models and the state-of-the-art
models ETL-Span (Yu et al. 2019), CasRel (Wei et al. 2019), RIN (Sun et al.
2020), TPLinker (Wang et al. 2020), CGT (Ye et al. 2021), CasDE (Ma, Ren, and
Zhang 2021), RIFRE (Zhao et al. 2021), StereoRel (Tian et al. 2021),PRGC
(Zheng et al. 2021), R-BPtrNet (Chen et al. 2021), BiRTE (Ren et al. 2022),
OneRel (Shang, Huang, and Mao 2022) and SPAN (Sui et al. 2023). Most results
of these baselines are copied from their original papers directly.

### Experimental Results

#### Overall Results

Tab. 3 shows the comparison results of our model against 13 baselines on NYT
and WebNLG in terms of Partial Match and Exact Match. Our BitCoin method
outperforms them in respect of almost all evaluation metrics. Especially on
WebNLG*, BitCoin obtains the best performance in terms of all three evaluation
metrics. These results verify our motivation.

We attribute the outstanding performance of BitCoin to its two advantages:
Firstly, BitCoin uses the novel supervised contrastive learning method with a
penalty term to make the pre-trained model more suitable for the RTE task. We
can get better features from the encoder through this method. Secondly, we
design taggers in two directions, and we can extract triples from
subject$\rightarrow$object and from object$\rightarrow$subject. The two
directions complement each other, and the extraction results can be cross-
validated. Also, we notice the fundamental property of a triple, which is the
interdependency and indivisibility of its entity pairs and relations. We
designe the relation prediction model to fully exploit the relational
information within the sentence, facilitating easier and more accurate triple
extraction.

Compared with the tagging-based method CasRel which inspires us to treat
relations as functions mapping subjects to objects instead of treating
relations as discrete labels on entity pairs, BitCoin achieves 3.5 and 2.6
absolute gain in F1-score on NYT* and WebNLG*, respectively. Such results
confirm that contrastive learning and bidirectional framework are effective
for RTE tasks.

#### Detailed Results on Complex Scenarios

To verify the ability of our method on complex scenarios, we evaluate
BitCoin’s ability for extracting triples from sentences that contain
overlapping triples and multiple triples. This ability is widely discussed in
existing models and is an important metric for evaluating the robustness of a
model. For a fair comparison, we follow the settings of some previous models
that classify sentences according to the degree of overlapping and the number
of triples contained in a sentence and conduct two extended experiments on
different subsets of NYT* and WebNLG*. We select five powerful models as
baselines, and the detailed results are shown in Tab. 4.

It can be observed that BitCoin has excellent superiority in handling complex
sentences and achieves the best F1-score on 10 of the 16 subsets. Moreover,
BitCoin achieves more performance improvement when handling the sentences of
the SEO class. This is mainly because a single entity in an SEO sentence may
be associated with multiple triples. Thus the existing models are more likely
to suffer from the problem that once the extraction of an entity in some SEO
triples is failed, all the associated triples of this entity would not be
extracted either. However, the bidirectional framework in BitCoin can
effectively overcome such deficiency, and the mentioned issue almost does not
affect it when handling the SEO sentences. This is also why BitCoin performs
well on sentences containing multiple triples. In general, these two further
experiments adequately show the advantages of our model in complex scenarios.

#### Ablation Study

Here we make five kinds of detailed evaluations on BitCoin, and the results
are shown in Tab. 5.

Model | Prec. | Rec. | F1
---|---|---|---
NYT* | BitCoin | 92.8 | 93.4 | 93.1
\- Contrastive Learning | 92.7 | 92.5 | 92.6
\- Direction from o2s | 92.3 | 92.2 | 92.4
\- Direction from s2o | 91.6 | 92.7 | 92.2
\- Relation Prediction | 92.5 | 92.3 | 92.4
WebNLG* | BitCoin | 94.4 | 94.5 | 94.4
\- Contrastive Learning | 93.9 | 93.2 | 93.6
\- Direction from o2s | 93.7 | 92.3 | 93.0
\- Direction from s2o | 94.0 | 91.7 | 92.8
\- Relation Prediction | 94.1 | 94.1 | 94.1
NYT | BitCoin | 93.1 | 92.6 | 92.8
\- Contrastive Learning | 92.9 | 92.1 | 92.5
\- Direction from o2s | 90.9 | 91.9 | 91.4
\- Direction from s2o | 91.6 | 91.3 | 91.5
\- Relation Prediction | 92.6 | 92.4 | 92.5
WebNLG | BitCoin | 91.9 | 90.5 | 91.2
\- Contrastive Learning | 90.1 | 88.5 | 89.3
\- Direction from o2s | 90.6 | 87.8 | 89.1
\- Direction from s2o | 90.8 | 86.4 | 88.6
\- Relation Prediction | 91.7 | 87.5 | 89.5

Table 5: Precision(%), Recall (%) and F1-score (%) for ablation study of
BitCoin.

First, we evaluate the effectiveness of supervised contrastive learning. To
this end, we implement a variant of BitCoin without the supervised contrastive
learning method. Tab. 5 show that this performance drops on all datasets,
indicating that our supervised contrastive learning is suitable for RTE tasks.
We can get better word embeddings through this method. Additionally, Tab. 5
show that the performance of the one-directional tagging framework is much
better than the CasRel model, which again indicates the effectiveness of our
supervised contrastive learning method.

Second, we evaluate the effectiveness of the bidirectional tagging framework.
To this end, we implement the following two variants of BitCoin: (1)
$\mathrm{BitCoin_{s2o}}$, a variant that only uses the
subject$\rightarrow$object direction to extract triples; (2)
$\mathrm{BitCoin_{o2s}}$, a variant that only uses the
object$\rightarrow$subject direction to extract triples. Tab. 5 show that the
performance of both variants drops on all datasets, which shows the
superiority of the proposed bidirectional tagging framework. Primarily, we
take the union set of triples extracted from two directions as the final
results. Both variants achieve lower perceptions and recalls, which indicates
that the information from both directions can interact with each other, and
the extracted triples can be cross-validated by the results of two-way
taggers.

Third, we evaluate the effectiveness of the relation prediction module. To
this end, we implement a variant of BitCoin that neglect the potential
relations obtained from the relationship prediction module. Tab. 5 show that
the performance drops on all datasets, which indicates that entity pairs and
relations are interdependency and indivisibility, and we can obtain reliable
results by fully utilizing the relational information present in the sentence
during the extraction.

## Conclusion

In this paper, we propose a novel bidirectional tagging and supervised
contrastive learning based joint model named BitCoin for RTE tasks. BitCoin
provides a general contrastive learning method that considers multiple
positives per anchor and designs a penalty term to prevent excessive
similarity between subject and object. Different from existing methods,
taggers in our method are conducted in two directions, enabling the extraction
of triples from subject$\rightarrow$object and object$\rightarrow$subject.
Extensive experiments on four widely used datasets demonstrate the
effectiveness of our method.

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# Individual addressing of trapped ion qubits with geometric phase gates

R. T. Sutherland<EMAIL_ADDRESS>Department of Electrical and
Computer Engineering, University of Texas at San Antonio, San Antonio, Texas
78249, USA  R. Srinivas  Department of Physics, University of Oxford,
Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, UK  D. T. C. Allcock
Oxford Ionics Limited, Begbroke Science Park, Begbroke, OX5 1PF, UK
Department of Physics, University of Oregon, Eugene, Oregon 97403, USA

###### Abstract

We propose a new scheme for individual addressing of trapped ion qubits,
selecting them via their motional frequency. We show that geometric phase
gates can perform single-qubit rotations using the coherent interference of
spin-independent and (global) spin-dependent forces. The spin-independent
forces, which can be generated via localised electric fields, increase the
gate speed while reducing its sensitivity to motional decoherence, which we
show analytically and numerically. While the scheme applies to most trapped
ion experimental setups, we numerically simulate a specific laser-free
implementation, showing cross-talk errors below $10^{-6}$ for reasonable
parameters.

