# The Impact of Quantum Memory Quality on Entanglement Assisted Communication

Stephen DiAdamo and Janis Nötzel
Emmy-Noether Gruppe Theoretisches Quantensystemdesign
Technische Universität München
{stephen.diadamo<EMAIL_ADDRESS>
###### Abstract

This work explores entanglement-assisted communication, where quantum
entanglement resources enable the transmission of classical information at an
enhanced rate. We consider a scenario where entanglement is distributed ahead
of time based on network traffic levels, and simulate a setting where idle
nodes generate and store entanglement to later transmit messages at an
accelerated rate. We analyze this communication model using noise models for
quantum memory in various scenarios, and extend our investigation to a
quantum-enhanced distributed computing environment, where entanglement storage
enhances data transmission rates for cooperative data processing. We propose a
protocol and demonstrate a distributed version of unsupervised clustering. Our
results show that, for qubit channels, high rates of entanglement generation
and modest storage requirements can surpass the classical limit with
entanglement assistance.

###### Index Terms:

Entanglement-assisted communication, entanglement storage, quantum networks,
quantum memory, distributed computing.

##  I Introduction

The field of quantum networking is becoming an increasingly important topic as
quantum hardware technologies begin to perform what has been predicted
theoretically. Theoretically, networked quantum technologies offer a variety
of use cases that promise to outperform current and potentially any future
classical implementation. Such use-cases are in distributed quantum computing
[1, 2, 3], distributed quantum sensing [4], quantum key distribution [5], and
especially related to this work, quantum communication [6, 7, 8]. On the
hardware side, experimentally we have seen non-local control gates between
physically separated quantum processors [9], deployed quantum key distributed
networks [10], and recently an entanglement-assisted communication experiment
[11]. These results all give promise to the future goal of deploying networked
quantum technologies.

In this work, we focus on entanglement-assisted classical communication over
quantum channels. At a high level, a quantum channel is a communication
channel that can transport quantum systems from one physical location to
another. Possible realizations of quantum channels are fiber optic cables or
free space. Quantum systems can have a unique property which is not seen in
purely classical states called entanglement. Two or more quantum systems being
entangled means, irrespective of physical distance, the systems are correlated
in their measurement outcomes. In particular, the exact form of the
correlation can depend on local actions performed by the parties holding parts
of the entangled systems. The effect of this behavior to network coding
situations has been investigated in recent works [12, 13, 14, 15].
Entanglement-assisted communication is the communication scenario where,
between communicating parties, entanglement resources are established before
messages are transmitted, and at the time of message transmission, messages
are encoded into sending party’s part of the entangled system to then be
transmitted over a quantum channel. The receiving party decodes the incoming
quantum system with the assistance of their part of the entangled resource by
measuring the transmitted system from the sender and their entangled part
together in a particular way. Entanglement-assisted communication has been
proven in a variety of communication scenarios to increase the classical
capacity of a quantum channel [16, 17]. The effect of this additional resource
on the link layer of a communication link has been also investigated in [18].

Most important to any entanglement-assisted communication scenario is the
ability to distribute entanglement so that when a message is transmitted, the
required entanglement resources are available at both ends of the
transmission. One can consider three possible ways to achieve this: 1)
Generate entanglement at the time it is needed and then use it immediately to
transmit a message, 2) Generate entanglement before it is needed and store it
for later use, or 3) A hybrid of the two. On one hand, generating entanglement
at the time it is needed adds a layer of communication complexity to the
overall protocol, but reduces the need for highly robust quantum storage
devices. On the other hand, establishing the entanglement ahead of time
reduces the communication complexity at the time of transmission, but storing
entanglement resources requires a robust quantum memory—a highly-challenging
engineering problem to build.

In this work, we, therefore, target the possibility of implementing a
classical communication system with the assistance of quantum entanglement. We
focus on the physical layer of the system, and specifically on the effects
noisy quantum memories have on the transmission and error rates of the
communication. We start with a point-to-point setting and further analyze a
network of devices. As a final part of our analysis, we consider using
entanglement-assisted channels for performing distributed computing. We find
that with qubit channels, an advantage can be seen with noisy memories, but
with relatively long coherence times. For a scheme to move from theory to
practice, the hardware parameters must be considered to determine if there are
advantages to gain. In this work, we aim to add clarity to the questions
regarding what quality of quantum memory is necessary to perform entanglement-
assisted communication.

##  II Review of Related Work

Past work related to the topics covered here is that which is related to
quantum channels that have a queuing aspect, co-dependent queues, entanglement
distribution protocols, or trading off entanglement resources for achieving
entanglement-assisted capacities. In this section, we review various results
and discuss their relation to the present work. Overall, this article offers a
unique perspective to analyzing how building-up entanglement resources, while
the network traffic rates are low, provides a communication advantage in
various network settings.

We begin by reviewing the findings of Mandayam et al. [19]. Their study
focuses on a communication scenario involving a sequentially processed stream
of qubits. Before processing, these qubits are stored in a quantum memory,
where processing entails performing quantum operations on the qubits or
transmitting them over a quantum channel. During the period between when
qubits enter the memory and when they are processed, they experience
decoherence, resulting in the loss of information over time. The authors model
the system after a single server queue and derive an expression for the
classical capacity in terms of the qubit’s waiting time. Their analysis is
conducted on a queuing model that allows for unlimited storage space, with
arriving qubits processed in a First-Come-First-Served order. The waiting
qubits’ decoherence is modeled as a quantum channel, which depends on their
waiting time.

The main outcome of their work is a formula for the classical capacity of the
quantum-queue channel. However, there are a few distinctions between their
findings and our present study. Firstly, the queued quantum systems we
consider are exclusively used for storing entanglement. For actual data, we
employ a classical message buffer that stores classical bits of information,
which are subsequently encoded in an entanglement-assisted manner using the
stored entangled quantum system. The encoded quantum system is then
transmitted over a quantum channel and decoded by the receiver. In our
approach, we utilize the entanglement-assisted classical capacity of the
channel to achieve accelerated communication rates. This type of capacity
differs from the stricter classical capacity discussed in the previous study.

