Design and Analysis of Communication Protocols for Quantum Repeater Networks

Cody Jones,1 Danny Kim,1 Matthew T. Rakher,1 Paul G. Kwiat,2 and Thaddeus D. Ladd1, ∗

1HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, USA2Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA

We analyze how the performance of a quantum-repeater network depends on the protocol em-ployed to distribute entanglement, and we find that the choice of repeater-to-repeater link protocolhas a profound impact on communication rate as a function of hardware parameters. We developnumerical simulations of quantum networks using different protocols, where the repeater hardwareis modeled in terms of key performance parameters, such as photon generation rate and collectionefficiency. These parameters are motivated by recent experimental demonstrations in quantum dots,trapped ions, and nitrogen-vacancy centers in diamond. We find that a quantum-dot repeater withthe newest protocol (“MidpointSource”) delivers the highest communication rate when there is lowprobability of establishing entanglement per transmission, and in some cases the rate is orders ofmagnitude higher than other schemes. Our simulation tools can be used to evaluate communicationprotocols as part of designing a large-scale quantum network.

I. INTRODUCTION

Quantum information technology applies quantum-mechanical effects to implement beyond-classical appli-cations. For example, quantum key distribution (QKD)provides the tamper-evident establishment of a secureprivate key [1–4]. In QKD, fundamental properties ofquantum mechanics prevent an eavesdropper from in-tercepting the transmission of a secret key without re-vealing their interference to authenticated parties. Theunique capabilities of quantum-secure communicationand other applications [4–6] motivate research into de-veloping quantum networks [4, 7–12].

As with classical networks, quantum networks mustaddress engineering concerns like synchronization and la-tency, though there are further challenges for storage andtransmission of quantum information. Quantum commu-nication depends on the faithful transmission of quantumbits (qubits), which are inherently fragile. Famous resultslike the no-cloning theorem [13] exclude the possibility ofcopying or “amplifying” quantum signals, as one coulddo with classical signals. Instead, quantum communi-cation that is robust to loss and error can be achievedby using quantum repeaters [7, 14–17]. Quantum re-peaters can transmit, store, and perform logic on qubits,and these operations allow repeaters to herald success-fully transmitted signals and to “distill” purified quan-tum information states using error correction [16, 18–26]. These error-suppression protocols provide robust-ness against both imperfect transmission and the med-dling of an eavesdropper.

The purpose and scope of this paper are as follows. Wefocus on designing the communication protocol betweentwo neighboring repeater nodes in order to maximize net-work performance. To make our analysis concrete, weassume that repeaters are connected with optical fiber,though free-space transmission is a simple extension of

∗ tdladd@hrl.com

our methods. Furthermore, we focus on quantum tech-nologies that couple controllable quantum memory withsingle photons and transfer entanglement through two-photon interference, such as trapped ions [27, 28], di-amond nitrogen-vacancy centers [29–31], and quantumdots [32–34]. We explain our reasoning for targetingthese technologies in Section II; succinctly, managingphoton loss is crucial for designing quantum networks,and the interference and detection of two single-photonsignals enables reliable determination of whether a signalwas received or lost in transmission while also being morerobust to path-length fluctuations than single-detectionschemes [27, 35].

The paper begins with some preliminary considera-tions for distributing entanglement in Section II. Sec-tion III examines three protocols for establishing entan-glement between repeaters, and we simulate the perfor-mance of these protocols in Section IV using hardwareparameters representative of recent experimental work.Section V summarizes our results and discusses relatedcommunication schemes that we chose not to examine,though they are appropriate for future work.

II. PRELIMINARIES

We begin by listing a few features common to any ofthe repeaters we consider. As shown in Fig. 1, each re-peater has some number of controllable memory qubitsthat must have long coherence times (of order 10 ms)and low-error gates to act on these memory qubits (errorper gate below 0.1%). The memory qubits may be pro-tected with error correction [16, 23–26] to extend theircoherence time or suppress gate error. Furthermore,there is an interface for generating an entangled state be-tween a memory qubit and a single-photonic qubit. Sincethere is no ambiguity in this paper, we will simply sayphoton to mean a photonic qubit. For example, mem-ory/photon entangled states have been demonstrated forions, nitrogen-vacancy centers in diamond, and quantumdots [28, 30–34]. The memory-photon entanglement is a

c©2015 HRL Laboratories, LLC. All Rights Reserved. 1

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Repeater RepeaterRepeater-to-repeater link

Another link Another link

Memory qubits

FIG. 1. Schematic of a link between two quantum repeaters,showing the basic components. Every repeater interfaces withtwo or more links, and the repeater apportions qubits to eachof its links. In this case, two repeaters share a link (gray box),and they participate in links facing left and right (dashedlines) with other repeaters not shown.

resource for generating entanglement between repeaters,by swapping entanglement using the photons (describedbelow).

Each quantum repeater in a network has multiple opti-cal links to its neighbors. In this paper, we focus on pro-tocols for efficiently distributing entanglement across thelink between two repeaters. As in Fig. 1, each repeaterwill devote some of its memory qubits to each active link.Repeaters establish entangled qubit pairs with multipleneighbors to mediate network-wide entanglement [6].

Designing a protocol to distribute entanglement in aquantum network is a non-trivial engineering problem,and several fundamental challenges must be addressed.Signals between two repeater nodes can only travel atthe speed of light, but the two repeaters need to makecoordinated, synchronous actions. We take the speed oflight in fiber to be c/n, where n ≈ 1.5 is the index ofrefraction for silica fiber. We assume that both quantumand classical signals propagate at this speed (ignoringany other networking delays, for simplicity). Two quan-tities for delay times are important to analyzing theseprotocols. The first is τlink = nL/c, which is the commu-nication delay between two repeaters separated by linkdistance L. The second is τclock, which is the minimumtime needed to either reset a memory qubit or allow de-tectors to recover from a prior detection event. The valueof τclock depends on both the employed hardware and thechoice of protocol.

Each repeater requires a clearly defined protocol formanaging its resources. As a guiding principle, we seekto minimize the amount of time that memory qubits are“locked up” while the information needed for the nextaction is unavailable, and delays from multiple back-and-forth communications should be avoided wherever possi-ble. To realize the highest communication rate, the state-machine protocol local to each node needs to infer whatthe repeater at the other end of the link is doing, andwhat information it has available. For some events, animmediate action is executed (such as processing or eras-ing a memory qubit), while in other cases the protocolwaits for more information. For simplicity, we assumethat these protocols are synchronized by a distributed

clock.

A defining feature of the protocols we consider is thatthey distribute entanglement using single-photon qubitsand two-photon interference [15, 27, 28, 35–37]. Specifi-cally, we perform Bell-state measurement in an appara-tus known as a Bell-state analyzer (BSA) [15, 27, 35–39].The type of BSA used in our schemes employs linearoptics and single-photon detectors, meaning that it nec-essarily succeeds for at most 50% of attempts [40]. Weimplement a BSA that can reliably measure two of fourBell states, which is sometimes called a “partial BSA.” Asuccessful Bell-state measurement is indicated by two co-incident single-photon detection events, so detectors withhigh efficiency and low dark count rate are critical to theentanglement-distribution schemes we consider.

The Bell-state measurement performs entanglementswapping [15, 27, 35–37, 41, 42], so that if the interfer-ing photons were entangled to memory qubits in two re-peaters, the memory qubits are projected into an entan-gled state on successful Bell-state measurement. Whenthis happens, classical messages are sent to both re-peaters. We design protocols that assume either zeroor one photons will enter the BSA at each input port. Iftwo photons enter one port and lead to detection eventsmarked as successful, this is an error that degrades fi-delity of entanglement. We assume that the probabilitya successful BSA outcome results from multiple-photonemission or detector dark counts is sufficiently small (onthe order of 1% or less) to be suppressed through entan-glement distillation [18–23].

