# Complex Hyperbolic Knowledge Graph Embeddings
with Fast Fourier Transform

Huiru Xiao  Xin Liu  Yangqiu Song
Hong Kong University of Science and Technology, Hong Kong SAR
<EMAIL_ADDRESS>
Ginny Y. Wong  Simon See
NVIDIA AI Technology Center (NVATIC), NVIDIA, Santa Clara, USA
<EMAIL_ADDRESS>

###### Abstract

The choice of geometric space for knowledge graph (KG) embeddings can have
significant effects on the performance of KG completion tasks. The hyperbolic
geometry has been shown to capture the hierarchical patterns due to its tree-
like metrics, which addressed the limitations of the Euclidean embedding
models. Recent explorations of the complex hyperbolic geometry further
improved the hyperbolic embeddings for capturing a variety of hierarchical
structures. However, the performance of the hyperbolic KG embedding models for
non-transitive relations is still unpromising, while the complex hyperbolic
embeddings do not deal with multi-relations. This paper aims to utilize the
representation capacity of the complex hyperbolic geometry in multi-relational
KG embeddings. To apply the geometric transformations which account for
different relations and the attention mechanism in the complex hyperbolic
space, we propose to use the fast Fourier transform (FFT) as the conversion
between the real and complex hyperbolic space. Constructing the attention-
based transformations in the complex space is very challenging, while the
proposed Fourier transform-based complex hyperbolic approaches provide a
simple and effective solution. Experimental results show that our methods
outperform the baselines, including the Euclidean and the real hyperbolic
embedding models.

##  1 Introduction

Figure 1: The summary of embedding spaces for hierarchical patterns and multi-
relation properties.

Knowledge graph (KG) representation learning is important to the KG inference
as well as the downstream tasks Nickel et al. (2016). It has been noticed that
the embedding space has significant effects on the performance of KG
completion tasks. Previous works have proposed the KG embedding models in
Euclidean space Bordes et al. (2013); Nickel et al. (2011); Yang et al.
(2015), complex Euclidean space Trouillon et al. (2016); Sun et al. (2019),
hyperbolic space Balazevic et al. (2019); Chami et al. (2020).111To avoid
wordiness, in this paper, we use hyperbolic space to refer to real hyperbolic
space, hyperbolic geometry to refer to real hyperbolic geometry, and
hyperbolic embeddings to refer to real hyperbolic embeddings. These models
learn the embeddings of the KG entities in the selected geometric spaces and
parameterize the relation representations as the geometric transformations,
such as translation, rotation, matrix multiplication, etc.

The Euclidean and complex Euclidean embedding models can capture relation
properties including symmetry/anti-symmetry, inversion, and composition, but
they cannot handle the transitive relations such as hypernymy. Generally, the
transitive relation forms a tree-like structure, for which hyperbolic geometry
has a more powerful representation capacity than Euclidean geometry because
the hyperbolic space can be regarded as a continuous approximation to trees
Krioukov et al. (2010).

However, most real-world graphs with transitivity do not necessarily form
exact tree structures since the transitive relations can lead to a globally
hierarchical structure with varying local structures, such as multitree
structures Griggs et al. (2012) and taxonomies Suchanek et al. (2007). Thus,
the hyperbolic geometry which resembles tree metrics still has limitations on
capturing various and flexible hierarchical structures. To tackle the
limitation of hyperbolic embeddings, a recent work Xiao et al. (2021) proposed
to explore the complex hyperbolic geometry to learn the embeddings of
hierarchical graphs. Due to the variable negative curvature Goldman (1999),
the complex hyperbolic space is more flexible in handling varying structures
while the tree-like properties are still retained. Despite the remarkable
improvements in single transitive relation inference, the complex hyperbolic
geometry has not been utilized for multi-relational embeddings.

In this paper, we are motivated to make use of the complex hyperbolic
geometry’s representation superiority in KGs. There are two main challenges in
extending the complex hyperbolic embeddings to multiple relations. First, the
geometric transformations in complex hyperbolic geometry are complicated and
challenging to optimize due to the numerical instabilities, making it
difficult to apply the complex geometric transformations for different
relations. Second, it is hard to build the neural network unit or layer in the
complex domain. Missing the complex attention mechanism would restrict the
parameterization capability and make the complex domain-based model difficult
to generalize to further downstream tasks.

To address the above problems, we propose a complex hyperbolic KG embedding
approach with the fast Fourier transform. Our approach can utilize the
representation capacity of the complex hyperbolic geometry as well as the
well-developed attention-based geometric transformations as relation
parameterization, while we borrow the fast Fourier transform (FFT) and inverse
fast Fourier transform (IFFT) to provide the conversion between the real and
complex hyperbolic space. We regard the complex hyperbolic embeddings in the
unit ball model (a projective geometry-based model to identify the complex
hyperbolic space) Goldman (1999) and the hyperbolic embeddings in the Poincaré
ball model (a model of the real hyperbolic space) Cannon et al. (1997) as
frequency domain and spatial domain respectively. Then FFT and IFFT enable us
to convert the embeddings between the two geometric spaces, accomplishing the
leverage of real hyperbolic transformations to the complex hyperbolic model.
The framework is simple and effective in learning the complex hyperbolic KG
representations.

Figure 1 summarizes the comparison among embedding spaces for hierarchical
patterns and multi-relation properties. In experiments, we evaluate our
approach on the KG link prediction task with two popular benchmarks—WN18RR
Bordes et al. (2013) and FB15k-237 Toutanova and Chen (2015). Empirical
results show that our Fourier transform-based complex hyperbolic KG embedding
approach outperforms the baseline models in other geometric spaces.

The code and data of our work are available at https://github.com/HKUST-
KnowComp/ComplexHyperbolicKGE.

##  2 Related Work

##### Euclidean KG embeddings.

The traditional KG embedding models first started with the Euclidean geometry
because of its convenient vectorial structure and closed-form computations
such as distance formula and inner-product. After the occurrence of the
translation-based models Bordes et al. (2013) and bilinear models Nickel et
al. (2011); Yang et al. (2015), several extensions Wang et al. (2014); Lin et
al. (2015); Ji et al. (2015) have been made to further develop the Euclidean
methods.

##### Complex Euclidean KG embeddings.

The follow-up works Trouillon et al. (2016); Hayashi and Shimbo (2017); Sun et
al. (2019) extended the traditional Euclidean models to complex hyperbolic
geometry. Specifically, ComplEx Trouillon et al. (2016) found that the
Hermitian dot product can effectively capture anti-symmetric relations while
retaining the efficiency benefits of the dot product. RotatE Sun et al. (2019)
defined each relation as a rotation in the complex vector space to infer
various relation patterns (symmetry/anti-symmetry, inversion, composition).
The effectiveness of these models revealed the potential of the complex
geometry.

##### Hyperbolic embeddings.

In recent years, the hyperbolic space attracted much attention for
representation learning since it can naturally characterize tree structures.
The hyperbolic embedding methods have developed from the single transitive
relation graphs Nickel and Kiela (2017, 2018); Sonthalia and Gilbert (2020) to
multi-relational KGs Balazevic et al. (2019); Chami et al. (2020). MurP
Balazevic et al. (2019) embedded the hierarchical multi-relational data in the
Poincaré ball model and learned relation-specific parameters by Möbius
operations. The state-of-the-art hyperbolic KG embedding models are a series
of hyperbolic transformation-based models RefH, RotH, and AttH Chami et al.
(2020), which utilize the geometric tree-like property to capture the
hierarchical structure naturally while using different geometric
transformations as well as attention mechanism to parameterize other relation
properties.

##### Lightweight Euclidean-based models.

Based on the hyperbolic embedding model RotH Chami et al. (2020), Wang et al.
(2021) developed two lightweight Euclidean-based models RotL and Rot2L, which
simplified the hyperbolic operations while keeping the flexible normalization
effect.

##### Complex hyperbolic embeddings.

Since many real-world hierarchically structured data such as taxonomies Miller
(1995); Suchanek et al. (2007) and multitree networks Griggs et al. (2012)
have varying local structures, they do not ubiquitously match the hyperbolic
geometry. Therefore, Xiao et al. (2021) explored the complex hyperbolic space
to embed a variety of hierarchical structures. The complex hyperbolic
embedding approach improved over the hyperbolic embedding models, but it only
focused on the representation of single-relational graphs instead of multi-
relational KGs.

##### Fourier Transform.

Fourier transform Heideman et al. (1984) converts a finite-sequence signal
from its temporal or spatial domain to the frequency domain. FFT Cooley et al.
(1969) is a practical algorithm that computes the discrete Fourier transform
(DFT) of a sequence. FFT and inverse FFT are widely used for many applications
Rockmore (2000); Burgess (2014) for their usefulness in signal processing as
well as computation efficiency. They are also used to efficiently perform
operations such as convolutions Smith et al. (1997); Kipf and Welling (2017)
and cross-correlations Bracewell and Bracewell (1986); Wang et al. (2018).
Hayashi and Shimbo (2019) also introduced the Fourier transform in KGE, where
the main idea was to use the block circulant matrices to parameterize
relations. While in our work, the Fourier transform is used to transform the
entity embeddings between different geometric spaces.

##  3 Preliminaries

###  3.1 Hyperbolic Geometry

The hyperbolic space is a homogeneous space with constant negative curvature
Cannon et al. (1997). In the hyperbolic space, the volume of a ball grows
exponentially with its radius. Contrastively, in the Euclidean space, the
curvature is constantly $0$, and the volume of a ball grows polynomially with
its radius. The exponential volume growth rate enables the hyperbolic space to
have powerful representation capability for tree structures since the number
of nodes grows exponentially with the depth in a tree, while the Euclidean
space is too flat and narrow to embed trees.

##### The Poincaré Ball Model.

To describe the hyperbolic space in mathematical language, there are several
models, among which the Poincaré ball model is popular for graph
representation Nickel and Kiela (2017); Chami et al. (2020) due to the
relatively convenient computations.

Denote the Poincaré ball model with constant negative curvature $-c$ as
$\mathcal{P}_{\mathbb{R}}^{N}=\\{\mathbf{x}\in\mathbb{R}^{N}:\|\mathbf{x}\|^{2}<\frac{1}{c}\\}$,
which represents the open $N$-dimension ball in the ambient Euclidean space
($\|\cdot\|$ is the Euclidean $L_{2}$ norm). By the framework of gyrovector
space Ungar (2008), the hyperbolic space can be formalized as an approximated
vectorial structure, where the Möbius addition Ganea et al. (2018) is used as
the vector addition in $\mathcal{P}_{\mathbb{R}}^{N}$:
$\mathbf{x}\oplus_{c}\mathbf{y}=\frac{(1+2c\mathbf{x}\mathbf{y}+c\|\mathbf{y}\|^{2})\mathbf{x}+(1-c\|\mathbf{x}\|^{2})\mathbf{y}}{1+2c\mathbf{x}\mathbf{y}+c^{2}\|\mathbf{x}\|^{2}\|\mathbf{y}\|^{2}}.$
(1) Then the distance function in $\mathcal{P}_{\mathbb{R}}^{N}$ is given by
$d_{\mathcal{P}}(\mathbf{x},\mathbf{y})=\frac{2}{\sqrt{c}}\text{artanh}(\sqrt{c}\|-\mathbf{x}\oplus_{c}\mathbf{y}\|).$
(2)

The practical computations in the hyperbolic space are often implemented using
the tangent space. For $\mathbf{x}\in\mathcal{P}_{\mathbb{R}}^{N}$, the
associated tangent space
$\mathcal{T}_{\mathbf{x}}\mathcal{P}_{\mathbb{R}}^{N}$ is an $N$-dimension
Euclidean space containing all tangent vectors passing through $\mathbf{x}$ do
Carmo (1976). The manifold of the Poincaré ball model and the tangent space
have closed-form maps to each other, which are defined as the exponential map
$\exp_{\mathbf{0}}^{c}(\mathbf{v}):\mathcal{T}_{\mathbf{0}}\mathcal{P}_{\mathbb{R}}^{N}\mapsto\mathcal{P}_{\mathbb{R}}^{N}$
and the logarithmic map
$\log_{\mathbf{0}}^{c}(\mathbf{y}):\mathcal{P}_{\mathbb{R}}^{N}\mapsto\mathcal{T}_{\mathbf{0}}\mathcal{P}_{\mathbb{R}}^{N}$:
$\displaystyle\exp_{\mathbf{0}}^{c}(\mathbf{v})=\tanh(\sqrt{c}\|\mathbf{v}\|)\frac{\mathbf{v}}{\sqrt{c}\|\mathbf{v}\|},$
(3)
$\displaystyle\log_{\mathbf{0}}^{c}(\mathbf{y})=\text{artanh}(\sqrt{c}\|\mathbf{y}\|)\frac{\mathbf{y}}{\sqrt{c}\|\mathbf{y}\|}.$
(4)

###  3.2 Complex Hyperbolic Geometry

The complex hyperbolic space is a homogeneous space of variable negative
curvature Goldman (1999). The complex hyperbolic space also maintains the
tree-like exponential volume growth property. From the properties of the
complex hyperbolic geometry, we see that the complex hyperbolic space can
naturally handle data with diverse local structures because of the non-
constant curvature while preserving the tree-like properties to better capture
the transitivity Xiao et al. (2021).

