# Multiplexed quantum repeaters
based on dual-species trapped-ion systems

Prajit Dhara1 Norbert M. Linke2 Edo Waks3 Saikat Guha1 Kaushik P.
Seshadreesan1,4<EMAIL_ADDRESS>1Wyant College of Optical Sciences,
University of Arizona, Tucson, AZ 85721, USA  2Joint Quantum Institute and
Department of Physics, University of Maryland, College Park, MD 20742, USA
3Joint Quantum Institute and Department of Electrical and Computer
Engineering, University of Maryland, College Park, MD 20742, USA  4Department
of Informatics and Networked Systems, School of Computing and Information,
University of Pittsburgh, Pittsburgh, PA 15260, USA

(August 27, 2024)

###### Abstract

Trapped ions form an advanced technology platform for quantum information
processing with long qubit coherence times, high-fidelity quantum logic gates,
optically active qubits, and a potential to scale up in size while preserving
a high level of connectivity between qubits. These traits make them attractive
not only for quantum computing, but also for quantum networking. Dedicated,
special-purpose trapped-ion processors in conjunction with suitable
interconnecting hardware can be used to form quantum repeaters that enable
high-rate quantum communications between distant trapped-ion quantum computers
in a network. In this regard, hybrid traps with two distinct species of ions,
where one ion species can generate ion-photon entanglement that is useful for
optically interfacing with the network and the other has long memory
lifetimes, useful for qubit storage, have been proposed for entanglement
distribution. We consider an architecture for a repeater based on such dual-
species trapped-ion systems. We propose and analyze a protocol based on
spatial and temporal mode multiplexing for entanglement distribution across a
line network of such repeaters. Our protocol offers enhanced rates compared to
rates previously reported for such repeaters. We determine the ion resources
required at the repeaters to attain the enhanced rates, and the best rates
attainable when constraints are placed on the number of repeaters and the
number of ions per repeater. Our results bolster the case for near-term
trapped ion systems as quantum repeaters for long distance quantum
communications.

##  I Introduction

Quantum information processing is set to revolutionize computation,
communication and sensing technologies Deutsch2020-yg ; Dowling2003-uf .
Applications of these technologies range from quantum speedups with NISQ
processors Bharti2021-es  to quantum key distribution Xu2020-zr , to quantum-
enhanced distributed sensors Zhang2020-py . Quantum technologies are currently
being actively developed across different physical platforms—from solid-state
systems such as superconducting circuits Kjaergaard2020-mq  and nitrogen
vacancies Pezzagna2021-zf  in diamond, to trapped ions Bruzewicz2019-tu , to
nano-photonic systems Flamini2019-me . Quantum networks capable of faithfully
transferring quantum states between nodes, including the capability to
distribute quantum entanglement Horodecki2009-lt , are being developed both
over short distances to scale up quantum computers in a modular fashion, as
well as over long-distances to connect remote quantum computers, or a local
area network of computers across physical platforms towards building a global
quantum internet Wehner2018-hg ; Pirandola2016-kc ; Kimble2008-yx .

Given that photons are the best transmitters of quantum information that can
be used to implement scalable quantum communications, the primary challenge in
quantum networking is the fundamental rate-loss trade-off. This trade-off
exists for quantum communications over a lossy optical communication channel
that models imperfections such as photon collection, coupling and detection
inefficiencies, as well as transmission losses. The entanglement distribution
capacity of the pure-loss optical channel with unlimited signal power and
unlimited local quantum operations and classical communications (LOCC) is
given by $C(\eta)=-\log_{2}(1-\eta)$ ebits per channel-use pirandola2017a ,
where $\eta$ is the channel transmissivity, and an ebit denotes a pair of
maximally entangled qubits. In the limit of low transmissivity $\eta\ll 1$,
this quantity scales as $\propto\eta$ Takeoka2014-ya . As a result, in long-
distance communications, say, over an optical fiber link whose transmissivity
decreases exponentially with distance as $e^{-\alpha l}$, ($\alpha$ being the
fiber loss coefficient per unit length), the entanglement distribution
capacity drops exponentially with distance independent of the presence or
absence of other imperfections. Quantum repeaters Briegel1998-va ;
Munro2015-hv  help overcome this challenge. They are special-purpose quantum
computers typically consisting of quantum sources, detectors, elementary logic
gates and quantum memories. Quantum repeater architectures based on different
physical platforms Ghalaii2020-qc ; Dias2020-lf ; Seshadreesan2020-zl ;
Loock2020-wd ; Sangouard2011-zm ; Childress2005-xe  along successive
generations of improved protocols Muralidharan2016-wp ; dhara2021  have been
proposed that can achieve enhanced entanglement distribution rates beyond the
direct transmission capacity.

Establishing a large-scale quantum network typically calls for setting up
long-distance core networks. Among the large variety of physical systems that
can be utilized to realize quantum repeaters for the core quantum network
backbone, trapped-ion based systems form an excellent, robust choice, due to
their inherently long memory coherence times Monroe2007-us . Moreover,
trapped-ions are known to be an advanced qubit technology Bruzewicz2019-tu ,
and one of the front runners in the race for scalable universal quantum
information processing Monroe2013-no . Repeater networks consisting of single-
species trapped-ion nodes have been considered and analyzed in-depth in Ref.
Sangouard2009-zy . More recently, Santra et al. Santra2019-di  analyzed
repeaters based on ion traps consisting of two species of ions with
complementary properties—a communication ion species with good optical
properties that enables the network nodes to communicate with one another, and
a memory ion species having a long coherence time, and therefore suitable for
information storage and efficient local quantum processing. Examples of such
complementary pairs of ion species include 138Ba+ and 171Yb+, and 9Be+ and
25Mg+. In the former, e.g., the 138Ba+ ion can emit a photon in the visible
part of the spectrum ($493$ nm) that is entangled with the atomic state of the
ion. Entanglement can be heralded between the atomic states of two such 138Ba+
ions by performing an optical Bell-state measurement Grice2011-hw  on the
photons they emit. Such optically-mediated entanglement, when heralded between
adjacent repeater nodes, can be faithfully transferred on to 171Yb+ ions
present at the respective nodes, whose atomic states have extremely long
coherence times, thus allowing the storage of entanglement between the nodes
which can later be processed using efficient quantum gate operations
Inlek2017-tp ; Tan2015-nq . Santra et al. Santra2019-di  presented a repeater
architecture based on such dual-species trapped ion (DSTI) modules, and
discussed a set of logic gates needed to implement repeater protocols. They
analyzed the quantum communication rates attainable over a line network of
such DSTI repeaters using a multiplexed protocol. The rates were shown to
exceed those possible with direct transmission.

In the present article, we explore a protocol based on spatial and temporal
multiplexing for the trapped-ion repeater architecture involving DSTI modules
that is more general than the one considered in Ref. Santra2019-di . In
spatial multiplexing, multiple communication ions attempt to generate remote
entanglement between every pair of adjacent repeater nodes at each time step
of a well-defined clock cycle. In time multiplexing, remote entanglement is
heralded from entanglement generation attempts across a block of multiple time
steps. While both spatial and time multiplexing were also considered in Ref.
Santra2019-di , the latter was only considered implicitly with a fixed clock
cycle duration for the photon-ion entanglement generation at the repeater
nodes determined by the distance between adjacent nodes. Here, we treat the
clock cycle duration as a free parameter, so that ion-photon entanglement
generation at the nodes can be attempted at rates independent of the
internodal spacing. This enables higher quantum communication rates than the
rates supported by the protocol of Ref. Santra2019-di . Our protocol warrants
suitably larger number of communication ions and memory ions at the repeater
nodes for ion-photon entanglement generation and for storing the unheralded
ion qubits, respectively. We determine the enhanced rates enabled by such a
general protocol for different spatial multiplexing, numerically optimized
over the number of repeaters and temporal multiplexing. We identify the number
of repeaters required and the number of communication and memory ions required
per repeater for the optimal implementation of the protocol, and discuss how
the rates deteriorate from their optimum values when the number of repeaters,
or the number of ions per repeater is constrained.