A network of nodes that can build up and re-distribute entanglement among
themselves is analyzed in [7]. In this work, various network topologies are
considered, and a method of entanglement generation is proposed to maintain
synchronized quantum buffers. This work deals with the specific case of
noiseless qubit storage and superdense coding between the network nodes.
Moreover, the work focuses mainly on the network layer of the stack. In this
work, we take a similar approach, building on the results of [18], where we
use the protocol proposed but and focusing on the physical layer aspects of
the system in this case. In [18], we proposed a link-layer approach for
storing and using entanglement depending on the required rate of transmission.
In that work, we did not consider in depth how the physical layer effects the
communication rate. Here we focus on those aspects and in particular on how
memory noise of the quantum storage affects the communication rate.

A method for using quantum channels to transmit a mixture of classical
information and entanglement is analyzed in [20]. Here the method of “trade-
off” coding is introduced which, for a given entanglement-consumption rate,
some of the transmission uses entanglement-assisted coding and the remaining
part of the transmission without entanglement. This method of transmitting
with and without entanglement in one transmission outperforms a more common
approach known as time-sharing, where some transmissions are only without
entanglement assistance, and some are with it. [20] takes the perspective of
finding the ultimate rates, but consideration for what hardware specifications
are needed to execute such a protocol is not considered. In this work, our
queuing model takes a time-sharing approach as a first step as our focus here
is to benchmark quantum hardware for their ability to perform entanglement-
assisted communication. We aim to clarify how good quantum hardware should be
to perform entanglement-assisted communication before moving to the more
complex coding scenario of trade-off coding.

For a communication channel using an auxiliary channel for entanglement
generation along with another quantum channel for data transmission,
Djordjevic has considered the transmission rates with a bosonic model [21].
What is shown is that for such a channel, using the auxiliary channel, not for
entanglement generation, but rather message transmission, the communication
rates will always be higher compared to entanglement-assisted communication
rates. Indeed, this work also does not consider that the entanglement can be
built up and stored ahead of time. Storing quantum states for very long
periods is beyond the ability of current technologies, and not making such
considerations shows us that without such technology, entanglement-assisted
communication may never provide an advantage over traditional methods. What we
determine in this work is how much of a communication advantage is there when
such memories exist, and if it is enough of a reason that additional efforts
for building robust quantum memories for communication will lead to a large
enough payoff. Prior work [22] has pointed out the potentially infinite-fold
benefits as well as the current possibilities of harvesting them [17, 23],
which points to to optimistic outlook.

The effects on entanglement concerning memory coherence times have been
investigated in a simulation setting in [24]. In this work, the authors
simulate entanglement sharing between two end nodes for entanglement
distribution and consider an entanglement management scheme for optimizing
performance. In the work, the application tested is the generation of
entanglement over a linear chain of network nodes using an entanglement
swapping procedure. The key metric considered is the entanglement generation
rate over a system with length-dependent loss in the channel. The present work
differs in that we consider a specific application requiring the consumption
of entanglement for classical message transmission. In this initial work, we
concern ourselves only with how the effects of memory influence the
communication of classical messages.

##  III The Communication Model

Our communication scenario is based on the second scenario discussed in the
introduction, which involves generating entanglement by preparing and storing
it for future use. Initially, we focus on a point-to-point communication
model, as illustrated in Fig. 1, where both classical and entanglement units
are stored. In this model, there is a sender and a receiver who are connected
by a quantum channel. They also share a synchronized quantum memory for
storing entanglement resources. Additionally, the sender possesses a classical
message buffer, which receives fixed-length bit-strings with a certain
probability per time interval.

Using this model, we can establish a communication process that prioritizes
message transmission while utilizing idle periods to generate entanglement. At
a high level, the process operates as follows: within a specific time
interval, a classical message either arrives or does not. If a message
arrives, the sender attempts to add it to a buffer of length $L$. If the
buffer is too full, the message is dropped. Subsequently, the sender tries to
transmit a message. In the absence of messages in the buffer, instead of
remaining idle, the sender generates multiple entanglement resources, denoted
as $G$, and shares half of them with the receiver, who stores them. The sender
also stores the remaining half locally in an $E$-length buffer, if feasible.
When both entanglement resources are available in storage and there are bits
to transmit, the sender employs those resources to transmit at the
entanglement-assisted capacity $C_{E}$ of the channel. In cases where no
entanglement resources are available at the time of transmission, or if $E=0$,
the sender resorts to an entanglement-free transmission, utilizing the
classical capacity $C_{C}$ of the channel. The protocol flow is illustrated in
Fig. 2, and we refer to this process, described in [18], as the Generate
Entanglement While Idle (GEWI)protocol.

Based on this setup, we have devised an implementation of the system using
hardware components specifically designed for quantum communication. The first
requirement is the generation and storage of entanglement, which can be
achieved through various methods [25], each with its own advantages and
disadvantages. In our case, we assume that entanglement is generated at the
sender’s side, with half of it transmitted through a communication channel
while the other half is stored. For storing the quantum systems, various
proposals for quantum memories have been put forth in existing works [26].
Once we have the entanglement generation and storage components in place, the
remaining components involve qubit generation, establishing a channel between
the two parties, and the manipulation of outgoing and incoming qubits. We
depict all these components collectively in Fig. 3.

Figure 1: A communicating party in (a) has a classical message buffer of
length $L$ which binary messages of length $J$ arrive and are stored. Further,
at (a) is a quantum memory with $E$ slots used for entanglement buffering.
Connecting parties (a) and (b) is a quantum channel, where classically encoded
quantum messages are sent, depending on the state of the entanglement buffer.
At (b) is another entanglement buffer that is synchronized with the one at
(a). Depending on the state of the entanglement buffer, messages are either
decoded with entanglement assistance or not. Figure 2: A flow diagram for the
GEWI processes which governs an buffered entanglement channel’s transmission
and internal entanglement generation. The process repeats indefinitely. Figure
3: The sender an receiver node configuration using in the NetSquid
simulations. The sender and receiver each have a quantum memory, which they
maintain synchronization between. The sender has two quantum sources, one for
generating qubits and one for generating EPR pairs. The sender has a classical
message buffer which feeds data to the quantum processor which prepares fresh
qubits from the source or qubits removed from the memory for transmission.
Once transmitted, the receiver can process the qubits or store them, and
outputs a classical message accordingly.