Photon loss is the primary concern in our analysis, andtwo-photon detection enables “loss heralding,” where theprotocol can reliably determine if both photons arrivedat the BSA. When a photon propagates through fiberover distances of 10 to 100 km, it has substantial prob-ability of being lost due to material absorption or otherimperfections. To address this problem, we exploit theproperty that a photon is quantized, so it either arrivesat its destination or is lost. In contrast to entanglementschemes employing bright coherent states, a detection ap-paratus that is expecting a single photon to arrive withina narrow time window can mark the absence of detec-tion as a failed transmission attempt, while a successfullytransmitted qubit is stored in quantum memory. The bitof information indicating whether the qubit was lost issent to the repeaters with classical communication, al-lowing the repeaters to coordinate entanglement distri-bution through the protocols we examine in Section III.

As a final consideration, we assume that all entangledmemory qubits are used at the end of one round of entan-glement distribution, which we explain more explicitly inthe next section. Simply put, two repeaters act only oninformation available through their shared link and donot hold qubits in storage while waiting for informationto arrive from elsewhere in the network. While morecomplex protocols may be required in some applications,this assumption simplifies our analysis by keeping pro-tocols contained to a single link and serves as a good

c©2015 HRL Laboratories, LLC. All Rights Reserved. 2

approximation for the implementations of QKD that wesimulate. The performance of each entanglement pro-tocol therefore depends on how frequently entanglementcan be attempted and how quickly information confirm-ing entanglement or photon loss is available.

III. PROTOCOLS FOR DISTRIBUTINGENTANGLEMENT

This section considers three different protocolsfor distributing entanglement between two neigh-boring repeaters using single photons, which welabel as MeetInTheMiddle, SenderReceiver, andMidpointSource. We chose these labels for conceptualclarity and consistent presentation, though versions ofthese schemes appear in several proposals in the liter-ature. To analyze these protocols, we examine boththe arrangement of hardware components and the time-dependent behavior of signal transmission, memory man-agement, and distributed decision making. Implement-ing a protocol requires control logic in the repeater torespond to new information (classical signals or detectoroutcomes) as it arrives. For each of the link protocols,we describe and analyze a sequence of operations to im-plement the associated communication scheme.

A. Meet-In-The-Middle

The MeetInTheMiddle protocol has two repeaternodes, at both ends of an optical link, transmit pho-tons to a BSA at the midpoint of the link. Each pho-ton is entangled with a memory qubit in the sendingnode. The arrangement of hardware components forMeetInTheMiddle is shown in Fig. 2. When the BSAsucceeds in swapping entanglement, the pair of mem-ory qubits, one in each repeater node, is projected intoan entangled state. This protocol was introduced inRefs. [27, 36, 37].

The simplicity of MeetInTheMiddle makes it a goodstarting point for explaining entanglement-distributionprotocols, and elements of MeetInTheMiddle will reap-pear in the more complex protocols considered later. Asin Figure 2, the link places repeaters Alice and Bob atdistance L apart and a BSA at the midpoint. In thissymmetric arrangement, both repeaters have the samenumber of memory qubits connected to the link. Aliceand Bob generate memory/photon entangled qubit pairstimed such that the photons interfere in the BSA. Thecorresponding memory qubits are projected into an en-tangled state when the BSA succeeds, though neitherAlice nor Bob can use this entanglement until a confir-mation signal returns from the BSA after speed-of-lightdelay of τlink. We say that the memory qubit is “lockedup” during this waiting period.

The sequence of transmissions is as follows. Entangle-ment is attempted in “rounds,” where each round has

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Repeater: Alice

Repeater: BobBSA

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Optical fiberL/2 L/2

FIG. 2. Hardware arrangement for MeetInTheMiddle. Re-peaters Alice and Bob are separated by distance L, which istypically tens of kilometers. Memory/photon pairs are gener-ated simultaneously at each repeater node, and the photonsare coupled into optical fiber. The photons interfere in a BSAlocated at the channel midpoint, and a successful entangle-ment swap, which is communicated to both repeaters withclassical signals, projects the corresponding memory qubitsinto an entangled state. The scheme can be modified to anasymmetric arrangement where the BSA is located at anyposition in the optical channel. In this case, the photons aregenerated at such times that they arrive simultaneously atthe BSA.

duration

τround = τlink +Nτclock, (1)

as explained below. Alice and Bob synchronize the emis-sion of a photon entangled to memory such that Alice’sphoton and Bob’s photon arrive at the BSA at the sametime (if they are not lost). Each repeater has N memoryqubits. If N > 1, the repeaters generate photons at regu-lar intervals synchronized to τclock, allowing detectors inthe BSA to recover if necessary. This time-division multi-plexing is what makes the round duration τlink +Nτclock.Each attempt at generating entanglement is assigned anidentifying number i ∈ [1, N ] for this round, which corre-sponds to both the photon’s position in the communica-tion sequence and the location of the entangled memoryqubit in the repeater. As photons arrive at the channelmidpoint, the BSA will transmit a message Mi to bothrepeaters containing one of two statements: “Bell-statemeasurement succeeded for transmission i,” or the op-posite “did not succeed.” The control protocol in eachrepeater processes these messages and determines whatnext action to take.

The control protocol for MeetInTheMiddle is shown inFig. 3 as a state machine. To understand this diagram,the symbols have the following meaning: arrows indicatetransitions between states; a rectangular box is an actionthat is executed when the protocol arrives in that state;a diamond is a query for information that branches basedon the answer; and a circle is a synchronize query. Thesynchronize query enforces clocking behavior, meaningthat it will exit through the “waiting” branch until aninternal clock steps into the next indicated time period(e.g. synchronized to τclock or τround). After receiving themessages {Mi} from the BSA indicating entanglement

c©2015 HRL Laboratories, LLC. All Rights Reserved. 3

Inner loop

Prepare qubit iand emit photonSynchronize

τclock

Ready

Waiting

Check messagesfrom BSA,use verified entanglement

Yes

Ready

No

Waiting

Start

Synchronize τround

Memory qubit: i ← 1

i ← i + 1 i < N?

FIG. 3. Control protocol for MeetInTheMiddle. Loop variablei follows the definition in the text. The statements “i← 1”and “i← i+ 1” simply mean “assign value 1 to i” and “in-crement i by 1,” respectively. If the repeater has multiplememory qubits (N > 1), transmission attempts are synchro-nized to τclock to allow the BSA to recover. After using allof the memory qubits to send photons, the protocol waits forresponses from the BSA to complete the round.

success or failure, the repeaters use all memory qubits atindices {i|Mi ≡ success}. Having completed the round,the repeaters reset the memory qubits and repeat theprocess in the next round.

We have made two assumptions to keep the protocolsimple, but these can be relaxed in a more complex proto-col for either expanded network functionality or increasedperformance. First, we assume that the repeater waitsfor all messages before using any memory qubits, whichis reasonable for Nτclock < τlink. Otherwise, the first sig-nals confirming entanglement arrive before all memoryqubits transmit photons, so there may be an opportu-nity to reduce memory lock-up time by using memoryqubits as soon as entanglement is confirmed. Second, weassume that all memory qubits are reset at the end ofeach round, though other protocols may need to preserveentangled memory qubits for multiple rounds. We leaveanalysis of these scenarios to future work.