To achieve universal error-corrected quantum computation, we require one- and
two-qubit gates with infidelities below $\sim 10^{-5}$ and $\sim 10^{-4}$,
respectively Nielsen and Chuang (2010); Lidar and Brun (2013). Perhaps more
challenging, we must also perform them in a system that can scale to thousands
of qubits. Trapped ions are one of the most promising platforms for achieving
these requirements Cirac and Zoller (1995); Monroe  _et al._ (1995); Wineland
_et al._ (1998); Häffner  _et al._ (2008); Blatt and Wineland (2008), due to
their high gate fidelities, long coherence times, and all-to-all connectivity
Ladd  _et al._ (2010); Harty  _et al._ (2014); Ballance  _et al._ (2016);
Gaebler  _et al._ (2016); Srinivas  _et al._ (2021); Clark  _et al._ (2021).
Even so, there are challenges to address before the platform can reliably
operate with enough qubits to perform useful computations. The now
demonstrated quantum charge-coupled device (QCCD) Home  _et al._ (2009); Pino
_et al._ (2020), where electrodes are used to move ions between separated trap
‘zones’, each with designated functions, could provide the modularization
needed to meet this challenge Wineland  _et al._ (1998); Kielpinski  _et al._
(2002). The question remains, however, as to the best way of generating the
gate fields themselves.

Figure 1: Illustration of the proposed single-qubit geometric phase gate. (a)
An E-field, with Rabi frequency $\Omega_{E}$, is combined with a similarly
detuned spin-dependent force, with Rabi frequency $\Omega_{p}$, producing
constructive and destructive interference with the $\ket{+}$ (blue) and
$\ket{-}$ (red) eigenstates of the gate’s Pauli operator
$\hat{\sigma}_{\alpha}$, respectively. (b) This interference causes the
eigenstates to have distinct phase-space trajectories. The difference in areas
(purple) of the closed trajectories ${\Phi\equiv\Phi_{+}-\Phi_{-}}$
corresponds to the gate’s rotation angle.

There are two general approaches to creating these fields, either with lasers
or magnets. Laser-based approaches have achieved one-qubit and two-qubit gates
with infidelities of $\sim 10^{-5}$, and $\sim 10^{-3}$ Ballance  _et al._
(2016); Gaebler  _et al._ (2016); Clark  _et al._ (2021), respectively, and
are straightforwardly focused for individual addressing Nägerl  _et al._
(1999). Unfortunately, they suffer from photon scattering Ozeri  _et al._
(2007), as well as phase and amplitude noise. Laser-free gates, by contrast,
do not suffer from these issues Wineland  _et al._ (1998); Mintert and
Wunderlich (2001); Ospelkaus  _et al._ (2008); Johanning  _et al._ (2009);
Ospelkaus  _et al._ (2011); Timoney  _et al._ (2011); Harty  _et al._ (2014,
2016); Weidt  _et al._ (2016); Lekitsch  _et al._ (2017); Srinivas  _et al._
(2019); Sutherland  _et al._ (2019); Zarantonello  _et al._ (2019); Sutherland
_et al._ (2020); Srinivas  _et al._ (2021). Similarly, they can operate at
infidelities of $\sim 10^{-6}$ for one-qubit gates Harty  _et al._ (2014), and
$\sim 10^{-3}$ for two-qubit gates Harty  _et al._ (2016); Zarantonello  _et
al._ (2019); Srinivas  _et al._ (2021). Laser-free approaches typically use
microwaves and magnetic field gradients, which cannot be localized to
individual ions; this makes single-qubit addressing more difficult.

The QCCD architecture makes this somewhat easier, as it enables gate
implementations in zones that are separated by 100s of microns. However, at
these separations there can still be significant cross-talk because magnetic
fields from current-carrying electrodes only decrease polynomially with
distance, and there can potentially be inductive coupling to other electrodes.
While cross-talk can be mitigated somewhat with active cancellation fields
Aude Craik  _et al._ (2014, 2017), schemes that require fields resonant with
qubit transitions will likely become more complicated with qubit number $N$.
This has been avoided by separating the ions in qubit frequency space using
magnetic field gradients Wang  _et al._ (2009); Johanning  _et al._ (2009);
Warring  _et al._ (2013a); Weidt  _et al._ (2016); Srinivas  _et al._ (2021).
Similarly, one could create differential qubit frequency shifts between
different zones in a QCCD architecture, but this is limited in two ways.
First, the shifts should be stable/repeatable, requiring significant book-
keeping for larger systems. Secondly, the Rabi frequencies of the gates must
be smaller than the separations of their qubit frequencies which could create
a speed limit for larger $N$.

In this work, we propose the first geometric phase gate Mølmer and Sørensen
(1999); Sørensen and Mølmer (2000); Milburn  _et al._ (2000); Leibfried  _et
al._ (2003) operation for directly implementing single-qubit
rotations—decreasing hardware complexity by enabling single qubit gates to be
implemented with the same fields used for two-qubit gates. Like its two-qubit
parallel, our scheme enables the separation of ions in motional frequency
space by differentiating zone potentials with locally tuned electrodes. Thus,
the scheme can be implemented exclusively with fields that are detuned from
all qubit transitions without modifying their frequencies.

Our scheme operates via the interference of spin-dependent and spin-
independent forces, both similarly detuned from the motion. Here, both
eigenstates of the spin operator return to the origin in phase-space, but
accumulate different geometric phases (see Fig. 1). The scheme has an
effective Rabi frequency $\propto\sqrt{\Omega_{E}\Omega_{p}}$, where
$\Omega_{E}$ is the Rabi frequency of an applied E-field, and $\Omega_{p}$ is
that of an applied B-field gradient. The dependence on
$\sqrt{\Omega_{\text{E}}}$ enables much faster single-qubit gates compared to
equivalent two-qubit gates; it is much easier to generate large E-field
couplings relative to those from magnetic field gradients. Compared to a
$\propto\Omega_{p}$ two-qubit gate, we show that infidelities are decreased by
at least a factor of $\sim\Omega_{p}/\Omega_{E}$ for static motional frequency
shifts and motional dephasing, as well as $\sim(\Omega_{p}/\Omega_{E})^{3/2}$
for heating. Lastly, we perform numerical simulations using experimental
parameters similar to Ref. Srinivas  _et al._ (2021), showing we are able to
reduce cross talk to below $10^{-6}$.

We want to generate a single-qubit rotation:

$\displaystyle\hat{U}_{g}=e^{-i\theta\hat{\sigma}_{\alpha}/2},$ (1)

where $\theta$ is the rotation angle of the gate while
$\hat{\sigma}_{\alpha}\equiv\hat{n}\cdot\vec{\sigma}$, such that $\hat{n}$ is
a unit vector, and
$\vec{\sigma}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$ is a
vector comprising the Pauli matrices. We will generate this effective
interaction with accumulated geometric phases, where the eigenstates
$\ket{\pm}$ of $\hat{\sigma}_{\alpha}$ (eigenvalues $\pm 1$) acquire distinct
phases $\Phi_{\pm}$ (see Fig. 1(b)). Representing the system in the
$\ket{\pm}$ eigenbasis, this gives a time-propagator:

$\displaystyle\hat{U}_{p}\ket{\psi}$ $\displaystyle=$ $\displaystyle
c_{+}e^{i\Phi_{+}}\ket{+}+c_{-}e^{i\Phi_{-}}\ket{-}$ (2) $\displaystyle=$
$\displaystyle
e^{i(\Phi_{+}+\Phi_{-})/2}\Big{(}c_{+}e^{-i\Phi/2}\ket{+}+c_{-}e^{i\Phi/2}\ket{-}\Big{)},~{}~{}~{}~{}~{}~{}$

where $\Phi\equiv\Phi_{-}-\Phi_{+}$, showing, up to a global phase, that
$\hat{U}_{g}$ is equivalent to $\hat{U}_{p}$ when $\theta=\Phi$.