##  IV Simulation Setup and Parameter Selection

Throughout this work, we simulate the communication scenarios using the system
in Fig. 3 in various configurations, namely a point-to-point, unidirectional
setting, a 4-node network, and a bi-directional point-to-point setting. The
settings we use for simulation are a quantum channel with a classical buffer,
quantum memory, and a parameter for the rate of incoming messages. For message
transmission, classical information is encoded into qubits, and when possible,
using pre-established entanglement, transmitted using superdense coding. When
there is no entanglement available, qubits are transmitted directly with no
entanglement assistance. The simulations contain a quantum channel that is
modeled after a lossless $1~{}km$ fiber with a fixed delay time of $5~{}\mu
s$, roughly the time light takes to travel through a $1~{}km$ fiber. We make
the choice to use a lossless link in this case to better understand the
ultimate rates for short distance communication. The qubits stored in the
quantum memories experience time-dependent decoherence properties and decohere
in the memories according to the memory relaxation time $T_{1}$ and dephasing
time $T_{2}$. These parameters are integrated into the simulation setup
directly, and the qubits stored undergo time-dependent decoherence while in
storage, within the simulation setting.

These effects act on a quantum state $\rho$ as follows, as defined in [27, Eq.
1] and we write it here for convenience. Assuming a state is affected by the
environment for $\Delta t$ seconds, the effects are defined as follows.
Firstly a quantum channel is applied to $\rho\mapsto E_{0}\rho
E_{0}^{\dagger}+E_{1}\rho E_{1}^{\dagger}$, where
$E_{1}\coloneqq\outerproduct{0}{0}+\sqrt{1-p_{1}}\outerproduct{1}{1}$ and
$E_{1}\coloneqq\sqrt{p_{1}}\outerproduct{0}{1}$, with $p_{1}\coloneqq
1-e^{-\Delta t/T_{1}}$. Then, a subsequent channel
$\rho\mapsto(1-p_{2})\rho+p_{2}Z\rho Z$ is applied, where
$p_{2}\coloneqq\tfrac{1}{2}(1-e^{\Delta t/T_{2}}e^{\Delta t/(2T_{1})})$. Here
we are concerned with the quality of storage required to implement the
protocol and do not apply noise effects to the writing and reading steps. Loss
effects for reading and writing to quantum memories will add a straightforward
rate reduction proportional to the probability of loss, so we do not consider
this effect here, as we are only considering storage efficiency.

The simulations we perform are concerned with how the physical
layer—specifically the quantum memory—properties affect the throughput and
error rate during communication. To simplify this as much as possible, we
program the simulation to be independent of any link-layer protocols, but in a
real communication setting it would be required for the receiver to
distinguish qubits to be stored and qubits to measure using classical control
information. An initial link-layer approach for this particular communication
setting has been considered in [28], where a data frame of qubits is
transmitted behind a frame header, indicating how the following qubits should
be treated at the receiver. Here, we ignore the link-layer protocol to focus
on the ultimate rate, which can only diminished with additional link-layer
processing.

To develop the simulations, we use the quantum network simulator NetSquid
[27]. NetSquid is a Python-based software library used for simulating—for
one—various physical layer effects in quantum networks. Incorporated into the
software are the physical models of various quantum technologies such as
quantum memories, fiber optical channels, and photon sources and detectors.
NetSquid is developed in a modular way, so that network nodes can contain any
variation of the built-in hardware models. We construct a sender and receiver
using these models.

##  V Point to Point Channels

In this section, we develop and present simulations of our communication model
using varying quantum memory parameters and analyze the achievable throughput
and decoding error rates for various scenarios. We investigate how the $T_{1}$
and $T_{2}$ times of the stored qubits and the number of memory positions
available affect the throughput when using the communication setting
previously defined.

To begin, each party starts with empty entanglement buffers and the sender’s
message buffer is empty. With some probability $r$, a bit-string of length $J$
fills a classical buffer of capacity $L$ if possible, otherwise, the message
is dropped. If no message arrives, the sender uses its qubit source to
generate an EPR pair, storing half of the pair in its memory and transmitting
the other half over the channel. We test two approaches here, namely, if the
quantum memory is full to replace the oldest qubit or to simply discard the
newly generated qubit. To transmit messages, when there is no entanglement
stored in the memory, 1 bit of the message is encoded into a single qubit
(mapping $0$ to $|0\rangle$ and $1$ to $|1\rangle$) generated by the secondary
qubit source which produces ground-state qubits and is sent over the fiber
after encoding in the quantum processor. When entanglement is instead present
in the memory, 1 qubit is removed from the memory using the selected queuing
practice. Here we test two practices which are using a First-In-Last-Out
(FILO) priority and a First-In-First-Out (FIFO) priority. The qubit removed
from memory is then put into the quantum processor to be encoded and
transmitted using the particular queuing practice.

We compare the transmitted messages to those received to determine the
percentage of the messages that were transmitted with and without error. To
compute the average throughput, we determine the fraction of the number of
bits transmitted in the total iterations multiplied by the decoding error
rate. We simulate 15 different settings grouped into three different
simulations. Firstly, we replicate the simulation from the previous section,
using $L=J=4$ varying the $T_{1},T_{2}$ times of the model and fixing $E=200$
memory slots. Next, we repeat the process using $J=4$ and $L=5J$. Finally, we
fix the $T_{1},T_{2}=1100,1000$ ns and vary the number of memory slots in the
quantum memory. The sender attempts to send a message every $10$ ns—simulating
a maximum of $400$ mb/s link at most—or else generates entanglement using an
entanglement source. We further assume the entanglement generation rates are
equally as high.

We analyze the results of the simulations, the first of which are plotted in
Fig. 4. In the upper plot is the average throughput results for varying
$T_{1}$ and $T_{2}$ times of the buffer against a varying message arrival
rate. As expected, the longer the $T_{1}$ and $T_{2}$ times, the smaller the
error rate and thus a better throughput can be achieved. We also observe that
under a certain threshold for the $T_{1}$ and $T_{2}$ times, superdense coding
performs far worse than in a purely classical way, where the dashed line
represents the entanglement-free case. We can see when the $T_{1},T_{2}$ times
of the memory are $11,10$ ns respectively, since the messages are 4 bits long,
until $r\approx 0.38$, the transmission produces completely random outputs
with a decoding error rate of $1-1/2^{4}=0.9375$. The point is exactly that at
which the majority of the messages are sent using classical means over
entanglement-assisted.