The article is organized as follows. In Sec. II, we present a general
architecture for the trapped-ion repeaters, along with the node operations and
an associated error model. We also summarize the different timing parameters
of the repeaters here. In Sec. III, we outline the concepts of spatial and
time multiplexing-based quantum repeater protocols. Section IV contains our
proposed protocol based on spatial and time multiplexing for the DSTI
repeaters for the case of 138Ba+ and 171Yb+ ions, along with numerical
results. We conclude with a discussion and summary in Sec. V.

Figure 1: General architecture of a repeater node based on dual-species
trapped ion (DSTI) modules to support entanglement distribution protocols
based on mode multiplexing. The lines denote optical fibers.

##  II Trapped-Ion Repeater Architecture

The general architecture of the trapped-ion repeaters and the overall network
analyzed in this work is depicted in Fig. 1. The repeaters consist of multiple
DSTI modules containing i) 138Ba+ ions, which serve as the communication ions,
and ii) 171Yb+ ions, which serve as the memory ions. The DSTI modules may thus
be thought of as consisting of two independent ion ensembles. Each repeater
node is equipped with lasers and light collection apparatuses for i) ion-
photon entanglement generation using the communication ions, and for ii)
performing qubit logic gates and measurement readouts on the memory ions.
Since the qubits states in the two ion species have different transition
frequencies, the above functions involve different lasers and thus do not
affect each other.

The photons emitted by the communication ions, upon collection, are first
frequency converted to telecom wavelengths for inter-nodal transmissions
Hannegan2021-mm ; Siverns2019-qf , and then coupled into optical fibers. The
communication ions in a DSTI module are assumed to be well spaced out so that
the rate of resonant re-absorption of the photon emitted from one
communication ion by another is low. Light collection is assumed to be
spatially resolved so that light from different communication ions can be
fiber coupled and transmitted over distinct spatial modes. The repeater nodes
are assumed to be linked by fiber bundles capable of transmitting multiple
single photons in distinct spatial modes to support spatial multiplexing.

The qubit logic gate operations on the memory ion qubits in a DSTI module are
effected using highly collimated laser beams that address individual ions.
However, for technical simplicity of operation of the DSTI modules, the
measurement readouts of the memory ions in a module are considered to be
global. This is because, readouts are effected by stimulating state-dependent
fluorescence that can cause high levels of cross talk among the memory ions in
the module even when individual ions are addressed for readout. It must be
noted, however, that strategies to circumvent this problem have been
successfully demonstrated, such as separating and shuttling selective ions
into a separate zone for readout Crain2019-zv , which requires a complex trap
geometry, or using another species for readout Negnevitsky2018-ci , which is
challenging to combine with networking as it involves two species already.

Each repeater node also carries a quantum multiplexer (MUX) Lee2020-nb ;
Krastanov2020-no , whose functionalities include: i) optical switching, and
ii) linear optical Bell-state measurements (BSMs). With photons transmitted
between nodes in well-indexed spatial modes, independent Bell state
measurements (discussed under repeater operations below) over the different
multiplexed spatial modes can be effected using the MUX. The detectors are
assumed to be ultrafast so that they can be reset and used over successive
time steps. In other words, it is assumed that optical BSMs can be effected on
successive time-bin modes using the same MUX to support temporal multiplexing.
The nodes are assumed to share a common clock reference.

Repeater operations: The basic repeater operations between and at the DSTI
modules that we consider were proposed in Santra et al. Santra2019-di , and
are summarized below. Interactions between DSTI modules, both within a
repeater node as well as between adjacent repeater nodes are optically
mediated. Photons emitted by the communication ions are duly collected,
coupled into optical fiber, and interfered and measured to realize Bell state
measurements. The simplest linear optical Bell state measurement for photonic
qubits succeeds probabilistically. When a successful optical Bell state
measurement is performed on photons from two ions, it results in entanglement
being heralded between the ions. The atomic states of the communication ions
are transferred or “swapped” to the memory ions by ion-ion gates based on
Coulomb interactions such as the Molmer-Sorensen gate
molmer_multiparticle_1999  to store entanglement over the long coherence times
of the memory ions. The action of the swap gate $S_{c\rightarrow m}$ (where
$c$ and $m$ labels denote the communication and memory ions, respectively) on
a pair of entangled communication ions is given by

$\displaystyle\begin{split}S_{c_{1}\rightarrow m_{1}}\otimes
S_{c_{2}\rightarrow
m_{2}}&|\psi\rangle_{m_{1}}|\beta\rangle_{c_{1}c_{2}}|\psi\rangle_{m_{2}}\\\
&=|\psi\rangle_{c_{1}}|\beta\rangle_{m_{1}m_{2}}|\psi\rangle_{c_{2}}.\end{split}$
(1)

Here, the various quantum states are, $\ket{\beta}_{c_{1}c_{2}}$ for the
entangled communication ions, $\ket{\psi}_{m_{i}}$ for the memory ions before
the linear optical entanglement swap, and $\ket{\psi}_{c_{i}}$ for the
communication ions after the ion-ion swapping operation.

Finally, the entanglement swap operation between two entangled memory-ion
pairs $|\beta\rangle_{m_{1}m_{2}}$ and $|\beta\rangle_{m_{3}m_{4}}$, when ions
$m_{2},m_{3}$ are in the same DSTI module, is accomplished by a CNOT gate
operation followed by $Z$ and $X$ basis measurements, where the latter may be
effected using Hadamard gates followed by $Z$ basis measurements. This
operation extends the range of entanglement by establishing entanglement
between $m_{1}$ and $m_{4}$. With regard to implementing the measurements as
part of the entanglement swap operations, firstly the gates constituting the
entanglement swaps (CNOTs and Hadamards) are applied on individual memory ions
in a DSTI module. Subsequently, the $Z$ basis measurements, as mentioned
earlier, are effected via a global measurement readout of all the memory ions
in the module. This forms an important consideration in our repeater protocol,
wherein all the memory ions in a DSTI modules end up being measured in the $Z$
basis periodically as dictated by the protocol.

Error model: The success probability of optically-mediated heralded
entanglement generation between communication ions present at two adjacent
repeater nodes is given by $p=\frac{1}{2}\eta_{c}^{2}\eta_{d}^{2}e^{-\alpha
L_{0}}$. Here, $\eta_{c}$ is the collection and coupling efficiency for the
optical elements, $\eta_{d}$ is the efficiency of the detectors used in the
Bell state measurement circuit, $\alpha$ is the fiber attenuation parameter,
which is typically 0.2 dB/km at 1550 nm, and $L_{0}$ is the inter-repeater
spacing (detector dark counts and frequency conversion inefficiencies are
neglected in the present analysis, and will be considered in future works).
When the communication ions are present at the same repeater node, but in
different DSTI modules, the success probability is given by
$p^{\prime}=\frac{1}{2}\eta_{c}^{2}\eta_{d}^{2},$ where it is assumed that the
losses in transmission are negligible. Moreover, the entangled state of the
communication ions is generally modeled by a Werner state of fidelity
parameter $F_{0}$, given by

$\displaystyle\rho_{c_{1},c_{2}}=F_{0}\Phi^{+}+\frac{1-F_{0}}{3}\left(\Phi^{-}+\Psi^{+}+\Psi^{-}\right),$
(2)

where $\Phi^{\pm}=|\Phi^{\pm}\rangle\\!\langle\Phi^{\pm}|$ and
$\Psi^{\pm}=|\Psi^{\pm}\rangle\\!\langle\Psi^{\pm}|$ are maximally entangled
qubit Bell state density operators, with
$|\Phi^{\pm}\rangle=\left(|0,1\rangle\pm|1,0\rangle\right)/2$ and
$|\Psi^{\pm}\rangle=\left(|0,0\rangle\pm|1,1\rangle\right)/2$, and
$\\{|0\rangle,|1\rangle\\}$ being the computational $Z$ basis eigenstates of
the qubits. The Werner state model accounts for errors in the communication
ions that may be caused by the presence of dephasing noise in the photonic
qubits that undergo optical Bell state measurement.