To generate $\hat{U}_{p}$, we add an E-field to the typical geometric phase
gate interaction:

$\displaystyle\hat{H}_{p}$ $\displaystyle=$
$\displaystyle\hbar\Omega_{p}\hat{\sigma}_{\alpha}\hat{a}^{\dagger}e^{i\Delta
t}+\hbar\Omega_{E}\hat{a}^{\dagger}e^{i[\Delta t+\phi]}+c.c.,$ (3)

where $\Delta$ is the gate detuning, $\hat{a}^{\dagger}(\hat{a})$ is a
creation(annihilation) operator, and $\phi$ is the phase of the E-field
relative to the spin-motion coupling. We have considered Eq. (3) in a frame
rotating with respect to the qubit frequency, mode frequency $\omega_{r}$, and
have assumed the rotating wave approximation for terms oscillating near these
frequencies. It is here worth noting that Refs. Turchette  _et al._ (1998);
Leibfried (1999); Warring  _et al._ (2013b, a) consider spin-dependent
interactions coupled to the traps’ rf micromotion, similarly operating via
spin-motion and E-field terms; this scheme is disadvantageous compared to Eq.
(3), because it leads to temperature dependent, and non-zero higher-order,
terms in the Magnus expansion sup . Furthermore, using the micromotion
requires pushing ions off the rf null to induce transitions. Aside from being
sensitive to stray E-fields which can move the ions and lead to unwanted cross
talk, this requires an extra transport operation, which can be slow. We can
exactly describe $\hat{U}_{p}$ with the Magnus expansion Magnus (1954) up to
$2^{\text{nd}}$-order:

$\displaystyle\hat{U}_{p}\\!=\exp\\!\Big{(}\\!-\frac{i}{\hbar}\int^{t_{g}}_{0}\\!\\!\\!\\!dt^{\prime}\hat{H}_{p}(t^{\prime})-\frac{1}{2\hbar^{2}}\\!\\!\int^{t_{g}}_{0}\\!\\!\\!\\!\int^{t^{\prime}}_{0}\\!\\!\\!\\!\\!dt^{\prime}dt^{\prime\prime}\Big{[}\hat{H}_{p}(t^{\prime}),\hat{H}_{p}(t^{\prime\prime})\Big{]}\Big{)},$

where higher-order terms vanish. We ensure the phase space trajectories of
$\ket{\pm}$ close by setting $t_{g}=2\pi K/\Delta$, where $K$ is the number of
loops traversed in phase-space. Up to a global phase this gives:

$\displaystyle\hat{U}_{p}$ $\displaystyle=$
$\displaystyle\exp\Big{(}\frac{4\pi
iK\Omega_{p}\Omega_{E}}{\Delta^{2}}\cos[\phi]\hat{\sigma}_{\alpha}\Big{)},$
(5)

showing the gate is most efficient when the two interactions in $\hat{H}_{p}$
are aligned, i.e. $\phi\in\\{0,\pi\\}$; we henceforth assume $\phi=0$. Setting
$\Delta=(8\pi K\Omega_{p}\Omega_{E}/\theta)^{1/2}$, we see
$\hat{U}_{p}=\hat{U}_{g}$, up to a global phase, with a gate time $t_{g}=(\pi
K\theta/2\Omega_{p}\Omega_{E})^{1/2}$, and an effective Rabi frequency of
$\Omega_{\text{eff}}\equiv\theta/t_{g}=(2\Omega_{p}\Omega_{E}\theta/\pi
K)^{1/2}$. Because its speed scales $\propto\sqrt{\Omega_{E}\Omega_{p}}$, it
can potentially operate orders of magnitude faster than than a
$\propto\Omega_{p}$ two-qubit gate. All of the interactions used to generate
the gate are tuned to the frequency of the addressed well, affecting a gate
with potentially no fields on-resonant with any spectator qubit transitions;
this enables the elimination of cross-talk with pulse-shaping.

While this derivation considered a single ion in a well, it can be extended
straightforwardly to include multiple ions in a well. This could, in theory,
be used to address individual ions in the same well sup . This requires the
ability to address a complete set of $N$ orthogonal modes of motion, which
could make the technique challenging to implement when many ions are in the
same well. In contrast to B-field gradients, addressing some modes with an
E-field requires a differential component; for example, the coupling strength
to the stretch mode of a two-ion crystal is proportional to the difference in
the E-field amplitudes at the position of each ion.

Figure 2: Infidelity $\mathcal{I}$ versus (a) heating rate $\dot{\bar{n}}$ (b)
dephasing rate $\eta$ and (c) static motional frequency shift magnitude
$|g_{d}|$ for analytic (dashed) and numerical (solid) calculations, when
$\Omega_{p}/2\pi=2~{}\text{kHz}$ and $\Omega_{E}/2\pi=100~{}\text{kHz}$. In
parts (a-c), we show calculations initialized to the motional ground state
$n=0$ with $K=1$ loops in phase space. In parts (a) and (b), we illustrate
that increasing $K$ decreases the effect $\dot{\bar{n}}$ or $\eta$ has on
$\mathcal{I}$, showing calculations where $K=4$ and $n=0$. In parts (b) and
(c), we illustrate the scheme’s temperature dependence versus $\eta$ and
$g_{d}$, showing calculations where $K=1$ and $n=10$. Note that when
$\mathcal{I}\ll 1$, $\mathcal{I}$ is symmetric about $g_{d}=0$.

Trapped ion quantum computers traditionally operate in a regime where the
infidelity is significantly lower for one-qubit gates than for two-qubit
gates. This is because two-qubit gates are slower than one qubit gates, and
they require a temporal window where the qubits are entangled to the motion,
rendering them sensitive to motional decoherence as well as added control
errors Mølmer and Sørensen (1999); Sørensen and Mølmer (2000); Sutherland  _et
al._ (2022). As we are now proposing a scheme for one-qubit gates that also
requires spin-motion entanglement, it is important to ensure that motional
decoherence affects $\mathcal{I}$ significantly less than its two-qubit
counterpart. We here consider the contribution of heating, motional dephasing,
and static motional frequency shifts to the gate infidelity $\mathcal{I}$. In
order to evaluate $\mathcal{I}$, we follow a technique that dates back to NMR
Haeberlen and Waugh (1968), but has been used more recently to evaluate the
contribution of various sources of noise to $\mathcal{I}$ for two qubit gates
Haddadfarshi and Mintert (2016); Martínez-García  _et al._ (2021); Sutherland
_et al._ (2022). We consider a Hamiltonian in the presence of an error term:

$\displaystyle\hat{H}_{t}=\hat{H}_{p}+\hat{H}_{e},$ (6)

where $\hat{H}_{p}$ is given by Eq. (3). Here, $\hat{H}_{e}$ represents the
error term to be considered where $\hat{H}_{e}=2\hbar g_{h}\cos(\omega
t)(\hat{a}^{\dagger}e^{i\omega_{r}t}+\hat{a}e^{-i\omega_{r}t})$ for heating
and $\hat{H}_{e}=\hbar g_{d}\cos(\omega t)\hat{a}^{\dagger}\hat{a}$ for
dephasing, where $g_{h(d)}$ are Rabi frequencies. For each source of error, we
evaluate the infidelity by first transforming into the interaction picture
with respect to either $\hat{H}_{e}$ (heating) or $\hat{H}_{p}$ (motional
dephasing and motional frequency shifts), in order to produce a factorized
time propagator $\hat{U}_{t,\omega}=\hat{U}_{p}\hat{U}_{e,\omega}$, making the
final equation for infidelity:

$\displaystyle\mathcal{I}_{\omega}$ $\displaystyle=$
$\displaystyle\sum_{n^{\prime}}|\bra{\psi(0)}\bra{n^{\prime}}\hat{U}_{e,\omega}\ket{\psi(0)}\ket{n}|^{2},$
(7)

We evaluate $\hat{U}_{e,\omega}$ either by Taylor expanding the exponential
(heating) or using perturbation theory (motional dephasing), both up to
$2^{\text{nd}}$-order. Finally, we determine $\mathcal{I}$ by averaging over
$S_{\omega}$, the normalized spectral power density:

$\displaystyle\mathcal{I}=\int^{\infty}_{0}d\omega
S_{\omega}\mathcal{I}_{\omega},$ (8)

subsequently making the Born-Markov approximation Sutherland  _et al._ (2022);
sup . As shown in Ref. Sutherland  _et al._ (2022), this is mathematically
equivalent to the Lindblad formalism when $\mathcal{I}\ll 1$. We also
determine $\mathcal{I}$ for static motional frequency shifts, i.e. the limit
$S_{\omega}\rightarrow\delta(\omega)$ where $\delta(\omega)$ is the Dirac
delta function, negating the need for the Born-Markov approximation Sutherland
_et al._ (2022); sup .