The diagram in Fig. 3 is designed to show how efficientMeetInTheMiddle is at distributing entanglement. Focuson the “inner loop” indicated by the grey region. Whilethe protocol is traversing the inner loop, the repeater isactively attempting entanglement distribution. After therepeater has filled all of its memory qubits, the “synchro-nize τround query” forces the repeater to wait and checkfor messages from the BSA. Let us define a measure of ef-ficiency: the link utilization factor for MeetInTheMiddleis

F =Nτclockτround

, (2)

which is simply the ratio of time spent in the inner loopto total round time. If we define p as the probability ofsuccessfully projecting two memory qubits into an entan-

gled state, then the average entanglement-distributionrate can be expressed as

R =Np

τround= F

p

τclock. (3)

The quantity Rub = p/τclock is an important quantity,because both p and τclock are fundamental propertiesof the link distance and hardware. Rub is average rateachieved if entanglement is attempted every clock cycle(the protocol is always in its inner loop), which is an up-per bound to any achievable rate. The efficiency of theMeetInTheMiddle protocol is captured by F , the frac-tion of time spent in the inner loop of Fig. 3, becauseR = FRub.MeetInTheMiddle realizes distributed decision mak-

ing rather easily, but it makes inefficient use of memoryqubits because each is locked up for at least τlink whilewaiting on the signal from the BSA, which succeeds withprobability p. In an asymmetric scheme where Bob hasa shorter distance to the BSA, his qubits are locked upfor a shorter duration, and he could potentially use fewermemory qubits, which is the basis of the next protocolwe consider.

B. Sender-Receiver

The SenderReceiver protocol is similar toMeetInTheMiddle, but the BSA has been movedinto a repeater at one endpoint of the optical link.This rearrangement has interesting consequences forrepeater design, and the modification is a precursorto the third protocol studied here, MidpointSource.SenderReceiver was discussed in Ref. [17].

To understand how SenderReceiver was de-rived, consider what limits the performance ofMeetInTheMiddle. The bottleneck to communica-tion rate in MeetInTheMiddle is τlink, the time for aphoton to reach the BSA and the return trip for theclassical signal that is the result of Bell-state measure-ment. When loss heralding is employed, a quantummemory must be locked up while waiting for this result.MeetInTheMiddle places the BSA in the middle tocreate symmetric delay for both repeaters, but if theBSA were closer to one of the repeaters, that repeaterwould be able to make a quicker decision as to whethereach local memory qubit was entangled to a qubit inthe other repeater. Taking this concept to its limit,SenderReceiver places the BSA inside Bob’s repeater,allowing Bob to make near-instantaneous decisions(limited only by the speed of detectors and controlelectronics) about whether or not his memory qubitholds useful entanglement. As the name suggests, wecall Alice the sender and Bob the receiver in Fig. 4.

The “sender” or “receiver” behavior applies to the in-terface between memory qubits and photons. A repeaterhas at least two interfaces in order to distribute entan-glement. This concept is illustrated in Fig. 5, which

c©2015 HRL Laboratories, LLC. All Rights Reserved. 4

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Repeater: Alice Repeater (memory and BSA): Bob

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L

FIG. 4. Hardware arrangement for SenderReceiver. Thearrangement is similar to MeetInTheMiddle, but the BSA isnow located inside Bob. As explained in the text, this proto-col is most effective when Alice has more memory qubits thanBob (three shown here), because of the asymmetry in delaysfollowing BSA interference. Alice has to wait for round-tripsignal propagation to learn if entanglement was established.Conversely, the BSA is internal to Bob, so he prepares a mem-ory/photon pair just as one of Alice’s photons arrives, and hecan determine almost instantly whether entanglement was es-tablished.

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Another link Another link

Receiver

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Another link Another linkL/2 L/2

L

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(a)Sender Sender Sender

Sender SenderReceiver

BSA

FIG. 5. Link interfaces for (a) MeetInTheMiddle and(b) SenderReceiver. In this paper, the type of every linkinterface is either sender or receiver.

compares the link interfaces for MeetInTheMiddle andSenderReceiver. Whereas MeetInTheMiddle has everylink interface act as a sender, SenderReceiver alternatesroles between sender and receiver, such that each re-peater is a sender in one direction and receiver in theother [17].

The sender operates essentially the same as inMeetInTheMiddle, but the receiver has new behavior.The receiver works by attempting to “latch” an incom-ing photon from Alice into memory. Latching here isfundamentally the same as in MeetInTheMiddle, exceptthat Bob can reset his memory immediately on BSA fail-ure. When a photon arrives, Bob attempts entangle-

ment swapping by producing a memory/photon entan-gled pair and interfering Alice’s photon and his photonin the BSA. If entanglement swapping succeeds, Alice’sphoton is transferred into Bob’s memory. We note that,to enable loss heralding, the latching process works in-directly through entanglement swapping using the BSA;Alice’s photon is not directly absorbed into a memoryqubit. A key feature of the SenderReceiver protocolis that loss heralding occurs inside the receiver, indicat-ing immediately to that repeater whether latching wassuccessful.

Knowing almost instantly when entanglement is es-tablished changes how the repeater operates. Supposethat Alice has NA � 1 memory qubits, allowing her tosend many transmissions to Bob, who has fewer memoryqubits NB < NA. By knowing immediately whether en-tanglement was established, Bob can process each incom-ing photon sequentially with the same memory qubit byresetting his qubit after failing to latch Alice’s photon.The inter-transmission time τclock for SenderReceiveris the maximum of the times for detector recovery andmemory reset. If τclock � τlink, then Bob can use fewermemory qubits than Alice because Bob can update hismemory based on the latching outcome almost instantly.However, the memory qubits in Alice are locked up fortotal round time

τ ′round = 2τlink +NAτclock, (4)

where we use the prime (′) to denote quantities associatedwith SenderReceiver. The factor of 2 before τlink ac-counts for Alice’s photon propagating distance L to Bob,followed by classical signals from Bob indicating whichof her transmissions that Bob latched into memory (shelearns the result τlink after Bob does). The messagingis similar to MeetInTheMiddle, except that the locationsin memory for stored entanglement are not the same forAlice and Bob. For each arriving photon i ∈ [1, NA],Bob will transmit a messageMi to Alice that says either“Transmission i failed” or “Transmission i is entangledwith memory qubit j in Bob,” for some j ∈ [1, NB ]. Asbefore, we assume that both repeaters wait until the endof the round to use entanglement and that all memoryqubits are reset before starting the next round.

The control protocol for SenderReceiver is differentfor Alice and Bob. Alice has essentially the same con-trol as MeetInTheMiddle (see Fig. 3): send a sequenceof photons into the channel and wait for a response mes-sage for each after 2τlink, because she is now distance Lfrom the BSA. However, Bob acts much faster. His pro-tocol, shown in Fig. 6, resets a memory qubit when latch-ing fails, allowing the next incoming photon to attemptlatching into that memory. After latching succeeds, Bobwill either try to latch subsequent incoming photons intothe next memory qubit or reject them if his memory isfull.

The asymmetry of SenderReceiver forces one to con-sider how many memory qubits should be allocated be-tween Alice and Bob. Some quantum communication

c©2015 HRL Laboratories, LLC. All Rights Reserved. 5

Inner loop

Yes

Yes

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Ready

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Use verified entanglement

Alice_index: i ← 1Bob_index: j ← 1

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Start

Synchronize τround

Store qubit, send message indicating latch of (i,j)

j ≤ N ?

B

i ≤ N ?

A

Send message that latch failedi ← i + 1

j ← j + 1

Prepare qubit jand emit photon

FIG. 6. Control protocol for the receiver (Bob) inSenderReceiver. Loop variables i and j follow the defini-tions in the text.

protocols like entanglement distillation [18–22] requireBob and Alice to operate on multiple pairs of entan-gled qubits simultaneously, so Bob may need more thanone memory qubit to access multiple entangled qubitsat the same time. In this case, SenderReceiver worksbest when the ratio of memory sizes in Bob and Alice isNB/NA ≈ p, where p is probability of successful latchingat Bob (assumed to be the same as MeetInTheMiddle).By satisfying this ratio, Alice is expected to entangle withall of Bob’s memory qubits on average, while utilizing allof her memory qubits. If the ratio is much greater or lessthan p, then one of the repeaters will not be utilizing allof its memory.

