Protocol Design for Quantum Repeater Networks
Luciano Aparicio

Rodney Van Meter

Hiroshi Esaki

Graduate School of Information
Science and Technology
The University of Tokyo, Japan
7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-8656, Japan+81-3-5841-7465

Faculty of Environment
and Information Studies
Keio University, Japan
5322, Endo, Fujisawa,
Kanagawa, 252-0882 Japan+81-4-6649-1100

Graduate School of Information
Science and Technology
The University of Tokyo, Japan
7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-8656, Japan+81-3-5841-7465

lucho@hongo.wide.ad.jp

rdv@sfc.wide.ad.jp

hiroshi@wide.ad.jp

ABSTRACT
When built, quantum repeater networks will require classicalnetwork protocols to control the quantum operations. However, existing work on repeaters has focused on the quantum operations themselves, with less attention paid to thecontents, semantics, ordering and reliability of the classicalcontrol messages. In this paper we define and describe our
implementation of the classical control protocols. The statemachines and packet sequences for the three protocol layers
are presented, and operation confirmed by running the protocols over simulations of the physical network. We also showthat proper management of the resources in a bottleneck linkallows the aggregate throughput of two end-to-end flows to
substantially exceed that of a single flow. Our layered architectural framework will support independent evolution of
the separate protocol layers.

Categories and Subject Descriptors
C.2.1. [Computer Communication Networks]: Network
Architecture and Design; C.2.2 [Computer Communication Networks]: Protocol Architecture

General Terms
Design, Theory.

Keywords
Quantum communication, Quantum repeater, Quantum networks

1. INTRODUCTION

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Applications that use distributed quantum properties, such
as QKD (Quantum Key Distribution) [14, 12], quantum
Byzantine agreement [2], possibly improved optical interferometers for telescopes [15], and other forms of distributed
computation [8, 3], have the limitation that the fidelity ofquantum states and the probability of success decrease with
distance, making the use of these systems over long distances almost impossible. Therefore, researchers have proposed the design of quantum repeater networks [11] which
would maintain distributed quantum states across greater
distances.
Networks of quantum repeaters utilize three concepts (explained in more detail in Section 2) to execute a distributed
algorithm that creates entangled quantum states between
nodes that are far apart: a basic entanglement mechanism
which depends on the physical implementation, error management (in this work, we study a method known as purification), and finally a quantum state propagation layer (here
we implement entanglement swapping, which builds multihop connections from single-hop connections). Some researchers are investigating approaches that are substantiallydifferent from entanglement swapping [18, 16, 13]. Here
we focus on swapping, but the layered architecture approachis broadly applicable, allowing other implementations to replace only a single layer in the protocol stack.
Previous work primarily focused on the physical andmathematical tools for building repeaters. Classical information is also needed to enable teleportation and swapping,as many quantum operations are not deterministic, and results of quantum measurements need to be reported to distant partners before further operations can proceed. Also,
operations in the middle of the network must be coordinatedto route and swap properly. This requires classical messages
to make operations robust, but message propagation times
penalize performance. Even though this delay is usually included in repeater simulations, prior work has not defined
the protocols in detail, especially with respect to how all of
the nodes make consistent decisions in a timely fashion.
In this work, we introduce a protocol stack for networks
of quantum repeaters that considers all the necessary clas-Protocol Architecture

Proposed Protocols

Application

Application (AP)End-to-End
Error Management

Repeated atintermediate
distances

Quantum State Propagation

Classical
messages
only

Purification Control (PC)
Entanglement Swapping
Control (ESC)

Error Management

Purification Control (PC)

Link Entanglement Control

Acknowledged Entanglement
Control (AEC)

Only one hop
Physical Entanglement

Quantum
effects

Qubus PhysicalEntanglement (QbPE)

Figure 1: Protocol Stack Architecture and Proposed Protocols
sical messages and which can be easily adapted for different approaches at all three protocol layers. We simulated aqubus mechanism as the physical protocol, Deutsch purification as the protocol for error management, and entanglement
swapping as the protocol responsible for making the entanglement span from end to end, in order to increase our confidence in the behavior of our network protocols. By adjusting the fidelity thresholds required for entanglement swapping, we show that some configurations boost the aggregate
throughput for multiple flows significantly above the maximum for a single flow, taking advantage of resources thatwould otherwise sit idle. The operation of such complex networks and such delicate tuning of the system without formal
protocol definitions would not be possible.
In Section 2, we introduce some concepts of quantum information science that are used in this work. In Section 3, we
begin by describing the division of work, separating some of
the functions of a quantum repeater into protocol layers thatexchange messages with a partner. We also explain the finite
state machines that control individual qubits or Bell pairs ina distributed fashion. Section 4 presents the results of oursimulations of the protocol. We conclude with future workand conclusions in Section 5.