Error type | Effect on gate fidelity
---|---
heating | $\mathcal{I}=\dot{\bar{n}}\sqrt{\frac{\Omega_{p}\theta^{3}}{8\pi K\Omega_{E}^{3}}}\lambda^{2}_{\hat{\sigma}_{\alpha}}$
motional dephasing | $\mathcal{I}=\eta\sqrt{\frac{\Omega_{p}\theta^{3}}{32\pi K\Omega_{E}^{3}}}\Big{(}2n+1+\frac{3\theta\Omega_{E}}{2\pi K\Omega_{p}}\Big{)}\lambda^{2}_{\hat{\sigma}_{\alpha}}$
motional frequency | $\mathcal{I}=\frac{g_{d}^{2}\theta^{2}}{16\Omega_{E}^{2}}\Big{(}2n+1+\frac{2\theta\Omega_{E}}{\pi K\Omega_{p}}\Big{)}\lambda_{\hat{\sigma}_{\alpha}}^{2}$
shifts
Table 1: Table of infidelities $\mathcal{I}$ for systems with motional states
with an initial Fock state $n$, due to common sources of motional decoherence.

The results of these calculations—recorded in Table 1—show that we are
proposing one qubit gates that are significantly less sensitive to motional
decoherence than their two-qubit counterparts. Compared to the analytic
formulae for two-qubit gates in Ref. Sutherland  _et al._ (2022), we find that
$\mathcal{I}$ is reduced by at least a factor of
$(\Omega_{p}/\Omega_{E})^{3/2}$ for heating, and of $\Omega_{p}/\Omega_{E}$
for motional dephasing and static motional frequency shifts. Since
$\Omega_{E}$ is due simply to an E-field interacting with a charge, it can be
several orders-of-magnitude larger than $\Omega_{p}$ (potentially limited by
the validity of the rotating wave approximation), leading to an increase in
speed and insensitivity to motional errors. This insensitivity to motional
decoherence is further illustrated in Fig. 2, where we show $\mathcal{I}$
versus heating rate $\dot{\bar{n}}$, motional decoherence rate
$\eta=2/\tau=\pi\varepsilon^{2}S_{0}/2$, where $\tau$ is the coherence time
between neighboring Fock states, and static motional frequency shift magnitude
$|g_{d}|$. Here, we compare the equations presented in Table 1 to direct
numerical integration; this figure clearly demonstrates the accuracy of our
analytic formulae in the relevant parameter regimes. The decrease in
sensitivity to these sources of motional decoherence is larger than expected
from the $\propto\sqrt{\Omega_{E}/\Omega_{p}}$ increase to the gate speed.
This is due to the fact that the increased speed comes from the interference
of the gradient with the external E-field, increasing the gate speed while
simultaneously decreasing the time-averaged spin-motion entanglement.

Figure 3:  (a) Numerically calculated infidelity $\mathcal{I}$ of target qubit
(red) and spectator qubit (grey) versus ramping time $t_{r}$ for an effective
$\theta=\pi/2$ gate. Here, the the motional frequency of the target qubit is
set to $\omega_{r,t}/2\pi=6.5~{}\text{MHz}$, while the motional frequency of
the spectator qubit is set to $\omega_{r,s}=6.3~{}\text{MHz}$. The inset shows
the probability of measuring $\ket{\uparrow}$ for both qubits
$\mathcal{P}_{\uparrow}=|\braket{\uparrow}{\psi_{t,s}(t)}|^{2}$ when
$t_{r}=10~{}\mu\text{s}$. (b) Infidelity of spectator qubit versus E-field
amplitude $\Omega_{E,s}$ relative to the E-field seen by the target qubit
$\Omega_{E,\tau}$ when $t_{r}=15~{}\mu\text{s}$. This is shown for the same
motional frequency of the target qubit, and the motional frequency of the
spectator qubit is set to $\omega_{r,s}/2\pi=6.1~{}\text{MHz}$ (purple
dashed), $\omega_{r,s}/2\pi=6.3~{}\text{MHz}$ (red solid),
$\omega_{r,s}/2\pi=6.35~{}\text{MHz}$ (green dotted), and
$\omega_{r,s}/2\pi=6.4~{}\text{MHz}$ (blue dot dashed).

Finally, we evaluate our scheme’s resilience to cross-talk with a specific
physical implementation Sutherland  _et al._ (2019); Srinivas  _et al._ (2019,
2021), noting that any protocol capable of generating a two-qubit geometric
phase gate—and the necessary E-field—is capable of implementing it. We here
consider two ions in separate wells, where one ion acts as the ‘target’ qubit,
and the other as the ‘spectator’ qubit. The target well is designated a
motional frequency $\omega_{\tau}$, while the spectator mode is designated a
detuned motional frequency $\omega_{s}$. Both are driven with a pair of
bichromatic microwave fields with Rabi frequency $\Omega_{\mu}$, symmetrically
detuned around the qubit frequency, and an rf B-field gradient with frequency
$\omega_{g}/2\pi=5~{}\text{MHz}$. The resulting interaction is

$\displaystyle\hat{H}_{\mu}$ $\displaystyle=$ $\displaystyle
2\hbar\sum_{\gamma=\tau,s}\Omega_{\mu}\hat{\sigma}_{+,\gamma}\cos(\delta
t)+\Omega_{g}\hat{\sigma}_{z,\gamma}\cos(\omega_{g}t)\hat{a}^{\dagger}_{\gamma}e^{i\omega_{\gamma}t}\\!\\!$
(9)
$\displaystyle+\Omega_{E,\gamma}\sin([\omega_{\tau}-\Delta]t)\hat{a}^{\dagger}_{\gamma}e^{i\omega_{\gamma}t}+c.c.,$

where $\hat{\sigma}_{\alpha,\gamma}$ is a Pauli operator acting on qubit
$\gamma$, $\Omega_{E,\gamma}$ is the E-field Rabi frequency seen by qubit
$\gamma$, and $\delta$ is the detuning of the microwave fields from the qubit
frequency. The above equation is in the rotating frame with respect to the
motional and qubit frequencies, and we have made the rotating wave
approximation for terms oscillating near the qubit frequency. Similar to Ref.
Sutherland  _et al._ (2020), we can set
$\delta=(\omega_{\tau}-\omega_{g})-\Delta$, and the time-propagator resulting
from Eq. (9) can be evaluated in the interaction picture with respect to the
$\propto\Omega_{\mu}$ term. Dropping fast rotating terms, this gives:

$\displaystyle\hat{H}_{\mu,I}\simeq
i\hbar\Big{(}\Omega_{g}J_{1}\Big{[}\frac{4\Omega_{\mu}}{\delta}\Big{]}\hat{\sigma}_{y,\tau}+\Omega_{E,\tau}\Big{)}\hat{a}_{\tau}^{\dagger}e^{i\Delta
t}+\text{c.c.},~{}~{}$ (10)

taking the form of Eq. (3) acting on the target qubit.