Errors in the swap gate are compactly modeled jointly for a pair of instances
of swap gates acting on two entangled communication ions to store the
entanglement in two memory ions. The model is a two-qubit Pauli channel acting
on the initial entangled state of the two communication ions resulting in a
noisy mapping onto two memory ions (see Santra et al. (Santra2019-di, , Eq. 3)
for details), and is described as

$\displaystyle\rho_{m_{1}m_{2}}$
$\displaystyle=(1-\epsilon_{g})\rho_{c_{1}c_{2}}+\frac{\epsilon_{g}}{16}\sum_{k,k^{\prime}=0}^{3}\sigma_{k^{\prime}}\otimes\sigma_{k}\rho_{c_{1}c_{2}}\sigma_{k^{\prime}}\otimes\sigma_{k},$
(3)
$\displaystyle=F_{i}\Phi^{+}+\frac{1-F_{i}}{3}\left(\Phi^{-}+\Psi^{+}+\Psi^{-}\right),$
(4) $\displaystyle F_{i}$
$\displaystyle=(1-\epsilon_{g})F_{0}+\frac{\epsilon_{g}}{4},$ (5)

where $\\{\sigma_{k}\\}_{k=0,1,2,3}$ are the Pauli matrices $I,X,Y$ and $Z$,
the state $\rho_{c_{1}c_{2}}$ is the Werner state of (2) and $\epsilon_{g}$ is
the error parameter associated with the swap gate.

Finally, imperfections in the entanglement swap operation are modeled in two
parts, namely, errors associated with i) the CNOT gate, and ii) the $X,Z$
measurement. i) The action of a noisy CNOT gate, also of error parameter
$\epsilon_{g}$, acting on two qubits $m_{2},m_{3}$ initially in state
$\rho_{m_{2}}\otimes\rho_{m_{3}}$, is modeled as

$\displaystyle\rho^{\prime}_{m_{2}m_{3}}$
$\displaystyle=(1-\epsilon_{g})\textrm{CNOT}\left(\rho_{m_{2}}\otimes\rho_{m_{3}}\right)\textrm{CNOT}^{\dagger}$
$\displaystyle+\epsilon_{g}\frac{I_{m_{2}m_{3}}}{4},$ (6)

where $m_{2},m_{3}$ are the control and the target qubits, respectively, for
the CNOT operation and $I_{m_{2}m_{3}}$ is the two-qubit identity operator.
Note that $\epsilon_{g}$ is an overestimate for the error in this gate, since
it involves fewer Coulomb gates than the swap operation. ii) Errors associated
with the $X$ and $Z$ measurements on the control and target qubits of the CNOT
gate are functions of the gate error parameter $\epsilon_{g}$ and the initial
fidelity of the entangled Werner state of two communication ions. When
entanglement swaps are performed across a chain of $n\in\mathbb{Z}^{+}$
repeater nodes, the final noisy entangled state heralded between memory ions
at the end nodes of the chain can be described also as a Werner state of the
form in (2) with a fidelity parameter given by $F_{f}=1-\frac{3}{2}Q(n)$,
where

$\displaystyle
Q(n)=\frac{1}{2}\left(1-\left(1-2\epsilon_{g}-\frac{4}{3}(1-F_{0})\right)^{n}\right).$
(7)

Timing Parameters: The proposed repeaters have a few characteristic timing
parameters that are summarized in Table 1. Firstly, the clock cycle duration
$\tau$ (or sometimes denoted as $\tau_{\textrm{rep}}$) is the primary time
unit, which denotes the rate at which the repeater nodes attempt ion-photon
entanglement generation. This is a tunable parameter for the repeater
operation. Secondly, there are the gate and measurement times $\tau_{g}$.
Typical values for this time are of the order of microseconds. For example,
high fidelity swap from 138Ba+ to 171Yb+ has been demonstrated in 100$\mu$s
Inlek2017-tp ; Tan2015-nq . Thirdly, there are the lifetimes of the
communication and memory ions $\tau_{o},\tau_{m}$. A typical value for the
former is $100\mu$s (as has been reported for 138Ba+ ions), while the latter
can be taken to be long—the lifetime of 171Yb+ transitions have been
engineered to run in the order of minutes wang_single-qubit_2017 . The
parameters $\tau_{o},\tau_{m},\tau_{g}$ are governed by the choice of ions and
gate implementation and hence take fixed values in any given physical
realization of the repeaters.

Timing Parameter | Associated Meaning
---|---
$\tau$ | Clock cycle duration
$\tau_{g}$ | Ion-ion gate and measurement times
$\tau_{o}$ | Communication ion lifetime
$\tau_{m}$ | Memory ion lifetime
Table 1: Timing parameters associated with trapped-ion repeaters.

##  III Repeater Protocols based on Spatial and temporal Multiplexing

In this section, we provide a brief background on multiplexing-based repeater
protocols. To begin with, due to the no-cloning theorem Wootters1982-ph ,
unlike classical communication, the simple strategy of ‘amplify and re-
transmit’ is not physically viable for entanglement generation between two
remote parties Alice and Bob. The rates for direct transmission of qubits over
a $M\in\mathbb{Z}^{+}$ quantum channels with a source repetition rate of
$1/\tau$ are limited by the repeaterless bound on the entanglement generation
capacity, which for pure loss channels of transmissivity $\eta$, is given by
pirandola2017a

$\displaystyle C_{\text{direct}}(\eta,M,\tau)=-\frac{M}{\tau}\log_{2}(1-\eta)\
\textrm{ebits/s},$ (8)

referred to as the PLOB bound hereinafter. The PLOB bound tends to be
$\propto\eta$ for $\eta\ll 1$.

There are multiple paradigmatic approaches using quantum repeaters to beat
this bound. A widely recognized classification of these approaches is in terms
of the so-called one-way versus two-way repeaters. One-way quantum repeaters
encode the transmitted qubits using error correcting codes and the task for
quantum repeaters is to decode, correct for transmission errors, re-encode and
transmit from pre-determined locations on the channel. Entanglement can be
distributed using these repeaters by encoding and transmitting one share of a
logically-encoded ebit through the repeater links. This is a similar strategy
as repeaters for one way classical communication. Two-way quantum repeaters,
on the other hand, rely on the generation of local entanglement on smaller
segments of the network. These locally shared ebits are concatenated with the
aid of entanglement swaps to eventually achieve shared entanglement between
the end parties on the channel. Such protocols could potentially be
interspersed with entanglement purification to improve the quality of the
shared ebit that is ultimately generated.

In this work, we focus on two-way repeaters. Two-way repeaters are equipped
with sources of photonic entangled pairs, quantum memory (QM) registers
(trapped memory ions for our proposed designs in this work) and additional
circuitry to perform quantum logic on the qubits stored in the QM register
(including entanglement swaps on the memory ions). We begin with the
assumption that the sources can produce perfect Bell pairs on demand every
$\tau$ seconds. For simplicity, we initially assume arbitrarily large QM
registers and infinitely long coherence times for the qubits stored in the QM.
This is a necessary consideration for a constraint-free analysis of
multiplexing.

