Fermat's Library | q-2021-02-15-397 annotated/explained version.

Resource requirements for eﬃcient quantum communication

using all-photonic graph states generated from a few matter

qubits

Paul Hilaire, Edwin Barnes, and Sophia E. Economou

Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

Quantum communication technologies

show great promise for applications rang-

ing from the secure transmission of secret

messages to distributed quantum comput-

ing. Due to ﬁber losses, long-distance

quantum communication requires the use

of quantum repeaters, for which there ex-

ist quantum memory-based schemes and

all-photonic schemes. While all-photonic

approaches based on graph states gen-

erated from linear optics avoid coher-

ence time issues associated with memo-

ries, they outperform repeater-less pro-

tocols only at the expense of a pro-

hibitively large overhead in resources.

Here, we consider using matter qubits to

produce the photonic graph states and an-

alyze in detail the trade-oﬀ between re-

sources and performance, as character-

ized by the achievable secret key rate

per matter qubit. We show that fast

two-qubit entangling gates between mat-

ter qubits and high photon collection and

detection eﬃciencies are the main ingre-

dients needed for the all-photonic proto-

col to outperform both repeater-less and

memory-based schemes.

The ability to share entangled states over long

distances is a major milestone for the realiza-

tion of a fully-functional quantum internet [1, 2].

Beyond secure communications via quantum key

distribution (QKD) [3–5], the implementation of

such a quantum internet would also have various

applications ranging from distributed quantum

computing [6], secure access to a remote quan-

tum computer [7, 8], accurate clock synchroniza-

tion [9], and improved telescope observations [10].

However, enabling world-wide quantum com-

munication requires addressing the major prob-

Paul Hilaire: paulhilaire@vt.edu

lem of photonic losses, which signiﬁcantly reduces

the range of quantum information transfer. Even

though direct ampliﬁcation of a quantum state

is made impossible due to the no-cloning theo-

rem [11, 12], this exponential photon loss can still

be overcome through the realization of quantum

repeaters (QR) [13, 14].

Several QR approaches have been proposed

and can be divided into two main categories de-

pending on the method used to propagate quan-

tum information between two adjacent repeater

nodes [15]. The ﬁrst approach proposed, re-

ferred to here as a "memory-based" approach,

relies on heralded-entanglement generation [16–

24] between two quantum memories (QM) situ-

ated at adjacent repeater nodes. Examples of

memory-based schemes are depicted schemati-

cally in Figs. 1(a),(b). The heralded entangle-

ment generally succeeds when a detector situated

at an intermediate node measures photons emit-

ted by the QMs [25–27]. A classical signal car-

rying the outcome of an entanglement generation

attempt must be sent back to the QMs, thus lim-

iting the repetition rate of these protocols. When

successful, entanglement swapping transfers the

entanglement through these repeater nodes.

The second category of repeater [28–34] relies

on logically-encoded multi-photon states [32, 35],

resistant both to photonic losses and errors, to

transfer quantum information across a network.

One particularly promising approach of this type

was put forward in Ref. [31], which proposed an

all-optical QR protocol based on repeater graph

states (RGS). An example of an RGS with logical

encoding is shown in Fig. 1(c), while the corre-

sponding communication protocol is summarized

in Fig. 1(d) and will be detailed later. Ref. [31]

also proposed a method to construct these states

probabilistically using single-photon sources, lin-

ear optics, and detectors [36]. However, this

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 1

arXiv:2005.07198v4 [quant-ph] 11 Feb 2021

Alice Bob

QR node

Measurement

node

(a) 2-quantum-memory protocol

Classical channel

Photon

Entanglement

Swapping

HEG

Multiplexed QR

(b) Multiplexed QR

m=3

(d) RGS protocol

Measurement

node

RGS node

Matter qubit

Alice Bob

Matter qubit

Figure 1: (a) 2-quantum-memory QR protocol. (b) Multiplexed quantum repeater composed of 2N memories per

repeater with heralded entanglement generation (HEG) at measurement nodes. (c) Left side: Example of a repeater

graph state. Vertices represent qubits (blue: physical qubits, orange: logical qubits) while edges represent CZ gates.

Right side: Example of logical encoding using a depth-3 tree graph. (d) Quantum communication scheme using the

RGS protocol. Matter qubits are also included.

generation procedure requires about 10

single-

photon sources per repeater node to just barely

outperform direct ﬁber transmission [37]. These

ﬁndings suggest that the RGS protocol may be

well beyond the reach of current and near-future

technological capabilities.

However, it remains unclear whether the RGS

protocol can become more feasible if alternative

methods to create the states are used. Ref. [38]

proposed a deterministic generation procedure to

produce a linear graph state using one quantum

emitter which emits spin-entangled photons [39];

this procedure was later demonstrated experi-

mentally [40]. Ref. [41] showed theoretically

that emitters that undergo entangling gates can

also be used to generate two-dimensional cluster

states. Refs. [42, 43] built on this protocol and

demonstrated that entanglement between emit-

ters can be harnessed for the generation of more

complex photonic graph states. These ideas have

been extended further to general prescriptions

and to protocols tailored to speciﬁc physical sys-

tems [44–47]. Ref. [42] introduced a protocol for

producing an arbitrary-sized RGS using only a

few matter qubits, thus signiﬁcantly decreasing

the resource overhead required for RGS genera-

tion, making it deterministic in principle. Us-

ing this generation technique, the RGS protocol

only necessitates a few-qubit processor at each

repeater node, which would ease its practical im-

plementation compared to other error-correction-

based proposals that generally require several

hundreds of qubits per repeater node [29, 30]

(with the notable exception of Ref. [48] which

uses techniques introduced in Ref. [42] and re-

quires deterministic spin-photon Bell measure-

ments). Although the RGS protocol with de-

terministic state generation seems promising, a

systematic and detailed evaluation of its perfor-

mance and resource requirements has not been

carried out.

In this paper, we compare the resource-

eﬃciency—characterized by the achievable secret

key rate per matter qubit—of this protocol to di-

rect ﬁber transmission and to QR schemes based

on memories and heralded entanglement gener-

ation. We ﬁrst show that the rate per matter

qubit has a fundamental upper bound in the case

of memory-based QRs. We then review the RGS

protocol and how RGSs can be generated using a

few matter qubits. We evaluate the performance

of this scheme, show that its rate per matter qubit

does not have a theoretical upper-bound, and ﬁnd

the conditions under which it outperforms both

the repeater-less and the memory-based QR ap-

proaches. These conditions depend sensitively on

the speed with which two-qubit gates between the

matter qubits can be executed and on the col-

lection and detection eﬃciencies of the photons

emitted by these matter qubits.

1 Upper bound on rate for memory-

based repeater schemes

In this section, we show that there is a theoret-

ical upper bound, R

(QM)

max

, on the rate per mat-

ter qubit for protocols based on quantum mem-

ories and heralded entanglement generation. In

such protocols, the total distance L between Al-

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 2

ice and Bob is divided into smaller distances L

by N

= L/L

− 1 repeater nodes. Quantum

memories at adjacent repeater nodes are entan-

gled via a heralded entanglement procedure (see

Fig. 1(a)). When a repeater node shares entan-

glement connections with its two adjacent nodes,

entanglement swapping is performed on the two

memory qubits within that node to create a di-

rect entanglement connection between memories

on the adjacent nodes. This procedure can be

repeated until Alice and Bob share an entangled

qubit pair.