We now determine the validity of Eq. (10), showing that with pulse shaping it
becomes a very good approximation. As was shown in Ref. Sutherland  _et al._
(2019), if we perform this gate while smoothly ramping $\Omega_{\mu(g)}$ on
and off before and after the gate, respectively, then the time-propagator
given by Eq. (10) converges to the time-propagator for Eq. (9). This enables
high-fidelity gates Srinivas  _et al._ (2021), despite the Rabi flopping due
to the $\propto\Omega_{\mu}$ term. There is no pulse shaping for the E-fields
in our simulations. In Fig. 3, we show this convergence for a $\theta=\pi/2$
gate, where $\omega_{\tau}/2\pi=6.5~{}\text{MHz}$,
$\Omega_{g}/2\pi=2~{}\text{kHz}$, $\Omega_{E,\tau}/2\pi=100~{}\text{kHz}$. The
$\propto\Omega_{\mu(g)}$ fields are simultaneously ramped on and off according
to $\sin^{2}(t/t_{r})$, ideally leading to $\hat{U}_{g}\rightarrow
e^{i\pi\hat{\sigma}_{y,\tau}/4}$ for the target qubit and
$\hat{U}_{g}\rightarrow\hat{I}$ for the spectator qubit. Figure 3(a) shows
that for values over $t_{r}\simeq 13~{}\mu\text{s}$, $\Omega_{E,s}=0$, and
$\omega_{s}/2\pi=6.3~{}\text{MHz}$, $\mathcal{I}$ can be suppressed to below
$10^{-6}$ for both the target and spectator qubits. We have here optimized
$t_{g}$ to maximize the fidelity of the target qubit operation when
$t_{r}=15~{}\mu\text{s}$, which makes $t_{g}\simeq 40.2~{}\mu\text{s}$ when
including the $2t_{r}$ of pulse shaping time. We found no limit to the degree
to which the cross-talk can be reduced via increasing $t_{r}$ in our
simulations, indicating a trade-off between reducing cross-talk and $t_{g}$.
The inset of Fig. 3 illustrates the time dynamics of the target and spectator
qubits, showing that while both qubits experience high-frequency oscillations
during the gate, after the pulse shaping sequence, both qubits converge to the
final states predicted by Eq. (10). Finally, while $\Omega_{E,s}$ will likely
decrease rapidly with spatial separation, it will not vanish entirely. Because
of this, Fig. 3(b) shows $\mathcal{I}$ versus $\Omega_{s,E}$; we here plot
values of $\omega_{s}/2\pi=6.1-6.4~{}\text{MHz}$, showing that when
$\omega_{s}$ is further detuned from $\omega_{\tau}$, one should expect less
cross-talk (keeping all other parameters fixed) due to the fact that all
spectator transitions are more off-resonant.

In conclusion, we have proposed a new one-qubit geometric phase gate scheme
that is much faster and more robust to noise than its two-qubit counterpart,
while also enabling the suppression of cross-talk to an arbitrary degree—even
with global qubit control fields. We first developed the theory for this gate
sequence, showing it requires a standard set of interactions present in most
trapped ion laboratories. We then showed, analytically and numerically, that
our proposed one-qubit geometric phase gates are significantly less sensitive
to motional decoherence compared to their two-qubit gate counterparts.
Finally, we provided a numerical simulation of one (of many) possible physical
implementations of our scheme, showing that when it is combined with pulse
shaping it can reduce cross-talk to an arbitrary degree, even when both qubits
experience the same microwave fields and gradient. This work is important to
the prospects of scalable, laser-free trapped ion architectures because it
shows how to perform targeted operations without spatially localized qubit
control fields.

The authors would like to thank C. J. Ballance, and D. H. Slichter for helpful
discussions.

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##  Appendix A Addressing with rf micromotion

As in Refs. Leibfried (1999); Warring  _et al._ (2013a), we consider a trapped
ion system with a sideband interaction and an E-field force, potentially from
the rf micromotion:

$\displaystyle\hat{H}_{\text{lab}}$ $\displaystyle=$
$\displaystyle\hbar\omega_{r}\hat{a}^{\dagger}\hat{a}+\frac{\hbar\omega_{0}}{2}\hat{\sigma}_{z}+2\hbar\Omega_{\text{rsb}}\cos([\omega_{0}-\omega_{r}+\Delta]t)\hat{\sigma}_{x}\Big{(}\hat{a}^{\dagger}+\hat{a}\Big{)}+2\hbar\Omega_{e}\cos([\omega_{r}-\Delta]t)\Big{(}\hat{a}^{\dagger}+\hat{a}\Big{)},$
(11)

where $\omega_{0}$ is the qubit frequency, and $\omega_{r}$ is motional mode
frequency. Transforming into the interaction picture with respect to the
motional and qubit frequencies, as well as dropping counter-rotating terms,
gives:

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hbar\Omega_{\text{rsb}}\Big{(}\hat{\sigma}_{-}\hat{a}^{\dagger}e^{i\Delta
t}+\hat{\sigma}_{+}\hat{a}e^{-i\Delta
t}\Big{)}+\hbar\Omega_{e}\Big{(}\hat{a}^{\dagger}e^{i\Delta t}+ae^{-i\Delta
t}\Big{)}.$ (12)

We now plug Eq. (12) into the Magnus expansion Magnus (1954) while setting the
gate time $t_{g}=2\pi N/\Delta$, observing that the $1^{\text{st}}$-order term
in the Magnus expansion sinusoidally integrates to zero. Dropping a global
phase, this gives:

$\displaystyle\hat{U}$ $\displaystyle=$
$\displaystyle\exp\Big{(}\frac{it_{g}}{\Delta}\Big{[}\Omega_{\text{rsb}}^{2}\hat{\sigma}_{z}\Big{\\{}\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\Big{\\}}+\Omega_{\text{rsb}}\Omega_{e}\hat{\sigma}_{x}\Big{]}\Big{)},$
(13)

up to $2^{\text{nd}}$-order. Because
$[\hat{H}(t^{\prime}),[\hat{H}(t^{\prime\prime}),\hat{H}(t^{\prime\prime\prime})]]\neq
0$, the expansion cannot be exactly expressed in a straightforward manner,
which could limit the fidelity—even in ideal conditions. Equation (13) also
reveals a temperature dependent $\propto\hat{a}^{\dagger}\hat{a}$ term,
further limiting the approach’s potential for high-fidelity operations.

##  Appendix B Single-qubit addressing for ions in the same well

Assuming a rotating frame with respect to the motional and qubit frequencies,
we consider a Hamiltonian describing the $j^{\text{th}}$ mode of a chain of
$N$ ions driven with a geometric phase gate interaction and a similarly
detuned E-field:

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hbar\Omega_{p,j}\hat{S}_{\alpha,j}\Big{(}\hat{a}^{\dagger}_{j}e^{i\Delta
t}+\hat{a}_{j}e^{-i\Delta
t}\Big{)}+\hbar\Omega_{e,j}\Big{(}\hat{a}_{j}^{\dagger}e^{i\Delta
t}+\hat{a}_{j}e^{-i\Delta t}\Big{)},$ (14)

where $\Omega_{p,j}$ and $\Omega_{e,j}$ are Rabi frequencies, and
$\hat{S}_{\alpha,j}\equiv\sum_{n}e_{n,j}\hat{\sigma}_{\alpha,n}$ is a
collective Pauli operator. Importantly, the contribution $e_{n,j}$ of the
$n^{\text{th}}$ ion to $\hat{S}_{\alpha,j}$ is proportional to its projection
onto mode $j$. Plugging Eq. (14) into the Magnus expansion Magnus (1954), and
evaluating at a time $t_{g}=2\pi K/\Delta$, gives a time propagator:

$\displaystyle\hat{U}_{p}^{\prime}=\exp\Big{(}\frac{it_{g}}{\Delta}\Big{[}\Omega_{p,j}^{2}\hat{S}_{\alpha,j}^{2}+2\Omega_{p,j}\Omega_{e,j}\hat{S}_{\alpha,j}\Big{]}\Big{)}.$
(15)