For the simplest network topology, namely a line network connecting two
communicating parties, the total link distance $L$ is divided into $n+1$
elementary links. The repeater stations occupy the nodes at either end of each
elementary link in this segmented network. The core strategy of the protocol
is to generate shared entanglement on the elementary links before attempting
entanglement swaps (between QMs) internally in the repeater stations. This is
achieved by performing a linear optical Bell state measurement (BSM) between
the transmitted qubits from neighbouring repeater stations. The simplest
linear optical BSM is a probabilistic operation, which has a probability
$p\leq 1/2$ of succeeding. Note that $p$ is dependent on the length of the
elementary links, i.e. increasing the length of the elementary links
deteriorates $p$. Since loss on fiber scales exponentially with distance,
$p\propto\exp(-\alpha L/(n+1))$. Note that increasing $n$, i.e., reducing the
elementary link length boosts $p$. However, this also means that a larger
number of elementary links must simultaneously succeed, which shows a clear
trade-off. Futher we assume that the QM entanglement swap can succeed with a
probability $q\leq 1$, where $q=1$ is possible with high fidelity entanglement
swapping gates for trapped ion qubits. The achievable rate is given by,

$\displaystyle R_{0}(n)=\frac{p^{n+1}\times q^{n}}{\tau}.$ (9)

Since $R_{0}(n)<e^{-\alpha L}/\tau$, we perform worse than with direct
transmission.

However, with the aid of multiple parallel attempts, i.e., multiplexing, the
entanglement generation rate can be engineered to surpass direct transmission.
This is the natural strategy to consider when individual links can only be
generated in a probabilistic manner; instead of independent single attempts
succeeding simultaneously, we perform multiple parallel attempts for each
elementary link and concatenate successful links. Spatial (or equivalently
spectral) multiplexing is the easiest modification to this protocol which is
based on this paradigm. Here, the design incorporates parallel channels
spatially, i.e., with separate optical fibers. With this modification, instead
of a single BSM attempt every time slot, we can perform $M$ attempts and look
for one success. With a spatial multiplexing size of $M$, and by considering
the probability of at least one success per elementary link, the end-to-end
entanglement generation rate is now given by,

$\displaystyle R_{1}(n)=\frac{(1-(1-p)^{M})^{n+1}\times q^{n}}{\tau}.$ (10)

It has been shown that with optimal choice of $n$ and suitable $M$, the rate
equation in Eqn. (10) can surpass the direct transmission PLOB bound at a
given link length. In fact, the rate envelope for Eqn. (10) has been derived
in guha2015 , and has been shown to scale as $R_{1}\propto e^{-s\alpha L}$
with $s<1$, which allows the protocol to surpass rates possible with direct
transmission.

Another strategy for multiplexing is to accumulate successes from
$m\in\mathbb{Z}^{+}$ attempts over blocks of the fundamental time slot of
$\tau$ seconds. This is called time multiplexing and mimics the effect of
using multiple channels without the necessity for additional physical
channels. Hence, we can perform the entanglement swap between different QMs at
a repeater node only after every $m$ time slots. An achievable entanglement
generation rate for the time multiplexed protocol is given by

$\displaystyle R_{2}(n,m)=\frac{(1-(1-p)^{m})^{n+1}\times q^{n}}{m\tau}.$ (11)

It has been shown in razavi2009 ; dhara2021  that a time-multiplexed protocol
can achieve a sub-exponential rate-vs.-distance scaling i.e. $R_{2}\propto
e^{-t\sqrt{\alpha L}}$ with $t<1$. This is an improved performance over
spatial multiplexing, and it has been shown that a protocol may surpass the
PLOB bound with just time multiplexing. However, in a practical
implementation, time multiplexing requires highly reliable QMs and large
switching trees that scale as $\log_{2}(m)$. Imperfections in these components
can lead to the loss of the sub-exponential advantage dhara2021 . In general,
with the incorporation of both spatial and temporal multiplexing, a two-way
repeater protocol can achieve the rate $R(L,n,m)$ given as

$\displaystyle R(L,n,m)=\frac{(1-(1-p)^{mM})^{n+1}\times q^{n}}{m\tau}.$ (12)

It is important to note that there is a key difference between the
multiplexing degrees in the spatial $(M)$ and time $(m)$ strategies.
Increasing $M$ in a spatially multiplexed protocol requires the use of
additional channels, which may be highly constrained (i.e. we may be limited
by the number of physical optical fibers). In fact it is generally something
that the network architect cannot modify, and hence it is not practical to
optimize the rate with respect to $M$. Rather, given a certain maximum value
of $M$, the rate envelope, as derived in Ref. guha2015 , gives us an idea
about the viability of the protocol to surpass the PLOB bound. Increasing the
time multiplexing degree $m$ is only governed by the lifetime of the QM. As
long as the lifetime surpasses a certain threshold governed by the protocol
design, we can modify $m$ without the need for additional resources. Unlike
spatial multiplexing, time multiplexing can boost the probability of link
creation on the elementary link seemingly arbitrarily, by increasing $m$.
However, by increasing $m$, the effective time step increases from $\tau$ to
$m\tau$ which degrades the rate (see Eqn. (11)). The boost in the success
probability of the link, along with the optimization of the number of quantum
repeater (QR) nodes $n$, overcompensates degradation, and an optimal value of
$m$ for a given $L$ achieves the sub-exponential scaling.

Note that the repeater protocol considered in this work belongs to the second
generation in the classification of successive generations of repeater
protocols sreraman2016 . This is so, because they do not include intermediate,
iterative entanglement distillation steps, but instead are based on multiple
redundant entanglement generation attempts across the elementary links over
spatial or spectral, and temporal modes, which can be perceived as the use of
a repetition-based error correcting code for elementary entanglement
generation. This is followed by identifying the latest successful heralded
elementary links across each time multiplexing block and synchronous
entanglement swapping of these elementary links at the repeater nodes.

##  IV Multiplexing-based protocol for trapped-ion repeaters: Protocol Design
and Evaluation

In this section, we present our proposed protocol based on spatial and time
multiplexing for entanglement distribution across a line network of trapped-
ion repeaters described in Sec. II, followed by numerical performance
analyses. The protocol design is independent of the number of repeater nodes,
and it is assumed that the ion resource parameters of the repeater nodes are
unlimited and can be chosen to be as large as necessary to support any choice
of values of the clock cycle duration $\tau$ and multiplexing parameters
$M,m$. However, when analyzing the performance of the protocol, we will also
consider constraints on these resources since in practice they are often
constrained.

The unit distance (inter-repeater spacing) and multiplexing parameters used in
defining our protocol, and the repeater resource parameters are listed in
Tables 2 and 3, respectively. For simplicity, similar to Ref. Santra2019-di ,
we will consider the case of one DSTI module per node, i.e., $s=1$. However,
the protocol leverages the multiple communication ions present within the DSTI
modules for multiplexed entanglement generation attempts across elementary
links in the network. The fundamental time step $\tau$, or in other words, the
clock cycle duration, multiples of which are used as time multiplexing blocks,
is chosen as a free parameter, and not tied to the physical distance between
the repeaters. This makes our protocol more general than the one presented in
Ref. Santra2019-di .

###  IV.1 Protocol Design and Rates

For $n$ equally spaced repeaters and a total distance $L$ (between the end
nodes), consider a $(m,M)$ repeater protocol with spatial multiplexing
$M\in\mathbb{Z}^{+}$ and time multiplexing $m\in\mathbb{Z}^{+}$. The inter-
repeater spacing is given by $L_{0}=L/(n+1)$. For a given $L_{0}$, the time it
takes for the heralding information of success or failure of optically-
mediated entanglement generation across adjacent repeater nodes to arrive at
the nodes is $T=L_{0}/c$, where $c$ is the speed of light in the optical fiber
used for inter-repeater node transmissions (henceforth referred to as the
heralding time). The protocol aims to successfully herald one elementary link
entanglement in each elementary link from $m\times M$ total attempts spread
over $m\tau$ seconds. The heralding time and the gates and measurement time
together add up to dictate the rate of generating the elementary link
entanglements. Since all the memory ions are in one DSTI module, entanglement
swapping across these elementary link entangled memory ions, which can
performed deterministically using CNOT gate followed by $X$ and $Z$
measurement, distributes entanglement between the end nodes.