It is clear that creating an entangled pair

between Alice and Bob requires generating an

entanglement connection between each adjacent

pair of repeaters. This means that the overall pro-

tocol rate R

(QM)

is limited by the entanglement

generation rate hT

ent

−1

between two adjacent re-

peaters. Here, we use this fact to derive an upper

bound on the rate per matter qubit for QR pro-

tocols that use quantum memories and heralded

entanglement generation.

To determine an upper bound on hT

ent

−1

, we

focus for concreteness on the protocol presented

in Fig. 1(a), which uses two quantum memories

per node. A quantum memory should emit a pho-

ton that is maximally entangled with one of its

degrees of freedom. Two photons generated at

adjacent repeater nodes arrive at the same mea-

surement node situated halfway between the two

repeaters, where they are measured in a Bell state

basis. Because a photon Bell state measurement

using only linear optics succeeds with probabil-

ity at best 1/2 (without ancillary qubits or QND

measurements [49–54]), the overall success prob-

ability of the distant heralded entanglement gen-

eration is P

ent

≤ 1/2. It is worth mentioning

that a method for achieving heralded entangle-

ment generation with higher success probability

has been proposed [55], but its eﬃcacy is re-

stricted to qubits separated by a short distance,

so we exclude this from our analysis (see Supple-

mentary Materials for more details). A classical

signal must then inform the repeater nodes of the

success or failure of the Bell state measurement.

Because the distance from the measurement node

to the repeater nodes is L

/2, this means that

the overall heralded entanglement generation at-

tempt takes total time T

trial

≥ L

/c. This in-

cludes the time L

/(2c) for the single-photon

transfer from the repeater to the measurement

Notation Deﬁnition

L Total distance between Alice and Bob.

Distance between adjacent nodes.

Number of repeater nodes.

2m Number of arms of the RGS.

b Branching vector of an error-correction

tree (

b = (b

, b

, . . . , b

n−1

)).

CZ gate time.

(l) Transmission of a ﬁber of length l (η

(l) =

exp(−l/L

att

)).

, η

In-ﬁber collection and detection eﬃcien-

cies of photons.

att

, c Attenuation distance of the ﬁber and

speed of light in a ﬁber. (L

att

≈ 20km)

att

Average time of ﬂight of photons in the

ﬁber (t

att

= L

att

/c).

RGS

Generation time of an RGS.

R, R

Rate and rate per matter qubit of the pro-

tocol.

 Single-photon error rate.

Table 1: Table of notations

node and also the time L

/(2c) for the classical

signaling in the opposite direction. Here, we are

neglecting the time it takes to prepare and pump

the quantum memories. Throughout this proto-

col, a QM can be maximally entangled with at

most one other qubit (either a photon or another

QM). Therefore, it cannot emit another spin-

entangled photon before receiving the classical

signal carrying the information about the success

or failure of the Bell measurement, hence limiting

the repetition rate of the protocol. To generate

an entanglement connection between the two ad-

jacent nodes, the procedure must be repeated on

average P

−1

ent

times. The entanglement generation

rate is therefore hT

ent

−1

= P

ent

trial

< c/(2L

We derived this result for a speciﬁc heralded en-

tanglement generation protocol but it also holds

for all known protocols [25–27] (see Supplemen-

tary Materials).

From these results, we can show that the rate

per matter qubit (where the number of matter

qubits is N

= 2(N

+ 1) = 2L/L

) has a the-

oretical upper bound, R

(QM)

max

(QM)

≤

ent

−1

≤

= R

(QM)

max

. (1)

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 3

This theoretical upper bound also holds if there

are more than two QMs at each repeater node

(see Fig. 1(b)) as the rate would linearly in-

crease with the number of matter qubits. There-

fore, we have derived a general theoretical upper

bound for memory-based protocols based on her-

alded entanglement generation. It is worth not-

ing that the fundamental reason for this upper

bound comes from the need for classical signal-

ing in these protocols. Such classical signaling

is not required for RGS protocols, enabling them

to surpass this limit, as we show below. In the

Supplementary Materials, we also show that a

tighter bound, R

(QM)

max

= c/7L, can be obtained

for memory-based schemes in which there are two

quantum memories per repeater node, and her-

alded entanglement swapping is used.

We emphasize that the upper bound derived in

this section holds for QR protocols that are based

on quantum memories and distant heralded en-

tanglement generation. This corresponds to the

ﬁrst and second generations of QRs, as catego-

rized in Ref. [15]. Consequently, in the following,

the RGS-based protocol will be compared only

to these categories of QRs, for which the perfor-

mance is limited by classical signaling.

2 Rate of the RGS protocol with de-

terministic graph state generation

In this section, we review the RGS protocol as

introduced in Ref. [31] and the deterministic gen-

eration of RGSs using a few matter qubits as

proposed in Ref. [42]. We show how the rate of

the RGS protocol depends on various parameters

in the case where deterministic state generation

methods are used.

2.1 RGS protocol and rate

An RGS is a quantum state |Gi that can con-

veniently be represented in the form of a graph

G = (V, E) with V vertices and E edges. Each

vertex corresponds to a photonic qubit prepared

in the |+i state, and each edge corresponds to the

application of a CZ gate between the two qubits

it connects:

|Gi =

(i,j)∈E

|+i

⊗V

. (2)

An example of the graph representing an RGS

is shown in Fig. 1(c). These states include 2m

inner photonic qubits that are referred to as the

ﬁrst-leaf qubits. All the ﬁrst-leaf qubits are fully

connected to each other and each of them is also

connected to one additional qubit, referred to

as a second-leaf qubit. The ﬁrst-leaf qubits are

logically-encoded using tree graph states; further

details on this are given below.

In an RGS protocol (see Fig. 1(d)), the distance

L separating Alice and Bob is also divided into

smaller steps L

by N

= L/L

−1 source nodes

where the RGSs are created. The RGS is divided

into two equal parts, each containing m arms,

and one part is sent to the left adjacent measure-

ment node and the other to the right. Thus, half

of one RGS meets half of another RGS at each

measurement node, where each second-leaf qubit

from one of the half-RGSs undergoes a Bell mea-

surement with its counterpart from the other half-

RGS. Further details about this entanglement

swapping procedure at the measurement node are

given later, but it is important to note that the

RGS protocol does not use quantum memories at

all and thus cannot store the information. This

implies that an entanglement connection between

Alice and Bob should be realized in only one trial

with a probability P

much higher than direct

ﬁber transmission: P

 η

(L). The genera-

tion of an entanglement connection between Alice

and Bob requires the realization of successful en-

tanglement connections between all the adjacent

RGSs. It is important to note that the measure-

ments performed at each measurement node do

not require information from other measurement

nodes so that, in contrast to memory-based ap-

proaches, the RGS protocol does not require any

classical signaling while entanglement is being ex-

tended through the network. Classical signaling

is needed only once at the end of the protocol to

recover the Pauli frame of the Bell pair shared

by Alice and Bob, i.e. to determine which local

Pauli rotations Alice and Bob’s qubits should un-

dergo. This means that in the RGS protocol, it

is not necessary to wait for any classical signaling

before proceeding to generate the next batch of

RGSs needed to create the next Bell pair shared

between Alice and Bob (see Supplementary Ma-

terials for more details). Consequently, unlike the

memory-based scheme, the rate of the RGS pro-

tocol does not depend on the time it takes for the

photon to get from one node to the next. It is

limited only by the generation time T

RGS

of an

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 4

RGS:

(RGS)

RGS↔RGS

L/L

RGS

, (3)

with P

RGS↔RGS

the probability to generate

an entanglement connection between two RGSs.