##### The Unit Ball Model.

The complex hyperbolic space’s ambient Hermitian vector space
$\mathbb{C}^{n,1}$ is the complex Euclidean space $\mathbb{C}^{n+1}$ endowed
with some Hermitian form
$\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{z},\mathbf{w}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$,
where $\mathbf{z},\mathbf{w}\in\mathbb{C}^{n+1}$. Different choices of the
Hermitian form
$\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{z},\mathbf{w}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
correspond to different models of complex hyperbolic geometry, such as the
unit ball model and the Siegel domain model Parker (2003). Here we choose the
standard Hermitian form which derives the unit ball model:
$\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{z},\mathbf{w}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=z_{1}\overline{w_{1}}+\cdots+z_{n}\overline{w_{n}}-z_{n+1}\overline{w_{n+1}},$
(5) where $\overline{w}$ denotes the complex conjugate of $w$. Then via the
projective geometry Goldman (1999), the formula of the unit ball model
is:222Here we denote the dimension as $n$ instead of $N$ as in Section 3.1
since the dimensions of the two models can differ in Fourier transform, which
we will see in Section 4.2.
$\mathcal{B}_{\mathbb{C}}^{n}=\\{(z_{1},\cdots,z_{n},1)||z_{1}|^{2}+\cdots+|z_{n}|^{2}<1\\}.$
(6) The metric on $\mathcal{B}_{\mathbb{C}}^{n}$ is Bergman metric, which
takes the formula below in 2-d case:
$ds^{2}=\frac{-4}{\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{z},\mathbf{z}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}^{2}}\det\begin{bmatrix}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{z},\mathbf{z}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}&\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}d\mathbf{z},\mathbf{z}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\\\
\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{z},d\mathbf{z}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}&\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}d\mathbf{z},d\mathbf{z}\mathclose{\hbox{\set<EMAIL_ADDRESS>(7) Then the distance function on $\mathcal{B}_{\mathbb{C}}^{n}$ can be
derived from the metric tensor:
$d_{\mathcal{B}}(\mathbf{z},\mathbf{w})=\text{arcosh}\Big{(}2\frac{\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{z},\mathbf{w}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{w},\mathbf{z}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}{\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{z},\mathbf{z}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{w},\mathbf{w}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}-1\Big{)}.$
(8)

Although the unit ball model and the Poincaé ball model look similar in
mathematical formulae, they have many differences in properties since the
complex hyperbolic geometry and hyperbolic geometry are intrinsically
different geometries. Not only the variable/constant negative curvature but
also their distance functions and the geometric computations vary with each
other.

Figure 2: The inference process for a query $(h,r)$ in our proposed complex
hyperbolic KG embedding framework. $\mathbf{h}$ and $\tilde{\mathbf{h}}$ are
the head entity embeddings in different spaces. $\mathbf{q}$ and
$\tilde{\mathbf{q}}$ are the query embeddings in different spaces.
$\mathcal{B}_{\mathbb{C}}^{n}$ denotes the $n$-dimension unit ball model in
the complex hyperbolic space while $\mathcal{P}_{\mathbb{R}}^{2(n-1)}$ denotes
the $2(n-1)$-dimension Poincaré ball model in the hyperbolic space.

##  4 Approach

Given the KG with entity set $V=\\{e_{j}\\}_{j=1}^{m}$, relation set
$R=\\{r_{j}\\}_{j=1}^{k}$, and triplet set $F=\\{(h,r,t)|h,t\in V,r\in\ R\\}$,
the link prediction task aims to predict the tail entity $t$ for each test
query $(h,r)$. To train the models for a higher-quality inference, we learn
the entity embeddings $\\{\mathbf{e}_{j}\\}_{j=1}^{m}$ in the unit ball model,
while parameterizing the relations by the hyperbolic transformations RotH,
RefH, and the hyperbolic attention-based model AttH Chami et al. (2020). We
construct the conversion between the two geometries through Fourier transform.
Figure 2 presents the overview of our framework. In this section, we first
present the attention-based hyperbolic transformations as relation
parameterization, then introduce the Fourier transform as the conversion
between complex hyperbolic domain and hyperbolic domain, followed by the
details of our framework.

###  4.1 Relation Parameterization by Hyperbolic Transformations and Attention

In our work, we adopt the attention-based hyperbolic transformations developed
by Chami et al. (2020) as the relation parameterization in the Poincaré ball
model. Here we present the models RotH, RefH, and AttH.

RotH and RefH represent rotations and reflections in the hyperbolic space
respectively. They can be modeled using the Givens transformation matrices,
which take the following formula:
$G^{\pm}(\theta)=\begin{bmatrix}\cos(\theta)&\mp\sin(\theta)\\\
\sin(\theta)&\pm\cos(\theta)\end{bmatrix}.$ (9) Let
$\Theta_{r}\equiv(\theta_{r,j})_{j\in\\{1,\dots,\frac{N}{2}\\}}$ and
$\Phi_{r}\equiv(\phi_{r,j})_{j\in\\{1,\dots,\frac{N}{2}\\}}$ be the relation
parameters, where $N$ denotes the dimension of the embedding space, then the
hyperbolic rotation and reflection are parameterized by the block-diagonal
Givens matrices:
$\displaystyle\text{Rot}(\Theta_{r})=\text{diag}(G^{+}(\theta_{r,1}),\dots,G^{+}(\theta_{r,\frac{N}{2}})),$
(10)
$\displaystyle\text{Ref}(\Phi_{r})=\text{diag}(G^{-}(\phi_{r,1}),\dots,G^{-}(\phi_{r,\frac{N}{2}})).$
(11) Thus, given a query $(h,r)$, RotH and RefH apply hyperbolic rotation and
reflection with relation-specific parameters $\mathbf{r}$ to the head
embeddings $\tilde{\mathbf{h}}\in\mathcal{P}_{\mathbb{R}}^{N}$, and then get
the query embeddings:
$\text{RotH}(\tilde{\mathbf{h}},\mathbf{r})=\text{Rot}(\Theta_{r})\tilde{\mathbf{h}},\text{RefH}(\tilde{\mathbf{h}},\mathbf{r})=\text{Ref}(\Phi_{r})\tilde{\mathbf{h}}.$
(12)

In order to handle multiple relation properties, AttH combines the above two
representations using the hyperbolic attention and adding a hyperbolic
translation $\mathbf{r}_{r}$ by Möbius addition (Eq. (1)):
$\text{AttH}(\tilde{\mathbf{h}},\mathbf{r})=\text{Att}(\text{RotH}(\tilde{\mathbf{h}},\mathbf{r}),\text{RefH}(\tilde{\mathbf{h}},\mathbf{r});\mathbf{a}_{r})\oplus_{c_{r}}\mathbf{r}_{r},$
(13) where $c_{r}$ is the curvature parameter of $r$. RotH, RefH, and AttH
leverage the trainable curvature so that each relation has its own curvature
parameterization. The hyperbolic attention is constructed from the exponential
map (Eq. (3)) of the average in the tangent space Chami et al. (2019); Liu et
al. (2019). More details about the hyperbolic attention mechanism can be
referred to Chami et al. (2020).

###  4.2 Conversion by Fourier Transform

The orthonormal Discrete Fourier Transform (DFT) $\mathcal{F}$ and its inverse
(IDFT) $\mathcal{F}^{-1}$ between two finite complex-valued sequences
$\\{x_{p}\\}_{p=0}^{N-1}$ and $\\{z_{q}\\}_{q=0}^{N-1}$ take the following
formulae:
$\displaystyle
z_{q}=\mathcal{F}\\{\mathbf{x}\\}_{q}=\frac{1}{\sqrt{N}}\sum_{p=0}^{N-1}x_{p}\cdot
e^{-i\frac{2\pi}{N}pq},$ (14) $\displaystyle
x_{p}=\mathcal{F}^{-1}\\{\mathbf{z}\\}_{p}=\frac{1}{\sqrt{N}}\sum_{q=0}^{N-1}z_{q}\cdot
e^{i\frac{2\pi}{N}pq}.$ (15)

In our models, we transform the unit ball embeddings
$\mathbf{z}\in\mathcal{B}_{\mathbb{C}}^{n}$ to the Poincaré ball embeddings
$\mathbf{x}\in\mathcal{P}_{\mathbb{R}}^{N}$ back and forth. Note that the
Poincaré ball embeddings $\mathbf{x}=\\{x_{0},\dots,x_{N-1}\\}$ are all real
numbers, then $\mathcal{F}\\{\mathbf{x}\\}$ is symmetric:
$z_{q}=\overline{z_{-q\text{ mod }N}},\forall q\in\\{0,\dots,N-1\\}$. The
dimension $N$ is even because of the construction of diagonal Givens
transformations (Eqs. (10) and (11)). Then it follows that $z_{0}$ and
$z_{\frac{N}{2}}$ are real-valued, and the remainder of
$\mathcal{F}\\{\mathbf{x}\\}$ is completely specified by just $\frac{N}{2}-1$
complex numbers. Therefore, in practical algorithms, we set $N=2(n-1)$, i.e.,
we use the first $\frac{N}{2}+1$ elements $\\{z_{0},\dots,z_{\frac{N}{2}}\\}$
as the transformed unit ball embeddings.

We notice that the Fourier transform is not simply a conversion technique
between complex and real domains. Performing circular convolutions in one
domain equals the multiplication in another domain Rader (1972); Smith et al.
(1997):
$\displaystyle\\{x\star y\\}[n]$
$\displaystyle\triangleq\sum_{p=0}^{N-1}x_{p}\cdot y_{(n-p)\text{ mod }N}$
$\displaystyle=\mathcal{F}^{-1}\\{\mathcal{F}\\{\mathbf{x}\\}\cdot\mathcal{F}\\{\mathbf{y}\\}\\}_{n}.$
(16) Its effectiveness provides useful transforms, while its practicability is
guaranteed by FFT and IFFT, where the fast Fourier algorithms can reduce the
computing complexity from $O(N^{2})$ to $O(N\log N)$ Cooley and Tukey (1965).

###  4.3 Complex Hyperbolic Embeddings with Fourier Transform

For a query $(h,r)$, Figure 2 briefly describes the inference process in our
framework. The head embedding $\mathbf{h}\in\mathcal{B}_{\mathbb{C}}^{n}$ is
in the $n$-dimension unit ball model. We apply inverse Fourier transform (Eq.
(15)) to $\mathbf{h}$ and get the transformed head embeddings in the
$2(n-1)$-dimension Poincaré ball model
$\tilde{\mathbf{h}}\in\mathcal{P}_{\mathbb{R}}^{2(n-1)}$:
$\tilde{\mathbf{h}}=\mathcal{F}^{-1}\\{\mathbf{h}\\}$.

Then we can apply RotH, RefH (Eq. (12)), and AttH (Eq. (13)) to get the query
embedding $\tilde{\mathbf{q}}$:
$\tilde{\mathbf{q}}_{\text{ModelH}}=\text{ModelH}(\tilde{\mathbf{h}},\mathbf{r})$
(17) where $\text{ModelH}=\\{\text{RotH},\text{RefH},\text{AttH}\\}$
represents the corresponding hyperbolic embedding model.

The query embedding $\tilde{\mathbf{q}}$ then gets transformed back to the
unit ball model by Fourier transform (Eq. (14)):
$\mathbf{q}=\mathcal{F}\\{\tilde{\mathbf{q}}\\}$.