In the plot of the error rates, the lower figure in Fig. 4, we note that the
red curve representing the $T_{1},T_{2}$ times $1100,1000$ ns is not non-
increasing, and at around the point $r\approx 0.38$ raises slightly to a local
maximum before beginning to steadily fall. This is because the system uses
FILO ordering for entanglement consumption, leaving some entangled pairs to
decohere for long periods. With the red dashed line in the same plot, we show
the error rates for a FIFO order. When using FIFO there is no increase in
error rate as observed for FILO, but the error rates are much worse overall in
comparison to FILO. The reason is that FIFO wil choose the entangled pairs
that have been stored the longest when performing entanglement-assisted
communication. Especially when small amounts of data arrive, inter-arrival
times for data become long and therefore the initially entangled quantum stats
will typically have decohered. Another point of note is when we apply the
approach of discarding old entangled pairs, replacing them with fresh pairs
when the memory is full with a FILO order. We call this strategy entanglement
replacing. The throughput and error rates in this case are shown with dotted
lines. Here we see significant improvement at low traffic rates, eventually
converging to the same trend at higher rates, as there are fewer opportunities
to replace aging pairs.

In Fig. 5, we repeat the simulation, but increase the message queue size so
that at most 5 jobs are in the message queue at once and observe similar
trends but with the plots squeezing more tightly to the smaller values of $r$.
Because fewer jobs are dropped with a larger message queue size, the system is
idle less often and therefore has fewer opportunities to generate
entanglement. Indeed in this scenario, the average throughput in the classical
case (the dashed black line) outperforms the three cases of noisy memory,
which is in contrast to the single job buffer. Moreover, comparing the
classical trend to the trend for the perfect quantum memory, there is only a
small portion of the incoming job probability domain in which superdense
coding with GEWI outperforms the classical case. For the error rates, we again
compare the FILO and FIFO consumption of entanglement. We see again that FIFO
ordering performs poorly, no better than random decoding, where FILO ordering
reduces the error rates. Again, the dotted trend represents the case where
entanglement bits are refreshed with a FILO ordering, and significant
improvement is observed at low message arrival rates.

The final set of simulations we performed for this section is comparing the
performance when varying the size of the quantum memory. In Fig. 6 are the
trends for four memory sizes which all share the properties for
$T_{1},T_{2}=1100,1000$ ns. What we observe is that with this choice of
$T_{1},T_{2}$ times, of all the entanglement buffer sizes that we tested, when
the rate of incoming messages is low, a smaller quantum memory size performs
better than a larger one, and indeed when $E=10$, until $r\approx 0.35$,
outperforms the cases when $E=200$. This is due to the aging of the entangled
pairs and using a FILO consumption without replacing the older stored pairs.
Since the rate of incoming messages is too low, the stored EPR pairs decohere
before they can be used, whereas in the $E=10$ case, the buffer is emptied
more often and so the EPR pairs are younger on average. When $r\gtrsim 0.38$,
the rate at which entanglement is consumed increases enough until both $E=10$
and $E=200$ perform equivalently. With the entanglement replacing strategy, we
see that for low traffic rates, replacing the entanglement with fresh units
improves performance significantly, reducing error rates significantly and
thus improving the overall throughput.

Overall, from these results we can conclude the following: Firstly not only do
the $T_{1},T_{2}$ times greatly affect throughput, but also the order in which
the entanglement is consumed plays a large role. By testing three orders of
magnitude in this case, we can already observe $T_{1},T_{2}$ values that
perform sub-classically, and those which perform super-classically. Moreover,
common orderings like FIFO and FILO are suboptimal orderings and so future
work will therefore include a deeper analysis into how the ordering of
entanglement consumption and the replacing of aging entanglement units
optimize the throughput of the system. Second, with suboptimal entanglement
consumption orderings, we see that the size of the entanglement memory plays
an important role as well. With shorter $T_{1},T_{2}$ times, using a smaller
entanglement buffer will ensure the entanglement buffer is emptied more often,
thereby maintaining fresher entanglement units on average and hence better
decoding error rates. Lastly, in all cases, replacing stored EPR pairs with
fresh pairs will improve the decoding error rates, thereby improving the total
throughput.

Figure 4: Two nodes separated by a fibre $1~{}km$ long with a fixed delay of
$5000~{}ns$. In the first simulation, Each node has an entanglement buffer of
200 memory slots which have a specific $T_{1}$ and $T_{2}$ times associated
with them.  Figure 5: The same configuration as Fig. 4 where here the size of
the message buffer here is increased to $L=5J$, allowing the message buffer to
hold 5 jobs at most. Figure 6: The same configuration as Fig. 4 where $T_{1}$
and $T_{2}$ times here are fixed at $1100~{}ns$ and $1000~{}ns$ respectively.
The size of the entanglement buffer is varied, and the message buffer is fixed
at $L=J$, allowing one job in the buffer at once.

##  VI Networks of Buffered-Entanglement Channels

In the previous section, we analyzed the properties of a single link under the
process of GEWI with superdense coding, and so a next naturally arising
question is: what are the properties of a network of such links? These types
of questions have arisen in other queuing theory contexts, specifically,
Jackson and Kelly networks for various queuing models [29, 30], where the
shape of the throughput can be determined analytically for some queue types.

The configuration in this simulation is the following: For the nodes with
outgoing connections, transmission is performed using the process shown in
Fig. 2, for each connection. When qubits are sent to a node, they are measured
and re-prepared to be transmitted again depending on the state of the
entanglement buffer. Each connection therefore builds up and consumes its own
entanglement, with one pair of synchronized entanglement buffers per
connection. In Fig. 7, the nodes other than the source $s$ have a larger
message buffer so that messages do not get dropped en route. Decoding and re-
encoding at the mid-way nodes eliminates the need for entanglement swapping,
simplifying the technological requirement for a physical realization in this
first stage. The sink node $d$ distinguishes messages arriving from its two
connections, and decodes entanglement-assisted messages based on the relay
node it receives the message from.

We use a similar settings as in the single link case: the source node can
store just one message at a time with $J=L=4$ bits. We repeat the simulation
with $L=5J$, allowing 5 messages in a buffer at once, but we do not show the
results here due to space constraints, and that they have similar trends as
the $J=L$ case. Each node has the capability to store $E=200$ entanglement
units with varying $T_{1},T_{2}$ times for the simulations. In this case, the
time that the source polls for new messages is every $10~{}ns$, thereby
simulating a maximum of $400$ mb/s network throughput.