Rate Formulas under ideal repeater operations. Assuming gate operations at the
repeaters to be ideal and the optical fibers to be pure loss channels (no
dephasing errors) for the moment, the rate in ebits per second attained by the
protocol is given by the general formula

$\displaystyle R=\frac{\left(1-(1-p)^{Mm}\right)^{n+1}}{\mathbf{T}},$ (13)

where the numerator denotes the probability of successfully heralding at least
one entangled ion-ion pair across each of the $n+1$ elementary links ($p$
being the success probability of optical Bell swap discussed in Sec. II), and
the $\mathbf{T}$ in the denominator is the time it takes to complete $m$ time
steps of entanglement generation attempts across the elementary links. In
order to attain optimal rates at any distance $L$, an optimal number of
repeaters $n_{\textrm{opt}}$ would be required to be placed along the
distance. Too few repeaters would result in excessive errors due to photon
loss, whereas too many repeaters would result in excessive operational errors
at the repeater nodes.

Parameter | Associated Meaning
---|---
$L_{0}$ | Inter-repeater spacing
$M$ | Degree of spatial multiplexing
$m$ | Degree of time multiplexing
Table 2: Unit distance and multiplexing parameters. Parameter | Associated Meaning
---|---
$s$ | DSTI modules per repeater
$N_{o}$ |  138Ba+ ions per DSTI module
$N_{m}$ |  171Yb+ ions per DSTI module
Table 3: Resource parameters.

Notice that the rate in Eqn. (13) is a function of the parameters $m,M,$ and n
along with physical system parameters such as collection and detection
efficiencies $\eta_{c},\eta_{d}$ and the total distance $L$ that enter the
formula through $p=\frac{1}{2}\eta_{c}^{2}\eta_{d}^{2}e^{-\alpha L_{0}}$,
where $L_{0}=L/(n+1)$. The denominator $\mathbf{T}$ is a function of the time
multiplexing block length $m$ and the clock cycle duration $\tau$, but also
depends on the ion-ion gate and measurement times $\tau_{g}$ and the heralding
time $T$, which is in turn a function of $L_{0}$. The dependence on $\tau_{g}$
is due to the fact that it takes a non-zero amount of time to perform the
essential entanglement swap operations at the repeater nodes, which is
$2\tau_{g}$ seconds ($\tau_{g}$ for the CNOT gate and $\tau_{g}$ for the $X,Z$
measurements).

Table A | Criterion: $T\geq\tau_{o}>\tau_{g}$ | Required $N_{o}$ | Required $N_{m}$ | Rate
---|---|---|---|---
|  | $2Mj$ | $\leq 2Mm$ | $\frac{\left(1-(1-p)^{Mm}\right)^{n+1}}{(k+m+2j-1)\tau}$
Table B | Criterion: $\tau_{o}>T\geq\tau_{g}$ | Required $N_{o}$ | Required $N_{m}$ | Rate
Case 1 | $T+\tau_{g}>\tau_{o}>T$ | Same as Table A
Case 2 | $\tau_{o}\geq T+\tau_{g}>T$ | $2(Mk+j)$ | $2m$ | $\frac{\left(1-(1-p)^{Mm}\right)^{n+1}}{(k+m+3j-1)\tau}$
Table C | Criterion: $\tau_{o}>\tau_{g}>T$ | Required $N_{o}$ | Required $N_{m}$ | Rate
Case 1 | $T+\tau_{g}>\tau_{o}>\tau_{g}$ | $2Mj$ | $\leq 2Mm$ | $\frac{\left(1-(1-p)^{Mm}\right)^{n+1}}{(m+3j-1)\tau}$
Case 2 | $\tau_{o}\geq T+\tau_{g}>\tau_{g}$ | Same as Table B, Case 2
Table 4: Rates and ion requirements for $s=1$ operation of the $(m,M)$
multiplexed repeater protocol. The rate expressions correspond to ideal gate
operations, and the optical fibers are assumed to be a pure loss channel. For
realistic gate operations, the overall rate is modified by the reverse
coherent information of the end-to-end state, which is a function of the gate
error parameter $\epsilon_{g}$ and the initial fidelity of the entangled
Werner state $F_{0}$. See Appendix A for detailed timing charts and
corresponding timing diagrams.

The precise formula for the rate attainable with an $(m,M)$ repeater protocol
over $n$ repeaters placed along a total distance $L$, along with the ion
requirements to support the protocol are tabulated in Table 4. The rate
depends on the clock cycle duration for ion-photon entanglement generation
attempts at the nodes $\tau$, and the relative values of the heralding time
$T$, ion-ion gate times $\tau_{g}$ and the communication ion lifetime
$\tau_{o}$. The value $\tau$ is allowed to be chosen independently of the
heralding time, which distinguishes our protocol from the one presented in
Ref. Santra2019-di . Consider $T=k\tau$ and the $\tau_{g}=j\tau$, where and
$j,k\in\mathbb{R}^{+}$. Values of $k,j<1$ clearly lead to sub-optimal rates,
since they imply ample idle time at the repeater nodes, when the ions are not
attempting entanglement distribution. Thus, we focus on operations that
correspond to $j,k\geq 1$. For simplicity of analysis, let us consider
$j,k\in\mathbb{Z}^{+}$. Values of $j=\tau_{g}/\tau>1$ can in principle be
realized with multi-sector traps. These are traps with distinct sectors of
ions (say $j$ of them) with each containing a batch of communication ions,
such that at every time step, ions in one of the sectors are collectively
excited, and the different sectors being excited in a cyclical fashion. It is
implicitly and reasonably assumed that i) $\tau_{o}>\tau_{g}$ so that a
communication ion’s quantum state can be faithfully transferred to a memory
ion with ion-ion gates before it irrecoverably decoheres, and ii) $\tau_{m}\gg
m\tau$ for a large range of values $m$, so that the memory ions can be
considered to be noise free. Among the $6={3\choose{2}}$ orderings of the
relative values of $T,\tau_{g}$ and $\tau_{o}$, due to the reasonable
assumption $\tau_{o}>\tau_{g}$, we are left with 3 possible orderings, namely:
$T\geq\tau_{o}>\tau_{g}$, $\tau_{o}>T\geq\tau_{g}$ and $\tau_{o}>\tau_{g}>T$.
Table 4 discusses the rates and the ion requirements achieved by the repeater
protocol for each of these cases. Timing charts and timing diagrams that
describe the protocol including the operations at the repeater nodes from
time-step to time-step, under these different conditions are elucidated in
Appendix A.

As an example, consider the case $T\geq\tau_{0}>\tau_{g}$ described in Table 5
in the Appendix. In this scenario, at every time step, $2M$ communication ions
generate ion-photon entanglement, with $M$ of the photons being directed
towards the left of the node and the other $M$ towards the right of the node.
The moment these photons are generated, an ion-ion gate is initiated on each
of the communication ions, to swap their atomic state into memory ions. For
gate time $\tau_{g}=j\tau,j\in\mathbb{Z}^{+}$, at time $t=j\tau,$ the
communication ions that were used to generate ion-photon entanglement at time
step $t=0$ are freed up due to the completion of the ion-ion gate, and hence
are ready to be reused. At this point, the first $2M$ atomic states have been
loaded into memory ions. At time step $t=k\tau,k\in\mathbb{Z}^{+}$, the
information about which two (one to the left of the node and one to the
right), if any, of the $2M$ entanglement generation attempts at time $t=0$
actually heralded an elementary link entanglement, is received, at which
point, the other $2(M-1)$ memory ions are freed up and ready for reuse. At
time $t=(k+m-1)\tau$, similarly, all potentially successfully heralded
elementary link entanglements across the time multiplexing block length of $m$
are stored in the memory ions. At this point, the repeater nodes choose the
latest successful heralded link to the left and to the right and perform
entanglement swap on those corresponding memory ions. Performing the
entanglement swap involves measuring these memory ions, which takes a time
duration $2\tau_{g}$, i.e., $2j\tau$. Thus, the rate of distributing 1 ebit
across the end nodes of the trapped-ion repeater chain is $\propto
1/(k+m-1+2j)\tau$. Since we consider global measurements that measure all ions
in the DSTI, all the other accumulated entanglement resources at the nodes are
also cleared in the process. The protocol then starts once again from time
step $t=0$.