Finally, we use the following score function Balazevic et al. (2019) to
measure the likelihood of a triplet $(h,r,t)$:
$s(h,r,t)=-d_{\mathcal{B}}(\mathbf{q},\mathbf{t})^{2}+b_{h}+b_{t},$ (18) where
$d_{\mathcal{B}}(\mathbf{q},\mathbf{t})$ is the unit ball model distance (Eq.
(8)) between the tail embeddings $\mathbf{t}$ and the query embeddings
$\mathbf{q}$ computed by the above procedures. $b_{h}$ and $b_{t}$ are bias
terms of the head and tail entity. The model learns the embeddings by
maximizing the score functions of the training triplets, i.e., making the
query embeddings of $(h,r)$ close with its ground truth tail embeddings. The
score function is also used to predict the test data.

In summary, our model parameters include the entity parameters:
$\\{\mathbf{e}_{j}\\}_{j=1}^{m}\in\mathcal{B}_{\mathbb{C}}^{n}$ (embeddings),
$\\{b_{j}\\}_{j=1}^{m}$ (biases); and the relation parameters: $\Theta_{r}$
(rotations), $\Phi_{r}$ (reflections), $\mathbf{r}_{r}$ (translations),
$\mathbf{a}_{r}$ (attention), $c_{r}$ (curvature). The FFT and IFFT can be
computed very efficiently, so our models have almost the same computation cost
with the base models RotH, RefH, and AttH, while we utilize a more powerful
representation geometry to improve the embedding quality.

##  5 Experiments

In this section, we evaluate our approaches on the KG link prediction task. We
show that our complex hyperbolic embedding models outperform the baseline
methods based on other geometric spaces.

###  5.1 Experimental Settings

####  5.1.1 Data

| $|V|$ | $|R|$ | $|F|$ | $\xi_{G}$
---|---|---|---|---
WN18RR | 40,943 | 11 | 93,003 | -2.54
FB15k-237 | 14,541 | 237 | 310,079 | -0.65
Table 1: Data statistics. $|V|$, $|R|$, $|F|$ denote # entities, # relations,
# triplets. $\xi_{G}$ is the global graph curvature.

We use two widely-used KG benchmarks to evaluate the embedding models. The
data statistics are provided in Table 1. The global graph curvature $\xi_{G}$
Gu et al. (2019) is provided in Chami et al. (2020), which is a distance-based
measure to estimate the tree-likeness of graphs. A lower $\xi_{G}$ corresponds
to a more tree-like graph.

##### WN18RR.

WN18RR Bordes et al. (2013) is a knowledge graph dataset created from WN18,
which is a subset of WordNet Miller (1995). WordNet is a large lexical
database with hypernymy relation, so WN18RR inherits the underlying
hierarchical structure.

##### FB15k-237.

FB15k-237 Toutanova and Chen (2015) is a knowledge graph dataset created from
FB15k, which is derived from Freebase. Compared with WN18RR, FB15k-237 has
much more relations and various relation properties, resulting in a more
flexible structure, which can be reflected by the larger $\xi_{G}$.

We follow the train-valid-test data splitting of previous works Chami et al.
(2020); Wang et al. (2021), where # train-valid-test triplets are
$86,845$-$3,034$-$3,134$ for WN18RR and $272,115$-$17,535$-$20,466$ for
FB15k-237. The data can be obtained in the public repository of Chami et al.
(2020).333https://github.com/HazyResearch/KGEmb.

####  5.1.2 Baselines

The following KG embedding baselines are compared with our approaches
(FFTRefH, FFTRotH, FFTAttH): complex Euclidean embedding models ComplEx-N3
Lacroix et al. (2018) and RotatE Sun et al. (2019); hyperbolic embedding
models MuRP Balazevic et al. (2019), RefH, RotH, AttH Chami et al. (2020); the
Euclidean analogues of the hyperbolic methods MuRE, RefE, RotE, AttE; the
lightweight Euclidean-based models RotL, Rot2L Wang et al. (2021).

####  5.1.3 Training and Evaluation

For the baselines, we either take the results from the original papers Chami
et al. (2020); Wang et al. (2021) (Table 2) or use their released best
hyperparameters as well as their open-source codes to train their models
(Table 3, 4, and 5). For our approaches, we tune the hyperparameters by grid
search on each validation set in $32$-dimension complex hyperbolic space,
which are given in Appendix A. Our embedding models are trained by optimizing
the full cross-entropy loss with uniform negative sampling. We conduct all the
experiments on four NVIDIA GTX 1080Ti GPUs with 11GB memory each.

We use the mean reciprocal rank (MRR) and the proportion of correct types that
rank no larger than N (Hits@N) as our evaluation metrics, which are widely
used for evaluating link prediction. We follow the filtered evaluation setting
Bordes et al. (2013) to filter out the true triplets during evaluation. In all
experiments, each running is executed five times and the mean values of
results are reported.

###  5.2 Overall Results

|  | WN18RR | FB15k-237
---|---|---|---
Geometry | Model | MRR | Hits@1 | Hits@3 | Hits@10 | MRR | Hits@1 | Hits@3 | Hits@10
$\mathbb{C}^{n}$ | ComplEx-N3 | 0.420 | 0.390 | 0.420 | 0.460 | 0.294 | 0.211 | 0.322 | 0.463
RotatE | 0.387 | 0.330 | 0.417 | 0.491 | 0.290 | 0.208 | 0.316 | 0.458
$\mathbb{R}^{n}$ | MuRE | 0.458 | 0.421 | 0.471 | 0.525 | 0.313 | 0.226 | 0.340 | 0.489
RefE | 0.455 | 0.419 | 0.470 | 0.521 | 0.302 | 0.216 | 0.330 | 0.474
RotE | 0.463 | 0.426 | 0.477 | 0.529 | 0.307 | 0.220 | 0.337 | 0.482
AttE | 0.456 | 0.419 | 0.471 | 0.526 | 0.311 | 0.223 | 0.339 | 0.488
RotL | 0.469 | 0.426 | - | 0.550 | 0.320 | 0.229 | - | 0.500
Rot2L | 0.475 | 0.434 | - | 0.554 | 0.326 | 0.237 | - | 0.503
$\mathcal{P}^{n}_{\mathbb{R}}$ | MuRP | 0.465 | 0.420 | 0.484 | 0.544 | 0.323 | 0.235 | 0.353 | 0.501
RefH | 0.447 | 0.408 | 0.464 | 0.518 | 0.312 | 0.224 | 0.342 | 0.489
RotH | 0.472 | 0.428 | 0.490 | 0.553 | 0.314 | 0.223 | 0.346 | 0.497
AttH | 0.466 | 0.419 | 0.484 | 0.551 | 0.324 | 0.236 | 0.354 | 0.501
$\mathcal{B}^{n}_{\mathbb{C}}$ | FFTRefH | 0.463 | 0.412 | 0.480 | 0.547 | 0.325 | 0.234 | 0.359 | 0.508
FFTRotH | 0.484 | 0.437 | 0.502 | 0.572 | 0.319 | 0.228 | 0.352 | 0.500
FFTAttH | 0.476 | 0.432 | 0.494 | 0.558 | 0.331 | 0.239 | 0.365 | 0.517
Table 2: Evaluation of link prediction task in $32$-dimension embedding spaces. The best results are shown in boldface. The second best results are underlined. Relation | $Khs_{G}$ | $\xi_{G}$ | # Triplets | RefH | RotH | AttH | FFTRefH | FFTRotH | FFTAttH
---|---|---|---|---|---|---|---|---|---
member meronym | 1.00 | -2.90 | 253 | 0.316 | 0.383 | 0.383 | 0.366 | 0.411 | 0.402
hypernym | 1.00 | -2.46 | 1,251 | 0.218 | 0.268 | 0.257 | 0.249 | 0.283 | 0.268
has part | 1.00 | -1.43 | 172 | 0.259 | 0.303 | 0.294 | 0.287 | 0.347 | 0.335
instance hypernym | 1.00 | -0.82 | 122 | 0.471 | 0.480 | 0.471 | 0.496 | 0.503 | 0.499
member of domain region | 1.00 | -0.78 | 26 | 0.417 | 0.417 | 0.404 | 0.436 | 0.423 | 0.410
member of domain usage | 1.00 | -0.74 | 24 | 0.424 | 0.451 | 0.445 | 0.431 | 0.458 | 0.424
synset domain topic of | 0.99 | -0.69 | 114 | 0.352 | 0.417 | 0.406 | 0.436 | 0.475 | 0.444
derivationally related form | 0.07 | -3.84 | 1,074 | 0.960 | 0.964 | 0.965 | 0.968 | 0.969 | 0.967
also see | 0.36 | -2.09 | 56 | 0.664 | 0.640 | 0.649 | 0.684 | 0.675 | 0.676
similar to | 0.07 | -1.00 | 3 | 1.000 | 1.000 | 0.944 | 1.000 | 1.000 | 1.000
verb group | 0.07 | -0.50 | 39 | 0.974 | 0.974 | 0.970 | 0.974 | 0.974 | 0.970
Table 3: Results of Hits@10 for WN18RR relations in $32$-dimension embedding
spaces. Higher $Khs_{G}$ and lower $\xi_{G}$ correspond to more tree-like. #
Triplets means the triplet count of each relation in test set. The best
results are shown in boldface.

Table 2 presents the results in $32$-dimension embedding spaces. We strictly
follow the experimental setting and data splitting of the previous works Chami
et al. (2020); Wang et al. (2021). The results of the baselines are taken from
the original papers, where RotL and Rot2L do not report the Hits@3 scores,
thus we leave them blank.

The results show that our Fourier transform-based complex hyperbolic
approaches have the best performance on the link prediction task,
demonstrating the powerful representation capacity of the complex hyperbolic
geometry and the effectiveness of Fourier transform. Specifically, FFTRotH
achieves the best results on WN18RR, while FFTAttH outperforms other methods
on FB15k-237. The relations in WN18RR typically have transitivity property, in
which case the hyperbolic rotation takes more advantages. FB15k-237 is a more
challenging link prediction dataset since it has more relations and varying
structures as well as a larger scale of triplets. Therefore, the attention
mechanism helps to generalize the hyperbolic transformations to multiple
relation properties.

From Table 2, we see that the traditional complex Euclidean models (ComplEx-N3
and RotatE) do not have competitive performance with the hyperbolic KG
embedding models or their Euclidean analogues. The hyperbolic methods (MuRP,
RefH, RotH, and AttH) have better results than their Euclidean analogues
(MuRE, RefE, RotE, and AttE), revealing the improvements of the hyperbolic
geometry over Euclidean geometry in low-dimensional KG representation. RotL
replaced the Möbius addition of RotH with a new flexible addition operation,
while Rot2L further utilizes two stacked rotation-translation layers in the
Euclidean space. The two Euclidean-based methods outperform their base model
RotH by adapting a lightweight architecture. However, they still cannot
achieve as promising results as the complex hyperbolic embedding approaches.

###  5.3 Exploring the Relations

|  $8$-dimension |  $16$-dimension |  $32$-dimension |  $64$-dimension
---|---|---|---|---
Model | MRR | Hits@1 | MRR | Hits@1 | MRR | Hits@1 | MRR | Hits@1
RefH | 0.190 | 0.140 | 0.401 | 0.360 | 0.447 | 0.408 | 0.475 | 0.433
RotH | 0.220 | 0.154 | 0.417 | 0.370 | 0.472 | 0.428 | 0.488 | 0.442
AttH | 0.158 | 0.102 | 0.404 | 0.356 | 0.466 | 0.419 | 0.476 | 0.430
FFTRefH | 0.369 | 0.319 | 0.447 | 0.408 | 0.463 | 0.412 | 0.469 | 0.425
FFTRotH | 0.411 | 0.358 | 0.468 | 0.423 | 0.484 | 0.437 | 0.488 | 0.442
FFTAttH | 0.387 | 0.330 | 0.459 | 0.415 | 0.476 | 0.432 | 0.479 | 0.435
Table 4: Results of MRR and Hits@1 in different embedding dimensions on WN18RR. The best results are shown in boldface. |  $8$-dimension |  $16$-dimension |  $32$-dimension |  $64$-dimension
---|---|---|---|---
Model | MRR | Hits@1 | MRR | Hits@1 | MRR | Hits@1 | MRR | Hits@1
RefH | 0.267 | 0.188 | 0.288 | 0.204 | 0.312 | 0.224 | 0.328 | 0.237
RotH | 0.269 | 0.187 | 0.289 | 0.204 | 0.314 | 0.223 | 0.323 | 0.231
AttH | 0.276 | 0.194 | 0.298 | 0.212 | 0.324 | 0.236 | 0.333 | 0.240
FFTRefH | 0.281 | 0.198 | 0.304 | 0.217 | 0.325 | 0.234 | 0.337 | 0.242
FFTRotH | 0.287 | 0.201 | 0.306 | 0.217 | 0.319 | 0.228 | 0.323 | 0.231
FFTAttH | 0.295 | 0.209 | 0.314 | 0.224 | 0.331 | 0.239 | 0.339 | 0.245
Table 5: Results of MRR and Hits@1 in different embedding dimensions on
FB15k-237. The best results are shown in boldface.