As with MeetInTheMiddle, we can understand the ef-ficiency of SenderReceiver by the fraction of time thatthe protocol spends in the inner loop. The sender andreceiver have different protocols, but they spend aboutthe same amount of time in their respective inner loops,because Bob attempts to latch every photon that Alicesends, until his memory is full. Determining the averagecommunication rate R′ for a link using SenderReceiveris a little more complicated since Bob may reject some ofAlice’s photons if his memory fills up:

R′ =1

τ ′round

NA∑x=0

min(x,NB)

(NA

x

)px(1− p)NA−x (5)

<pNA

τ ′round. (6)

In Eqn. (5) the term min(x,NB) accounts for the possi-bility that Bob rejects incoming photons after filling uphis memory, and the upper bound is given by replacingthe minimum function with summation variable x (equiv-alent to assuming NB = NA).

To make a fair comparison with MeetInTheMiddle, letus say that the number of memory qubits connected toa link is fixed, meaning 2N = NA + NB , which is moti-vated by the following line of reasoning. We assume forthe moment that number of memory qubits is the limit-ing resource for repeater technology, so we will comparethe two protocols when this quantity is fixed. Considera linear chain of repeaters, where each has 2N mem-ory qubits and is connected to two links. The repeaterscould implement MeetInTheMiddle, where each repeaterassigns N qubits to a link. Alternatively, the repeaterscould implement SenderReceiver, where each repeateris a sender in one direction and a receiver in the other. Ifwe assume the links have identical parameters, then NA

and NB will be the same for all links, and the number ofqubits assigned to a link from both connected repeatersis 2N , as illustrated with the example N = 3 in Fig. 5.Under these conditions, NA < 2N and

R′ <pNA

2τlink +NAτclock<

pN

τlink +Nτclock= R. (7)

In other words, SenderReceiver has a lower average ratethan MeetInTheMiddle! However, a few considerationsshould be made. First, SenderReceiver places the BSAinside a repeater rather than at the link midpoint, whichcould be quite important for practical concerns related toinstalling a network. Second, one can show that R andR′ are similar if NB/NA ≈ p. Third, in SenderReceiver,the receiver spends nearly the same amount of time in itsinner loop as the sender despite using far fewer memoryqubits. We omit analytical derivation for optimal valuesof NA, NB , or link utilization F ′, because such analysisis complicated and not particularly illuminating; instead,we estimate these quantities with numerical simulationin Section IV. Ultimately, both MeetInTheMiddle andSenderReceiver are limited by the delays for memorylock up at the sender interface(s), so the final protocolconsiders what happens when all link interfaces are re-ceivers.

C. Midpoint-Source

The MidpointSource protocol, shown in Fig. 7, is themost complex entanglement-distribution scheme that weconsider, and it uses more optical network componentsthan the other two. However, our analysis shows thatthis extra complexity is justified in many scenarios be-cause MidpointSource is more robust to photon loss.While MeetInTheMiddle and SenderReceiver have verysimilar performance, MidpointSource has a fundamen-tally different average rate as a function of transmissionprobability p; the former protocols have rate proportionalto p, but MidpointSource has rate that scales like

√p,

which can be a dramatic improvement when p � 1.MidpointSource was introduced in Ref. [43].

In SenderReceiver, the receiver node is able to makeefficient use of its memory qubits by using loss herald-

c©2015 HRL Laboratories, LLC. All Rights Reserved. 6

Two-detectioncoincidence

Q

Repeater (memory and BSA): Bob

Classical signals

Optical fiber

Two-detectioncoincidence

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Repeater (memory and BSA): Alice

Entangled photonsource

L/2 L/2

FIG. 7. Hardware arrangement for MidpointSource. A source of entangled photons in the middle of the channel splits a pairof entangled qubits, sending one to each repeater. The source can generate photon pairs at a fast clock rate, so long as therepeaters are synchronized with it. For each arriving photon, the repeater uses its protocol to determine if the photonic qubitwas latched into memory using teleportation. Each repeater node has an internal BSA, allowing rapid reset of a memory qubitif latching fails.

ing to know instantly whether a photon was latched intomemory. In MidpointSource, both repeaters connectedto a link use the receiver interface to achieve efficientmemory utilization. Since both repeaters latch incomingphotons into memory, entanglement is distributed usinga source of entangled photons placed somewhere in theoptical channel. MidpointSource exploits the fact thatsources of entangled photons are relatively mature tech-nology compared to quantum memories, and experimentshave demonstrated sources of high-fidelity entangled pho-tons available at up to MHz clock rates [44–46]. Simi-lar entanglement-distribution schemes have been studiedthat place a source of entangled photons at the link mid-point [2, 4, 10, 14, 47]. The MidpointSource protocolgoes further, using fast discrimination of lost photons toachieve rapid entanglement distribution that is less sen-sitive to loss [43].

To set the scene for a control protocol, consider thatthe latching process for a receiver in MidpointSourcedoes not carry the same information as it does inSenderReceiver. In SenderReceiver, a successful latchindicates that a photon traversed the entire channel, buta latch in MidpointSource only indicates that a photonfrom the entangled pair source has been stored in this re-peater, without any indication of whether the other pho-ton from that entangled pair was latched into the distantreceiver. Having less information per latch event mightappear to put MidpointSource at a disadvantage, butthe protocol redeems itself if the hardware can attemptlatching at a very fast clock rate. Each receiver indepen-dently tries to latch each arriving photon. After a latchattempt succeeds, the receiver holds this qubit in mem-ory and sends a signal to the other receiver indicating thelatch. Subsequent photons may be rejected for a shortperiod, as described below. If both receivers latch a pho-ton from the same entangled-photon pair, then the cor-responding memory qubits are entangled, which is con-firmed by both repeaters using classical messages that re-quire time τlink to propagate. Otherwise, a stored qubitis discarded when a repeater learns that the other pho-

ton was not latched, which happens after delay of at mostτlink.

There are several ways to design a protocol forMidpointSource. The one implemented in Fig. 8 op-erates in discrete rounds, for direct comparison withMeetInTheMiddle and SenderReceiver. Operating ina free-running mode with asynchronous memory resetcould be more efficient, but it also requires more com-plex control; the case of having one qubit per receiverwas solved in Ref. [43], and specifying an asynchronousMidpointSource protocol for more than one memoryqubit is a matter for future work. For the proto-col in Fig. 8, each repeater has N ′′ memory qubits,where double prime (′′) denotes quantities associatedwith MidpointSource. The round consists of a seriesof N ′′ communication time “bins,” followed by one delayof τlink to confirm entanglement, so

τ ′′round = τlink +N ′′τ ′′bin. (8)

Each bin is associated with a memory qubit. Duringbin i, the protocol attempts to latch incoming photonsinto memory qubit i. If latching succeeds, the proto-col rejects subsequent photons until the start of the nextbin, when the protocol attempts to latch into the nextqubit. Within a communication bin, every entangled-photon pair from the midpoint source has a unique iden-tifier k ∈ [1,K], where K will be determined below. Aswith SenderReceiver, each receiver generates a messageMi,k for each bin i and incoming photon k in that bin.The message says simply “Photon k was latched intomemory i” or “Photon k in bin i was rejected.” Af-ter attempting entanglement in all N ′′ bins, the protocolwaits τlink to confirm entanglement.

Because the arrangement of hardware is different fromthe other two protocols, we need to define some newquantities for probability of latching a photon into mem-ory. Let p′′m be the probability that an entangled-photonpair is generated at the midpoint source. If a pair is gen-erated, let p′′l (p′′r ) be the probability that the left (right)photon is latched into the repeater that receives it. If

c©2015 HRL Laboratories, LLC. All Rights Reserved. 7

Inner loop

Prepare qubit iand emit photon

Yes

Synchronize τclock

Ready

Waiting

Store qubit, send message indicating latch of (i,k)

Local BSAsuccess

?

Check messagesfrom distant BSA,use verified entanglement

Memory qubit: i ← 1

No

Ready

Waiting

Start

Synchronize τround

k ≤ K?