We can see that this operation leaves an extraneous
$\propto\hat{S}^{2}_{\alpha,j}$ term, creating unwanted entanglement between
the qubits; because $\hat{\sigma}_{\alpha}^{2}=\hat{I}$, this term corresponds
to a global phase for one qubit systems. This entanglement term can be
eliminated by dividing the operation into two $K/2$ loop operations while
flipping the sign of the detuning $\Delta\rightarrow-\Delta$, and the sign of
$\hat{S}_{\alpha,j}\rightarrow-\hat{S}_{\alpha,j}$ with a spin-echo sequence.
Doing so results in a total time propagator:

$\displaystyle\hat{U}_{p}$ $\displaystyle=$
$\displaystyle\exp\Big{(}\frac{-it_{g}}{2\Delta}\Big{[}\Omega_{p,j}^{2}\hat{S}_{\alpha,j}^{2}-2\Omega_{p,j}\Omega_{e,j}\hat{S}_{\alpha,j}\Big{]}\Big{)}\exp\Big{(}\frac{it_{g}}{2\Delta}\Big{[}\Omega_{p,j}^{2}\hat{S}_{\alpha,j}^{2}+2\Omega_{p,j}\Omega_{e,j}\hat{S}_{\alpha,j}\Big{]}\Big{)}$
(16) $\displaystyle=$
$\displaystyle\exp\Big{(}\frac{2i\Omega_{p,j}\Omega_{e,j}}{\Delta}\hat{S}_{\alpha,j}t_{g}\Big{)}$
$\displaystyle=$
$\displaystyle\exp\Big{(}-i\frac{\theta}{2}\hat{S}_{\alpha,j}\Big{)},$

giving a single-qubit gate with an effective rotation angle of
$\theta\equiv-2\Omega_{p,j}\Omega_{e,j}/\Delta$.

Because $e_{n,j}$ is proportional to each ion’s projection onto each mode, the
ability to perform this operation on a complete set of $N$ modes should, in
theory, give experimentalists the ability to perform individual qubit
addressing. In a well with two ions, for example, if we can address a center-
of-mass mode
$\hat{S}_{\alpha,c}=\hat{\sigma}_{\alpha,2}+\hat{\sigma}_{\alpha,1}$ and a
stretch mode
$\hat{S}_{\alpha,s}=\hat{\sigma}_{\alpha,2}-\hat{\sigma}_{\alpha,1}$, we can
address qubit $2$ by generating Eq. (16) with an angle $\theta$ for each
rotation:

$\displaystyle\hat{U}_{2}$ $\displaystyle=$
$\displaystyle\exp\Big{(}-i\frac{\theta}{2}\hat{S}_{\alpha,c}\Big{)}\exp\Big{(}-i\frac{\theta}{2}\hat{S}_{\alpha,s}\Big{)}$
(17) $\displaystyle=$
$\displaystyle\exp\Big{(}-i\theta\hat{\sigma}_{\alpha,2}\Big{)},$

corresponding to a $2\theta$ rotation on qubit 2. Similarly, performing two
operations with opposite signed values of $\theta$ gives:

$\displaystyle\hat{U}_{2}$ $\displaystyle=$
$\displaystyle\exp\Big{(}i\frac{\theta}{2}\hat{S}_{\alpha,s}\Big{)}\exp\Big{(}-i\frac{\theta}{2}\hat{S}_{\alpha,c}\Big{)}$
(18) $\displaystyle=$
$\displaystyle\exp\Big{(}-i\theta\hat{\sigma}_{\alpha,1}\Big{)},$

giving a $2\theta$ rotation on qubit $1$. This process can, theoretically, be
extended to chains of $N$ ions. In practice, however, it may be difficult to
produce an E-field differential pronounced enough to generate large values of
$\Omega_{e,j}$ in wells with many ions. We therefore focus the main body of
this work on systems with one ion in a well, likely the most relevant to
scalable QCCD architectures.

##  Appendix C Infidelities from motional decoherence

In this section, we discuss the effect of three sources of motional
decoherence on the fidelity of the single-qubit geometric phase gates we
proposed in the main body. First, in a frame rotating with respect to the
qubit and trap frequencies, we examine the geometric phase gate Hamiltonian
acting in the presence of an error Hamiltonian:

$\displaystyle\hat{H}_{t}$ $\displaystyle=$
$\displaystyle\hat{H}_{p}+\hat{H}_{e},$ (19) $\displaystyle=$
$\displaystyle\hbar\hat{A}\Big{(}\hat{a}^{\dagger}e^{i\Delta
t}+\hat{a}e^{-i\Delta t}\Big{)}+\hat{H}_{e},$

where we have introduced
$\hat{A}=\Omega_{p}\hat{\sigma}_{\alpha}+\Omega_{e}\hat{I}$, and have assumed
we can apply the rotating wave approximation for $\hat{H}_{e}$.

### Static motional frequency shifts

We first consider static motional frequency shifts $\hat{H}_{e}\equiv\hbar
g_{d}\hat{a}^{\dagger}\hat{a}$, which gives:

$\displaystyle\hat{H}_{t}$ $\displaystyle=$
$\displaystyle\hbar\hat{A}\Big{(}\hat{a}^{\dagger}e^{i\Delta
t}+\hat{a}e^{-i\Delta t}\Big{)}+\hbar g_{d}\hat{a}^{\dagger}\hat{a}.$ (20)

For static motional frequency shifts and motional dephasing, we analyze the
gate fidelity by following a technique laid out in Ref. Sutherland  _et al._
(2022). We first transform into the interaction picture with respect to
$\hat{H}_{p}$ using:

$\displaystyle\hat{U}_{p}$ $\displaystyle=$
$\displaystyle\exp\Big{(}-\frac{i}{\hbar}\int^{t}_{0}dt^{\prime}\hat{H}_{p}(t^{\prime})-\frac{1}{2\hbar^{2}}\int^{t}_{0}\int^{t^{\prime}}_{0}dt^{\prime}dt^{\prime\prime}[\hat{H}_{p}(t^{\prime}),\hat{H}_{p}(t^{\prime\prime})]\Big{)}$
(21) $\displaystyle=$
$\displaystyle\exp\Big{(}\hat{A}[\alpha(t)a^{\dagger}-\alpha^{*}(t)\hat{a}]+i\hat{A}^{2}f(t)\Big{)},$

where $\alpha(t)\equiv(1-e^{i\Delta t})/\Delta$ and $g(t)\equiv(t-\sin(\Delta
t)/\Delta)/\Delta$; the latter commutes with $\hat{H}_{e}$, having no effect
on the transformation. The interaction picture Hamiltonian is then:

$\displaystyle\hat{H}_{I}$ $\displaystyle=$ $\displaystyle\hbar
g_{d}\Big{(}\hat{a}^{\dagger}\hat{a}+\hat{A}[\alpha(t)\hat{a}^{\dagger}+\alpha^{*}(t)\hat{a}]+\hat{A}^{2}|\alpha(t)|^{2}\Big{)}.$
(22)

We begin the next section with Eq. (22) because $\hat{H}_{e}$ for motional
dephasing deviates only through a $g_{d}\rightarrow g_{d}\cos(\omega t)$
substitution. We use time-dependent perturbation theory to evaluate the time
propagator $\hat{U}_{I}$ for $\hat{H}_{I}$ up to $2^{\text{nd}}$-order:

$\displaystyle\hat{U}_{I}\simeq\hat{I}-\frac{i}{\hbar}\int^{t_{g}}_{0}\hat{H}_{I}(t^{\prime})-\frac{1}{\hbar^{2}}\int^{t_{g}}_{0}\int^{t^{\prime}}_{0}dt^{\prime}dt^{\prime\prime}\hat{H}_{I}(t^{\prime})\hat{H}_{I}(t^{\prime\prime}).$
(23)

Plugging this into Eq. (7) in the main text, dropping all terms higher-order
than $\mathcal{O}(g_{d}^{2}$), and assuming system is initialized to a state
with $n$ phonons, gives:

$\displaystyle\mathcal{I}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{d}^{2}\Omega_{p}^{2}t_{g}^{2}}{\Delta^{2}}\lambda^{2}_{\hat{\sigma}_{\alpha}}\Big{(}2n+1+\frac{16\Omega_{e}^{2}}{\Delta^{2}}\Big{)}$
(24) $\displaystyle=$
$\displaystyle\frac{g_{d}^{2}\theta^{2}}{16\Omega_{e}^{2}}\lambda_{\hat{\sigma}_{\alpha}}^{2}\Big{(}2n+1+\frac{2\theta\Omega_{e}}{\pi\Omega_{p}K}\Big{)},$

where we have substituted $\Delta\equiv\sqrt{8\pi
K\Omega_{p}\Omega_{e}/\theta}$ in the second line.