The main notations used in this work are deﬁned

in Table 1.

A successful entanglement link can be gener-

ated if at least one of the Bell measurements at

a measurement node succeeds. In that case, the

two ﬁrst-leaf qubits attached to the second-leaf

qubits that underwent the successful Bell mea-

surement are measured in the X basis, while the

remaining 2m − 2 ﬁrst-leaf qubits are measured

in the Z basis. The X measurements transfer

the entanglement connection to the next two ad-

jacent measurement nodes, while the Z measure-

ments disentangle all the excess qubits associated

with failed Bell measurements. All these ﬁrst-leaf

qubit measurements must be successful in order

to reliably create an entanglement link. There-

fore, the probability to successfully create an en-

tanglement link between two RGSs is given by:

RGS↔RGS

= (1 − (1 − P

Bell

)

× Pr(M

X,`

)

Pr(M

Z,`

)

2m−2

(4)

where P

Bell

= P

/2 is the probability of a

successful Bell measurement. This depends on

= η

/2), which is the probability

that a single photon is emitted and collected

into the ﬁber (η

), is transmitted to the mea-

surement node (η

/2)) and detected (η

Pr(M

X,`

) and Pr(M

Z,`

) are the probabilities that

the logical X and Z measurements on the ﬁrst-

leaf qubits succeed. Note that if the ﬁrst-leaf

qubits were not logically encoded, we would have

Pr(M

)

Pr(M

)

2m−2

= P

≤ η

), and it

would be impossible to have an advantage over

direct ﬁber transmission. Therefore, the loss-

tolerance of logically-encoded qubits is crucial for

this protocol. Next, we review this encoding,

which was introduced in Refs. [35] and [31].

2.2 Loss-tolerance with tree graph states

We review how the probabilities Pr(M

X,`

) and

Pr(M

Z,`

) depend on the single-photon transfer

probability P

and on the shape of the tree graph

state used for the logical encoding. Ref. [35]

demonstrated that this encoding remains loss-

tolerant as long as P

is above 50%.

We consider the calculation of the probabili-

ties of successful measurements of the logically

encoded qubits in the presence of loss errors on

a tree graph state. A tree is characterized by

its branching vector

b = (b

, b

, ..., b

n−1

) (see

Fig. 1(c)), which describes the connectivity be-

tween the diﬀerent levels of the tree. To perform

a Z measurement M

Z,k

on a qubit at level k, it is

possible to either perform a direct measurement

on this qubit (with success probability P

) or, if

it fails (with probability 1 − P

), perform an in-

direct measurement (with probability r

). Thus,

the overall success probability of a Z measure-

ment at level k is:

Pr(M

Z,k

) = P

+ (1 − P

. (5)

To perform an indirect measurement on a qubit

(call it A) at level k, one can use the stabilizing

property of a graph state [56]. It is possible to

deduce the outcome of the Z measurement on

A by performing an X measurement on another

qubit (B) at level k + 1 and a Z measurement on

all the qubits, C

, that are in the neighborhood

of B at level k + 2 (see Fig. 2(a)). This works

because of the invariance of graph states when

they are acted upon by their stabilizers:

|Gi = X

⊗

j∈N(B)

|Gi

= X

⊗ Z

⊗

i∈{1,b

k+1

}

|Gi ,

(6)

so we have:

|Gi = X

⊗

i∈{1,b

k+1

}

|Gi . (7)

A single indirect measurement has a success

probability s

. Note, however, that the tree

structure allows b

indirect measurement at-

tempts, and only one needs to succeed to indi-

rectly measure a qubit at level k. So the proba-

bility that at least one indirect measurement suc-

ceeds is

= 1 − (1 − s

)

, (8)

with

= P

Pr(M

Z,k+2

)

k+1

. (9)

It is thus possible to derive the success probabil-

ity of a measurement recursively, given that the

qubits at the lowest level can only be measured

directly: Pr(M

Z,n

) = P

. Logical measurements

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 5

Level k

Level k+1

Level k+2

(a) (b)

Figure 2: (a) Indirect measurement of qubit A at the level k using a stabilizer based on qubit B at level k + 1. (b)

Protocol for the deterministic generation of an RGS with logical encoding using matter qubits.

in the X or Z basis are given by [31]:

Pr(M

X,`

) = r

Pr(M

Z,`

) = (P

+ (1 − P

)

= Pr(M

Z,1

)

(10)

It is interesting to note that logical encoding with

tree graph states can also correct single-qubit er-

rors, as shown in Ref. [31] and described in the

Supplementary Materials. This will be used later

when we evaluate the sensitivity of the RGS pro-

tocol to errors.

2.3 Generation of an RGS

The achievable rate between Alice and Bob also

depends on the repetition rate of the protocol,

which is given by the generation time of the RGS.

Because it is impossible to realize determinis-

tic two-qubit gates on photons with linear op-

tics, such a graph state can either be generated

probabilistically by the recursive fusion of smaller

graphs using linear optics and Bell state measure-

ments as shown in Ref. [31] and [37], or deter-

ministically using a few matter qubits as shown

in Ref. [42]. We now review the latter.

An arbitrary-sized RGS can be generated de-

terministically by following a given sequence

based on four operations on matter qubits: the

emission of a photon maximally entangled with

the matter qubit E

, the Hadamard gate H,

measurements in the Pauli bases M

, M

and the CZ gate. The generation of an RGS with

2m arms and a tree graph encoding with branch-

ing vector

b = (b

, b

, ..., b

n−1

) requires n+1 mat-

ter qubits Q

, ..., Q

n+1

, and is given by the se-

quence (see also Fig. 2(b))

Y,Q

X,Q

ph,Q

)

with G

= M

Z,Q

ph,Q

k−1

k+1

k−2

and G

n+2

= E

ph,Q

n+1

(11)

where, for simplicity, we have omitted the single

photonic qubit rotations.

The overall generation time of an RGS using

this procedure is therefore

RGS

= 2m



1 + f(

b, n − 1)



+ T

+ 2m



2 + f(

b, n − 2)



+ T

)

+ 2mf(

b, n − 2)T

(12)

with f (

b, k) =

i=0

j=0

and with T

, T

and T

the times for photon emission, mat-

ter qubit measurement, Hadamard and CZ gates,

respectively.