In Section 5.2, we see that the overall results of Fourier transform-based
complex hyperbolic methods surpass their corresponding hyperbolic methods.
Here we explore their performance on each relation of WN18RR. For each
relation, we give their statistics of Krackhardt hierarchy score ($Khs_{G}$)
Balazevic et al. (2019) and estimated graph curvature $\xi_{G}$ Chami et al.
(2019). Higher $Khs_{G}$ and lower $\xi_{G}$ mean more tree-like, i.e., the
relation is more transitive. We report the Hits@10 scores in Table 3.

We find that for most relations, FFT complex hyperbolic methods outperform
hyperbolic methods significantly. For the transitive relations such as member
meronym, hypernym, has part, etc, rotation has much better results than
reflection. This phenomenon is consistent with the analysis of previous work
Chami et al. (2020), where they found hyperbolic rotations work better on
anti-symmetric relations while hyperbolic reflections encode symmetric
relations better. Transitivity fulfills anti-symmetry naturally, so rotation
gains higher scores (RotH$>$RefH, FFTRotH$>$FFTRefH). For the symmetric
relation such as also see, reflection outperforms rotation (RefH$>$RotH,
FFTRefH$>$FFTRotH). Since most relations in WN18RR exhibit transitivity, the
rotation models have better performance than the reflection models in overall
results (Table 2). Regardless of the relation properties, our approaches
improve the corresponding hyperbolic methods largely, except for the relations
with few test triplets such as similar to and verb group, where they all have
close-to-$1$ Hits@10 results.

###  5.4 Exploring the Embedding Dimensions

In this section, we explore the performance of Fourier transform-based complex
hyperbolic approaches and the corresponding hyperbolic methods in various
embedding dimensions. The results are presented in Table 4 and 5. We find that
when the embedding dimension is small, the complex hyperbolic approaches
outperform the hyperbolic base models by a large margin. Remarkably, FFTRotH
improves over RotH by around $100\%$ in $8$-dimension on WN18RR. With the
increase of the embedding dimension, their predictions get more and more
similar and gradually converge. The results reveal the effectiveness of our
approaches especially in small dimensions, demonstrating the strong
representation capacity of complex hyperbolic geometry.

##  6 Conclusion and Future Work

In this work, we explore the complex hyperbolic geometry for multi-relational
KG embeddings. The whole framework utilizes the Fourier transform as the
efficient conversion between geometric spaces. With the aid of the Fourier
transform, the complex hyperbolic embeddings can be transformed into the real
domain and be capable of applying real hyperbolic transformations, which
enables our approach to take the advantages of both the powerful complex
hyperbolic geometry and the attention-based real hyperbolic transformations.
Experiments show that the Fourier transform-based complex hyperbolic embedding
models can effectively learn the KG embeddings and outperform the baseline
models of other spaces in the link prediction task. We believe our proposed
approach not only provides a novel and interesting representation learning
framework for KGs but also potentially inspires the learning algorithms for
more general multi-relational data and contributes to improvements on more
downstream tasks.

## Limitations

##### Limited improvements in high dimensions.

Although our approaches can significantly outperform the baselines in low-
dimensional KG embedding setting, we find that our approaches would get
converge and have close results with the hyperbolic base models in
sufficiently high dimensions. For example, in Table 4, FFTRotH and RotH have
the same results in $64$-dimension embedding spaces on WN18RR.

This issue has been observed previously Nickel and Kiela (2017); Chami et al.
(2020), though their comparisons are established between hyperbolic space and
Euclidean space. The representation capacity gap between geometric spaces is
distinctly revealed in low dimensions. The gap may get eliminated to some
extent by increasing the dimension. The complex hyperbolic geometry and
hyperbolic geometry usually converge their results in much lower dimensions
than Euclidean geometry because of the exponential growth property, resulting
in the limited improvements in high dimensions.

## Acknowledgements

The authors of this paper were supported by the NSFC Fund (U20B2053) from the
NSFC of China, the RIF (R6020-19 and R6021-20) and the GRF (16211520) from RGC
of Hong Kong, the MHKJFS (MHP/001/19) from ITC of Hong Kong and the National
Key R&D Program of China (2019YFE0198200) with special thanks to HKMAAC and
CUSBLT, and the Jiangsu Province Science and Technology Collaboration Fund
(BZ2021065). We also thank the support from the UGC Research Matching Grants
(RMGS20EG01-D, RMGS20CR11, RMGS20CR12, RMGS20EG19, RMGS20EG21). We also thank
the support from NVIDIA AI Technology Center (NVAITC).

## References

  * Balazevic et al. (2019) Ivana Balazevic, Carl Allen, and Timothy M. Hospedales. 2019.  Multi-relational poincaré graph embeddings.  In _NeurIPS_ , pages 4465–4475.
  * Bordes et al. (2013) Antoine Bordes, Nicolas Usunier, Alberto García-Durán, Jason Weston, and Oksana Yakhnenko. 2013.  Translating embeddings for modeling multi-relational data.  In _NIPS_ , pages 2787–2795.
  * Bracewell and Bracewell (1986) Ronald Newbold Bracewell and Ronald N Bracewell. 1986.  _The Fourier transform and its applications_ , volume 31999.  McGraw-Hill New York.
  * Burgess (2014) Richard James Burgess. 2014.  _The history of music production_.  Oxford University Press.
  * Cannon et al. (1997) James W Cannon, William J Floyd, Richard Kenyon, Walter R Parry, et al. 1997.  Hyperbolic geometry.  _Flavors of geometry_ , 31(59-115):2.
  * Chami et al. (2020) Ines Chami, Adva Wolf, Da-Cheng Juan, Frederic Sala, Sujith Ravi, and Christopher Ré. 2020.  Low-dimensional hyperbolic knowledge graph embeddings.  In _ACL_ , pages 6901–6914. Association for Computational Linguistics.
  * Chami et al. (2019) Ines Chami, Zhitao Ying, Christopher Ré, and Jure Leskovec. 2019.  Hyperbolic graph convolutional neural networks.  In _NeurIPS_ , pages 4869–4880.
  * Cooley and Tukey (1965) James W Cooley and John W Tukey. 1965.  An algorithm for the machine calculation of complex fourier series.  _Mathematics of computation_ , 19(90):297–301.
  * Cooley et al. (1969) JW Cooley, P Lewis, and P Welch. 1969.  The finite fourier transform.  _IEEE Transactions on audio and electroacoustics_ , 17(2):77–85.
  * do Carmo (1976) Manfredo P. do Carmo. 1976.  _Differential geometry of curves and surfaces_.  Prentice Hall.
  * Ganea et al. (2018) Octavian-Eugen Ganea, Gary Bécigneul, and Thomas Hofmann. 2018.  Hyperbolic neural networks.  In _NeurIPS_ , pages 5350–5360.
  * Goldman (1999) William Mark Goldman. 1999.  _Complex hyperbolic geometry_.  Oxford University Press.
  * Griggs et al. (2012) Jerrold R. Griggs, Wei-Tian Li, and Linyuan Lu. 2012.  Diamond-free families.  _J. Comb. Theory, Ser. A_ , 119(2):310–322.
  * Gu et al. (2019) Albert Gu, Frederic Sala, Beliz Gunel, and Christopher Ré. 2019.  Learning mixed-curvature representations in product spaces.  In _ICLR (Poster)_. OpenReview.net.
  * Hayashi and Shimbo (2017) Katsuhiko Hayashi and Masashi Shimbo. 2017.  On the equivalence of holographic and complex embeddings for link prediction.  In _ACL (2)_ , pages 554–559. Association for Computational Linguistics.
  * Hayashi and Shimbo (2019) Katsuhiko Hayashi and Masashi Shimbo. 2019.  A non-commutative bilinear model for answering path queries in knowledge graphs.  In _EMNLP/IJCNLP (1)_ , pages 2422–2430. Association for Computational Linguistics.
  * Heideman et al. (1984) Michael Heideman, Don Johnson, and Charles Burrus. 1984.  Gauss and the history of the fast fourier transform.  _IEEE ASSP Magazine_ , 1(4):14–21.
  * Ji et al. (2015) Guoliang Ji, Shizhu He, Liheng Xu, Kang Liu, and Jun Zhao. 2015.  Knowledge graph embedding via dynamic mapping matrix.  In _ACL (1)_ , pages 687–696. The Association for Computer Linguistics.
  * Kipf and Welling (2017) Thomas N. Kipf and Max Welling. 2017.  Semi-supervised classification with graph convolutional networks.  In _ICLR (Poster)_. OpenReview.net.
  * Krioukov et al. (2010) Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. 2010.  Hyperbolic geometry of complex networks.  _Physical Review E_ , 82(3):036106.
  * Lacroix et al. (2018) Timothée Lacroix, Nicolas Usunier, and Guillaume Obozinski. 2018.  Canonical tensor decomposition for knowledge base completion.  In _ICML_ , volume 80 of _Proceedings of Machine Learning Research_ , pages 2869–2878. PMLR.
  * Lin et al. (2015) Yankai Lin, Zhiyuan Liu, Maosong Sun, Yang Liu, and Xuan Zhu. 2015.  Learning entity and relation embeddings for knowledge graph completion.  In _AAAI_ , pages 2181–2187. AAAI Press.
  * Liu et al. (2019) Qi Liu, Maximilian Nickel, and Douwe Kiela. 2019.  Hyperbolic graph neural networks.  In _NeurIPS_ , pages 8228–8239.
  * Miller (1995) George A. Miller. 1995.  Wordnet: A lexical database for english.  _Commun. ACM_ , 38(11):39–41.
  * Nickel and Kiela (2017) Maximilian Nickel and Douwe Kiela. 2017.  Poincaré embeddings for learning hierarchical representations.  In _NIPS_ , pages 6338–6347.
  * Nickel and Kiela (2018) Maximilian Nickel and Douwe Kiela. 2018.  Learning continuous hierarchies in the lorentz model of hyperbolic geometry.  In _ICML_ , volume 80 of _Proceedings of Machine Learning Research_ , pages 3776–3785. PMLR.
  * Nickel et al. (2016) Maximilian Nickel, Kevin Murphy, Volker Tresp, and Evgeniy Gabrilovich. 2016.  A review of relational machine learning for knowledge graphs.  _Proc. IEEE_ , 104(1):11–33.
  * Nickel et al. (2011) Maximilian Nickel, Volker Tresp, and Hans-Peter Kriegel. 2011.  A three-way model for collective learning on multi-relational data.  In _ICML_ , pages 809–816. Omnipress.
  * Parker (2003) John R Parker. 2003.  Notes on complex hyperbolic geometry.  _preprint_.
  * Rader (1972) Charles M. Rader. 1972.  Discrete convolutions via mersenne transrorms.  _IEEE Transactions on Computers_ , 21(12):1269–1273.
  * Rockmore (2000) Daniel N. Rockmore. 2000.  The FFT: an algorithm the whole family can use.  _Comput. Sci. Eng._ , 2(1):60–64.
  * Smith et al. (1997) Steven W Smith et al. 1997.  The scientist and engineer’s guide to digital signal processing.
  * Sonthalia and Gilbert (2020) Rishi Sonthalia and Anna C. Gilbert. 2020.  Tree! I am no tree! I am a low dimensional hyperbolic embedding.  In _NeurIPS_.
  * Suchanek et al. (2007) Fabian M. Suchanek, Gjergji Kasneci, and Gerhard Weikum. 2007.  Yago: a core of semantic knowledge.  In _WWW_ , pages 697–706. ACM.
  * Sun et al. (2019) Zhiqing Sun, Zhi-Hong Deng, Jian-Yun Nie, and Jian Tang. 2019.  Rotate: Knowledge graph embedding by relational rotation in complex space.  In _ICLR (Poster)_. OpenReview.net.
  * Toutanova and Chen (2015) Kristina Toutanova and Danqi Chen. 2015.  Observed versus latent features for knowledge base and text inference.  In _Proceedings of the 3rd workshop on continuous vector space models and their compositionality_ , pages 57–66.
  * Trouillon et al. (2016) Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier, and Guillaume Bouchard. 2016.  Complex embeddings for simple link prediction.  In _ICML_ , volume 48 of _JMLR Workshop and Conference Proceedings_ , pages 2071–2080. JMLR.org.
  * Ungar (2008) Abraham Albert Ungar. 2008.  _Analytic hyperbolic geometry and Albert Einstein’s special theory of relativity_.  World Scientific.
  * Wang et al. (2018) Chen Wang, Le Zhang, Lihua Xie, and Junsong Yuan. 2018.  Kernel cross-correlator.  In _AAAI_ , pages 4179–4186. AAAI Press.
  * Wang et al. (2021) Kai Wang, Yu Liu, Dan Lin, and Michael Sheng. 2021.  Hyperbolic geometry is not necessary: Lightweight euclidean-based models for low-dimensional knowledge graph embeddings.  In _EMNLP (Findings)_ , pages 464–474. Association for Computational Linguistics.
  * Wang et al. (2014) Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen. 2014.  Knowledge graph embedding by translating on hyperplanes.  In _AAAI_ , pages 1112–1119. AAAI Press.
  * Xiao et al. (2021) Huiru Xiao, Caigao Jiang, Yangqiu Song, James Zhang, and Junwu Xiong. 2021.  Unit ball model for embedding hierarchical structures in the complex hyperbolic space.  _arXiv preprint arXiv:2105.03966_.
  * Yang et al. (2015) Bishan Yang, Wen-tau Yih, Xiaodong He, Jianfeng Gao, and Li Deng. 2015.  Embedding entities and relations for learning and inference in knowledge bases.  In _ICLR (Poster)_.