In each of the simulations, we use a simple routing approach. We route such
that a node chooses the outgoing link that has the most stored entanglement at
the time of transmission. The result of this is that entanglement-assisted
transmission is used as often as possible. Alternative approaches can be used,
for example in [7], where the stored entanglement over entire paths is
considered for routing. Given the single source of network traffic being
analyzed in this case, the alternative approach is equivalent. We again bypass
any link-layer protocols of the network to focus on the physical layer.

Figure 7: A network of buffered entanglement channels. Here $s$ is the source
node, where classical data arrives to be sent into the network and $d$ is the
sink, where classical data will be processed and removed from the network.

For the setting when $J=L=4$, the average throughput and decoding error rate
are displayed in Fig. 8. Compared to the single-link case, the average
throughput and the decoding bit error rates differ significantly compared to
those in Fig. 4. The main difference is, with two outgoing links from the
sender, there is always one link idle during any transmission—one link is
transmitting data, the other is idle. This implies that for each transmission
of data, one EPR pair is generated across the idle link and therefore
entanglement is always available across either the two links. Using superdense
coding is therefore always possible for the first hop and an average
throughput from the sender is maximized at two bits per unit time under
perfect memory settings. Simulation results showed that a quantum storage of
size $E=200$ is excessive and $E=1$ for these simulation produces the same
average throughput and error plots for the different storage parameters. We
therefore do not show plots for the throughput and decoding error rates for
varying $E$ in this case. In the latter half of the network, the middle nodes
will have the same property where one will always be idle, therefore having
the ability to generate entanglement with the receiver. In this case, since
$E=1$ will achieve the same throughput and average error rate, we further do
not need to consider entanglement queuing practices, since for a queue with
one position, all practices are equivalent. With dotted lines, we show the
trend for replacing old entanglement units, which demonstrate, as with the
point-to-point case in the previous section, a significant improvement in
error rates and thus overall throughput.

Because entanglement is always available hop-by-hop, entanglement-assisted
communication is always used, thereby leading to larger error rates with
respect to the memory noise. Here we see that for $T_{1},T_{2}=11,10$ ns, the
error rate never falls from a totally random decoding of $4$ bits, even when
old entangled pairs are replaced. On the other hand, with
$T_{1},T_{2}=1100,1000$ ns, a significant throughput advantage is observed,
outperforming the classical case with relatively low incoming message rate of
$r\approx 0.2$. For the $T_{1},T_{2}=110,100$ ns case, the non-assisted data
transmission rate is only surpassed at a high incoming rate of $r\approx 0.9$,
where entanglement will be stored for only short periods.

What we can say in this case is that with only two connections, the use of the
GEWI communication strategy can supersede a purely classical transmission
approach even with low storage times of $T_{1},T_{2}=110,100$ ns for high
rates of traffic, and with an order of magnitude larger,
$T_{1},T_{2}=1100,1000$ ns, we can observe higher than non-assisted rates even
with low incoming message rates. Here the size of the buffer is less important
as with two links and with one source and the method of routing selected,
there is more idle time on average between the two links in comparison to the
single link case. In this case, we selected a network topology motivated by
the results in [7], where the so-called “wide network” had the best advantage.
Future work will be to consider more complex topologies.

Figure 8: The throughput and error rates of a network as in Fig. 7.

##  VII Two-Way Communication for Distributed Computing

For the single-link setting, an important and natural extension to consider is
a two-way communication scenario, where there are idle periods between
responses between the communicating parties. Such scenarios where two parties
communicate between themselves with periods of idle are common in, for
example, distributed computing. In this setting, two parties send messages
back and forth between themselves and once messages are received, there is a
period of processing until a response message is sent. This downtime is an
opportunity that can be used to build up entanglement.

Here, we focus on (classical) distributed computation and simulate the setting
where two parties perform the unsupervised learning algorithm $k$-means
clustering [31] in a distributed fashion. The initial condition of the
distributed computation is that each party has access to the complete data set
as well as knowledge of the initial centroid locations. For the first
iteration of the algorithm, each party computes the distance to the centroids
for a fair share of the data (i.e. each node has an equal number of distances
to calculate), thereby labeling just their part of the data. The computed
labels are then sent to the other parties, and each party receives the labels
from the others to update the centroid locations. We only consider, as a
proof-of-concept, binary labeling, using superdense coding to transmit the
label information at an accelerated rate with two computing nodes. The
simulation setting we use in this case is much like the single link
simulations in Section V, except we duplicate the channel to accommodate two-
way communication.

The simulation setting, in this case, is the following: the nodes can
communicate back and forth over a quantum channel, where all messages are
transmitted via qubits. Each node can store $500$ EPR pairs in a quantum
memory that has decoherence properties based on a $T_{1}$, $T_{2}$ time model,
as described in earlier sections. The memory we select here is one with
$T_{1}=1100$ ns and $T_{2}=1000$ ns. We repeat the process for more robust
memories with $T_{1}=T_{2}=1$ ms and $T_{1}=T_{2}=10$ ms. The channel
connecting the nodes, in this case, is reduced to $20$ m to simulate a data
center distance, reducing the transmission latency in comparison to the
previous cases.

Figure 9: The synthetic data set used for the distributed clustering
simulation.

The dataset we consider is a synthetic dataset consisting of 500 data points
surrounding two clusters clustered around $(-1,0)$ and $(1,0)$, shown in Fig.
9. The data sets are generated with a standard deviation of $0.1$ around the
center. This simple configuration is chosen to focus on the communication
aspect of distributed computation rather than an overly complex computation
scenario. The computation and communication scenario is the following: For a
maximum of 10 iterations, the two nodes compute the distances for their fair
share of the 500 data points (i.e. 250 points each). During this time, they
transmit a varying number of EPR pairs between themselves. We assume that the
time between the last EPR transmission to the end of the data processing is 1
ms, a duration selected based on the performance of a modern laptop performing
one iteration of clustering on 250 data points. During this 1 ms, the qubits
decohere in the memory. After the 1 ms, the two parties share their respective
250 binary labels with the other party so that new centroid locations can be
computed. In this case, this is a binary string of 250 characters. Using the
preshared entanglement, they transmit several messages to the other party
using superdense coding as much as possible and then revert to non-assisted
message transmission when not. To mitigate errors, after each iteration of
clustering, the indexes of the data points labeled by each party in the
previous iteration swap, and so each party labels the data points of the other
in an alternating fashion.