In the following, we make the realistic assump-

tion that the CZ gate time T

is much longer

than the durations of the other operations, and so

we set T

= T

= 0 for simplicity. With

this assumption, the generation time T

RGS

only

depends on the number of CZ gates and their

duration:

RGS

= 2m





2 +

n−2

k=0

j=0





. (13)

This is for an RGS with 2m arms and a log-

ical tree encoding with branching vector

b =

, ..., b

n−1

). In the following, we will assume

a depth-two tree graph state (

b = (b

, b

)), so

that the full RGS can be generated with only

three matter qubits (n = 2). The total num-

ber of matter qubits in the network is therefore

= (L/L

− 1)(n + 1) = 3(L/L

− 1).

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 6

3 Performance comparisons

We now evaluate the performance of the RGS

protocol when it is generated deterministically

by a few matter qubits. We compare our re-

sults to the direct ﬁber transmission limit derived

by Refs. [57, 58] and to the memory-based upper

bound R

(QM)

max

= c/4L found in Sec. 1.

3.1 Optimizing the RGS protocol

(a) (b)

(d)(c)

Figure 3: Maximum achievable rate per matter qubit

(RGS)

(here normalized by the parameter T

−1

)

and optimal node separation L

for the RGS protocol

with depth-2 logical tree encoding with the total dis-

tance ﬁxed at L = 50L

att

for a range of RGS parameters

, b

, m. (a) Three-dimensional plot showing optimal

rate and node separation as a function of b

, b

and

m. For each point in the plot, the value of L

is op-

timized to maximize R

(RGS)

. Point sizes represent

maximized R

(RGS)

values while point colors repre-

sent the optimal values of L

(indicated with the color

scale). The RGS parameters that achieve the largest

value of R

(RGS)

in this case are indicated with

dashed lines. The corresponding maximal R

(RGS)

max

is given in Table 2. (b,c,d) Three diﬀerent orthogonal

two-dimensional slices of the plot shown in panel (a).

In the following, we show how the RGS proto-

col parameters can be optimized to maximize the

overall rate per matter qubit, R

(RGS)

, or in the

presence of errors, the secret key rate per matter

qubit. For the moment, we assume perfect pho-

ton collection and detection eﬃciencies (η

= 1)

and no single-photon errors ( = 0); we take these

eﬀects into account later on. From Eqs. (3), (4),

and (13), the achievable rate per matter qubit

(RGS)

of the RGS protocol for a total distance

L is inversely proportional to the CZ gate time

, but it depends non-linearly on the separa-

L/L

att

m b

(RGS)

max

10 0.23 11 8 4 2.4 × 10

−5

25 0.21 13 10 5 6.8 × 10

−6

50 0.19 14 10 5 2.8 × 10

−6

100 0.17 15 10 5 1.1 × 10

−6

150 0.15 16 10 5 6.8 × 10

−7

Table 2: Optimal RGS parameters m, b

, b

and node

separation L

for several diﬀerent total network dis-

tances L. Here, L

att

is the attenuation length of optical

ﬁbers.

tion distance, L

, between two RGS nodes and

the RGS shape (number of arms 2m and the tree

branching vector

b = (b

, b

)). Therefore, for each

choice of the total distance L, there is a certain

node separation and RGS shape that maximize

the achievable rate.

The optimization of the rate per matter qubit

for a total distance L = 50L

att

(≈ 1000 km) is

shown in Fig. 3. The position of each point cor-

responds to a speciﬁc RGS shape, the color of the

point indicates the node separation L

that op-

timizes the rate for that shape, and the size of

the point represents the maximal rate per mat-

ter qubit for these parameters. This optimization

converges, allowing us to extract the optimal RGS

shape and distance L

for this particular choice of

the total distance L. The optimization can be re-

peated for various choices of L, and the extracted

optimal parameters are recorded in Table 2.

We compare the RGS protocol to direct ﬁber

transmission in Fig. 4(a). The maximum achiev-

able rate per matter qubit for the RGS proto-

col, R

(RGS)

max

, is shown as a function of the total

distance L/L

att

. In the case of direct transmis-

sion, we show seven diﬀerent curves correspond-

ing to the achievable rate for seven diﬀerent val-

ues of the single-photon source repetition rate.

We see that the RGS protocol outperforms direct

transmission with the highest repetition rate for

L & 30L

att

. In the same ﬁgure, we also show how

well the protocol works if we keep the optimized

parameters ﬁxed and change the total distance

L. To demonstrate this, we ﬁx the RGS param-

eters and node separation L

to the values that

optimize the rate for a total distance L

tar

. We

then adjust L away from L

tar

without changing

the RGS parameters or L

, and we calculate the

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 7

new rate for each value of L. In Fig. 4(a), we

show the resulting rates as a function of L/L

for ﬁve diﬀerent choices of L

tar

. We see that the

RGS protocol continues to work well over a broad

range of total distance L when we use parameters

that are optimized for a large total distance L

tar

For long distances L in the range 50L

att

−

200L

att

, R

(RGS)

max

scales approximately as L

−1.27

and thus the scaling with distance is slightly

worse than the upper bound scaling for memory-

based schemes, R

(QM)

max

∼ L

−1

, obtained in Sec. 1.

However, contrary to the memory-based schemes,

there are no fundamental upper limits imposed by

classical signaling on the maximal achievable rate

(RGS)

max

, as the latter is inversely proportional to

the CZ gate time, T

. If T

is suﬃciently

small, then the RGS protocol should surpass the

upper bound of memory-based schemes. This is

illustrated in Fig. 4(b), where the red region of

the plot indicates the regime where the RGS pro-

tocol outperforms memory-based schemes.

From our calculations, it seems that the max-

imum achievable rate per matter qubit, R

(RGS)

max

is in principle unbounded since it is inversely pro-

portional to the CZ gate time. We should recall

however that these results are based on the as-

sumption that the CZ gate takes much longer

than the other operations O made on the mat-

ter qubits: T

 T

. Decreasing T

ini-

tially increases the rate, but if T

becomes small

enough, neglecting the durations of other opera-

tions eventually becomes invalid. Consequently,

to increase the rate further, not only the CZ

gate time but all the operation times should be

reduced simultaneously. If the durations of all

these operations could be made arbitrarily small,

then the rate per matter qubit would increase

to inﬁnity. In practice however, the operation

times could also have an intrinsic lower bound

that limits the performance of the RGS proto-

col, but these lower bounds would depend on the

speciﬁc system on which the protocol is imple-

mented, while the limit imposed by classical sig-

naling is much more stringent and general.

For distances below 200L

att

≈ 4000 km, a gate

time T

below 6 × 10

−4

att

≈ 60 ns is suﬃ-

ciently short to outperform any memory-based

scheme. For this range of total distance, the

optimal value of the node separation L

ranges

from 0.15 − 0.19L

att

≈ 3 − 3.8 km. As an

illustration of the RGS performance, for L =

1000 km and T

= 10 ns, the total rate is

R = 220 kHz for N

= 786 matter qubits used,

leading to R

(RGS)

max

= 276 Hz per matter qubit,

while R

(QM)

max

= 50 Hz per matter qubit for this

distance.