Note that typically with time-multiplexed repeaters the heralding time only
causes latency in the protocol without affecting the rates dhara2021 .
However, in the present scheme of trapped-ion repeaters, as mentioned above,
ion measurements are considered to be global, full-trap measurements that
measure all ions present in a DSTI, as opposed to measurement of individual
ions in a trap. As a result, all the other accumulated entangled resources at
the nodes are also cleared in the process, which negatively impacts the
entanglement distribution rates.

Rate Formulas under realistic (noisy) gate operations. When realistic noisy
operations are considered at the repeater nodes, the rate formulas in Table 4
get scaled by the distillable entanglement of the noisy end-to-end entangled
state $\rho_{AB}$ across the line repeater network. The noisy entangled state
is given by a Werner state of fidelity parameter $F=1-\frac{3}{2}Q(n)$, where
$Q(n)$ is as given in Eqn. 7. A lower bound on the distillable entanglement is
given by the reverse coherent information of the state $\rho_{AB}$, defined as
$I_{R}(\rho_{AB}):=H(B)_{\rho}-H(AB)_{\rho},$ where
$H(B)_{\rho}=-\operatorname{Tr}(\rho_{B}\log_{2}\rho_{B})$ is the von Neumann
entropy of $\rho_{B}$. For Bell diagonal states, and hence for Werner states,
$I_{R}$ can be easily computed, since $\rho_{B}$ is the maximally mixed state
of entropy $H(B)_{\rho}=1$ and the entropy
$H(AB)_{\rho}=-F\log_{2}F-(1-F)\log_{2}\frac{1-F}{3}$.

Ion Requirements: For the case $T\geq\tau_{0}\geq\tau_{g}$, the ion
requirements can be identified from the timing chart in Table 5. The
requirement on the number of communication ions is $2jM$, which is the value
at which freed ions begin getting reused and the number of loaded
$\textrm{Ba}^{+}$ ions saturates. In other words, $2jM$ $\textrm{Ba}^{+}$ ions
are sufficient to support the optimal $(m,M)$ repeater protocol. The maximum
number of memory ions required in this case is given by $2mM$. The actual
number could be smaller, depending on the value of $m$ and its relation to
$j,k$, which might allow for some freed memory ions to be reused. On the other
hand, for the cases where $\tau_{o}>(k+j)\tau$, the communication ion
requirement is $2(Mk+j)$, whereas the memory ion requirement is independent of
$M,$ and given by $2m$. This is because the large $\tau_{o}$ allows one to
wait for the heralding information and subsequently apply the swap gate only
between the successfully heralded communication ion and the corresponding
memory ions.

###  IV.2 Performance Evaluation: Numerical Results

Unconstrained Repeaters: Here we numerically analyze the performance of the
repeater protocol assuming there are no constraints on the number of repeater
nodes, or the number of ions per node. We begin with the rate-vs.-distance
trade-off. To illustrate the results, we choose operating parameters of the
repeater to be $\tau=1\mu$s, $\tau_{g}=1\mu$s (i.e., j=1), $\tau_{o}=50\mu$s,
$\eta_{c}=0.3,\eta_{d}=0.8$, and the inter-repeater transmissions are assumed
to be over optical fiber of attenuation $\alpha=0.2$ dB/km and refractive
index $1.47$. The operational errors in gates and measurements are chosen as
$\epsilon_{g}=1-F_{0}$, where $F_{0}$ is the Werner fidelity of the elementary
link states. The value of $F_{0}$ is varied from $0$ (for ideal repeaters) to
$10^{-4},10^{-3}$ (with noisy operations). Different values of spatial
multiplexing $M=1,5,10$ are considered. The rates are numerically optimized
over the time multiplexing block length $m$ and the number of repeaters $n$.
The maximum of the optimal values of the different rate expressions
corresponding to the different cases in Table 4 (which for the chosen set of
parameters happens to correspond to Table 4 B, Case 2), is plotted in Fig. 2.
The rates are found to show sub-exponential decay with respect to distance,
primarily owing to deterministic entanglement swapping and additionally due to
time multiplexing. The rates are higher for higher $M$, but the advantage over
the corresponding PLOB bounds calculated as per Eqn. 8 also occurs at
commensurately longer distances. In the presence of operational errors in the
repeaters, the degradation of the rates with distance is more pronounced with
increasing values of the noise parameter. Nevertheless, the rate-distance
trade-off still beats the PLOB bound.

The optimal time multiplexing block-length and the optimal number of repeaters
for different degrees of spatial multiplexing $M$ are plotted as functions of
the total distance in Figs. 3 and 4, respectively. Notice that the optimal
value of $m$ increases with distance, saturating at large distances. The
optimal value of $m$ in relation to noise in the gates and measurements is
found to behave non-monotonically, first decreasing then increasing with noise
at large distances. However, most importantly, the values of $m$ are higher
for the lower value of $M$. In fact, at any given distance $L$ and noise
parameter, when $M$ is varied, the optimal $m$ ( call it $m_{\textrm{opt}}$)
satisfies the same mode multiplexing product $\mathbf{m}=m\times M$. For a
total distance of $150$ km and noise parameter $\epsilon_{g}\leq 10^{-4}$, the
optimal product is found to be $\mathbf{m}_{\textrm{opt}}\approx 220$. The
reason we find an optimal value for the product is that the rate (in
ebits/sec) per spatial mode, i.e., the rate in Eqn. (12) divided by $M$, has a
direct dependence on the product of the two multiplexing parameters, i.e.,
$m\times M$. (The value $220$ itself is a function of the choice of system
parameters such as the communication ion lifetime, gate times, optical fiber
attenuation, and coupling and detection efficiencies.) As a result, for
$M=1,5,10$, we have $m_{\textrm{opt}}=220,44,22$, respectively. The optimal
number of repeaters is seen to grow with the total distance at a rate
proportional to $M$. It is found to naturally slow down with increasing gate
and measurement noise, as more QR nodes would add more operational noise to
the shared ebits. For example, for $M=10$ and a total distance of $L=150$ km,
the optimal number of repeaters for noise parameter values
$\epsilon_{g}=10^{-4},10^{-3}$, are found to be $87$ and $25$, respectively,
which amount to inter-repeater spacing values of $L_{0}\approx 1.7$ km and $6$
km, respectively.

Figure 2: Entanglement distribution rate as a function of total distance
optimized over the number of repeater nodes and the degree of time
multiplexing, for different values of spatial multiplexing $M$, noise
parameter $\epsilon_{g}$, and $\tau_{g}=\tau=1\mu$s. These rates are compared
against the direct transmission benchmark, namely the corresponding PLOB
bounds (dotdashed lines) given by $-\frac{M}{\tau}\log_{2}(1-\eta).$ Figure 3:
Optimal degree of time multiplexing as a function of total distance for
different values of spatial multiplexing $M$, noise parameter $\epsilon_{g}$,
$\tau_{g}=\tau=1\mu$s, and optimal number of repeaters. Figure 4: Optimal
number of repeater nodes as a function of total distance for different values
of spatial multiplexing $M$, noise parameter $\epsilon_{g}$,
$\tau_{g}=\tau=1\mu$s and optimal time multiplexing. Figure 5: Required number
of communication ions as a function of total distance for different values of
spatial multiplexing $M$, noise parameter $\epsilon_{g}$,
$\tau_{g}=\tau=1\mu$s and optimal number of repeaters and time multiplexing.
Figure 6: Required number of memory ions as a function of total distance for
different values of spatial multiplexing $M$, noise parameter $\epsilon_{g}$,
$\tau_{g}=\tau=1\mu$s and optimal number of repeaters and time multiplexing.