##  Appendix A Hyperparameters

Data | Model | Optimizer | Batch size | Negative samples | Learning rate | Double negative
---|---|---|---|---|---|---
WN18RR | FFTRefH | Adam | 500 | 100 | 0.0003 | True
FFTRotH | Adam | 500 | 100 | 0.0003 | True
FFTAttH | Adam | 500 | 100 | 0.0004 | True
FB15k-237 | FFTRefH | Adagrad | 500 | 250 | 0.02 | False
FFTRotH | Adam | 100 | 100 | 0.0002 | False
FFTAttH | Adagrad | 500 | 100 | 0.03 | False
Table 6: Hyperparameters of our approaches.

We tune our hyperparameters by grid search on each validation set in
$32$-dimension complex hyperbolic space, which are given in Table 6. For FFT
and IFFT algorithms, we use the package
torch.fft444https://pytorch.org/docs/stable/fft.html. and set the parameter
norm=“ortho”, which is consistent with the defined orthonormal Fourier
transform in Section 4.2.


$\dagger$$\dagger$footnotetext: These authors contributed equally to this
work.**footnotetext: Corresponding author<EMAIL_ADDRESS>
# A high-fidelity quantum matter-link between ion-trap microchip modules

M. Akhtar1,2,†, F. Bonus2,3,†, F. R. Lebrun-Gallagher1,2,†, N. I. Johnson1, M.
Siegele-Brown1, S. Hong1, S. J. Hile1, S. A. Kulmiya4, S. Weidt1,2 and W. K.
Hensinger1,2,∗ 1Sussex Centre for Quantum Technologies, University of Sussex,
Brighton, BN1 9RH, UK  2Universal Quantum Ltd, Brighton, BN1 6SB, UK
3Department of Physics and Astronomy, University College London, London, WC1E
6BT, UK  4Quantum Engineering Centre for Doctoral Training, University of
Bristol, Bristol, BS8 1TH, UK

(30th August 2024)

###### Abstract

System scalability is fundamental for large-scale quantum computers (QCs) and
is being pursued over a variety of hardware platforms. For QCs based on
trapped ions, architectures such as the quantum charge-coupled device (QCCD)
are used to scale the number of qubits on a single device. However, the number
of ions that can be hosted on a single quantum computing module is limited by
the size of the chip being used. Therefore, a modular approach is of critical
importance and requires quantum connections between individual modules. Here,
we present the demonstration of a quantum matter-link in which ion qubits are
transferred between adjacent QC modules. Ion transport between adjacent
modules is realised at a rate of 2424 s-1 and with an infidelity associated
with ion loss during transport below $7\times 10^{-8}$. Furthermore, we show
that the link does not measurably impact the phase coherence of the qubit. The
quantum matter-link constitutes a practical mechanism for the interconnection
of QCCD devices. Our work will facilitate the implementation of modular QCs
capable of fault-tolerant utility-scale quantum computation.

quantum information, scalable quantum computation, ion trapping, atomic and
molecular physics

##  I Introduction

Platforms using trapped atomic ions form an exceptional foundation on which
QCs and quantum simulators can be developed Bruzewicz2019 . Encoding qubits in
the internal electronic states of trapped ions offers the highest quantum gate
fidelities and the longest coherence times when compared to other physical
implementationsWang2021 ; Harty2014 ; Srinivas2021 ; Monz2011 ; Ladd2010 ;
Popkin2016 ; Clark2021 .

So far, small-scale trapped-ion QCs with up to 10s of qubits have been
demonstrated Wright2019 ; Pogorelov2021 ; Pino2021 ; Egan2021 . At this scale,
quantum logic can be realised using a single linear ion crystal as a qubit
register. Multi-qubit operations are mediated by the Coulomb interaction
within the crystal. However, as the crystal size increases, limitations on the
motional mode density make scaling a single register to larger qubit numbers
challenging Brown2016 . One architecture that allows for multiple qubit
registers in its design is the quantum charge-coupled device (QCCD) which
consists of an array of segmented electrodes Wineland1998 ; Kielpinski2002 .
Locations or zones within a single device can be allocated specific functions,
such as quantum information processing, memory and read-out. In this
configuration, a shared RF pseudopotential provided by a single pair of RF
electrodes delivers the necessary radial confinement, while confinement in the
axial direction is created by quasistatic potentials applied to the remaining
segmented electrodes. This allows for small qubit registers to be interfaced
to one another via mobile ions. For quantum information processing using a
shuttling-based trapped ion architecture, there are two sources of infidelity
to consider. The first source of infidelity is the decay of the quantum state
during the ion transport operation. Work by Kaufmann et al. has already
demonstrated that ion transport operations within a single device can give
rise to a state fidelity of $99.9994(^{+6}_{-7})$% per transport operation
Kaufmann2018 . The second possible source of infidelity is the infidelity
associated with ion loss during transport.

Furthermore, the QCCD architecture has been used to implement high-fidelity
quantum logic, and can be paired with laser-free gate schemesPino2021 ;
Srinivas2021 ; Lekitsch2017 . Recently, QCCD architectures have been used to
demonstrate key steps towards the fault-tolerant operation of a trapped ion
quantum computer Pino2021 ; Ryan-Anderson2021 ; Hilder2022  and stand-alone
near-term devices offer an attractive platform for the execution of restricted
noisy intermediate-scale quantum algorithms Bharti2022 ; Cerezo2021  and
simulations Altman2021 .

Figure 1: An illustration of the process for connecting two modules (Module A
and Module B) for the purposes of a modular ion-trap QC. a) Photonic links use
probabilistic, heralded entanglement which is generated from the interference
of emitted photons from each module. The red and blue spheres denote different
ion species. An N$\times$N optical switch links the modules, where N is the
number of modules. The emitted photons interfere at the beam splitter and a
single photon detector array is used to herald the generation of entanglement
between modules. In order to generate a high-fidelity link between modules a
distillation process is required. The dashed box represents a proposal for a
3-level distillation process that would take a 94% entanglement fidelity to
99.7% fidelity, more details can be found in Ref.Nigmatullin2016 . After
distillation, quantum teleportation can then be used to map the qubit state
between modules, thereby completing the information transfer. A similar
protocol involving measurement, single qubit rotations and classical
communications can also be used to execute a remote two-qubit gate. b) Linking
modules using ion qubit transport. DC voltage waveforms control the ion motion
such that the ion qubit is physically transported between the modules. The
dashed box contains a plot of the voltage shuttling waveforms used in this
work, each plot colour denotes the voltage evolution of a different DC
electrode. Here the modules are depicted as surface traps but other geometries
are also applicable. This method does not require quantum gates. Furthermore,
for information transfer in a QC architecture based on quantum matter-links,
multi-species gates are not necessary. Figure 2: An illustration of a small
section of a modular version of the QCCD quantum computer architecture. a) Two
modules (“Module A” and “Module B”) are shown as fully opaque and parts of ten
other modules are shown as partially transparent. As an example, each module
contains 4 X-junctions and is structured such that it tessellates with
neighbouring modules. A realistic implementation may feature many more
X-junctions per module. Inset b) shows a single X-junction on a module. For
this architecture, specific areas of the X-junction are associated with
certain functions and are connected by mobile ions. For instance, a qubit may
be located in a gate zone for quantum logic operations (blue box) while
another qubit may be stored in, or moved to, a memory zone (red box). Inset c)
depicts the gap separating the modules. Distinct RF electrode pairs extend out
to the edge of the modules to create a radially confining potential such that
ions can be transported to the neighbouring module.

To unlock many of the anticipated applications of a QC within the necessary
level of error correction, far larger qubit numbers will be required than are
available on current devices Gidney2021 ; Webber2022 ; Lee2021 . For example,
simulations of the FeMoco molecule could lead to a better understanding of
nitrogen fixation for the production of ammonia in fertilisers, but simulating
its ground state would require more than $10^{6}$ qubits Webber2022 . However,
incorporating these large numbers of qubits into a single QCCD does not appear
feasible given the size limitations of a single device. For instance, an ion
trap array occupying an area of 90 $\times$ 90 mm2 could be fabricated on a
standard 150-mm silicon wafer Lekitsch2017 . A 1296 X-junction array could
then be realised using X-junction trap structures with a footprint of 2.5
$\times$ 2.5 mm2 per X-junction. Considering a practical upper bound where up
to 20 trapped ion qubits are controlled per junction, $\sim 2.6\times 10^{4}$
qubits could be housed on a single device. While a larger number of qubits
could be incorporated onto wafers with sizes up to 450 mm, the fabrication of
these devices is increasingly difficult to implement with increasing wafer
size. A realistic ion-trap QC architecture must therefore be constructed from
a network of QC modules and offer inter-module connection rates that are
orders of magnitude faster than qubit decoherence times.

Thus far, the only experimentally demonstrated method used to connect trapped-
ion QC modules relies on photonic links Hucul2015 ; Stephenson2020 ;
Monroe2013 ; Monroe2014 . This optical interface permits the heralded,
probabilistic distribution of entanglement between remote modules. Figure 1a
briefly describes the processes involved in implementing a photonic link for
an ion-trap QC. Photonic interconnects between two QC modules have been
realised with an entanglement connection rate of $182\text{\,}\mathrm{s}^{-1}$
at an entanglement fidelity of 94% Stephenson2020 .

Large optical switch arrays and wavelength conversion schemes offer a pathway
towards connecting multiple modules together for large scale QCs. However,
implementing these solutions in a fault-tolerant QC, would currently achieve
an effective connection rate that is $\sim$2 orders of magnitude less than the
raw entanglement rate (see Methods). This reduces the rate to a level that
might be too slow for implementation in a practical quantum computer
($\sim$1\text{\,}\mathrm{s}$$). While techniques to improve photon collection
efficiencies such as on-chip cavities could increase the entanglement rate,
their integration within ion-trap modules remains an unsolved and difficult
challenge Awschalom2021 . By contrast, an alternative implementation relying
on the entanglement of trapped ion chains through heralded single-photon
transfer in shared overlapped cavities offers fast ($\mu$s) interaction rates
Ramette2022 . Nevertheless, the approach is likely to remain constrained to
tens of thousands of qubits after which slower photonic links may be relied
upon for further expansion.