The metrics for performance that we consider are the total number of message
transmissions used during the execution of the protocol, and the $F_{1}$ score
of the classification comparing the final labels of the two parties, which is
defined as, for a collection of binary labels $A$ and $B$,

$\displaystyle F_{1}(A,B)\coloneqq 2\cdot\frac{precision_{AB}\cdot
recall_{AB}}{precision_{AB}+recall_{AB}}.$ (1)

We run each simulation 200 times over, plotting the average. The standard
deviation in the $F_{1}$ score is $<2\%$ in each case, and for clarity, we do
not plot error bars in the figures.

In Fig. 10, we see the performance of this mixed computation-communication
setting. In each case, the number of transmissions follows the black dashed
line, where the number of transmissions is shown on the right side of the
plot. What we observe is that with a $T_{1},T_{2}=1100,1000$ ns (blue), the
fidelity of the entanglement is not high enough at the time of transmission.
As more entanglement units are used, more noise is introduced to the messages,
making the decoding process nearly random, and eventually so much so, that the
labels of one party and the other become completely random at the end of the
10 iterations. As one might expect, as the memory fidelity increases, the
$F_{1}$ score improves as well, implying better matching between sender and
receiver. Because superdense coding is being used to communicate, fewer and
fewer transmissions are made as more EPR pairs are generated between
iterations, but at the cost of a reduced $F_{1}$ score. When $T_{1},T_{2}$
times is $1$ ms, still the quality of the entanglement is not sufficient, and
low $F_{1}$ scores are observed for higher numbers of entanglement units. When
the $T_{1},T_{2}$ times is $10$ ms (red), ten times larger than the processing
time, the best $F_{1}$ score vs. total transmissions made trade-off is seen,
where there is a high $F_{1}$, $\geq 0.9$, score when the fewest total
transmissions are used.

Figure 10: The trends for the total number of transmissions made between two
computers over a quantum channel versus the accuracy they can achieve for the
binary classification. The red (up most) plot represents $T_{1}=T_{2}=10$ ms,
the green (middle), $T_{1}=T_{2}=1$ ms and the blue (lower most),
$T_{1},T_{2}=1100,1000$ ns.

##  VIII Conclusion

In conclusion, we investigated a communication scenario where, to boost
communication rates over a quantum channel, entanglement is generated during
times when communication is idle to be used when communication is later
required. We proposed a communication protocol that implements this and
investigated the quantum memory properties required to implement the
communication strategy. We considered three communication settings: a point-
to-point channel, a network of four nodes with a single source and a single
sink, and finally a two-party distributed computation setting. To analyze the
three settings, we implemented simulations that simulate physical models of
the quantum memory to determine various performance metrics for imperfect
memory coherence times. We further studied how extracting qubits in a
particular order affects the communication rates. What our results show is
that, using our GEWI communication protocol, it could be possible to
outperform the non-assisted communication settings where no entanglement is
used, for future communication systems and distributed computing
architectures.

Indeed, these results are a starting point for this type of analysis. When
considering any network protocol, it is important to also consider aspects
like multi-user networks , various traffic models, different topologies, and
network noise parameters can affect protocol performance. These aspects all
will play a role in the overall throughput for any entanglement assisted
communication network, and deeper investigation is the target for future
study.

## Acknowledgments

Funding from DFG Emmy-Noether program under grant number NO 1129/2-1 and the
support of the Munich Center for Quantum Science and Technology (MCQST) are
acknowledged. This research is part of the Munich Quantum Valley, which is
supported by the Bavarian state government with funds from the Hightech Agenda
Bayern Plus and received support by the BMBF through projects QuaPhySI, grant
number 16KIS1598K, and “Souveran. Digital. Vernetzt.” joint project 6G-life,
grant number: 16KISK002.

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# Design and characterization of individual addressing optics based on multi-
channel acousto-optic modulator for 171Yb+ qubits

Sungjoo Lim  Seunghyun Baek  Jacob Whitlow  Marissa D'Onofrio  Tianyi Chen
Samuel Phiri  Stephen Crain111Present address: IonQ, Inc., College Park, MD
20740 Kenneth R. Brown  Jungsang Kim<EMAIL_ADDRESS>Junki Kim
<EMAIL_ADDRESS>
###### Abstract

We present the design and characterization of individual addressing optics
based on a multi-channel acousto-optic modulator (AOM) for trapped
ytterbium-171 ions. The design parameters of the individual addressing system
were determined based on the tradeoff between the expected crosstalk and the
required numerical aperture of the projection objective lens, and we found the
optimal target beam diameter and separation to 1.90 µm and 4.28 µm,
respectively. The individual beams shaped by the projection optics were
characterized by two complementary methods: capturing the beams at an
intermediate image plane and scanning the ion positions while monitoring the
ion-field interactions. The resulting effective beam diameters and separations
were approximately 2.34–2.36 µm and 4.31 µm, respectively, owing to residual
aberration.

###### keywords:

Ion trap , quantum computing , quantum information , individual addressing
optics

[1]organization=SKKU Advanced Institute of Nanotechnology (SAINT) & Department
of Nano Science and Technology, Sungkyunkwan University, city=Suwon,
postcode=16419, country=Korea [2]organization=Department of Physics and
Astronomy, Seoul National University, city=Seoul, postcode=08826,
country=Korea [3]organization=Duke Quantum Center, Duke University,
city=Durham, NC, postcode=27701, country=USA [4]organization=Department of
Electrical and Computer Engineering, Duke University, city=Durham, NC,
postcode=27708, country=USA [5]organization=Department of Physics, Duke
University, city=Durham, NC, postcode=27708, country=USA [6]organization=IonQ,
Inc., city=College Park, MD, postcode=20740, country=USA
[7]organization=Department of Nano Engineering, Sungkyunkwan University,
city=Suwon, postcode=16419, Country=Korea

##  1 Introduction

Trapped ions are promising quantum information science and technology
platforms that exhibit long coherence times [1, 2] and high-precision quantum-
state control [3, 4, 5, 6, 7]. Quantum information is encoded in the internal
states of trapped ions, and their quantum states can be precisely manipulated
using a coherent laser or microwave field. To independently control each ion’s
internal state, exposing only the selected ions to the control field is
essential and it can be realized by multiple tightly focused control laser
beams. Individual addressing beam arrays for ion qubits have been achieved
using multi-channel acousto-optic modulators (AOMs)[8, 9], acousto-optic
deflectors [10], micro-electromechanical system (MEMS) tilting mirrors [11,
6], and waveguide optics [10, 12, 13, 14, 15]. Among these, the multi-channel
AOM approach enables independent control of the intensity, frequency, and
phase of individual beams and demonstrates parallel entangling gates [16] and
simultaneous sideband cooling [17].