3.2 Sensitivity to errors

So far, we have considered the generation of a

perfect RGS, which is a pure entangled state of

many photons. Generating such a perfect state

is not feasible experimentally, and so we need to

evaluate the sensitivity of the RGS protocol to

errors. To do so, we now consider single-photon

loss and single-photon errors in our optimization

process.

The single-photon loss (other than the ﬁber

losses) depends on the probability that an emit-

ted photon is neither collected into the ﬁber

nor detected; this corresponds to the case where

6= 1. Fig. 4(c) shows the optimized rate per

matter qubit for diﬀerent values of η

. Simi-

larly, Fig. 4(d) compares the performance of the

RGS protocol with the upper bound for memory-

based protocols, R

(QM)

max

, as a function of T

and

for a total distance L = 50L

att

. These re-

sults show that the photon losses must be below

15% (η

> 85%) in order for the RGS proto-

col to outperform both the direct ﬁber transmis-

sion and memory-based protocols. This rather

stringent requirement is mainly due to the loss-

tolerance of the tree graph state, which only

works when the total photon loss (including ﬁber

losses) is below 50%.

Apart from the photon losses, the RGS pro-

tocol also depends on other kinds of errors that

reduce the ﬁdelity F

of the ﬁnal entangled pho-

ton pair shared by Alice and Bob. These er-

rors include, but are not limited to, single-photon

measurement errors, photon Bell-measurement

errors, and depolarization errors. In the present

case where the RGS is generated using matter

qubits, the limited coherence time of the matter

qubits should also limit the ﬁdelity of the entan-

gled photons. For QKD applications, the ﬁnal

secret key rate depends on the total rate R and

the ﬁdelity F

[59, 60]:

skr

= R(1 − 2h(F

)), (14)

where h is the binary entropy function:

h(F ) = −F log

(F ) − (1 − F ) log

(1 − F ). (15)

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 8

max

(RGS)

(QM)

max

R-

max

(RGS)

(QM)

max

R+

(a)

(b)

(c)

(d)

=50

(f)

=50

max

(RGS)

)

max

(e)

Figure 4: (a) The maximum achievable secret key rate per matter qubit as a function of the total distance L for

the RGS protocol with depth-2 logical tree encoding compared to direct ﬁber transmission. The dashed red line

is the rate for the RGS protocol, with all RGS parameters and L

optimized for each value of L. The seven red

curves correspond to rates for direct ﬁber transmission [58] for seven diﬀerent choices of the single-photon source

repetition rate: 10

−1

(for i=0,...,6). The solid yellow, green, and blue curves show the performance of the RGS

protocol when L is changed while keeping all RGS parameters and L

ﬁxed to the optimal values obtained for total

distance L

tar

. (b) Relative diﬀerence between the maximum achievable rates of the RGS protocol, R

(RGS)

max

, and

memory-based protocols, R

(QM)

max

, as a function of gate time T

and total distance L. The red region corresponds

to R

(RGS)

max

> R

(QM)

max

. (c,d) Optimized rate of the RGS protocol for diﬀerent values of photon loss probability η

compared to (c) direct ﬁber transmission and (d) memory-based limit. (e,f) Similar to (c,d) but now showing results

for the single-photon error probability . In (c,e), the RGS rate is maximized for each value of L.

In the remainder of this paper, we now use

(RGS)

to refer to the secret key rate per matter

qubit, which coincides with the rate per matter

qubit in the absence of errors. We model these

imperfections by using a single-qubit error prob-

ability  that aﬀects all the photonic qubits. This

model works well for qubit measurement errors as

well as depolarization errors. In Ref. [38], it was

also shown that decoherence errors on quantum

emitters induce only local errors in the emitted

cluster state. The detailed derivation of F

given in the Supplementary Materials. Fig. 4(e)

compares the optimized secret key rate of the

RGS protocol against that of direct ﬁber trans-

mission as a function of the single-qubit error

probability and the total distance L. Fig. 4(f)

compares the performance of the RGS protocol to

the upper bound for memory-based protocols for

L = 50L

att

. We ﬁnd that the single-qubit error

should be below  < 10

−4

to outperform both the

direct ﬁber transmission and the memory-based

protocol upper bound, R

(QM)

max

One might expect a better tolerance to single-

qubit errors since they are also corrected by the

logical encoding of the 1

-leaf qubits. This is

not the case because the 2

-leaf qubits are not

logically encoded and thus are very sensitive to

errors. The dependence on single-qubit errors is

actually similar to the protocols using heralded

entanglement generation. It is also possible to

use entanglement puriﬁcation [61–63] to improve

the ﬁdelity F

and maximize the secret key rate.

From the maximum single-qubit error rate, we

can derive an order-of-magnitude estimate for

the coherence time requirements for the matter

qubits. To do so, we use two strong assumptions.

We ﬁrst assume that the ﬁdelity F

RGS

of an RGS

depends on the ratio between the generation time

RGS

of the RGS and the coherence time T

the matter qubits:

RGS

≈ e

−

RGS

(16)

The coeﬃcient 3 corresponds to the number of

matter qubits that degrade the RGS ﬁdelity.

Here, we have made the simplifying assumption

that the main source of inﬁdelity is the matter

qubit decoherence, and we have assumed that the

eﬀect of this decoherence is to reduce the overall

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 9

Matter

qubit

F T

Coupled

QDs

1ns [64]

[47, 65]

80%

[65]

3µs

[66, 67]

99.7%

[68]

defects

(diamond)

120ns

[69–71]

99.2%

[72]

0.6s

[73]

99%

[23]

Trapped

ions

10µs

[74]

99.9%

[74]

10min

[75]

89%

[76]

Table 3: State-of-the-art characteristics of candidate

quantum emitters. T

is the duration of a CZ gate,

is the coherence time, β is the single-mode photon

emission probability, and F the two-qubit gate ﬁdelity.

Blue (red) color means that the condition is (not) ful-

ﬁlled with this system, while orange means that it is

theoretically possible but has not been realized experi-

mentally. Note that all these results were not achieved

on the same system at the same time.

ﬁdelity of the graph state by e

−T

RGS

, which de-

cays exponentially with the time during which the

qubit is attached to the RGS (T

RGS

) divided by

its coherence time. This assumption may or may

not be justiﬁed depending on the system used

to implement the protocol, but we use it sim-

ply to get an order of magnitude estimate of the

typical coherence time that is required for this

protocol. We also assume that the errors are ho-

mogeneously distributed over the N

photons of

an RGS, such that F

RGS

= (1 − )

. This as-

sumption is consistent with Ref. [38], which shows

that errors generated by a quantum emitter and

imprinted onto the generated graph state are lo-

calized. With these assumptions, the coherence

time should satisfy T

≥ 2500T

4 Discussion

In the previous section, we derived a set of re-

quirements that need to be fulﬁlled in order for

the RGS protocol to have a clear advantage over

both memory-based QRs and repeater-less proto-

cols:

(i) T

≤ T

lim

(L) ≈ 60ns,

(ii) T

≥ 2500T

(iii) η

> 85%.