Send message that latch failed, k ← k + 1

Yes

No

Photon qubit: k ← 1

i ≤ N? i ← i + 1

Yes

Synchronize τbin

Waiting

No

Ready

FIG. 8. Control protocol for each receiver in MidpointSource. Loop variables i and k follow the definitions in the text.

there is just a single attempt at generating entanglement,the probability of success would be

p′′ = p′′l p′′mp′′r . (9)

Note that even if p′′m = 1, we expect that the probabilityof latching both qubits into memory would obey p′′l p

′′r <

p, because p for MeetInTheMiddle and SenderReceiveraccounts for one BSA while MidpointSource has two.We assume p′′l = p′′r throughout our analysis, though theprotocol can be modified for asymmetric designs.

We further decompose loss into the fundamental piecesof a link, which will aid our comparison of protocols. Weassume that the probability of successful BSA when twophotons arrive is pBSA, which is the same for all proto-cols. Furthermore, poptical is the product of probabilitiesfor successful transmission through the memory/photoninterface (with fiber coupling) and transmission throughoptical fiber over distance L/2 (half of the link distance).We can write the total link-transmission probability forMeetInTheMiddle and SenderReceiver as

p = pbsa (poptical)2. (10)

For MidpointSource, p′′l = p′′r = pbsapoptical, so

p′′ = p′′m (pbsapoptical)2. (11)

The added complexity of the entangled-photon sourceand second BSA are manifest in the additional termsthat reduce probability of successful entanglement distri-bution.

The latching process in MidpointSource is attemptedat a fast repetition rate having cycle time τclock that is

limited only by the maximum of three time quantities:(a) the clock period for generating entangled photons,(b) the recovery time for BSA detectors, and (c) the resettime for memory qubits. This clock cycle is independentof the signaling time across the channel, so the averagetime to latch a qubit can be less than τlink. We set thenumber of latch attempts per bin to be

K =τ ′′binτclock

=

⌈3

p′′l p′′m

⌉. (12)

The numerator 3 in Eqn. (12) is selected somewhat arbi-trarily to make the probability of latching a photon in atime bin near unity:

platch = 1− (1− p′′l p′′m)K > 0.95. (13)

We will use Eqn. (13) repeatedly to place bounds on theaverage-case performance of MidpointSource.

The key advantage of MidpointSource is that alatched qubit has already overcome at least half of theloss for one side of the link. In particular, the probabilitythat both repeaters latch a photon in time bin i is greaterthan 0.90 with the above parameters. Memory qubits atindex i for two receivers are entangled only if they bothlatched photons from entangled pair k. For the left re-peater, the probability that a latched memory qubit isentangled to its partner on the right side is a little com-plicated, since latching is only attempted for photon k iflatching failed for photons 1 to k−1. We can express theprobability that entanglement is established in time bin

c©2015 HRL Laboratories, LLC. All Rights Reserved. 8

i as the sum:

p′′ent =

K∑k=1

p′′ [1− p′′m(p′′l + p′′r ) + p′′]k−1

. (14)

Using Eqn. (9), this can be expressed as

p′′ent = p′′K∑

k=1

(1− p′′( 1

p′′l+

1

p′′r− 1)

)k−1

=p′′l

2− p′′l

[1−

(1− p′′( 2

p′′l− 1)

)K], (15)

where we assume as before that p′′l = p′′r . Using Eqn. (13)and the trivial fact that 1/p′′l > 1, we can derive boundson the entanglement probability:

0.95p′′l2< p′′ent <

p′′l2− p′′l

. (16)

If p′′l < 0.1, then we can say that p′′ent ≈ p′′l /2 with atmost 5% relative error.

The average rate of entanglement is simply the ex-pected number of entanglement events per round:

R′′ =N ′′p′′entτ ′′round

. (17)

We can compare this rate to MeetInTheMiddle (whichbounds SenderReceiver) using the decomposition intosuccess terms in Eqns. (10) and (11). We set N ′′ = Nand assume that Nτ ′′bin � τlink (“fast-clock” assumption,discussed below), so

R′′ ≈ Npbsapoptical2τlink

. (18)

Under the same fast clock assumption,

R ≈ Npbsa (poptical)2

τlink. (19)

The ratio of the two is

R′′

R≈ 1

2poptical. (20)

The probability poptical is associated with mem-ory/photon interface losses and fiber attenuation, andis common to all protocols. In particular, we can saythat R′′ is scaling like poptical while R and R′ are scaling

as (poptical)2, explaining the ∼ √p separation mentioned

at the beginning of this section. Another interpretationis that the MidpointSource protocol is less sensitive tothe underlying signal losses represented by poptical, mak-ing MidpointSource well-suited to early prototypes ofquantum repeaters.

The rate in Eqn. (19) has a remarkable feature. Theaverage communication rate is independent of the prob-ability that the entangled source generates an entangled

pair, even if p′′m � 1. As a result, MidpointSource worksvery well, even if a probabilistic source of entangled pho-tons is used, such as spontaneous parametric downcon-version. The explanation for this effect has two compo-nents. First, we choose a number of latching attemptsper round (K) according to Eqn. (12), which scales in-versely with p′′m. Under the fast-clock assumption, thetime for latching attempts is insignificant compared tototal round time, indicating how critical a fast clock isfor “absorbing” the impact of low signal transmissionprobability. Second, we presume that the entangled-pairsource does not emit single photons; the fact that a re-ceiver latches post-selects events where a photon was sentto the other receiver. We set K so that a receiver latcheswith near certainty, and when that receiver latches, theprobability that the other receiver latched a photon fromthe same pair is proportional to poptical. In additionto having less sensitivity to loss in optical components,MidpointSource has almost no dependence on p′′m, show-ing how robust the protocol is to signal loss. However, weemphasize that this robustness depends on a fast clockcycle at the receiver.

Another way to see how MidpointSource achieves highperformance is the fraction of time spent in its inner loop.In a time bin, this fraction is

F ′′bin =1

K

K∑k=1

kp′′l p′′m(1− p′′l p′′m)k, (21)

and we use Eqn. (13) to establish bounds 0.953 < F ′′bin <

13 . The link utilization for one round of MidpointSourceis

F ′′ =N ′′F ′′binτ

′′bin

τ ′′round. (22)

However, the connection between utilization and rate isnot as simple as before: R′′ 6= F ′′p′′/τ ′′clock, due to thefollowing difference in available information. When thereceiver in SenderReceiver exits its inner loop, it knowsthat any latch events correspond to established entangle-ment, whereas when MidpointSource exits its inner loop,there is only a probability about p′′l /2 that both receiverslatched the same photon. MidpointSource compensatesby having a rate of photon arrival from the entangled-photon source that is independent of the memory size ineither repeater, which is not true for MeetInTheMiddleor SenderReceiver.

We should consider for a moment the fast-clock as-sumption made above. This simplified our analy-sis, but satisfying this condition is also necessary forMidpointSource to yield higher communication rate.One can show that the performance of MidpointSourcedegrades to being worse than the other two protocols inthe other limit, Nτ ′′bin > τlink. Under the fast clock as-sumption, memory qubits are locked up for about τlinkin MeetInTheMiddle and MidpointSource. When thisassumption does not hold, the protocols as specified will

c©2015 HRL Laboratories, LLC. All Rights Reserved. 9

tend to produce entanglement at a rate independent ofN , because memory qubits are locked up for round timesthat scale with N . Worse yet, MidpointSource haslower success probability per entanglement attempt duethe additional entangled-photon source and BSA, and afast clock is essential to offset this complexity. In sum-mary, all protocols benefit from fast clock cycle τclock �τlink/N , and MidpointSource shows the greatest benefitwhen the clock period satisfies τclock � τlink/(NK).