In this paper, we report the design and characterization of a multi-channel
AOM-based individual addressing optical setup for up to 32 ions. We studied
the effect of design parameters on the intensity crosstalk between neighboring
ions and designed projection optics according to the optimized parameters. The
resulting individual beams were monitored using an image sensor in the
intermediate imaging plane and characterized using a trapped ion as a field
probe.

##  2 Setup design

We define two hyperfine clock states of 171Yb+ on a qubit basis, where
$|0\rangle$ is 2S1/2 $|F=0,m_{F}=0\rangle$ and $|1\rangle$ is 2S1/2
$|F=1,m_{F}=0\rangle$. Coherent transitions between qubit states can be
realized through stimulated Raman transitions driven by a mode-locked 355-nm
Nd:YAG laser (Coherent Paladin). To achieve the individual control of ion
qubits in a linear chain, switchable individual beams that are tightly focused
on each ion are necessary.

We built modular optical systems comprising upstream and downstream optical
submodules interconnected by single-mode photonic crystal fibers [18, 19]. The
upstream module is responsible for distributing the laser beams to downstream
modules, modulating the frequencies of laser beams, and switching the laser
beams using acoustic-opto modulators (AOMs). We built two downstream modules
to shape and project Raman laser beams onto the 171Yb+ ion qubits. One
downstream module was used for global illumination, whereas the other was used
for individual addressing beams of up to 32 ions [20, 9]. Raman transitions of
the ions can be realized by either two counter-propagating Raman beams from
the downstream modules (counter-propagating configuration) or by a single two-
tone Raman beam from one of the downstream modules (co-propagating
configuration). In this study, we focused on the design and characterization
of the individual addressing downstream optical submodules.

We employed a 32-channel AOM (L3 Harris) to control the individual Raman beams
for up to 32 ions. The input laser beam of the individual downstream module
was divided into 32 separate beams using a diffractive optical element (DOE)
and each beam was directed into the 32-channel AOM. By controlling the radio
frequency (RF) signal applied to each AOM channel, the individual beams
diffracted by the AOM can be switched and modulated. The nominal active
aperture size and separation between AOM channels were 140 µm and 450 µm,
respectively. To avoid UV-induced damage to the optics, the entire AOM and DOE
systems were purged with pure nitrogen gas.

The following projection optics relayed AOM output beams onto an ion chain
with target magnifications. We used different magnifications along the ion
chain axis and perpendicular radial direction because of differing design
criteria. Along the ion chain axis, the size of each beam affects the laser
intensity spillover onto neighboring ions and the required numerical aperture
(NA) of the focusing objective lens. Figure 1 shows the tradeoff between the
expected intensity crosstalk and the required NA of the projection lens. The
intensity crosstalk is defined by the laser intensity spillover normalized by
the peak intensity at the nearest ion position, assumed to be 5 µm. It induces
a gate error in neighboring ions and is minimized at small beam sizes.
However, a small beam requires a high-NA objective lens for tight focusing,
making alignment difficult and increasing susceptibility to aberrations. To
minimize the distortion by beam clipping, the NA of the objective lens should
be higher than twice of the target Gaussian beam divergence, i.e.,
NA$>2\sin(\lambda/\pi w_{0})$ where $\lambda$ is the wavelength and $w_{0}$ is
the target beam radius. With the objective NA of 0.24, we found the minimum
diameter of the Gaussian beam to be 1.88 µm and the corresponding minimum
crosstalk was $10^{-25}$.

Figure 1:  Expected intensity crosstalk and required NA as a function of the
axial beam size. The vertical dashed line indicates the lower bound of the
Gaussian beam diameter determined by the objective lens used in our system
(NA=0.24).

In the radial direction, perpendicular to the ion chain axis, the optical
clearance of the employed surface trap was considered. The optimal beam size
along the direction perpendicular to the chip surface can be defined by
minimizing the beam clipping on the trap surface. We used surface traps
fabricated by Sandia National Laboratory [21, 22] and given the geometry of
the traps, the optimal Gaussian beam diameter was approximately 5–10 µm.

Based on the design criteria above, we built a three-stage telescope system to
relay and shape the individual beams (Fig.2). The telescope system maps the
beams on AOM centers (object plane) to the ion chains (imaging plane). The
first telescope comprised cylindrical lenses with effective focal lengths of
75 mm and 15 mm (a pair of 30-mm lenses) to realize elliptical beams with an
aspect ratio of 1:5. The second telescope has $\sim$2x magnification using
spherical lenses, and the final stage used a custom-made 4x magnification
objective lens with an imaging NA of 0.24 to minimize spherical aberration.
The beam diameters at the AOM center were adjusted to 200 µm, to match the
target beam diameter and beam separations. The overall magnifications were
$105\times$ along the chain axis and $21\times$ in the radial direction, and
the resulting beam diameter at the ion position was expected to be 1.90 µm and
9.58 µm in the axial and radial directions, respectively. The estimated beam
separation was 4.28 µm; at this distance, there is a sufficient Coulomb
interaction between ions for a qubit entangling gate while maintaining an
acceptable tradeoff in crosstalk reduction. Using a ray-tracing simulation
(Zemax Optical Studio), we confirmed that the designed optical system was
diffraction-limited.

Figure 2: Schematics of the individual beam projection optics. The three-stage
telescope relays the beam array launched from the multi-channel AOM to the
target ions. The dimensions are not to scale. (a) Trajectory of the individual
beam arrays and (b) the Gaussian beam propagation along the ion chain axis. A
single-line (double-line) arrow represents a singlet (doublet) lens, and the
dashed arrow represents a cylindrical lens along the radial direction of the
ion chain. The telescopes are designed for a target magnification of
$105\times$ and $21\times$ in the axial and radial directions of the ion
chain, respectively.