It is important to note that these bounds were

obtained using RGSs with a depth-2 logical tree

encoding. We may expect an increase in the rate

per matter qubit if deeper trees were used but

the improvement would certainly not be large as

RGS

grows quickly with tree depth. Checking

this directly is challenging because the process

of optimizing the RGS protocol becomes signif-

icantly more complex. We leave this intensive

numerical analysis to future work.

Table 3 shows the current state-of-the-art for

three candidate systems that can be used to gen-

erate RGSs. So far, the characteristics of the cur-

rent systems are not good enough yet to exceed

the memory-based resource-eﬃcient upper bound

(QM)

max

. The main limitation comes from long

two-qubit gate times, since fast and high-ﬁdelity

gates are critical for the performance of the RGS

protocol. Currently, most of the two-qubit gates

that have been experimentally demonstrated are

slow because they rely on local interactions, such

as the hyperﬁne coupling, which are generally

weak. Eﬃcient light-matter interfaces using cav-

ities can also be used to realize deterministic,

photon-mediated CZ gates between two matter

qubits [70], using spin-photon gates [77–79]. The

crucial advantage of this technique is that it only

requires optical interactions, which are intrin-

sically much faster than hyperﬁne interactions.

Progress in the experimental realization of strong

spin-dependent phase shifts has been made re-

cently [23, 80–82]. The second condition (ii) will

also be easier to achieve if the CZ gate time is

reduced.

The third condition (iii) requires the eﬃcient

collection of single photons emitted by a single

quantum emitter. The collection of single pho-

tons needs to be facilitated by increasing the

emission of photons into a single optical mode.

This is generally realized using waveguides or

cavities with single-mode emission probabilities

β that can reach 99.7% [68]. This single mode

should then be eﬃciently coupled to a single-

mode ﬁber. Finally, superconducting photon de-

tectors can be used to achieve high detection ef-

ﬁciencies [83].

In conclusion, we have identiﬁed the main re-

quirements that must be met in order for all-

photonic repeater protocols based on determin-

istically generated graph states to outperform

memory-based protocols. We did this by ﬁnd-

ing the graph state structures that maximize the

secret key rate and by comparing this to an up-

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 10

per bound on the rate for memory-based schemes

that we derived. We found that two-qubit gate

times and the eﬃcient collection and detection

of emitted photons are the most important fac-

tors in determining whether the all-photonic ap-

proach is superior to memory-based ones. An in-

teresting future direction would be to improve the

protocol for the emitter-based generation of pho-

tonic graph states by incorporating atom-photon

CZ gates [44, 79, 84], which would both sim-

plify the procedure and decrease gate times, and

thus enhance the performance of deterministic,

all-photonic repeaters.

Acknowledgements

We thank Yuan Zhan and Shuo Sun for their

contribution in solving issues in the code. This

research was supported by the NSF (Grant No.

1741656) and by the EU Horizon 2020 programme

(GA 862035 QLUSTER).

Methods

The numerical model used for the results in this

article is available here:

https://github.com/Paulhilaire/

performance_repeater_graph_state.

References

[1] H. J. Kimble. The quantum internet.

Nature, 453(7198):1023–1030, 2008. DOI:

10.1038/nature07127.

[2] Stephanie Wehner, David Elkouss, and

Ronald Hanson. Quantum internet: A

vision for the road ahead. Science, 362

(6412):eaam9288, 2018. DOI: 10.1126/sci-

ence.aam9288.

[3] Charles H Bennett and Gilles Brassard.

An update on quantum cryptography. In

Workshop on the Theory and Application

of Cryptographic Techniques, pages 475–480.

Springer, 1984. DOI: 10.1007/3-540-39568-

7_39.

[4] Charles H. Bennett, François Bessette, Gilles

Brassard, Louis Salvail, and John Smolin.

Experimental quantum cryptography. Jour-

nal of Cryptology, 5:3–28, 1992. DOI:

10.1007/BF00191318.

[5] Thomas. Jennewein, C. Simon, Gregor

Weihs, Harald Weinfurter, and Anton

Zeilinger. Quantum cryprography using

entangled photons. Physical Review Let-

ters, 84:4729, 2000. DOI: 10.1103/Phys-

RevLett.84.4729.

[6] Anne Broadbent, Joseph Fitzsimons, and El-

ham Kasheﬁ. Universal blind quantum com-

putation. In 2009 50th Annual IEEE Sym-

posium on Foundations of Computer Sci-

ence, pages 517–526. IEEE, 2009. DOI:

10.1109/FOCS.2009.36.

[7] Lov K Grover. Quantum telecomputation.

arXiv preprint quant-ph/9704012, 1997.

[8] Naomi H Nickerson, Joseph F Fitzsimons,

and Simon C Benjamin. Freely scalable

quantum technologies using cells of 5-to-50

qubits with very lossy and noisy photonic

links. Physical Review X, 4(4):041041, 2014.

DOI: 10.1103/PhysRevX.4.041041.

[9] Peter Komar, Eric M Kessler, Michael

Bishof, Liang Jiang, Anders S Sørensen, Jun

Ye, and Mikhail D Lukin. A quantum net-

work of clocks. Nature Physics, 10(8):582,

2014. DOI: 10.1038/nphys3000.

[10] Daniel Gottesman, Thomas Jennewein, and

Sarah Croke. Longer-baseline telescopes

using quantum repeaters. Physical Re-

view Letters, 109(7):070503, 2012. DOI:

10.1103/PhysRevLett.109.070503.

[11] W. K. Wootters and W. H. Zurek. A single

quantum cannot be cloned. Nature, 299:802–

803, 1982. DOI: 10.1038/299802a0.

[12] D. Dieks. Communication by EPR devices.

Physics Letters A, 92:271–272, 1982. DOI:

10.1016/0375-9601(82)90084-6.

[13] H-J Briegel, Wolfgang Dür, Juan I Cirac,

and Peter Zoller. Quantum repeaters: the

role of imperfect local operations in quan-

tum communication. Physical Review Let-

ters, 81(26):5932, 1998. DOI: 10.1103/Phys-

RevLett.81.5932.

[14] W Dür, H-J Briegel, JI Cirac, and P Zoller.

Quantum repeaters based on entanglement

puriﬁcation. Physical Review A, 59(1):169,

1999. DOI: 10.1103/PhysRevA.59.169.

[15] Sreraman Muralidharan, Linshu Li,

Jungsang Kim, Norbert Lütkenhaus,

Mikhail D Lukin, and Liang Jiang. Optimal

architectures for long distance quantum

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 11

communication. Scientiﬁc reports, 6:20463,

2016. DOI: 10.1038/srep20463.

[16] L. M. Duan, M. D. Lukin, J. I. Cirac,

and P. Zoller. Long-distance quantum

communication with atomic ensembles and

linear optics. Nature, 414(6862):413–418,

November 2001. ISSN 0028-0836. DOI:

10.1038/35106500.

[17] Lilian Childress, JM Taylor, Anders Sønd-

berg Sørensen, and MD Lukin. Fault-

tolerant quantum communication based on

solid-state photon emitters. Physical Re-

view Letters, 96(7):070504, 2006. DOI:

10.1103/PhysRevLett.96.070504.