Since MidpointSource depends critically on the fastclock condition, we present a substitute for the entangled-pair source that can realize p′′mid = 0.5, thereby reduc-ing K and easing the requirements on τclock. Supposetwo triggered, deterministic single-photon sources emitphotons that are indistinguishable except for polariza-tion, where one emits horizontal and the other vertical.These photons interfere on a beamsplitter, and the pho-tons exiting the two output ports are collected. This ap-proach was presented in Ref. [43] as an alternative to anentangled-photon source for MidpointSource. Label theleft/right input modes of the beamsplitter as “a”/“b”,and label the left/right output modes as “c”/“d”. Whenthe single photons interfere at the beamsplitter, the stateis transformed as

|H〉a |V 〉b →1√2

∣∣Ψ−⟩−1

2|H〉c |V 〉c+

1

2|H〉d |V 〉d , (23)

where ∣∣Ψ−⟩ =1√2

(|H〉c |V 〉d − |V 〉c |H〉d) (24)

is a maximally entangled state of the two photons. Notethat the entangled state can trigger BSA success at bothreceivers, whereas the states |H〉c |V 〉c and |H〉d |V 〉d can-not. As a result, the interference of two single-photonsources is a “post-selected” source of entanglement [48]when used in the MidpointSource scheme, where thepost-selection occurs when both BSAs indicate success-ful latching. The entangled-photon state occurs in halfof the attempts.MidpointSource is more complex than both

MeetInTheMiddle and SenderReceiver in two ways.First, the link requires more optical hardware. Thereare two BSAs, meaning more single-photon detectors,and there is a source of entangled photons. However,both the preceding analysis and numerical simulation inSection IV show that the additional optical componentsallow MidpointSource to make more efficient use ofmemory qubits. Consequently, a network employingMeetInTheMiddle or SenderReceiver would requiremore memory qubits in one or both repeaters to matchthe performance of MidpointSource. Given thatmemory qubits are currently a less mature technologythan either entangled-photon sources or single-photondetectors, MidpointSource might be the best pro-tocol for early repeater networks because it requiresfewer memory qubits at the expense of more opticalcomponents. This trade-off in resources motivates our

numerical simulations, and we make that trade-offmore quantitative in Section IV. The second way thatMidpointSource is more complex is that its controlprotocol has much more information to process. Never-theless, the demands that the protocol in Fig. 8 placeson both local processors and network transmission seemmodest for classical information technology, and weargue that this additional complexity is justified byhigher rate of entanglement distribution for the samenumber of memory qubits.

D. Further Development of Protocols

The protocols developed in this manuscript were de-signed to demonstrate key features of the three waysof linking two repeaters (sender-sender, sender-receiver,receiver-receiver). However, they are not optimal, andfurther performance improvements are possible. For ex-ample, waiting τlink for classical messages to propagateafter all photon transmissions is not necessary in mostcases. Similarly, MidpointSource can operate in a “freerunning” mode that does not associate memory qubitswith time bins. Additional care must be taken to ensurethat memory qubits do not get stuck in a pathologicalpattern of locking up asynchronously, which would pre-vent entanglement distribution; the case for N = 1 wassolved in Ref. [43]. Our protocols were designed to besimple while capturing the essential way in which perfor-mance is limited by the quantum hardware. When thefast-clock assumption holds, these simple protocols de-liver nearly optimal performance since any entanglementmust be confirmed with classical signals requiring delayτlink.

There are interesting avenues to explore in developingbetter protocols. For example, the location of a BSAin the link can determine the frequency of the interfer-ing photons, thereby affecting the performance of single-photon detectors. Another approach is to consider asyn-chronous designs that do not have a fixed round time,which could perform much better when the fast-clock ap-proximation does not hold. Furthermore, a round timethat is longer than memory lifetimes would be unaccept-able, which could be relevant if the number of memoryqubits is very large. Finally, sending multiple signalsin parallel (such as with frequency-division multiplexing)can increase communication rate, though it might requirea more sophisticated protocol.

The protocols considered here are not an exhaustivelist, and one could search for new hardware arrangementsand control schemes not yet discovered. One way tofind a new protocol would be to combine elements fromthe three protocols above, then eliminate any unneces-sary components. Indeed, MidpointSource was derivedfrom SenderReceiver in the following way. Take a re-peater chain consisting of SenderReceiver protocol oneach link, but alternate the direction (much like the “but-terfly arrangement” in Ref. [17]). Both link interfaces for

c©2015 HRL Laboratories, LLC. All Rights Reserved. 10

every odd-numbered repeater are type sender in each di-rection, and the even-numbered repeaters have receiverinterfaces. Now consider a single sender-type node. Itsimultaneously sends photons in both directions to differ-ent receivers. Instead of holding the memory qubits whilewaiting for latching results from the receivers, the sendernode could perform an immediate entanglement swap ofthese memory qubits and send out the classical result ofthe Bell measurement. If the swap is executed immedi-ately, the sender node only requires two memory qubits,since they are reset before the next photon transmission.The sender node is simply acting as a source of entangledphotons. By replacing the sender node with a more con-ventional entangled-photon source that does not requirequantum memory at all, you have the MidpointSourcehardware arrangement. It may be possible to derive newcommunication protocols using similar techniques of mix-ing and replacing fundamental repeater elements.

IV. SIMULATION OF NETWORKPERFORMANCE

We develop numerical simulations of a quantum re-peater network based on the protocols in Section III, tocompare their performance using more complicated mod-els that do not admit simple analytical results. We per-form two types of comparisons. First, we compare theprotocols in a repeater network consisting of ten linksusing a common set of parameters that represent a ma-ture platform for repeater technology, including long-lived quantum memory and low-error gates for purifi-cation. Moreover, we choose parameters that satisfythe fast-clock condition to highlight the differences be-tween protocols. The repeaters store successfully trans-mitted entanglement in memory and perform purifica-tion, as explained below. The results are straightforward:MidpointSource is the best protocol when the fast-clockassumption is valid. If this assumption does not hold,the simpler MeetInTheMiddle may perform better.

In the second simulation, we evaluate the performanceof the protocols for near-term experiments using currentstate-of-the-art device parameters. Two nodes establishentanglement across a single link and perform immediatemeasurement, and we compare the rate of entanglementgeneration for MeetInTheMiddle and MidpointSource.The three repeater technologies we consider are trappedions, diamond NV centers, and quantum dots; these de-vices can store qubits in memory, apply operations, andinterface memory with single photons. For this set of sim-ulations, the fast-clock assumption does not necessarilyhold, and we can assess how well these protocols mightperform in practice.

A. Simulations to Compare Protocols

The model network in our simulations is a linear chainof ten links (eleven repeater nodes), which distribute en-tanglement across each link with initial fidelity of 0.95.We purify entangled pairs using the decoding circuit ofthe [[7, 1, 3]] Steane code [49, 50], where we assume forsimplicity that local gates and memory in the repeaterare error-free. In a more realistic setting, errors can besuppressed using quantum error correction [49, 50], but adetailed implementation here is a matter for future work.The error on each input Bell state to the purification pro-cedure is i.i.d. εin, so the error of a successfully purifiedstate is bounded by

εout ≤ 7ε3in(1− εin)4 + ε7in, (25)

and the probability of success is bounded by

psuccess > (1− εin)7. (26)

For εin = 0.05, we have εout < 10−3 and psuccess > 0.698.After entanglement purification across each link, the net-work establishes end-to-end entangled pairs using en-tanglement swapping, with total error over ten linksbounded by εtotal < 10εout < 10−2. The end-to-endentangled pairs with fidelity greater than 0.99 can beused for QKD. The communication rate of the networkis reported as the number of these end-to-end entangledqubit pairs (ebits) created per second. The communi-cation rate is plotted as a function of inter-repeater linkdistance. Each plotted point is the mean of 1000 samplestaken, and error bars show the 90% confidence intervalfor the sampled distribution.