2. BACKGROUND
Communication of quantum states depends on several
quantum operations and properties: key among these is entanglement, in which the states of two or more quantum bits
(qubits) are not independent. This operation is done by interacting qubits, producing this high correlation among theirquantum states. For communications, one useful, basic formof entanglement is a Bell pair. Bell pairs can be created overa distance using optical pulses that are coupled to a qubit(represented as e.g. the spin of a single electron held in aquantum dot) at each end of a waveguide. Due to losses in
the waveguide, this operation is probabilistic. Bell pairs canbe used for teleporting a qubit from one location to another.
The Bell pair is consumed in the process, so we must continually refresh the supply of available pairs. To cover distancesof more than one hop, a form of teleportation called entanglement swapping is used to splice two short Bell pairs into
one long one.
The fidelity of a quantum state describes how accurately
the state matches our desired one; F = 1.0 indicates thatthe state is perfect. After entanglement succeeds, usually
the fidelity is not high enough for distributed quantum computation, and entanglement swapping and memory decoherence further degrade the fidelity. Purification is an algorithm
which boosts the fidelity of a Bell pair by sacrificing a second pair. As purification is a non-deterministic operation,
many resources are needed in the process to obtain a highfidelity Bell pair.
Previous work on quantum repeaters [7, 9, 4, 5, 10] hasproposed different ways to produce entanglement via single
photons or via very weak laser pulses. These produce highfidelity Bell pairs, which makes purification almost unnecessary, but with a low probability of success. Other approaches
improve the probability of success at the cost of reducing theinitial fidelity [19]. The qubus mechanism used in this work
is described in Section 3.1.

3.

PROTOCOL DESIGN

The process of designing quantum networks is similar todesigning classical networks, as they require detailed protocol designs, including finite states machines to control physical resources and track logical state. A layered protocol
stack has previously been proposed [20]; here we provide
detailed designs for the individual layers. We give a brief
description of each and the functions which are simulated.
Fig. 1 shows the layers of this protocol.
One of the key purposes of the classical protocols is tokeep track of the fidelity of Bell pairs. The fidelity is theprobability that we measure the right quantum state, whichis due to non-deterministic quantum mechanics. Fidelity
can only be estimated and not measured. Measurement offidelity cannot be done directly, as it describes the probability of finding the system in the ”right” state. Moreover,measurement destroys the entanglement. Instead, fidelity is

Discarded byhigher layers

Discarded byhigher layers

Initializing /Increment Epoch
Uninitialized

Entanglement
Attempt (in)Unentangled
Uninitialized

Unentangled

Resetting
Entanglement
Failure (in)Time Out

Entanglement
Failure (out)Entanglement
Attempt (out)Entanglement
Success (out)Interim Entangled
To higher
layer (PC)

Entanglement
Success (in)Figure 3: Finite state machine for Receiver’s ACKed
Entanglement Control
To higher
layer (PC)simulations assume such a capability.
Figure 2: Finite state machine for Transmitter’s ACKed
Entanglement Control

tracked statistically for a given environment, by sacrificing
some of the created Bell pairs, measuring them and confirming the state, as part of ongoing link monitoring. Decisions
are made based on these estimated values in order to allow
Bell pairs to be swapped or sent to the Application layer foruse. The control of a qubit (a single-qubit buffer) is passed
from layer to layer until consumed by the Application layeror reinitialized to start over from the lowest layer.