Lekitsch et al. proposed an alternative approach to scaling trapped-ion QCs
where ion qubit transport between independent modules is mediated by electric
fields Lekitsch2017 . Figure 1b highlights the processes involved in using a
quantum matter-link for an ion-trap QC. In the illustration, surface-electrode
ion-trap modules are depicted with an electrode structure that spans to the
module’s edge. When the electrodes on the edge of each module are aligned with
respect to its neighbour, ions can be transported with translating potentials
from one module, across the inter-module gap, to the next Lekitsch2017 .
Figure 2 shows an example of how this method might be implemented on a large
scale. It illustrates a conceptual modular QCCD architecture within which
quantum information is not only distributed between zones within a single
module, but also between modules, where radial confinement is provided by
discontinuous RF electrode pairs. Using this architecture, it is then possible
to build a quantum network of tessellated QC modules. One of the first steps
towards realising the deterministic transport of individual trapped ions
between RF ion-trap modules was made by Stopp et al., where an ion was ejected
and recaptured by the same module, demonstrating a recapture fidelity of 95.1%
Stopp2021 .

In this article, we demonstrate the transfer of trapped ions between two
quantum computing modules at a transfer rate of $2424\text{\,}\mathrm{s}^{-1}$
over a total distance of $684\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and with
an infidelity associated with ion loss during transport of less than $7\times
10^{-8}$. Furthermore, we demonstrate that there is no measurable loss of
coherence of the qubit associated with the transport operation, therefore
realising a high-fidelity coherent quantum matter-link between adjacent
quantum computer modules.

##  II Experimental set-up

We address the challenge of connecting independent QC modules by using two
linear surface-electrode Paul traps. Both ion-trap microchip modules have been
fabricated on a silicon substrate and feature electrode structures that extend
to the edge of the inter-module gap (see Methods) while the substrate recedes
$\sim$ $75\text{\,}\mathrm{\SIUnitSymbolMicro m}$ away from the gap to
facilitate module alignment. When the modules are aligned, this electrode
configuration allows for the confining potential to extend over the inter-
module gap, creating an electric field interface between the two modules.

Figure 3: a) Picture of the two microfabricated ion-trap modules used to
demonstrate inter-module transport. Each module is 17.5 $\times$ 13.5 mm2 in
size. A dashed box overlay signifies the area depicted in b). b) A schematic
of 11 of the DC electrode pairs on the module Alice and 4 of the DC electrode
pairs on the module Bob. Electrodes pairs 1–8 are used for ion transport over
the $10(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ inter-module gap. The
dimensions of the trap electrodes and inter-module gap are not drawn to scale.
The DC electrodes are in yellow and the RF electrodes are in orange. Ions are
loaded into the trap in the Loading Zone by the photoionisation of neutral
ytterbium atoms from a hot atomic beam, whose trajectory is restricted by a
metal grating. This prevents contamination of the electrodes around Zones 1
and 2. In the Loading Zone, $369.5\text{\,}\mathrm{nm}$,
$935.1\text{\,}\mathrm{nm}$ and $399.0\text{\,}\mathrm{nm}$ laser beams are
overlapped. The $369.5\text{\,}\mathrm{nm}$ and $399.0\text{\,}\mathrm{nm}$
light is used for photo-ionisation, and the $369.5\text{\,}\mathrm{nm}$ and
$935.1\text{\,}\mathrm{nm}$ light is used for Doppler cooling. Once an ion is
loaded, the 399.0 nm light is turned off. Thereafter both the ion and the
remaining laser beams are translated to Zone 1. Detection of the ion occurs in
Zone 1. Zone 1 and Zone 2 form the start and end points of an inter-module
link.

One of the ion-trap modules (‘Alice’, left in Figure 3) is rigidly mounted to
the vacuum chamber via a heatsink circulating cryogenic helium gas such that
the trap operates at $36$–$42\text{\,}\mathrm{K}$ Lebrun2021 . The second ion-
trap module (‘Bob’, right in Figure 3) is cooled via a flexible copper braid
forming a thermal link between the two modules. Bob is mounted to an in-vacuum
three-axis piezo stage assembly (Physik Instrumente Ltd, P625.1 and P625.2).
When considering many ion trap QCCD modules tessellated into a large-scale
architecture as presented in Figure 2, small UHV- and cryogenic-compatible XYZ
piezo actuators will be required to be installed beneath each module
Lekitsch2017 . These may be engineered from a combination of compact shear
piezo actuators, providing the sufficient travel range to compensate for
module drift that can arise from temperature changes while maintaining high
module alignment accuracy.

In this work, the selection of piezo actuators was not constrained by the size
of the module as only two modules required alignment. Instead, the implemented
piezo stage assembly was chosen for its fine $5$-nm positioning accuracy and
large travel range of $600\text{\,}\mathrm{\SIUnitSymbolMicro m}$ which
provides additional experimental flexibility. However, since the measurement
of the alignment of the two modules is done optically, the precision of the
module alignment in the x–y plane is limited by the imaging system. To image
the modules a lens system with $\times$13 magnification is used in conjunction
with an sCMOS camera. The imaging system has a spatial resolution of
$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ which leads to an alignment error
of $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the x–y plane. In the z-axis,
the alignment is measured by scattering $369.5\text{\,}\mathrm{nm}$ laser
light off each of the modules’ surfaces. The beam is aligned parallel to the
plane of the modules and lowered onto either side of the inter-module gap. The
difference in the beam height at which scatter is maximised on each of the
modules is used to determine the alignment. This procedure leads to an
alignment error in the z-axis of $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
For comparison, 3-dimensional (3D) microfabricated ion traps constructed using
wafer stacking techniques achieve an alignment accuracy between symmetric
electrodes in the order of 10s of micrometers when aligned manually Pyka2014 ;
Blakestad2009 . Employing precision-machined self-aligning features directly
integrated within the wafers Ragg2019 ; Decaroli2021 , or semi-automatic
bonding processes Auchter2022 , have since reduced the alignment imprecision
to $\leq$2.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The present piezo
actuator solution therefore offers a level of alignment accuracy between
separate trap modules that is comparable to the level currently reached
between electrodes on stacked wafer trap designs. Following alignment of Bob
with respect to Alice, we observed no measurable drift that would require
periodic readjustment of the module position.

For the results presented in this work the separation between Alice and Bob in
each axis is $\Delta x=$10(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, $\Delta
y=$0(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and $\Delta
z=$0(3)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. From previous simulations
Lekitsch2017 , a misalignment in all three axes by
$\leq$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ should lead to a RF barrier
$\leq$0.2\text{\,}\mathrm{meV}$$ for a trap depth of
$\sim$100\text{\,}\mathrm{meV}$$ and an ion height of
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Overcoming an RF barrier of this
magnitude has already been shown to be compatible with high-fidelity ion
transport Blakestad2009 . Both Alice and Bob are driven by independent RF
circuits. These are calibrated to drive both modules at the same frequency,
amplitude and phase (see Methods). RF voltages are applied at an amplitude
$V_{0}=$101.75\text{\,}\mathrm{V}$$ and frequency
$\Omega_{RF}/2\pi=$19.32\text{\,}\mathrm{MHz}$$, which from trap simulation
using the Finite Element Method (FEM) yields a trap depth of
$53.9\text{\,}\mathrm{meV}$. FEM simulations also indicate changes in the
height of the pseudopotential minimum across the inter-module gap that are
limited to $<$100\text{\,}\mathrm{nm}$$ around a mean ion height of
$121.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

174Yb+ and 171Yb+ are used in this work. The qubit stored in the S1/2
hyperfine manifold of 171Yb+ is used to measure the effects of decoherence
mechanisms on the matter-link, while 174Yb+ is used to measure the infidelity
associated with ion loss during transport due to its higher fluorescence rate.
A detailed analysis of the isotope dependent fluorescence and the energy level
structure of 174Yb+ and 171Yb+ can be found in Ref.ejtemaee2010optimization .

To initialise the system, isotope selective loading occurs on Alice in the
Loading Zone (Figure 3b). Once loaded, the ions are shuttled from the Loading
Zone to Zone 1 over a distance of $1840\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, which is the starting point of all subsequent experiments. The axial trap
frequency is
$\nu_{\rm{ax}}=\omega_{\rm{ax}}/2\pi=$141(1)\text{\,}\mathrm{kHz}$$ and the
radial frequencies are $\nu_{\rm{rad}}=\omega_{\rm{rad}}/2\pi=1.15(3)$ and
$1.31(3)\text{\,}\mathrm{MHz}$. The details of the voltage control system can
be found in Methods.

##  III A Quantum Matter-Link

###  III.1 Inter-module Transport

Ion transport between modules is implemented by varying the voltages applied
to the 4 electrode pairs closest to the inter-module gap on both Alice and Bob
(1-8 in Figure 3b). Successive voltage updates sent to each electrode realise
a translating potential well at the ion (see Methods). Each ion transport from
Zone 1 $\rightarrow$ Zone 2, or Zone 2 $\rightarrow$ Zone 1 constitutes a
single matter-link between modules. Zones 1 (2) were chosen as the start or
end point of the link, since the ion can be confined independently on Alice
(Bob) without requiring potentials from the neighbouring module. Zones 1 and 2
are separated by a distance of $684\text{\,}\mathrm{\SIUnitSymbolMicro m}$. As
an initial verification step, the success of the inter-module link was
confirmed by imaging the scattered ion fluorescence in Zone 1 and in Zone 2,
before and after ion transport. Thereafter, the lasers and detection optics
were repositioned to detect ion fluorescence in Zone 1.

The infidelity associated with ion loss during inter-module transport was
measured by transporting a single 174Yb+ ion between Zone 1 and Zone 2. After
each set of $2\times 10^{5}$ links the presence of the ion was verified by the
detection of fluorescence using a photomultiplier tube (PMT). With a single
link duration of $412.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$, at an
equivalent link rate of $2424\text{\,}\mathrm{s}^{-1}$, $15\times 10^{6}$
consecutive links were completed successfully. The ion was subsequently lost
after completing a larger, number of transport operations in the following set
of transport operations, thereby placing an upper limit of $7\times 10^{-8}$
on the infidelity associated with ion loss during transport. The ion travelled
$10.26\text{\,}\mathrm{km}$ at an average transport speed of
$1.66\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$. Throughout these
transport measurements, the digital-to-analogue converters (DACs) were updated
at the fastest possible rate. This led to distortions in the transport
waveforms, resulting from the low cut-off frequency of the DC filtering
circuits. No difference in the ion lifetime for stationary and transported
ions was identifiable, therefore the infidelity associated with ion loss
during inter-module transport was not measurably affected by the distortions
in the DC waveforms. The limit on the probability of successful transport is
therefore attributed to laser instability and ion loss from collisions with
background gas molecules.

The ion-transport rate was limited by hardware constraints. Faster transport
times can be achieved by using DACs with a faster update rate, modifying the
DC filter circuits to have a higher cut-off frequency (see Methods) or by pre-
distorting the DC waveforms to compensate for the DC filter circuits.

###  III.2 Preserving Qubit Coherence

Figure 4: a) Ramsey fringes measured after 0, 2 and 100 links. The solid lines
represent a sinusoidal fit to the data. For each dataset $\tau=$ 100 ms, 100
averages were taken per data point. Error bars are standard deviations. The
Ramsey fringe contrasts are found to be 0.96(2), 1.00(2) and 0.97(2) for 0, 2
and 100 links respectively. b) The Ramsey fringe contrast as a function of
time for 0, 2 and 100 links. The error bars represent the standard deviation
in the measured contrast for a given time delay. The Gaussian fit to each
dataset is given by the solid lines and the associated shaded areas represent
the 1-sigma error in the fit. The dashed line indicates the 1/$e$ decoherence
threshold. The coherence times are $560(40)\text{\,}\mathrm{ms}$,
$560(40)\text{\,}\mathrm{ms}$ and $540(30)\text{\,}\mathrm{ms}$ for 0, 2 and
100 links respectively.

To show that the coherence of the qubit can be maintained throughout the
matter-link, the effect of the inter-module transport on qubit states is
investigated. Here the qubit is formed using two hyperfine levels of 171Yb+ in
the S1/2 manifold: $|0\rangle\equiv|F=0,m_{f}=0\rangle$ ,
$|1\rangle\equiv|F=1,m_{f}=0\rangle$. The two states are separated by
12,642,812,118+311$B^{2}$ Hz where $B$ is the magnetic field in Gauss Wang2021
. The ambient magnetic field at the qubit is $10.177(1)\text{\,}\mathrm{G}$ as
measured on the $|F=0,m_{f}=0\rangle$ to $|F=1,m_{f}=-1\rangle$ transition.
The first order magnetic field insensitivity of the qubit (compared to the
$|F=1,m_{f}=\pm 1\rangle$ states) increases its robustness against decoherence
from ambient magnetic field fluctuations with the qubit transition frequency
experiencing a magnetic field dependent linear slope of
$6.2\text{\,}\mathrm{kHz}\text{\,}{\mathrm{G}}^{-1}$ at an ambient magnetic
field of $10\text{\,}\mathrm{G}$. The magnetic field fluctuations are
therefore expected to result in a frequency shift of up to
$\pm$6\text{\,}\mathrm{Hz}$$, on timescales longer than the time taken for a
single fringe measurement, as indicated by near unity fringe contrast in the
measurements in Figure 4a.