### Motional dephasing

We take motional dephasing to be the broadband limit of $\hat{H}_{e}$ for
static motional frequency shifts. To evaluate the contribution to
$\mathcal{I}$, we let $\hat{H}_{e}=\hbar
g_{d}\hat{a}^{\dagger}\hat{a}\cos(\omega t)$, calculate the infidelity for
that frequency $\mathcal{I}_{\omega}$, then average over the normalized
spectral power density $S_{\omega}$. We may begin our evaluation with Eq.
(22), substituting $g_{d}\rightarrow g_{d}\cos(\omega t)$, which gives:

$\displaystyle\hat{H}_{I,\omega}$ $\displaystyle=$ $\displaystyle\hbar
g_{d}\cos(\omega
t)\Big{(}\hat{a}^{\dagger}\hat{a}+\hat{A}[\alpha(t)\hat{a}^{\dagger}+\alpha^{*}(t)\hat{a}]+\hat{A}^{2}|\alpha(t)|^{2}\Big{)}.$
(25)

Again, we evaluate the interaction picture time propagator using
$2^{\text{nd}}$-order time-dependent perturbation theory:

$\displaystyle\hat{U}_{I}\simeq\hat{I}-\frac{i}{\hbar}\int^{t_{g}}_{0}dt^{\prime}\hat{H}_{I,\omega}(t^{\prime})-\frac{1}{\hbar^{2}}\int^{t_{g}}_{0}\int^{t^{\prime}}_{0}dt^{\prime}dt^{\prime\prime}\hat{H}_{I,\omega}(t^{\prime})\hat{H}_{I,\omega}(t^{\prime\prime}).$
(26)

Plugging this into Eq. (7) of the main text, and dropping all terms higher-
order than $\mathcal{O}(g_{d}^{2})$ gives:

$\displaystyle\mathcal{I}_{\omega}$ $\displaystyle\simeq$
$\displaystyle\frac{2}{\hbar^{2}}\sum_{n^{\prime}}\int^{t_{g}}_{0}\int^{t^{\prime}}_{0}dt^{\prime}dt^{\prime\prime}\bra{\psi(0)}\bra{n^{\prime}}\hat{H}_{I,\omega}(t^{\prime})\hat{H}_{I,\omega}(t^{\prime\prime})\ket{\psi(0)}\ket{n}$
(27)
$\displaystyle-\frac{1}{\hbar^{2}}\sum_{n^{\prime}}\int^{t_{g}}_{0}\int^{t_{g}}_{0}dt^{\prime}dt^{\prime\prime}\bra{\psi(0)}\bra{n^{\prime}}\hat{H}_{I,\omega}(t^{\prime})\ket{\psi(0)}\ket{n}\bra{\psi(0)}\bra{n}\hat{H}_{I,\omega}(t^{\prime\prime})\ket{\psi(0)}\ket{n^{\prime}}.$

We calculate $\mathcal{I}$ by integrating over $S_{\omega}$, assuming it is
broad enough to warrant the Born-Markov approximation:

$\displaystyle\mathcal{I}$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}d\omega S_{\omega}\mathcal{I}_{\omega}$ (28)
$\displaystyle\simeq$
$\displaystyle\frac{2}{\hbar^{2}}\sum_{n^{\prime}}\int^{\infty}_{0}\int^{t_{g}}_{0}\int^{t^{\prime}}_{0}d\omega
dt^{\prime}dt^{\prime\prime}S_{\omega}\bra{\psi(0)}\bra{n^{\prime}}\hat{H}_{I,\omega}(t^{\prime})\hat{H}_{I,\omega}(t^{\prime\prime})\ket{\psi(0)}\ket{n}$
$\displaystyle-\frac{1}{\hbar^{2}}\sum_{n^{\prime}}\int^{\infty}_{0}\int^{t_{g}}_{0}\int^{t_{g}}_{0}d\omega
dt^{\prime}dt^{\prime\prime}S_{\omega}\bra{\psi(0)}\bra{n^{\prime}}\hat{H}_{I,\omega}(t^{\prime})\ket{\psi(0)}\ket{n}\bra{\psi(0)}\bra{n}\hat{H}_{I,\omega}(t^{\prime\prime})\ket{\psi(0)}\ket{n^{\prime}}.$

Similar to Ref. Sutherland  _et al._ (2022), upon plugging Eq. (25) into Eq.
(28), we are left with a sum of triple integrals that are proportional to:

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\\!\int^{\infty}_{0}\\!\\!\int^{t_{g}}_{0}\\!\int^{t_{s}}_{0}\\!d\omega
dt^{\prime}dt^{\prime\prime}\frac{S_{\omega}}{2}\Big{\\{}\\!\cos(\omega[t^{\prime\prime}\\!+\\!t^{\prime}])\\!+\\!\cos(\omega[t^{\prime\prime}\\!-\\!t^{\prime}])\Big{\\}},$

where $t_{s}\in\\{t_{g},t^{\prime}\\}$. In order to evaluate Eq. (C), we
perform the following manipulations:

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\\!\int^{\infty}_{-\infty}\\!\int^{t_{g}}_{0}\\!\int^{t_{s}}_{0}d\omega
dt^{\prime}dt^{\prime\prime}\frac{S_{\omega}}{4}\Big{\\{}\\!\cos(\omega[t^{\prime\prime}\\!+\\!t^{\prime}])\\!+\\!\cos(\omega[t^{\prime\prime}\\!-\\!t^{\prime}])\\!\Big{\\}}$
(30) $\displaystyle\simeq$
$\displaystyle\frac{S_{0}}{4}\\!\int^{\infty}_{-\infty}\\!\int^{t_{g}}_{0}\\!\int^{t_{s}}_{0}\\!d\omega
dt^{\prime}dt^{\prime\prime}\\!\Big{\\{}\\!\cos(\omega[t^{\prime\prime}\\!+\\!t^{\prime}])\\!+\\!\cos(\omega[t^{\prime\prime}\\!-\\!t^{\prime}])\Big{\\}}$
$\displaystyle=$ $\displaystyle\frac{\pi
S_{0}}{2}\int^{t_{g}}_{0}\int^{t_{s}}_{0}dt^{\prime}dt^{\prime\prime}\Big{\\{}\delta(t^{\prime\prime}+t^{\prime})+\delta(t^{\prime\prime}-t^{\prime})\Big{\\}},$

where we have assumed that we can approximate $S_{\omega}$ as $S_{0}$, pulling
it ouside the integral. We can now evaluate the required integrals, and, after
some algebra, obtain a final equation:

$\displaystyle\mathcal{I}$ $\displaystyle\simeq$
$\displaystyle\frac{2\eta\Omega_{p}^{2}t_{g}}{\Delta^{2}}\Big{(}2n+1+\frac{12\Omega_{e}^{2}}{\Delta^{2}}\Big{)}\lambda_{\hat{\sigma}_{\alpha}}^{2}$
(31) $\displaystyle=$
$\displaystyle\eta\sqrt{\frac{\Omega_{p}\theta^{3}}{32\pi
K\Omega_{e}^{3}}}\Big{(}2n+1+\frac{3\theta\Omega_{e}}{2\pi
K\Omega_{p}}\Big{)}\lambda_{\hat{\sigma}_{\alpha}}^{2},$

where we have defined $\eta\equiv\pi g_{d}^{2}S_{0}/2$ as the motional
dephasing rate, and set $\Delta\equiv\sqrt{8\pi K\Omega_{p}\Omega_{e}/\theta}$
as well as $t_{g}=2\pi K/\Delta$ in the second line.