The first simulation in Fig. 9 uses an “optimistic” set ofparameters, where the linear-optics BSA has maximumsuccess probability 50% (meaning perfect single-photondetectors), each memory-photon interface has transmis-sion probability 50%, and optical fiber has standard at-tenuation length of Latt = 22 km (0.2 dB/km). Thenumber of memory qubits is given by N = 100, andthe clock time is 1 ns, which was chosen to illustratethe impact of the fast-clock assumption. Each repeaterhas three of its memory qubits reserved for storing pu-rified entangled states waiting to be swapped, while therest participate in attempting to establish entanglement.Since the purification protocol requires seven qubits, thenumber of receiver qubits in SenderReceiver is chosen tobe 6 + d2Np/(p+ 1)e, where p = pBSA(poptical)

2 (see Sec-tion III B). The total simulation time is 103τlink, whichdepends only on inter-repeater distance L. We simulatethree values for the entangled-photon generation proba-bility in MidpointSource (p′′mid = 1, 0.1, and 0.02). Thefirst two values satisfy the fast-clock assumption of Sec-tion III since τlink ranges from 25 to 250 µs (L = 5 to50 km), while NKτclock for MidpointSource ranges from700 ns to 2 µs for p′′mid = 1, and this time is 10 and 50times larger for p′′mid = 0.1 and 0.02, respectively. Forp′′mid = 0.02, the fast-clock assumption is violated, and

c©2015 HRL Laboratories, LLC. All Rights Reserved. 11

0 10 20 30 40 50 6010

1

102

103

104

105

Link distance L (km)

Com

mun

icat

ion

rate

(ebi

ts/s

ec)

MeetInTheMiddleSenderReceiverMidpointSource (p

mid = 1)

MidpointSource (pmid

= 0.1)

MidpointSource (pmid

= 0.02)

FIG. 9. Communication rate for the link protocols with opti-mistic hardware parameters, meaning high transmission prob-ability, as a function of link distance L. There are ten links,so network length is 10L. The parameters are: N = 100,pBSA = 0.5, poptical = 0.5 exp(−L/2Latt), τclock = 1 ns.

the performance of MidpointSource is degraded. Never-theless, the robustness of MidpointSource to signal lossallows even the p′′mid = 0.02 instance to outperform theother two protocols for L ≥ 20 km.

The simulation results in Fig. 9 show several featuresthat are consistent with the analysis in Section III. First,MeetInTheMiddle and SenderReceiver have very sim-ilar performance, with the former being slightly better.Second, MidpointSource has a communication rate thatdecreases with smaller slope (less dependence on trans-mission probability) than MeetInTheMiddle, because thehigh clock rate of MidpointSource enables it to be lesssensitive to photon loss. Whether MidpointSource out-performs MeetInTheMiddle depends on the link distanceand the probability that an entangled-pair source gener-ates a photon pair, which determines in part whether thefast-clock condition is satisfied.

Another simulation is performed for a “pessimistic”set of parameters, where BSA and memory/photon in-terface each have transmission probability 0.10, with re-sults plotted in Fig. 10. Since photon loss is more severe,all protocols have lower communication rate than the pa-rameter set in Fig. 9. Notably, MidpointSource does notdecrease in performance as much as the other protocols,and the gap in performance between MidpointSourceand MeetInTheMiddle is generally larger than the resultsin Fig. 9.

Unlike the “optimistic” set of parameters, the commu-nication rates for MidpointSource protocols in Fig. 10do not properly satisfy the fast-clock condition due to thelower transmission probabilities. The result is that thecommunication-rate curves for different values of p′′mid arefurther apart in Fig. 10 than Fig. 9, because the way Kis chosen for these protocols (see Eqn. 12) leads to thelatching process accounting for most of the round dura-

0 10 20 30 40 50 6010

−2

10−1

100

101

102

103

104

Link distance L (km)

Com

mun

icat

ion

rate

(ebi

ts/s

ec)

MeetInTheMiddleSenderReceiverMidpointSource (p

mid = 1)

MidpointSource (pmid

= 0.1)

MidpointSource (pmid

= 0.02)

FIG. 10. Communication rate for the link protocols withpessimistic hardware parameters, meaning low transmissionprobability, as a function of link distance L. There areten links, so network length is 10L. The parameters are:N = 100, pBSA = 0.1, poptical = 0.1 exp(−L/2Latt), τclock =1 ns. The downward curve in rate for MeetInTheMiddle andSenderReceiver for L ≥ 20 km is a consequence of finitesimulation time for Markov-chain Monte Carlo. End-to-endentangled pairs require entanglement distribution across alllinks, and these protocols generate link-level pairs so slowlythat the number of entangled pairs in the repeater memorieshas not reached a steady-state distribution for the finite timeof the simulation, which is 103τlink (25 to 250 µs).

tion. Nevertheless, the fast clocking of MidpointSourcestill provides some ability to overcome signal loss, andthese protocols outperform MeetInTheMiddle in mostcases. Notice that MidpointSource still has effec-tive entanglement distribution at inter-repeater link dis-tances up to 50 km, while both MeetInTheMiddle andSenderReceiver drop to zero end-to-end ebits aroundL = 25 km due to finite simulation time (see caption ofFig. 10).

B. Hardware-Specific Simulations

The preceding simulations indicate thatMidpointSource is the best simulation if the fast-clock condition is satisfied. However, the clock cycle of1 ns is very fast, and existing proposals for quantumrepeater hardware do not yet operate at this speed. Wenow seek to determine what the best protocol wouldbe for “realistic” hardware parameters. We considertrapped ions [28, 51–53], diamond nitrogen-vacancy(NV) centers [30, 31, 54, 55], and quantum dots [32–34, 56, 57]. Note that self-assembled quantum dots aremore challenging than ions to integrate into coupledarrays, but such integration is needed to effectivelyintegrate the communication protocols. See Ref. [58] fora hardware proposal combining demonstrated quantumdot spin-photon methods with demonstrated methods

c©2015 HRL Laboratories, LLC. All Rights Reserved. 12

TABLE I. Timing Parameters for Memory-Photon Interfaces

Memory typeCycletime

Emissionfraction

Collectionefficiency

Trapped ion(171Yb+)

1 µs 1.00 0.05

Diamond NV 100 ns 0.05 0.50Quantum dot(InGaAs)

10 ns 1.00 0.50

for constructing multi-qubit arrays. For each hardwareplatform, we take the optimistic approach of finding thebest parameters from recent experimental results andpresuming that these may be realized in one system. Therelevant parameters for clocking and memory/photoninterface transmission probability are listed in Table I.

In addition to the quantum repeater hardware, alllink protocols considered here require a BSA, andMidpointSource requires two BSAs and an entangled-pair source. We assume that the BSA uses supercon-ducting nanowire detectors (SNSPDs) [59, 60]. In ourmodel, these detectors have quantum efficiency of 0.80and very low dark count rates, so the linear-optics BSAhas success probability pBSA = 0.24. The entangled-photon source could be one of several potential designs.A quantum dot could potentially emit entangled pho-tons with success probability approaching unity (p′′mid =1) [61]. Two deterministic single-photon sources mix-ing on a beamsplitter (see Section III C) would produceentangled photons with p′′mid = 0.5, post-selected bythe two BSAs. Finally, entangled-photon pairs can begenerated using four-photon scattering in optical fiberwith p′′mid = 0.02 [62] or other optical nonlinearities forparametric downconversion [44–46, 63, 64]. We assumethat these components operate at photon frequency near1550 nm for low-loss transmission in optical fiber. Im-portantly, none of the quantum memory technologies inTable I emit at this frequency, so some form of photonicfrequency conversion [65–68] is necessary for the inter-face to optical fiber. Although important, analyzing fre-quency conversion is outside the scope of our work, andfor this investigation we assume that signal losses associ-ated with frequency conversion are included in the “col-lection efficiency” for each technology.