A Ramsey-type experiment is used to probe the coherence of the qubit by
measuring the $T^{*}_{2}$ time. This experiment is performed by first
optically pumping the ion into the $|0\rangle$ state and subsequently applying
two $\pi/2$ Ramsey pulses, separated by a delay time $\tau$. The pulses are
applied using resonant microwave fields from an external microwave horn. The
probability of the qubit being in $|1\rangle$ is then read out using a state-
dependent fluorescence detection scheme Olmschenk2007 . The experiment is then
repeated with inter-module transport operations taking place during the delay
time $\tau$ (see Methods). Figure 4a shows an example of a stationary Ramsey
experiment in comparison to results using 2 and 100 links within the delay
time. The Ramsey fringe contrasts measured were 0.96(2), 1.00(2) and 0.97(2)
for 0, 2 and 100 links respectively. The measured contrasts indicate that
there is no measurable loss of qubit coherence during inter-module qubit
transport for $\tau=$100\text{\,}\mathrm{ms}$$.

Figure 4a shows phase offsets of $1.8690(1)\text{\,}\mathrm{rad}$ and
$3.7988(1)\text{\,}\mathrm{rad}$ for the 2 and 100 links respectively,
relative to the stationary measurement. The experimental set-up does not
include any magnetic field shielding or any active magnetic field
stabilisation. These phase offsets are attributed to miscalibrations in the
energy level splitting frequency of the qubit, uncompensated magnetic field
drifts and transport of the qubit through spatial magnetic field
inhomogeneities. Once shielding and/or active magnetic field stabalisation are
installed, magnetic field drifts can be arbitrarily reduced. Any phase
accumulation resulting from ion transport across quasistatic magnetic field
inhomogenities can be calibrated and compensated for using an additional phase
rotation after the transport operation.

Imperfections in the electrode-voltage signal chain, such as signal
distortions from the filter circuits can lead to heating. Therefore, for the
Ramsey sequence an inter-module transfer rate of
$1250\text{\,}\mathrm{s}^{-1}$ was used and the number of links executed by
the qubit within the delay time was restricted to 100. This ensures that the
measurement statistics of the bright state remain unaffected by the reduction
in fluorescence resulting from kinetic energy gain induced by ion-transport.

Figure 4a demonstrates that, within the available measurement accuracy, qubit
coherence is unaffected by inter-module transport at
$\tau=$100\text{\,}\mathrm{ms}$$. To investigate that qubit coherence is
maintained throughout transport operations between QC modules more generally,
the Ramsey-type experiment shown in Figure 4a was reproduced with longer delay
times up to $\tau=$500\text{\,}\mathrm{ms}$$. A Gaussian decay is then fitted
to the fringe contrast to calculate a coherence time $T^{*}_{2}$, where the
Gaussian decay in fringe contrast is indicative of low frequency noise
dominating the dephasing process home2006entanglement . Figure 4b shows the
coherence measurements and the resultant fits. For the stationary case,
$T^{*}_{2}=$560(40)\text{\,}\mathrm{ms}$$. With 2 links,
$T^{*}_{2}=$560(40)\text{\,}\mathrm{ms}$$, and with 100 links,
$T^{*}_{2}=$540(30)\text{\,}\mathrm{ms}$$. The main limiting factors of the
coherence time are expected to be magnetic field fluctuations over the
timescale of the experiment. Each of the 1-sigma errors of the measured
coherence times overlap, demonstrating that we cannot detect a loss of
coherence due to inter-module transport within the uncertainty of our
measurement.

Figure 4a can be used to determine an upper bound for the loss of coherence of
the qubit during transport. The lower bound of the fringe contrast for 100
links is 0.95. This translates to an upper bound on the infidelity per link of
$5\times 10^{-4}$. This confirms that the matter-link can be implemented to
facilitate information transfer with a high quantum-state fidelity. The actual
infidelity per link is expected to be orders of magnitude lower as shown in
the experimental demonstration for linear ion transport by Kaufmann et al
Kaufmann2018 .

##  IV Conclusion

The techniques used in this work demonstrate that inter-module ion transport
is a practical approach to interface QC modules. Two neighbouring surface-
electrode ion-trap modules were connected using ion-transport operations,
realising a fast, deterministic and high-fidelity quantum matter-link. This
method of linking modules naturally extends the QCCD architecture from one to
multiple modules. We therefore realise a key milestone towards the
implementation of a scalable QCCD architecture. Furthermore, the inter-module
link realised here is three orders of magnitude faster than the measured
$T^{*}_{2}$ time. We also note that modular QCCD implementations involving
different fabrication processes or ion trap designs, such as 3D
microfabricated ion traps, could be connected with this method. To allow for
successful multi-qubit operations following a link, it is necessary that ion
transport operations with low motional excitation are realised between
modules. These have already been experimentally demonstrated within linear
sections of trapped ion modules where trapped ions were transported over the
timescale of few secular oscillations, with sub-quanta motional excitation
bowler2012coherent ; walther2012controlling . Ion transport with low motional
excitation is critical in order to maximise the fidelity of multi-qubit
operations after transport events. Therefore, transport operations that
facilitate inter-module and intra-module transport must be constructed to
minimise motional excitation or must be combined with sympathetic cooling
mechanisms at a cost of increased experiment time. Following upgrades to the
filtering circuitry as well as the control hardware, future work will focus on
implementing and characterising transport with sub-quanta motional excitation
on the two-module system. Implementing this technique then holds the potential
for an additional order of magnitude increase in the inter-module shuttling
rate.

## Methods

### Effective connection rates for photonic interconnects

Scalable optical components are important for the implementation of a large-
scale QC using photonic interconnects. Large switch arrays at trapped-ion
wavelengths in the ultra-violet have yet to be developed, and high branching
ratios in the repumper transition typically make it inefficient to use longer
wavelength trapped ion transitions in the 700–$900\text{\,}\mathrm{nm}$
range.gerritsma2008precision  Alternatively, a large array of telecom
wavelength switches combined with an ion-to-telecom wavelength conversion
scheme could be used. Telecom wavelength switches with low optical losses of
2.1 dB on average have been demonstrated Kim2003 . Furthermore, conversions
from Sr+ or Yb+ wavelengths to the telecom bands have been shown with
conversion efficiencies of $\sim$9% Kasture2016 ; Wright2018 . Therefore,
assuming a raw entanglement rate $R$, frequency up-conversion results in an
up-converted rate of $0.09R$. Switch losses would then reduce this rate to
$0.06R$. In addition, a distillation process may be required to achieve an
entanglement fidelity within the fault-tolerant threshold. By following the
rate calculation outlined in Ref. Nigmatullin2016 , which compares the raw
entanglement rate of photonic interconnects to a distilled entanglement rate,
that produces entangled states of higher fidelity, a fidelity of 99.7% could
be achieved from the current raw entanglement fidelity of 94%, at the cost of
reducing the effective entanglement rate. Assuming all processes except the
production of entangled pairs via remote entanglement are instantaneous and
assuming a mixed-species gate can be achieved with 99.9% fidelity, the
effective entanglement rate would be reduced by a factor of 6. This results in
a distilled entanglement rate of $0.01R$. To complete the information transfer
between two modules an additional set of gates is required as can be seen in
Figure 1a. The time taken for this process is also assumed to be
instantaneous.

### Surface-electrode ion-trap modules

Within each module, electrodes were produced using standard photolithography
techniques. Dicing and isotropic silicon etching steps were subsequently used
to create a precision alignable edge. This combination of processes ensures
that the level of fabrication accuracy and surface quality is uniform across
the microchip module. The roughness of the electrode edge sidewalls is
measured to be submicron and does not hinder module-module alignment
capabilities. Each module features a pair of RF electrodes that are
$270\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide,
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick and are separated by
$90\text{\,}\mathrm{\SIUnitSymbolMicro m}$, resulting in the RF
pseudopotential capable of radially confining ions at a height of
$122\text{\,}\mathrm{\SIUnitSymbolMicro m}$ above the module’s surface.
Confinement in the axial direction is provided by outer segmented static
voltage electrodes with width $220\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
thickness $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and separated by a gap of
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The RF and DC electrodes are
linearly cut at the inter-module interface in line with the approach developed
by Lekitsch et al.Lekitsch2017  to connect QCCD modules via ion transport
operations. To prevent electrical discharges between electrodes, the
separation between the RF electrodes and surrounding DC electrodes is at least
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

### Ramsey experimental sequence

Figure 5: Pulse sequence diagram for the Ramsey-type experiment. The relative
times of the different processes are represented on the same time axis (not to
scale). From the top down the axes are: the on/off times for the
$369.5\text{\,}\mathrm{nm}$ Doppler cooling laser, the electro-optic modulator
(EOM) for optical pumping, the microwaves resonant with the qubit transition,
the PMT for detection and the DC system for shuttling.

A schematic of the Ramsey-type experimental sequence can be seen in Figure 5,
and can be broken down as follows: The ion is initially Doppler cooled for up
to $50\text{\,}\mathrm{ms}$. Thereafter the ion is optically pumped into
$|0\rangle$ over the course of $10\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The
optical pumping is followed by an on-resonance microwave $\pi/2$ pulse with
phase $\phi_{1}=$0\text{\,}\mathrm{rad}$$. The qubit is left to freely precess
for a time delay $\tau$, before a second $\pi/2$ pulse with phase offset of
$0\leq\phi_{2}\leq$2\text{\,}\pi\ \mathrm{rad}$$ is applied. For each phase
offset, the measurement is repeated 100 times. To measure the coherence time,
the experimental sequence was repeated for
$\tau=\\{5,100,200,300,400,500\\}\,$ms for the stationary and 2 link data
whereas the 100 link data spanned $\tau=\\{83,100,200,300,400,500\\}\,$ms.
When investigating the impact of the matter-link on the $T_{2}^{\ast}$ time, a
variable number $N=\\{2,100\\}$ of qubit transport operations can be
undertaken within the delay time $\tau$, such that $NT_{L}<\tau$, where
$T_{L}$ is the time taken for one link
($800\text{\,}\mathrm{\SIUnitSymbolMicro s}$).

### Trap RF set-up

Figure 6: Diagram of the RF delivery system for the Alice and Bob trap
modules. Each of the modules uses a separate RF source which is amplified and
impedance matched to the ion traps using a helical resonator. A capacitive
divider is used to measure the RF amplitude and phase applied to each trap on
an oscilloscope. A variable capacitor is used to match the resonant frequency
of the RF circuit for Bob with the RF circuit for Alice.

A diagram of the trap RF delivery system is shown in Figure 6. Both modules
are driven using separate direct digital synthesiser (DDS) channels from an
AD9910 Urukul card. Each DDS output is amplified and a helical resonator,
which has been designed following the methods presented in Ref.Siverns2012 ,
impedance matches the RF source to the ion trap module. While the helical
resonators are constructed to be identical, a variable capacitor in the RF
circuit of which Bob is a part of allows for the fine tuning of the resonant
frequency of the Bob RF system to match that of the Alice RF system. A 1:54
capacitive divider is used after each helical resonator to measure the signal
amplitude at which the module is driven on an oscilloscope. The RF output can
then be modified such that the amplitude at each module is the same within the
$130\text{\,}\mathrm{mV}$ error of the oscilloscope. In addition, the
capacitive dividers are also used to measure the relative phase of the RF
signals of each of the modules and is corrected to match to within
$35\text{\,}\mathrm{\SIUnitSymbolMicro rad}$. The capacitive coupling between
Alice and Bob is on the order of $1\text{\,}\mathrm{fF}$ such that parasitic
cross-coupling is found to be negligible at the RF drive voltages used in this
work. The absence of significant capacitive cross-coupling is further
confirmed by the lack of variation in both the S11 parameter of the helical
resonator, and the resonant frequencies of the RF circuits, when the modules
are aligned to one another.