[18] Lorenz Hartmann, Barbara Kraus, H-J

Briegel, and W Dür. Role of memory er-

rors in quantum repeaters. Physical Review

A, 75(3):032310, 2007. DOI: 10.1103/Phys-

RevA.75.032310.

[19] OA Collins, SD Jenkins, A Kuzmich,

and TAB Kennedy. Multiplexed memory-

insensitive quantum repeaters. Physical

review letters, 98(6):060502, 2007. DOI:

10.1103/PhysRevLett.98.060502.

[20] Nicolas Sangouard, Christoph Simon,

Hugues De Riedmatten, and Nicolas Gisin.

Quantum repeaters based on atomic en-

sembles and linear optics. Reviews of

Modern Physics, 83(1):33, 2011. DOI:

10.1103/RevModPhys.83.33.

[21] Scott E Vinay and Pieter Kok. Practi-

cal repeaters for ultralong-distance quan-

tum communication. Physical Review A,

95(5):052336, 2017. DOI: 10.1103/Phys-

RevA.95.052336.

[22] Filip Rozpędek, Raja Yehia, Kenneth

Goodenough, Maximilian Ruf, Peter C

Humphreys, Ronald Hanson, Stephanie

Wehner, and David Elkouss. Near-

term quantum-repeater experiments with

nitrogen-vacancy centers: Overcoming the

limitations of direct transmission. Physi-

cal Review A, 99(5):052330, 2019. DOI:

10.1103/PhysRevA.99.052330.

[23] Mihir K Bhaskar, Ralf Riedinger,

Bartholomeus Machielse, David S Lev-

onian, Christian T Nguyen, Erik N Knall,

Hongkun Park, Dirk Englund, Marko

Lončar, Denis D Sukachev, et al. Experi-

mental demonstration of memory-enhanced

quantum communication. Nature, 580

(7801):60–64, 2020. DOI: 10.1038/s41586-

020-2103-5.

[24] Sumeet Khatri, Corey T Matyas, Aliza U

Siddiqui, and Jonathan P Dowling. Prac-

tical ﬁgures of merit and thresholds for

entanglement distribution in quantum net-

works. Physical Review Research, 1(2):

023032, 2019. DOI: 10.1103/PhysRevRe-

search.1.023032.

[25] C Cabrillo, JI Cirac, P Garcia-Fernandez,

and P Zoller. Creation of entangled states

of distant atoms by interference. Phys-

ical Review A, 59(2):1025, 1999. DOI:

10.1103/PhysRevA.59.1025.

[26] Sean D Barrett and Pieter Kok. Eﬃ-

cient high-ﬁdelity quantum computation us-

ing matter qubits and linear optics. Phys-

ical Review A, 71(6):060310, 2005. DOI:

10.1103/PhysRevA.71.060310.

[27] L.-M. Duan and H. J. Kimble. Scalable

photonic quantum computation through

cavity-assisted interactions. Physical Re-

view Letters, 92(12):127902, 2004. DOI:

10.1103/PhysRevLett.92.127902.

[28] Liang Jiang, Jacob M Taylor, Kae Nemoto,

William J Munro, Rodney Van Meter, and

Mikhail D Lukin. Quantum repeater with

encoding. Physical Review A, 79(3):032325,

2009. DOI: 10.1103/PhysRevA.79.032325.

[29] William J Munro, Ashley M Stephens, Si-

mon J Devitt, Keith A Harrison, and Kae

Nemoto. Quantum communication with-

out the necessity of quantum memories.

Nature Photonics, 6(11):777, 2012. DOI:

10.1038/nphoton.2012.243.

[30] Sreraman Muralidharan, Jungsang Kim,

Norbert Lütkenhaus, Mikhail D Lukin,

and Liang Jiang. Ultrafast and fault-

tolerant quantum communication across

long distances. Physical review letters, 112

(25):250501, 2014. DOI: 10.1103/Phys-

RevLett.112.250501.

[31] Koji Azuma, Kiyoshi Tamaki, and Hoi-

Kwong Lo. All-photonic quantum repeaters.

Nature communications, 6:6787, 2015. DOI:

10.1038/ncomms7787.

[32] Fabian Ewert, Marcel Bergmann, and Pe-

ter van Loock. Ultrafast long-distance

quantum communication with static lin-

ear optics. Physical review letters, 117

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 12

(21):210501, 2016. DOI: 10.1103/Phys-

RevLett.117.210501.

[33] Yasushi Hasegawa, Rikizo Ikuta, Nobuyuki

Matsuda, Kiyoshi Tamaki, Hoi-Kwong Lo,

Takashi Yamamoto, Koji Azuma, and

Nobuyuki Imoto. Experimental time-

reversed adaptive bell measurement towards

all-photonic quantum repeaters. Nature

communications, 10(1):378, 2019. DOI:

10.1038/s41467-018-08099-5.

[34] Zheng-Da Li, Rui Zhang, Xu-Fei Yin, Li-

Zheng Liu, Yi Hu, Yu-Qiang Fang, Yue-Yang

Fei, Xiao Jiang, Jun Zhang, Li Li, et al. Ex-

perimental quantum repeater without quan-

tum memory. Nature Photonics, page 1,

2019. DOI: 10.1038/s41566-019-0468-5.

[35] Michael Varnava, Daniel E Browne, and

Terry Rudolph. Loss tolerance in one-

way quantum computation via counterfac-

tual error correction. Physical review letters,

97(12):120501, 2006. DOI: 10.1103/Phys-

RevLett.97.120501.

[36] Daniel E Browne and Terry Rudolph.

Resource-eﬃcient linear optical quantum

computation. Physical Review Letters, 95

(1):010501, 2005. DOI: 10.1103/Phys-

RevLett.95.010501.

[37] Mihir Pant, Hari Krovi, Dirk Englund, and

Saikat Guha. Rate-distance tradeoﬀ and

resource costs for all-optical quantum re-

peaters. Physical Review A, 95(1):012304,

2017. DOI: 10.1103/PhysRevA.95.012304.

[38] Netanel H Lindner and Terry Rudolph. Pro-

posal for pulsed on-demand sources of pho-

tonic cluster state strings. Physical Re-

view Letters, 103(11):113602, 2009. DOI:

10.1103/PhysRevLett.103.113602.

[39] C Saavedra, KM Gheri, P Törmä, JI Cirac,

and P Zoller. Controlled source of entan-

gled photonic qubits. Physical Review A,

61(6):062311, 2000. DOI: 10.1103/Phys-

RevA.61.062311.

[40] Ido Schwartz, Dan Cogan, Emma R

Schmidgall, Yaroslav Don, Liron Gantz,

Oded Kenneth, Netanel H Lindner, and

David Gershoni. Deterministic generation of

a cluster state of entangled photons. Sci-

ence, 354:434–437, 2016. DOI: 10.1126/sci-

ence.aah4758.

[41] Sophia E Economou, Netanel Lindner, and

Terry Rudolph. Optically generated 2-

dimensional photonic cluster state from cou-

pled quantum dots. Physical review letters,

105(9):093601, 2010. DOI: 10.1103/Phys-

RevLett.105.093601.