The number of communication and memory ions per repeater node $(N_{o},N_{m})$
required to support the optimal rates under the proposed mode-multiplexing
protocol are shown in Figs. 5 and 6, respectively. The value of $N_{o}$
decreases sub-exponentially with distance so long as the gate and measurement
noise $\epsilon_{g}$ is small. In such a scenario, the required numbers are
smallest for protocol with smaller $M$ and increases with $M$. The value of
$N_{o}$ tends to increase with distance for $\epsilon_{g}$ above a threshold.
This is because large $\epsilon_{g}$ drives down the optimal number of
repeaters and consequently drives up the inter-repeater spacing. With
increasing inter-repeater spacing, the protocol in Table 4 B, Case 2 warrants
higher number of communication ions given by $2(MT+\tau_{g})/\tau$. On the
other hand, the required number memory ions always increases with distance,
since the optimal time multiplexing blocklength $m$ increases, too, and the
required number of memory ions is proportional to $m$. It is higher for lower
values of $M$ (or in other words for higher values of $m$).

Figure 7: Entanglement distribution rate as a function of total distance
optimized over the number of repeater nodes and the degree of time
multiplexing, for different values of spatial multiplexing $M$, with realistic
noisy gate operations of infidelity $\epsilon_{g}=10^{-4},\tau=1\mu$s and
$\tau_{g}=10\tau=10\mu$s. These rates are compared against the direct
transmission benchmark, namely the corresponding PLOB bounds (dot dashed
lines) given by $-\frac{M}{\tau}\log_{2}(1-\eta).$

We note that the rates attained in Fig. 2 based on the protocol in Sec. IV are
higher than those reported in ref. (Santra2019-di, , Fig. 7(b)). For instance,
at a total distance of $150$ km, noise parameter $\epsilon_{g}=10^{-4}$,
$M=10$ spatial multiplexing, and all other parameters being identical, our
protocol attains $20000$ ebits/sec, whereas the protocol in ref. Santra2019-di
achieves 700 ebits/s. The rate enhancement does come at the cost of higher ion
number requirements. For the said parameter values, the required number of
communication ions and memory ions per repeater for our protocol are
$N_{o}=170$, and $N_{m}=55$, respectively, whereas the protocol in ref.
Santra2019-di  required only $N_{o}=10,N_{m}=2$.

To conclude this section, in Fig. 7, we plot the entanglement distribution
rates when $\tau=1\mu$s, and $\tau_{g}=10\mu$s, i.e., for $j=10$. In other
words, this refers to a scenario where the gate operations are an order of
magnitude slower compared to the clock-cycle duration for ion-photon
entanglement generation, which is still retained at $1\mu$s. The end-to-end
entanglement distribution rates are seen to decrease only marginally (compared
to the case $\tau_{g}=\tau=1\mu$s). This is made possible by an increased
requirement on the number of communication ions in the DSTI modules. For
example, for $M=10,\epsilon_{g}=10^{-4}$, while the required number of
communication ions in the case of $\tau_{g}=\tau=1\mu$s was 170, it is 220 for
the case $\tau_{g}=10\tau=10\mu$s. Further, the repeater operation in the
latter scenario warrants traps with distinct sectors of communication ions
($j=\tau_{g}/\tau=10$ number of sectors) that can be excited successively in a
cyclical manner.

Constrained Repeaters: Here we numerically analyze how restrictions on the
number of repeater nodes and on the amount of ion resources at the repeaters
affect the entanglement distribution rates supported by the proposed protocol.

To understand how the entanglement distribution rates deteriorate upon moving
away from the optimal number of repeaters, the rates for different but fixed
values of inter-repeater spacing $L_{0}$ are plotted in Fig. 8 for repeater
operations with $M=10$ and $\epsilon_{g}=10^{-4}$. For $L_{0}$ larger than the
distance-dependent $L_{0}^{\textrm{opt}}$, the rates are seen to be
significantly smaller. For example, with $L_{0}=20$ km, the rate at a total
distance of $150$ km drops to $\approx 25\%$ of its value corresponding to the
optimal value of $L_{0}\approx 1.7$ km. However, the rate-distance scaling
remains unchanged in the limit of large distances, which implies it is still
possible to operate at rates higher than direct transmission rates even with
fewer and farther-spaced repeaters at large distances.

Figure 8: Entanglement distribution rate as a function of total distance
optimized over the degree of time multiplexing, for spatial multiplexing
degree $M=10$, noise parameter $\epsilon_{g}=10^{-4}$, $\tau_{g}=\tau=1\mu$s,
and different values of inter-repeater spacing $L_{0}$. These rates are
compared against the direct transmission benchmark, namely the corresponding
PLOB bound given by $-\frac{M}{\tau}\log_{2}(1-\eta).$

With regard to the number of communication ions, packing too many of these
ions in a DSTI module can cause issues with ion-photon entanglement due to
resonant re-absorption of emitted photons by neighboring communication ions.
Hence, the effect of restrictions on the number of communication ions is an
important consideration. Given a limit on the maximum number of communication
ions $N^{\max}_{o}$, a fixed gate operation time $\tau_{g}$ and the minimum
allowed clock cycle duration $\tau^{\min}$, the rates can be optimized over
the number of repeaters and time multiplexing parameter for different
combinations of spatial multiplexing $M$ and $j$ where $j=\tau_{g}/\tau$ can
be varied by choosing different values for the clock cycle duration $\tau$.
For $N^{\max}_{o}=125$, $\tau_{g}=\tau^{\min}=1\mu s$ and noise parameter
$\epsilon_{g}=10^{-4}$, Fig. 9 shows the plots of the repeater performance for
a few different allowed $M,j$ combinations, when the number of optical ions is
constrained to 125. The solid lines show the repeater performance when $j=1$,
which appears to allow a maximum $M$ of 5. By changing $\tau$ to $10\mu s$,
i.e., by reducing $j=0.1$, the maximum allowable $M$ increases to 50. The
performance under this modified regime of timing parameters is shown by the
dashed line, which is clearly sub-optimal. Thus, a higher degree of spatial
multiplexing at the expense of slower clock rate $1/\tau$ does not appear to
yield higher rates. Instead, the optimal strategy appears to be to pick the
highest possible $j$ value and then optimally choosing $M$ such that the
constraint $N^{\max}_{o}$ is still satisfied.

Finally, with regard to the number of memory ions, given a limit on the
maximum number of these ions, the set of allowed values for the time
multiplexing parameter becomes restricted. The reader may refer to the steady
state value of ion occupancy values in Tables 5-7 to understand the dependence
of various constraints on the protocol and network parameters. As a result the
optimal rate drops faster with distance for smaller values of $N^{\max}_{m}$
as illustrated in Fig. 10. When $N^{\max}_{m}$ falls below a threshold, the
rate-distance scaling no longer beats the direct transmission benchmark.

Figure 9: Entanglement distribution rate as a function of total distance under
constrained resource availability, $N^{\max}_{o}=125$. The values of $M$ are
specified with realistic noisy gate operations of infidelity
$\epsilon_{g}=10^{-4}$ and $\tau_{g}=1\mu$s. The operational clock cycle
duration $\tau$ is different for the solid $(\tau=1\mu s)$ and dotted
$(\tau=10\mu s)$ lines. Figure 10: Entanglement distribution rate as a
function of total distance under constrained resource availability,
$N^{\max}_{m}$. For spatial multiplexing $M=5$, realistic noisy gate
operations of infidelity $\epsilon_{g}=10^{-4}$ and $\tau_{g}=\tau=1\mu$s, the
rates corresponding to different values of $N^{\max}_{m}$ are plotted.