The results of the “hardware-specific” simulations areshown in Fig. 11, where each of the panels corresponds toa particular repeater technology with parameters givenin Table I. The plots compare MeetInTheMiddle toMidpointSource with a selection of entangled-photonsources with different values for entanglement-generationprobability p′′mid. In general, the signal transmissionprobabilities are lower and clock rates are higher thanthe preceding simulations, indicating that further im-provements in repeater technology are needed to re-alize high performance networks. The simulations inFig. 11 are for a single link between just two nodes. Thenumber of memory qubits is N = 3, and no purifica-tion is performed. SenderReceiver has rate lower than

MeetInTheMiddle, so it is not simulated. The simula-tion runs for 104τlink to better resolve low communicationrates.

The device parameters in Table I were chosen to rep-resent possible near-term experiments showing entangle-ment distribution to validate repeater technology. We donot simulate memory errors, to separate comparison ofthe optical protocols from considerations of memory life-time and the implementation of error correction. We notethat independence of communication rate and memorylifetime can be realized in an entanglement-tomographyexperiment by immediately measuring a memory qubitafter either emitting a photon (MeetInTheMiddle) orlatching (MidpointSource), then post-selecting caseswhere the BSA measurements indicate entanglement suc-cess (sometimes known as a “delayed choice” experi-ment [33, 69, 70]).

For many combinations of device parameters,MidpointSource delivers the highest communicationrate, but not always. MeetInTheMiddle may performbetter when the fast-clock condition does not hold,and this condition becomes more difficult to satisfywith slower clock cycle or low values of p′′mid. Forexample, trapped ions have the slowest τclock, andMidpointSource only outperforms MeetInTheMiddlein Fig. 11(a) for high values of p′′mid and link distancegreater than about 20 to 30 km. Diamond NV centershave higher collection efficiency, but they only emitabout 5% of the time into the zero-phonon line (althoughthere are significant results showing Purcell enhance-ment [55], the cavity would need to be degenerate forthe two photonic qubit states), so poptical for NV centersis similar to that of ions. However, NV centers dooperate at a 10× faster clock rate in our model, soMidpointSource shows more substantial advantage inFig. 11(b) when p′′mid is high. Finally, quantum dots havethe fastest clock cycle and reasonably high transmissioninto optical fiber; indeed, two of the MidpointSourcecurves in Fig. 11(c) are indistinguishable, indicating thatthe fast-clock condition is satisfied for those parameters.Further evidence of the benefit MidpointSource derivesfrom a fast clock cycle is that the curve for p′′mid = 0.02is closer to p′′mid = 1 than in the other two panels.Quantum dots have the highest communication rates,which is a direct result of our model using a fast clockrate and high collection efficiency.

V. DISCUSSION

The key contributions of this paper are twofold. First,we provide detailed instructions for the time-dependentoperation of multiple quantum-networking protocols inone paper. Second, we simulate the performance of net-works using the quantum communication protocols. Westart with simulations using optimistic parameters, tocompare how the protocols might perform on mature re-peater technology. We then simulate the protocols on

c©2015 HRL Laboratories, LLC. All Rights Reserved. 13

0 10 20 30 40 5010

−2

10−1

100

101

102

103

104

Link distance L (km)

Com

mun

icat

ion

rate

(ent

angl

ed p

airs

/sec

)

MeetInTheMiddle

MidpointSource (pmid = 1)

MidpointSource (pmid = 0.5)

MidpointSource (pmid = 0.02)

0 10 20 30 40 50Link distance L (km)

MeetInTheMiddle

MidpointSource (pmid = 1)MidpointSource (p

mid = 0.5)MidpointSource (pmid = 0.02)

0 10 20 30 40 5010

−2

10−1

100

101

102

103

104

Link distance L (km)

MeetInTheMiddle

MidpointSource (pmid = 1) / (p

mid = 0.5)MidpointSource (pmid = 0.02)

Trapped ion Diamond NV center Quantum dot(a) (b) (c)

FIG. 11. Communication rate for the link protocols as a function of link distance L using experimentally motivated hardwareparameters for (a) trapped ions, (b) diamond NV centers, and (c) quantum dots. The parameters used in this simulationcorrespond to Table I. The simulated network consists of just two nodes sharing one link. The common parameters are:N = 3 and pBSA = 0.24. Hardware-specific parameters are (a) ion: poptical = 0.05 exp(−L/2Latt), τclock = 1 µs; (b) NV:poptical = 0.025 exp(−L/2Latt), τclock = 100 ns; (c) QD: poptical = 0.5 exp(−L/2Latt), τclock = 10 ns. Note that in (c), theMidpointSource protocols with p′′mid = 1 and p′′mid = 0.5 are indistinguishable, indicating that the fast-clock condition issatisfied.

multiple hardware platforms by selecting realistic per-formance parameters that are consistent with recent ex-perimental demonstrations. Taken together, this papercan be used as an engineering assessment for design-ing quantum networks and setting application-motivatedmilestones for the development of quantum hardware.

This paper examined three protocols for creating dis-tributed entanglement in a quantum repeater network.The performance of these protocols in terms of commu-nication rate was examined both analytically and withnumerical simulation. The different protocols offer com-plementary strategies for developing quantum networks.The simplest protocol, MeetInTheMiddle, works bestwhen signal transmission probability is relatively high.Alternatively, the more complex MidpointSource cancompensate for low signal transmission with a more so-phisticated protocol and faster clocking. Our simulationsshow that even hardware based on recent experimen-tal demonstrations could demonstrate an advantage forMidpointSource, such as quantum dots operated at afast clock rate. A further advantage of MidpointSourceis that the Bell-state measurement procedure is local toboth repeaters sharing a link, allowing local tracking ofclock-synchronization information [58].

To see why we chose to implement two-detectionschemes with loss heralding, one should consider the rel-ative merits of our approach and its alternatives. Forexample, other single-photon schemes exist that encodea qubit in the presence or absence of a photon [8, 10, 14].In this case, a single detection heralds entanglement, butthere are two significant problems. First, the single-detection BSA cannot distinguish between one photonsent by one repeater and two photons sent (one from

each repeater) where one is lost in transmission. Theprobability for each repeater to emit a photon must besufficiently small to suppress the likelihood of the double-emission event [10]. Second, single-detection schemes arevery sensitive to path-length fluctuations, which is prob-lematic for long-distance fiber transmission [8, 27, 35].

While two-photon detection schemes address problemswith single-photon detection, another concern is that lossheralding requires two-way communication with delays toconfirm entanglement. Relatively recent proposals con-sider only one-way communication with error correctionto overcome the effects of loss [24–26, 71]. One-way com-munication avoids the round-trip signaling delays, re-moving the need for long-lived quantum memory. How-ever, these proposals are very sensitive to photon loss fortwo reasons. First, one-way protocols require much moresophisticated hardware, because the error correction re-quires many-qubit entangled states stored in quantummemory, the optical channel, or both. The complex-ity overhead increases significantly with loss probabil-ity [25, 26]. Second, Bennett et al. showed that one-waycommunication is impossible (i.e. information capacityof the quantum channel is zero) if probability of qubitloss is 50% or greater [72]. This rather general boundrefers to the total loss during transmission between tworepeaters, and it places a upper bound on inter-repeaterdistance if one transmits qubits through standard opticalfiber [24–26].

Compared to alternatives, two-way protocols with lossheralding require less device complexity and can functioneven in settings with greater than 50% loss probability,making them suitable for near-term quantum repeatertechnology. The one-way protocols may prove to have

c©2015 HRL Laboratories, LLC. All Rights Reserved. 14

better network performance at later stages of technologymaturation, when repeaters that operate on hundreds ofqubits are achievable. We argue that the designs consid-ered here would use essentially the same hardware tech-nology as one-way communication protocols, so that de-veloping repeaters with loss-heralding protocols is a pre-cursor to implementing one-way protocols. In this way,developing repeaters based on loss-heralded protocols is

prudent for near-term technology development.

ACKNOWLEDGMENTS

We thank Jim Harrington for suggesting improvementsto the manuscript.

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