As the output beams and ion chains are both one-dimensional arrays, it is
important to match the beam orientation to the ion chain axis. Although the
beam array from the 32-channel AOM was oriented horizontally with respect to
the optical table, the ions were oriented vertically. To rotate the
orientation by $90^{\circ}$, we employed a periscope built with two mirrors
whose normal axes are perpendicular to each other. Fine adjustments of the
orientation were made using a dove prism mounted on the optical path. Before
the beam entered the vacuum chamber, a pair of waveplates was installed to
control the polarization of the laser beams.

##  3 Results & Discussion

Figure 3: CCD image of the individual beam arrays at an intermediate imaging
plane with (a) all 32 channels switched on and (b) only the 1st and 32nd
channels on. The average beam separation in the intermediate plane was 48.3
$\pm 0.12$ µm, whereas the expected separation was 44.5 µm.

Figure 3 shows an intermediate test of the individual beam intensity
distributions in the imaging plane of the first telescope. A CCD image sensor
captured the beams after they were directed by a temporary mirror and a unit-
magnification telescope. We confirmed that all 32 beams could be individually
switched on or off by toggling RF signals. The average separation of beams at
this intermediate plane was $48.3\pm 0.12$ µm, whereas that from the
simulation result was 44.5 µm. We believe the slight mismatch of the beam
separation and image aberration mostly comes from the temporary unit-
magnification telescope. The relative orientation between the principal axes
of the elliptical beams and the beam array axis was matched within the image
resolution limit of 15 mrad.

For a more comprehensive analysis, we measured the intensity profiles of
individual beams using a trapped 171Yb+ ion as a probe. When a trapped ion
experiences Raman transition in a co-propagation configuration, the Rabi
frequency of the transitions is proportional to the laser intensity at the ion
position. By adjusting the DC voltages providing an electrical trapping
potential to the ions, we scanned the ion position along the axial direction.
The ion position was monitored using an imaging system at $20\times$
magnification and measurement resolution of 0.19 µm.

Figure 4: Stimulated Raman transition for various pulse durations and ion
positions of two individual beams (a, c) and their model fit (b, d). The beam
A (B) corresponds to the 16th (17th) beam out of 32 individual beams. The red
dashed lines show the fit center and the white dashed lines show the Gaussian
fit radius. The Gaussian beam diameters of beams A and B were 1.86 µm and 1.88
µm, respectively.

Figure 4 shows the Rabi oscillations from two nearby Raman beams, and their
model fits for various ion positions. We investigated two central individual
beams of the 16th (beam A) and 17th (beam B) out of 32 beams due to the
limitations of the established control system. We applied individual Raman
beam pulses with various pulse durations while scanning the ion position along
the axial direction and measured the probability of finding the qubit in the
$|1\rangle$ state. The expected probability follows a resonant Rabi
oscillation trace, and assuming a Gaussian intensity profile for the
individual beams, the experimental data were fitted using the following model:

$P_{|1\rangle}(x,t)=\sin^{2}\left(\frac{\Omega_{0}t}{2}\exp(-2(x-x_{c})^{2}/w_{0}^{2})\right),$
(1)

where $\Omega_{0}$ is the peak Rabi frequency and $x_{c}$ and $w_{0}$ are the
Gaussian beam center and diameter, respectively. We found that the model of
fit agreed well with the data, although rapid oscillations were not captured
because of the relatively large sampling interval in the ion positions. From
the fitting, the Gaussian diameters of beams A and B were 1.86 µm and 1.88 µm,
respectively. The beam distance was 4.31 µm, which agrees with the simulation
within the ion position resolution. The peak Rabi frequencies of beams A and B
were 2$\pi\times$ 1.91 kHz and 2$\pi\times$ 2.79 kHz, respectively, where the
intensity imbalance originated from technical issues in the RF sources.

Figure 5: (a) Normalized Rabi frequency vs. ion positions for the two
individual beams A and B. Rabi frequencies are normalized by each peak value.
The solid line is the normalized Gaussian beam intensity distribution plotted
based on the parameters obtained in Fig.4. (b, c) Rabi oscillation traces at
the center of beam A (b) and beam B (c). The solid lines are sine-squared fits
of the data. No increase of $P_{|1\rangle}$ was observed within the measured
SPAM error, suggesting the intensity crosstalk is less than 0.6%.

We also analyzed the intensity spillover of each Raman beam (Fig. 5). Figure 5
(a) presents the spatial distribution of Rabi frequencies of each beam, where
the Rabi frequencies were extracted from the Rabi oscillation traces at each
position. From the achieved spatial Rabi frequency distribution, we calculated
the second-moment width (D4$\sigma$) of each beam as 2.34 µm (beam A) and 2.36
µm (beam B), which are approximately 10% larger than the Gaussiabn team
diameter. This mismatch is believed to be caused by imperfect optical
alignment and residual aberrations.

Finally, we compared the Rabi oscillations of the two beams at their peak
centers (Fig. 5 (b,c)). When the system operates, the ions should be pinned at
the centers of individual beams, and the effect of neighboring individual
beams should be negligible to minimize gate crosstalk. Compared to the vivid
oscillatory trace at the beam center, we did not observe any discernible
increase in $P_{|1\rangle}$ within 2.5 ms due to influence from the
neighboring beams. This absence of influence from neighboring beams suggests
the effective suppression of crosstalk from neighboring beams. With a state
preparation and measurement error of 1%, the data indicate an upper limit of
the crosstalk Rabi frequency at 2$\pi\times$ 12.8 Hz, corresponding to only
0.6% of the intensity crosstalk to the nearest ion qubits.

In conclusion, we designed and constructed an individual addressing optical
setup for Raman transitions of 171Yb+ qubits. The design parameters were
determined to suppress the gate errors induced by intensity crosstalk and
minimize beam aberration. The individual beams were characterized using both
intermediate CCD images and stimulated Raman transitions while scanning the
ion position, which agreed with prediction and suggested an intensity
crosstalk of less than 0.6%. The tailored individual addressing module
demonstrated the capability of providing reliable and programmable operations
for universal quantum computation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government(MSIT) (2021M3H3A103657312,
2022M3E4A105136512, 2022M3K4A109712211, 2022M3H3A106307412) and National
Science Foundation STAQ Project, PHY-1818914.

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