##  V Conclusion

To conclude, we presented a general architecture for a repeater node based on
DSTI modules, and discussed a repeater protocol based on spatial and time
multiplexing. For DSTI modules with 138Ba+ as communication ions and 171Yb+ as
memory ions, assuming reasonable values for operating parameters (operation
errors under $\epsilon_{g}<10^{-4}$, gate time $\tau_{g}=1\mu$s, clock cycle
duration $\tau=1\mu$s, communication ion lifetime $\tau_{o}=50\mu$s, coupling
efficiency $\eta_{c}=0.3$ and detection efficiency $\eta_{d}=0.8$), the
proposed repeater protocols based on spatial and time multiplexing can attain
entanglement distribution rate $\sim 20000$ ebits/s at a distance of 150 km,
with repeaters placed at $\approx 1.7$ km spacing, and each containing about
$170$ and $55$ 138Ba+ and 171Yb+ ions, respectively. This constitutes a nearly
$30\times$ improvement over the rate reported in the earlier work of
Santra2019-di  for the same set of operating parameters, but requires larger
number of ions at the repeater nodes. The larger ion number requirements can
potentially be met by bootstrapping several DSTI modules at the repeater
nodes. However, the modular interactions even within a repeater node would
then require probabilistic optically-mediated entanglement swapping
operations, which can cause a degradation of the entanglement distribution
rates. This calls for the design of more advanced protocol that can assuage
this degradation and optimally leverage multiple DSTI modules at repeater
nodes. In this regard, allowing for arbitrary optically-heralded intra-node
ion-ion logic across traps along with universal-capable logic in the traps, we
will explore protocols that incorporate block entanglement distillation codes
BDSW96  as part of future work. These protocols will take advantage of
multiple successful, elementary entanglement generation attempts, while
alleviating the drawbacks stemming from the constraint of having to measure
all the ions in a trap simultaneously and the probabilistic, optically
heralded intra-node logic.

###### Acknowledgements.

This work was supported by the National Science Foundation Convergence
Accelerator Program, award number 2040695. NML additionally acknowledges
support from the Army Research Office, award number W911NF1910296. SG and KPS
additionally acknowledge the NSF Center for Quantum Networks (CQN), grant
number EEC-1941583. KPS thanks Liang Jiang for helpful discussions.

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##  Appendix A Timing Analysis for Resource Count Calculation

There are multiple operating regimes for realistic operations with noisy and
non-instantaneous quantum gates. The reader may refer to Table 4 for a summary
of the various protocol types. We have examined the timing analysis of each
protocol type in depth in Tables 5-7. The corresponding timing diagrams are
shown in Figs. 11-14

Case 1 |  $Ba^{+}$ Occupancy |  $Yb^{+}$ Occupancy | Max. number of heralded ions
---|---|---|---
Time | Initialized | Freed | Loaded | Loaded | Freed
0 | $2M$ | - | $2M$ | - | - | -
$\tau$ | $4M$ | - | $4M$ | - | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$(j-1)\tau$ | $2jM$ | - | $\mathbf{2jM}$ | - | - | -
$j\tau$ | $2(j+1)M$ | $2M$ | $2jM$ | $2M$ | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$(m-1+j)\tau$ | $2(m+j)M$ | $2mM$ | $2jM$ | $\mathbf{2mM}$ | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$k\tau$ | $2(k+1)M$ | $2(k-j+1)M$ | $2jM$ | $2(k-j+1)M$ | $2(M-1)$ | $2$
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$\mathbf{(k+m-1)\tau}$ | $2(k+m)M$ | $2(k+m-j)M$ | $2jM$ | $2(k+m-j)M$ | $2m(M-1)$ | $\mathbf{2m}$

Case 2 |  $Ba^{+}$ Occupancy |  $Yb^{+}$ Occupancy | Max. number of heralded ions
---|---|---|---
Time | Initialized | Freed | Loaded | Loaded | Freed
0 | $2M$ | - | $2M$ | - | - | -
$\tau$ | $4M$ | - | $4M$ | - | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$(j-1)\tau$ | $2jM$ | - | $\mathbf{2jM}$ | - | - | -
$j\tau$ | $2(j+1)M$ | $2M$ | $2jM$ | $2M$ | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$k\tau$ | $2(k+1)M$ | $2(k-j+1)M$ | $2jM$ | $2(k-j+1)M$ | $2(M-1)$ | $2$
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$(m-1+j)\tau$ | $2(m+j)M$ | $2mM$ | $2jM$ | $\mathbf{2mM}$ | $2(m+j-k)(M-1)$ | $2(m+j-k)$
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$\mathbf{(k+m-1)\tau}$ | $2(k+m)M$ | $2(k+m-j)M$ | $2jM$ | $2(k+m-j)M$ | $2m(M-1)$ | $\mathbf{2m}$

Table 5: Timing chart for Table 4 Type A, i.e., when $T\geq\tau_{o}>\tau_{g}$, and Table 4 B Case 1, i.e., $(\tau_{o}>T\geq\tau_{g})\wedge(T+\tau_{g}>\tau_{o}>T)$. Both of these involve sub-cases 1 and 2 corresponding to $k-j+1\geq m$ and $k-j+1<m$, respectively, where $T=k\tau,\tau_{g}=j\tau$. The timing diagrams for the different cases of this protocol type are shown in Figs. 11 and 12 Figure 11: Timing diagram for Table 1 Type A when $k-j+1\geq m$.  Figure 12: Timing diagram for Table 1 Type A when $k-j+1<m$. Case 2 |  $Ba^{+}$ Occupancy |  $Yb^{+}$ Occupancy | Max. number of heralded ions
---|---|---|---
Time | Initialized | Freed | Loaded | Loaded | Freed
0 | $2M$ | - | $2M$ | - | - | -
$\tau$ | $4M$ | - | $4M$ | - | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$k\tau$ | $2(k+1)M$ | $2(M-1)$ | $2(kM+1)$ | - | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$(k+j)\tau$ | $2(k+j+1)M$ | $2(M-1)(j+1)+2$ | $\mathbf{2(kM+j)}$ | $2$ | - | $2$
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$\mathbf{(k+j+m-1)\tau}$ | $2(k+j+m)M$ | $2(M-1)(j+m)+2m$ | $2(kM+j)$ | $\mathbf{2m}$ | $-$ | $\mathbf{2m}$
Table 6: Timing chart for Tables 4 B and C, Case 2, i.e., when $(\tau_{o}>T\geq\tau_{g})\wedge(\tau_{o}\geq T+\tau_{g}>T)$ and $(\tau_{o}>\tau_{g}>T)\wedge(\tau_{o}\geq T+\tau_{g}>\tau_{g})$, respectively.The timing diagram for this protocol type is shown in Fig. 13. Figure 13: Timing diagram for Table 1 Type B. Case 2 |  $Ba^{+}$ Occupancy |  $Yb^{+}$ Occupancy | Max. number of heralded ions
---|---|---|---
Time | Initialized | Freed | Loaded | Loaded | Freed
0 | $2M$ | - | - | - | - | -
$\tau$ | $4M$ | - | - | - | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$k\tau$ | $2(k+1)M$ | - | - | - | - | -
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$j\tau$ | $2(j+1)M$ | $2M$ | $\mathbf{2jM}$ | $2M$ | $2(M-1)$ | 2
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
$\mathbf{(j+m-1)\tau}$ | $2(j+m)M$ | $2mM$ | $2jM$ | $\mathbf{2mM}$ | $2m(M-1)$ | $\mathbf{2m}$
Table 7: Timing chart for Table 4 C, Case 1, i.e., when
$(\tau_{o}>\tau_{g}>T)\wedge(T+\tau_{g}>\tau_{o}>\tau_{g})$. The timing
diagram for this protocol type is shown in Fig. 14. Figure 14: Timing diagram
for Table 1 Type C.

##  Appendix B Optimization Process

Since the protocol in Section IV is determined by relation between the timing
parameters, it is not directly apparent, which rate equation holds true for a
given set of protocol parameters (refer Table 2). The number of repeaters
plays a primarily role in determining the heralding time $T$. For the present
numerical analysis, we find the optimal parameter values for a given set of
conditions using standard optimization techniques. Depending on the optimal
values calculated, we now have to make a decision about which type of the rate
equation from Table 4 is actually applicable. This is done by traversing the
decision tree for the optimal parameter values shown in Fig. 15. One can note
that based on the conditions (red diamonds) that are satisfied, the end leaves
of the decision tree indicate which rate equation holds true (blue boxes).

Figure 15: Decision tree to determine relative timing parameter ordering and
associated rate equations.