3.1 Physical Entanglement Layer: Qubus
The physical entanglement layer represents the physicalinteraction that creates Bell pairs between two different stations. There are many proposals for this layer and at the
moment no clear winner. Our simulations model the qubus
mechanism [19] in which laser pulses of many photons generate low-fidelity Bell pairs with high probability. For a
distance of 20km, over an optical fiber with a 0.17 dB/km
loss, the probability of success of the entanglement is around36%, with an initial fidelity of 0.633. This probability is dueto the attenuation in the optical fiber, which has an exponential increase with distance. Thus, photons may be lost whentraveling, or may not be detected at the receiver. Once theBell pairs are produced, decoherence also decreases their fidelity as a function of time, as information leaks into thesurrounding environment.
The physical capabilities of different physical layers vary.
Some support only a single physical transceiver qubit, and so
can support only a single outstanding entanglement attempt.
Others support independent multiplexing of incoming lightpulses to local qubits, which is done by a classical herald
pulse (trigger) followed by one or more quantum pulses. Our

3.2

Link Entanglement Control: ACKed Entanglement Control

The second layer, AEC (ACKed Entanglement Control),is responsible for managing the single-hop physical entanglement process, selecting qubits to attempt entanglement
at each end of the link, and utilizing classical messages toreport the results. When a laser pulse is detected by the receiver, measurements are done, and a message will be sent
back to the transmitter, informing it which qubits on the receiver were entangled to which qubits on the transmitter.
The stationary qubits in a repeater are not destroyed whenthe qubit is measured or reinitialized, though the quantum
information held in the qubit is. Each qubit has an associatedepoch, a counter of the number of times it has been initialized, to prevent old messages from being misinterpreted.
Once entanglement succeeds, this layer will transfer control of the Bell pair to a higher protocol layer.
Fig. 2 shows the finite state machine which describes the
behavior of this layer for qubits in the transmitter, and Fig. 3
for qubits in the receiver.
Transitions represented in blue include an interaction witha remote station. OUT refers to messages sent, and IN tomessages received. Black transitions represent local operations. The different states are:
• Uninitialized. This can be reached from a higher protocol layer or after starting up the repeater. Qubits are
in an unknown state.
• Unentangled. During initialization of the qubits, they
are prepared to have a known quantum state, ready tostart entanglement, and the epoch is incremented. If
the qubit’s fidelity falls below a threshold, it becomes
unusable and will be returned to the Uninitialized state.

To bottom layer(AEC - Uninitialized)
Purification
Failure (in) Resetting
MyHalfPurified

To higher
layer (ESC)

From higher
layer (ESC)

From lower
layers (AEC/ESC)

ResettingPurification
Success (in)Qubitsselected

Purification
Success (out)Qubitsselected

Entangled

PurifyAttempt

Purification
Success (out)HerHalfPurified
ResettingPurification
Success (in)SacrificeAttempt

Resetting

To bottom layer(AEC - Uninitialized)

Purification
Failure (out)Figure 4: Finite state machine for Purification Control• Interim Entangled. This state is reached after entanglement is attempted. This operation is done by sending a laser pulse to the remote station after interactingwith the local qubits. The qubit will remain in this state
until it receives an answer from the remote station or a
local timer times out. This timer is based on the lifetime of the quantum memory used and in practice will
be set to limit the impact on the fidelity to around 1%.
If an Entanglement Failure message is received or noanswer is received before the timer expires, the qubit
will be moved to Uninitialized to start over again. If an
Entanglement Success message is received, the qubit
will be moved to the next higher protocol layer, in thiscase, Purification Control.