### Heating

Finally, we discuss how heating affects gate fidelity. As in Ref. Sutherland
_et al._ (2022), we represent $\hat{H}_{e}$ as an E-field with frequency
$\omega$:

$\displaystyle\hat{H}_{t}$ $\displaystyle=$
$\displaystyle\hat{H}_{p}+\hat{H}_{e}$ (32) $\displaystyle=$
$\displaystyle\hbar\hat{A}\Big{(}\hat{a}^{\dagger}e^{i\Delta
t}+\hat{a}e^{-i\Delta t}\Big{)}+2\hbar g_{h}\cos(\omega
t)\Big{(}\hat{a}^{\dagger}e^{i\omega_{r}t}+e^{-i\omega_{r}t}\Big{)}$
$\displaystyle\simeq$
$\displaystyle\hbar\hat{A}\Big{(}\hat{a}^{\dagger}e^{i\Delta
t}+\hat{a}e^{-i\Delta t}\Big{)}+\hbar
g_{h}\Big{(}\hat{a}^{\dagger}e^{i(\omega_{r}-\omega)t}+e^{-i(\omega_{r}-\omega)t}\Big{)},$

where we have made made the rotating wave approximation in the second line. We
now transform into the interaction picture with respect to $\hat{H}_{e}$ using
the transformation:

$\displaystyle\hat{U}_{e}=\exp\Big{(}\beta(t)\hat{a}^{\dagger}-\hat{\beta}^{*}(t)\hat{a}\Big{)},$
(33)

where
$\beta(t)\equiv\frac{g_{h}}{\omega_{r}-\omega}(1-e^{i[\omega_{r}-\omega]t})$.
This gives:

$\displaystyle\hat{H}_{I}$ $\displaystyle=$
$\displaystyle\hbar\hat{A}\Big{(}\hat{a}^{\dagger}e^{i\Delta
t}+\hat{a}e^{-i\Delta t}\Big{)}+\hbar\hat{A}\Big{(}\beta(t)^{*}e^{i\Delta
t}+\beta(t)e^{-i\Delta t}\Big{)}$ (34) $\displaystyle\equiv$
$\displaystyle\hat{H}_{I,p}+\hat{H}_{I,e}$

showing the effect of the extraneous E-field is an added $\propto\hat{A}$
shift to the Hamiltonian in this frame. Importantly,
$\hat{H}_{I,p}=\hat{H}_{p}$ and $[\hat{H}_{I,p},\hat{H}_{I,e}]=0$. This allows
us to write the time propagator as $\hat{U}_{p}\hat{U}_{I,e}$, where:

$\displaystyle\hat{U}_{I,e}=\exp\Big{(}-\frac{i}{\hbar}\int^{t_{g}}_{0}dt^{\prime}\hat{H}_{I,e}(t^{\prime})\Big{)}.$
(35)

We Taylor expand $\hat{U}_{I,e}$ up to $2^{\text{nd}}$-order, and plug it into
Eq. (7) of the main text, which gives:

$\displaystyle\mathcal{I}_{\omega}$ $\displaystyle\simeq$ $\displaystyle
1-\sum_{n^{\prime}}\Big{|}\bra{\psi(0)}\bra{n^{\prime}}\hat{I}-i\hat{A}\int^{t_{g}}_{0}dt^{\prime}\Big{(}\beta^{*}(t^{\prime})e^{i\Delta
t^{\prime}}+\beta(t^{\prime})e^{-i\Delta t^{\prime}}\Big{)}$ (36)
$\displaystyle-\frac{\hat{A}^{2}}{2}\int^{t_{g}}_{0}\int^{t^{\prime}}_{0}dt^{\prime}dt^{\prime\prime}\Big{(}\beta^{*}(t^{\prime})e^{i\Delta
t^{\prime}}+\beta(t^{\prime})e^{-i\Delta
t^{\prime}}\Big{)}\Big{(}\beta^{*}(t^{\prime\prime})e^{i\Delta
t^{\prime\prime}}+\beta(t^{\prime\prime})e^{-i\Delta
t^{\prime\prime}}\Big{)}\ket{\psi(0)}\ket{n}\Big{|}^{2}$ $\displaystyle=$
$\displaystyle\lambda^{2}_{\hat{A}}\int^{t_{g}}_{0}\int^{t_{g}}_{0}dt^{\prime}dt^{\prime\prime}\Big{(}\beta^{*}(t^{\prime})e^{i\Delta
t^{\prime}}+\beta(t^{\prime})e^{-i\Delta
t^{\prime}}\Big{)}\Big{(}\beta^{*}(t^{\prime\prime})e^{i\Delta
t^{\prime\prime}}+\beta(t^{\prime\prime})e^{-i\Delta
t^{\prime\prime}}\Big{)}.$

We can plug this into Eq. (8) from the main text, and integrate over the
normalized spectral density $S_{\omega}$:

$\displaystyle\mathcal{I}$ $\displaystyle=$
$\displaystyle\lambda^{2}_{\hat{A}}\int^{\infty}_{0}\int^{t_{g}}_{0}\int^{t_{g}}_{0}d\omega
dt^{\prime}dt^{\prime\prime}S_{\omega}\frac{4g_{h}^{2}}{(\omega_{r}-\omega)^{2}}\Big{(}\cos(\Delta
t^{\prime})-\cos\Big{[}(\omega_{r}-\omega-\Delta)t^{\prime}\Big{]}\Big{)}\Big{(}\cos(\Delta
t^{\prime\prime})-\cos\Big{[}(\omega_{r}-\omega-\Delta)t^{\prime\prime}\Big{]}\Big{)}$
(37) $\displaystyle=$
$\displaystyle\lambda^{2}_{\hat{A}}\int^{\infty}_{0}\int^{t_{g}}_{0}\int^{t_{g}}_{0}d\omega
dt^{\prime}dt^{\prime\prime}S_{\omega}\frac{4g_{h}^{2}}{(\omega_{r}-\omega)^{2}}\cos([\omega_{r}-\omega-\Delta]t^{\prime})\cos([\omega_{r}-\omega-\Delta]t^{\prime\prime})$
$\displaystyle\simeq$
$\displaystyle\lambda^{2}_{\hat{A}}\int^{\infty}_{0}\int^{t_{g}}_{0}\int^{\infty}_{-\infty}d\omega
dt^{\prime}dt^{\prime\prime}S_{\omega}\frac{2g_{h}^{2}}{(\omega_{r}-\omega)^{2}}\cos([\omega_{r}-\omega-\Delta]t^{\prime})\cos([\omega_{r}-\omega-\Delta]t^{\prime\prime})$
$\displaystyle=$
$\displaystyle\lambda^{2}_{\hat{A}}\int^{\infty}_{0}\int^{t_{g}}_{0}d\omega
dt^{\prime}S_{\omega}\frac{4\pi
g_{h}^{2}}{(\omega_{r}-\omega)^{2}}\cos([\omega_{r}-\omega-\Delta]t^{\prime})\delta(\omega_{r}-\omega-\Delta)$
$\displaystyle=$
$\displaystyle\frac{4\dot{\bar{n}}t_{g}}{\Delta^{2}}\lambda_{\hat{A}}^{2},$

where we have substituted $\dot{\bar{n}}\equiv\pi g_{h}^{2}S_{\omega_{r}}$ in
the last line. Finally, we can substitute $\Delta\equiv\sqrt{8\pi
K\Omega_{p}\Omega_{e}/\theta}$ and
$\lambda^{2}_{\hat{A}}=\Omega_{p}^{2}\lambda^{2}_{\hat{\sigma}_{\alpha}}$,
giving:

$\displaystyle\mathcal{I}\simeq\dot{\bar{n}}\lambda^{2}_{\hat{\sigma}_{\alpha}}\sqrt{\frac{\Omega_{p}\theta^{3}}{8\pi
K\Omega_{e}^{3}}}.$ (38)