### DC ion transfer protocol

A Sinara Kasli field-programmable gate array (FPGA) controller equipped with
three AD5432 digital-to-analog converter (DAC) Zotino cards is used to control
the DC waveforms applied to the ion-trap modules, for ion transport. The DAC
cards update at a rate of $139\text{\,}\mathrm{kHz}$ per channel. Each DAC
channel has an internal third-order Butterworth filter with 75 kHz cut-off
frequency. In addition, a second set of second-order RC filters with a 47 kHz
cut-off frequency is used prior to the vacuum chamber. Inside the vacuum
chamber each DC channel is connected to a final first-order RC filter with a
257 kHz cut-off frequency.

The ion transfer waveforms are numerically determined from a FEM electrostatic
simulation. The simulation includes electrode potentials for both ion-trap
modules. In order to calculate the potentials, a sequential least-squares
programming (SLSQP) optimiser is used (implemented using the wrapper of the
algorithm in Ref. kraft1988software  in the SciPy python library) to minimise
a cost function for a given ion position. The minimisation problem is
constructed from a cost function, which aims to minimise the sum of the
squares of the voltages across all electrodes. In addition, two constraint
functions are also aimed to be minimised. These consist of: 1. the electric
field at the ion position, and 2. the axial electric field curvature (and thus
the axial secular frequency). Each constraint function is weighted with
penalty factors $p_{1}$ and $p_{2}$ respectively such that the parameter that
is to be minimised is normalised. This is due to several orders of magnitude
in difference in the numerical values of each of these parameters. These
penalty factors remain constant throughout the optimisation.

Using this method, a set of voltage values is calculated. For the simulations
used in this work, potentials were calculated in
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ steps between Zones 1 and 2. This
provides trapping potentials that are linearly incremented along the ion
transport path. The evolution of the voltage on each electrode is further
post-processed using a second-order Savitzky–Golay filter savitzky1964  with a
moving filter window of 25 voltage values. This post-processing removes
numerical noise resulting from non-optimal solutions of the minimisation
problem, without distorting the waveform.

From the post-processed waveforms, the sets of voltage solutions were down-
sampled to provide $12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ incremented
solutions for ion transport. This axial separation between steady state
potential minima in the shuttling sequence was found to provide the best
trade-off between shuttling rate and shuttling fidelity. Each set of voltages
is applied at a constant time delay relative to the previous set of voltages.
The combination of constant spacing and constant update rate of the voltage
solutions leads to a constant velocity of the trapped ion during transport.

## Acknowledgements

The authors thank David Bretaud for providing valuable technical assistance
with the DC filtering system and Christophe Valahu for helpful discussions
with the planning of the Ramsey experiment. Ion-trap module microfabrication
was carried out at a number of facilities including the Center of
MicroNanoTechnology (CMi) at the École Polytechnique Fédérale de Lausanne
(EPFL), the London Centre for Nanotechnology (LCN), and the Scottish
Microelectronics Centre (SMC) at the University of Edinburgh. This work was
supported by the U.K. Engineering and Physical Sciences Research Council via
the EPSRC Hub in Quantum Computing and Simulation (EP/T001062/1), the U.K.
Quantum Technology hub for Networked Quantum Information Technologies (No.
EP/M013243/1), the European Commission’s Horizon-2020 Flagship on Quantum
Technologies Project No. 820314 (MicroQC), the U.S. Army Research Office under
Contract No. W911NF-14-2-0106 and Contract No. W911NF-21-1-0240, the Office of
Naval Research under Agreement No. N62909-19-1-2116, the Luxembourg National
Research Fund (FNR) (Project Code 11615035), the University of Sussex, the
Engineering and Physical Sciences Research Council (EP/SO23607/1,
EP/SO21582/1) via the Quantum Engineering Centre for Doctoral Training at the
University of Bristol & the Centre for Doctoral Training in Delivering Quantum
Technologies at the University College London.

## References

  * (1) Bruzewicz, C. D., Chiaverini, J., McConnell, R. & Sage, J. M.  Trapped-ion quantum computing: Progress and challenges.  _Applied Physics Reviews_ 6 (2019).
  * (2) Wang, P. _et al._ Single ion qubit with estimated coherence time exceeding one hour.  _Nature Communications_ 12, 233 (2021).
  * (3) Harty, T. P. _et al._ High-fidelity preparation, gates, memory, and readout of a trapped-ion quantum bit.  _Phys. Rev. Lett._ 113, 220501 (2014).
  * (4) Srinivas, R. _et al._ High-fidelity laser-free universal control of trapped ion qubits.  _Nature_ 597, 209–214 (2021).
  * (5) Monz, T. _et al._ 14-Qubit entanglement: creation and coherence.  _Physical Review Letters_ 106, 130506 (2011).
  * (6) Ladd, T. D. _et al._ Quantum computers.  _Nature_ 464, 45–53 (2010).
  * (7) Popkin, G.  Quest for qubits.  _Science_ 354, 1090–1093 (2016).
  * (8) Clark, C. R. _et al._ High-fidelity bell-state preparation with ${}^{40}{\mathrm{ca}}^{+}$ optical qubits.  _Phys. Rev. Lett._ 127, 130505 (2021).
  * (9) Wright, K. _et al._ Benchmarking an 11-qubit quantum computer.  _Nature Communications_ 10, 1–6 (2019).
  * (10) Pogorelov, I. _et al._ Compact ion-trap quantum computing demonstrator.  _PRX Quantum_ 2, 1 (2021).
  * (11) Pino, J. M. _et al._ Demonstration of the trapped-ion quantum CCD computer architecture.  _Nature_ 592, 209–213 (2021).
  * (12) Egan, L. _et al._ Fault-tolerant control of an error-corrected qubit.  _Nature_ 598, 281–286 (2021).
  * (13) Brown, K. R., Kim, J. & Monroe, C.  Co-designing a scalable quantum computer with trapped atomic ions.  _npj Quantum Information_ 2, 16034 (2016).
  * (14) Wineland, D. J. _et al._ Experimental issues in coherent quantum-state manipulation of trapped atomic ions.  _Journal of Research of the National Institute of Standards and Technology_ 103, 259 (1998).
  * (15) Kielpinski, D., Monroe, C. & Wineland, D. J.  Architecture for a large-scale ion-trap quantum computer.  _Nature_ 417, 709–711 (2002).
  * (16) Kaufmann, P., Gloger, T. F., Kaufmann, D., Johanning, M. & Wunderlich, C.  High-fidelity preservation of quantum information during trapped-ion transport.  _Physical Review Letters_ 120, 010501 (2018).
  * (17) Lekitsch, B. _et al._ Blueprint for a microwave trapped ion quantum computer.  _Science Advances_ 3, 1–12 (2017).
  * (18) Ryan-Anderson, C. _et al._ Realization of real-time fault-tolerant quantum error correction.  _Physical Review X_ 11, 41058 (2021).
  * (19) Hilder, J. _et al._ Fault-tolerant parity readout on a shuttling-based trapped-ion quantum computer.  _Physical Review X_ 12, 011032 (2022).
  * (20) Bharti, K. _et al._ Noisy intermediate-scale quantum algorithms.  _Reviews of Modern Physics_ 94, 015004 (2022).
  * (21) Cerezo, M. _et al._ Variational Quantum Algorithms.  _Nature Reviews Physics_ 3, 625–644 (2021).
  * (22) Altman, E. _et al._ Quantum Simulators: Architectures and Opportunities.  _PRX Quantum_ 2, 017003 (2021).
  * (23) Nigmatullin, R., Ballance, C. J., de Beaudrap, N. & Benjamin, S. C.  Minimally complex ion traps as modules for quantum communication and computing.  _New Journal of Physics_ 18 (2016).
  * (24) Gidney, C. & Ekerå, M.  How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits.  _Quantum_ 5, 1–31 (2021).
  * (25) Webber, M., Elfving, V., Weidt, S. & Hensinger, W. K.  The impact of hardware specifications on reaching quantum advantage in the fault tolerant regime.  _AVS Quantum Science_ 4, 013801 (2021).
  * (26) Lee, J. _et al._ Even more efficient quantum computations of chemistry through tensor hypercontraction.  _PRX Quantum_ 2, 1 (2021).
  * (27) Hucul, D. _et al._ Modular entanglement of atomic qubits using photons and phonons.  _Nature Physics_ 11, 37–42 (2015).
  * (28) Stephenson, L. J. _et al._ High-rate, high-fidelity entanglement of qubits across an elementary quantum network.  _Phys. Rev. Lett._ 124, 110501 (2020).
  * (29) Monroe, C. & Kim, J.  Scaling the ion trap quantum processor.  _Science_ 339, 1164–1169 (2013).
  * (30) Monroe, C. _et al._ Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects.  _Physical Review A_ 89, 022317 (2014).
  * (31) Awschalom, D. _et al._ Development of quantum interconnects (QuICs) for next-generation information technologies.  _PRX Quantum_ 2, 1–21 (2021).
  * (32) Ramette, J. _et al._ Any-To-Any Connected Cavity-Mediated Architecture for Quantum Computing with Trapped Ions or Rydberg Arrays.  _PRX Quantum_ 3, 010344 (2022).
  * (33) Stopp, F., Lehec, H. & Schmidt-Kaler, F.  A deterministic single ion fountain.  _Quantum Science and Technology_ 7, 034002 (2022).
  * (34) Lebrun-Gallagher, F. R. _et al._ A scalable helium gas cooling system for trapped-ion applications.  _Quantum Science and Technology_ 7, 024002 (2022).
  * (35) Pyka, K., Herschbach, N., Keller, J. & Mehlstäubler, T. E.  A high-precision segmented Paul trap with minimized micromotion for an optical multiple-ion clock.  _Applied Physics B: Lasers and Optics_ 114, 231–241 (2014).
  * (36) Blakestad, R. B. _et al._ High-fidelity transport of trapped-ion qubits through an X-junction trap array.  _Physical Review Letters_ 102, 1–4 (2009).
  * (37) Ragg, S., Decaroli, C., Lutz, T. & Home, J. P.  Segmented ion-trap fabrication using high precision stacked wafers.  _Review of Scientific Instruments_ 90, 103203 (2019).
  * (38) Decaroli, C. _et al._ Design, fabrication and characterization of a micro-fabricated stacked-wafer segmented ion trap with two X-junctions.  _Quantum Science and Technology_ 6, 044001 (2021).
  * (39) Auchter, S. _et al._ Industrially microfabricated ion trap with 1 eV trap depth.  _Quantum Science and Technology_ 7, 035015 (2022).
  * (40) Ejtemaee, S., Thomas, R. & Haljan, P.  Optimization of yb+ fluorescence and hyperfine-qubit detection.  _Physical Review A_ 82, 063419 (2010).
  * (41) Olmschenk, S. _et al._ Manipulation and detection of a trapped Yb+ hyperfine qubit.  _Physical Review A_ 76, 052314 (2007).
  * (42) Home, J.  _Entanglement of two trapped-ion spin qubits_.  Ph.D. thesis, University of Oxford (2006).
  * (43) Bowler, R. _et al._ Coherent diabatic ion transport and separation in a multizone trap array.  _Physical review letters_ 109, 080502 (2012).
  * (44) Walther, A. _et al._ Controlling fast transport of cold trapped ions.  _Physical review letters_ 109, 080501 (2012).
  * (45) Gerritsma, R. _et al._ Precision measurement of the branching fractions of the 4p 2p3/2 decay of ca ii.  _The European Physical Journal D_ 50, 13–19 (2008).
  * (46) Kim, J. _et al._ 1100 × 1100 port MEMS-based optical crossconnect with 4-dB maximum loss.  _IEEE Photonics Technology Letters_ 15, 1537–1539 (2003).
  * (47) Kasture, S. _et al._ Frequency conversion between UV and telecom wavelengths in a lithium niobate waveguide for quantum communication with Yb+ trapped ions.  _Journal of Optics_ 18 (2016).
  * (48) Wright, T. A. _et al._ Two-way photonic interface for linking the Sr+ transition at 422 nm to the telecommunication C band.  _Physical Review Applied_ 10, 1 (2018).
  * (49) Siverns, J. D., Simkins, L. R., Weidt, S. & Hensinger, W. K.  On the application of radio frequency voltages to ion traps via helical resonators.  _Applied Physics B_ 107, 921–934 (2012).
  * (50) Kraft, D.  A software package for sequential quadratic programming.  _Forschungsbericht- Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt_ (1988).
  * (51) Savitzky, A. & Golay, M. J.  Smoothing and differentiation of data by simplified least squares procedures.  _Analytical chemistry_ 36, 1627–1639 (1964).