3.3 Error Management Layer: Purification
In this work we use an error management method called
purification. In order to purify a Bell pair (boost its fidelity),an additional Bell pair is sacrificed in the process. The thirdlayer of the protocol, PC (Purification Control), is responsible for choosing two Bell pairs, and electing one pair tohave its fidelity boosted and the other to be sacrificed, assuring that both stations make the same decisions. Purification
is done between two arbitrary stations, and no other stations
need to be considered. Thus, PC does not need to make any
complex routing decisions, but it does need to be able toaddress any station in the network. After the PC layer confirms a sufficient fidelity for the Bell pairs, control is given
to the next higher protocol layer, which could be the Application layer, or the Entanglement Swapping Control (ESC),
depending on whether or not this round of purification was
done between end-to-end stations. If the physical layer produces high-fidelity Bell pairs, there is no need to execute any
purification, so this layer could be configured as a null layer.
In order to select qubits to purify, the possible values offidelity are grouped into fidelity bands. Two qubits in thesame band are chosen for purification. The node which starts
the purification process will choose the qubit with the lowest
memory address as the one to be purified, and the second
one to sacrifice. The node which receives the purification
attempt will make the same decisions based on the address ofthe qubits of the remote station, in this case the node whichstarted the purification. This way assures that both nodes
choose the same qubits for purification and for sacrifice.
Fig. 4 shows the finite state machine which describes the
behavior of this layer. The states are:
• Entangled. This state indicates that the qubit is entangled to another qubit in a distant station, but with notenough fidelity to start teleportation. It can be enteredfrom a lower layer like AEC (just after entanglement
is produced) or ESC (as fidelity is always reduced byswapping), from a higher layer ESC (if decoherence
affects the fidelity and the Bell pair needs to be purified again), or finally from the purification algorithm
itself after successful purification. Once a high levelof fidelity is reached, control of the qubit is transferred
into the next higher layer (ESC).
If the qubit remains in this state for a long time, thefidelity will drop due to decoherence. The qubit is moved to Uninitialized in order to attempt entanglement
again.
• PurifyAttempt. Once we have two Bell pairs withsimilar fidelity, one Bell pair is assigned to this state.
If purification succeeds, this pair will have its fidelity
boosted. After attempting purification, the qubit is moved to the state MyHalfPurify or HerHalfPurify, depending on whether this station starts the purification
before receiving any purification message from the remote station.

From lower
layer (PC)

From lower
layer (PC)

Swapping
Not entangled
to destination
Swapping

Entangled
High-Fidelity

Other Bellpair ready

Entangled
High-Fidelity

Swapping

Entangled to
To higher
destination
layer (AP)Swapping

Swapping
FidelityLow

To lower
layer (PC)To lower
layer (PC)To bottom layer(AEC - Uninitialized)

Figure 5: Finite State Machine for Entanglement Swapping
Control for a middle node
• SacrificeAttempt. The second Bell pair chosen for purification is assigned to be sacrificed to improve the fidelity of the first Bell pair. Regardless of the resultof the purification, this qubit will always be moved
to Uninitialized state after the purification process finishes.
• MyHalfPurify. If the station starts the purification
process, after sending a message to the remote station,
the qubit is moved to this state, until it receives an answer. On success it will be moved to Entangled, or toUninitialized on failure.
• HerHalfPurify. This state indicates that the station
received a message notifying it that the remote station
has started the purification process. If the operationon the local station succeeds, the qubit will be moved
to Entangled, after sending a message to the remotestation. If it fails, it will be moved to Uninitialized,
after sending a Purify Failure message to the remotestation.

3.4 Quantum State Propagation Layer: Entanglement Swapping Control
For networks which have more than two stations, further
steps are required. As mentioned in the introduction, in thiswork we focus on entanglement swapping. A node in the
middle of the network waits until it has two high-fidelity Bellpairs, one to each node it wants to couple. Then, this middle
node performs the Bell state measurement that splices thetwo short Bell pairs into a longer one. As a consequenceof this operation, the fidelity of the extended new Bell pairdrops, and further purification may be necessary. ESC and
PC are repeated until we have an end-to-end Bell pair ofsufficient fidelity for our application.

To lower
layer (PC)

Figure 6: Finite state machine for Entanglement Swapping
Control for an end nodeThis layer is responsible for administering the Bell pairs,
especially for networks with shared resources. Important decisions, such as whether to purify or swap first, or when toswap, need to be carefully taken. The finite state machine forthe stations that are in the middle and make the decision toswap is shown in Fig. 5. Finally, Fig. 6 shows the behaviorof the end stations which received the swapping report fromthe middle station.

3.5

Application Layer

As of the time of writing, the most important, existing application is QKD (Quantum Key Distribution). A distributed
quantum algorithm has also been proposed that will synchronize clocks to better-than-atomic-clock precision overa distance by using Bell pairs [17, 6]. In the future, if quantum computers are developed, distributed quantum computing will become important [14, 12, 2, 3, 8].

4.

SIMULATIONS

Our simulator consists of 2000 lines of code written in
C++ on top of Omnet++, a C++-based network simulator.
The operation of a dumbbell quantum network was simulated for 10 seconds of simulated operation, taking around
150 seconds to complete on an Intel Core 2 T7200 CPU running at 2 GHz with 1 GB of RAM.
We simulated a qubus mechanism, with 20km hops, as described in Section 3.1. The number of qubits in each transmitter is 50, and 16 in the receivers. In all of our simulations,
we use a target end-to-end fidelity of 0.98. We have run simulations for two cases: only one flow, and two flows competing for shared resources in the network shown in Fig. 7. Bothflows are over three-hop paths (AEFB and CEFD), with the
middle hop (EF) being a shared link and hence the throughput bottleneck. Bell pairs created on the EF link are assigned
randomly to be used for the AB or CD flows. Used naively,

A

F1

F1

F1

E

F2

F

B

F1

D

C
Transmitter with 50 qubits

F1 Fidelity threshold

Receiver with 16 qubits

F2 Fidelity threshold in shared link

Figure 7: Simulated dumbbell network
the first and third hops on each path will remain idle half ofthe time.
We hypothesized that careful tuning of the purificationthresholds might better balance the system. If we raise therequired fidelity on the under-utilized links, can we reduce
the fidelity penalty incurred by entanglement swapping andimprove aggregate performance? To test this hypothesis, we
ran simulations with the purification threshold of each link
set to several different values, while keeping the end-to-end
target F = 0.98. We used two values for the first and thirdhops, F1 = 0.98 and F1 = 0.99, with the second case requiring an additional round of purification on those single
hops. We then varied the fidelity threshold of the middle
hop, F2 = 0.86, 0.94, 0.98, altering the number of purification rounds required over that single hop. The aggregate
throughput of the flows for each of the twelve simulations isshown in Fig. 8. With F1 = 0.99, the performance of two
flows doubled that of a single flow, as seen in the green andpurple bars on the right of the figure. Fig. 9 plots the final,delivered fidelity of the same cases. In the F2 = 0.98 cases,
the tradeoff for higher throughput is that the final fidelity justbarely clears our established end-to-end target of F = 0.98.
This gain in throughput occurs because making F1 = 0.99
produces end-to-end Bell pairs with a fidelity above the fixed
threshold just after the final swapping is done. Therefore,
no additional end-to-end purification steps are required andthese Bell pairs can be used immediately by the application
layer, providing a throughput improvement. As no further
purification is done, the end-to-end fidelity is lower than forthe other cases, but it is high enough to be used.
In the lower performance cases, end-to-end Bell pairs do
not have enough fidelity to be used by the application layer,
so the network waits until another Bell pair is available inorder to attempt purification with them. This clearly increases the final end-to-end fidelity of the Bell pair at a costof throughput reduction.

5. FUTURE WORK AND CONCLUSIONSWe provide a layered protocol architecture for quantum
repeater networks and the design of one protocol stack for
purify-and-swap repeaters. Our stack consists of ACKed En-

tanglement Control, Purification Control, and Entanglement
Swapping Control. The latter two can be combined recursively as necessary to span arbitrary numbers of hops. We
intend to make the simulator open source, and we hope forour protocol implementations to be used in actual experiments with quantum repeaters.
We studied the behavior of a dumbbell network, applying statistical multiplexing to manage the shared resources.
Careful tuning of the fidelity thresholds in this topology
allowed us to utilize idle resources and effectively workaround a bottleneck link. We obtained a higher total throughput of high-fidelity Bell pairs (F ≥ 0.98) when two flows
shared the network, compared to only one flow. In related
work [1], we have tested these protocols in more complex
topologies with more flows and found that the best multiplexing scheme was statistical multiplexing, which we use
in this work.
For more complex topologies, routing in quantum networks should also be addressed as it will directly affect thequantum state propagation layer. Other remaining work includes optimization of the total throughput in these topologies with more flows competing for resources, and trying tofind a relationship between the fidelity threshold, number ofhops and the number of flows.
This proposed architecture can be used as a reference model much as the OSI model is for classical networks. Researchers are continuing to work on many different technologies for the physical layer and other approaches to error
management and quantum state propagation, and we expect
these advances to integrate smoothly with this layered architecture.

Acknowledgment
LA acknowledges funding from MEXT. This work was supported by JSPS KAKENHI 21500020.

6.

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