[42] Donovan Buterakos, Edwin Barnes, and

Sophia E Economou. Deterministic gen-

eration of all-photonic quantum repeaters

from solid-state emitters. Physical Review

X, 7(4):041023, 2017. DOI: 10.1103/Phys-

RevX.7.041023.

[43] Antonio Russo, Edwin Barnes, and Sophia E

Economou. Generation of arbitrary all-

photonic graph states from quantum emit-

ters. New Journal of Physics, 21(5):055002,

2019. DOI: 10.1088/1367-2630/ab193d.

[44] Hannes Pichler, Soonwon Choi, Peter Zoller,

and Mikhail D Lukin. Universal photonic

quantum computation via time-delayed feed-

back. Proceedings of the National Academy

of Sciences, 114(43):11362–11367, 2017.

DOI: 10.1073/pnas.1711003114.

[45] Antonio Russo, Edwin Barnes, and Sophia E

Economou. Photonic graph state genera-

tion from quantum dots and color centers for

quantum communications. Physical Review

B, 98(8):085303, 2018. DOI: 10.1103/Phys-

RevB.98.085303.

[46] Ming Lai Chan. Optimized protocol

to create repeater graph states for all-

photonic quantum repeater. arXiv preprint

arXiv:1811.10214, 2018.

[47] Mercedes Gimeno-Segovia, Terry Rudolph,

and Sophia E Economou. Determinis-

tic generation of large-scale entangled pho-

tonic cluster state from interacting solid

state emitters. Physical review letters,

123(7):070501, 2019. DOI: 10.1103/Phys-

RevLett.123.070501.

[48] Johannes Borregaard, Hannes Pichler, Tim

Schröder, Mikhail D Lukin, Peter Lodahl,

and Anders S Sørensen. One-way quan-

tum repeater based on near-deterministic

photon-emitter interfaces. Physical Review

X, 10(2):021071, 2020. DOI: 10.1103/Phys-

RevX.10.021071.

[49] S Lloyd, MS Shahriar, JH Shapiro, and

PR Hemmer. Long distance, uncondi-

tional teleportation of atomic states via com-

plete bell state measurements. Physical Re-

view Letters, 87(16):167903, 2001. DOI:

10.1103/PhysRevLett.87.167903.

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 13

[50] Yoon-Ho Kim, Sergei P Kulik, and Yan-

hua Shih. Quantum teleportation of a

polarization state with a complete bell

state measurement. Physical Review Let-

ters, 86(7):1370, 2001. DOI: 10.1103/Phys-

RevLett.86.1370.

[51] Yoon-Ho Kim, SERGEI KULIK, and Yan-

hua Shih. Quantum teleportation with a

complete bell state measurement. Journal of

Modern Optics, 49(1-2):221–236, 2002. DOI:

10.1080/09500340110087633.

[52] Warren P Grice. Arbitrarily complete bell-

state measurement using only linear optical

elements. Physical Review A, 84(4):042331,

2011. DOI: 10.1103/PhysRevA.84.042331.

[53] Fabian Ewert and Peter van Loock. 3/4-

eﬃcient bell measurement with passive lin-

ear optics and unentangled ancillae. Physical

review letters, 113(14):140403, 2014. DOI:

10.1103/PhysRevLett.113.140403.

[54] Stephen Wein, Khabat Heshami, Christo-

pher A Fuchs, Hari Krovi, Zachary Dut-

ton, Wolfgang Tittel, and Christoph Si-

mon. Eﬃciency of an enhanced linear opti-

cal bell-state measurement scheme with re-

alistic imperfections. Physical Review A,

94(3):032332, 2016. DOI: 10.1103/Phys-

RevA.94.032332.

[55] Leigh S Martin and K Birgitta Whaley.

Single-shot deterministic entanglement be-

tween non-interacting systems with linear

optics. arXiv preprint arXiv:1912.00067,

2019.

[56] Marc Hein, Jens Eisert, and Hans J Briegel.

Multiparty entanglement in graph states.

Physical Review A, 69(6):062311, 2004. DOI:

10.1103/PhysRevA.69.062311.

[57] Masahiro Takeoka, Saikat Guha, and

Mark M Wilde. Fundamental rate-loss trade-

oﬀ for optical quantum key distribution. Na-

ture communications, 5(1):1–7, 2014. DOI:

10.1038/ncomms6235.

[58] Stefano Pirandola, Riccardo Laurenza, Carlo

Ottaviani, and Leonardo Banchi. Funda-

mental limits of repeaterless quantum com-

munications. Nature communications, 8(1):

1–15, 2017. DOI: 10.1038/ncomms15043.

[59] Peter W Shor and John Preskill. Simple

proof of security of the bb84 quantum key

distribution protocol. Physical review let-

ters, 85(2):441, 2000. DOI: 10.1103/Phys-

RevLett.85.441.

[60] Valerio Scarani, Helle Bechmann-

Pasquinucci, Nicolas J Cerf, Miloslav

Dušek, Norbert Lütkenhaus, and Momtchil

Peev. The security of practical quantum key

distribution. Reviews of modern physics,

81(3):1301, 2009. DOI: 10.1103/RevMod-

Phys.81.1301.

[61] Jian-Wei Pan, Christoph Simon, Časlav

Brukner, and Anton Zeilinger. Entangle-

ment puriﬁcation for quantum communica-

tion. Nature, 410(6832):1067, 2001. DOI:

10.1038/35074041.

[62] M Zwerger, W Dür, and HJ Briegel.

Measurement-based quantum repeaters.

Physical Review A, 85(6):062326, 2012.

DOI: 10.1103/PhysRevA.85.062326.

[63] M Zwerger, HJ Briegel, and W Dür.

Measurement-based quantum communica-

tion. Applied Physics B, 122(3):50, 2016.

DOI: 10.1007/s00340-015-6285-8.

[64] Alex Greilich, Samuel G. Carter, Danny

Kim, Allan S. Bracker, and Daniel Gammon.

Optical control of one and two hole spins in

interacting quantum dots. Nature Photon-

ics, 5(11):702–708, November 2011. DOI:

10.1038/nphoton.2011.237.

[65] Danny Kim, Samuel G Carter, Alex Greilich,

Allan S Bracker, and Daniel Gammon.

Ultrafast optical control of entanglement

between two quantum-dot spins. Na-

ture Physics, 7(3):223–229, 2011. DOI:

10.1038/nphys1863.

[66] A. Greilich, A. Shabaev, D. R. Yakovlev,

Al. L. Efros, I. A. Yugova, D. Reuter, A. D.

Wieck, and M. Bayer. Nuclei-induced fre-

quency focusing of electron spin coherence.

Science, 317(5846):1896–1899, 2007. DOI:

10.1126/science.1146850.

[67] Xiaoya Judy Wang, Stefano Chesi, and

W. A. Coish. Spin-echo dynamics of a

heavy hole in a quantum dot. Phys. Rev.

Lett., 109:237601, 2012. DOI: 10.1103/Phys-

RevLett.109.237601.

[68] Daniel Najer, Immo Söllner, Pavel Sekatski,

Vincent Dolique, Matthias C Löbl, Daniel

Riedel, Rüdiger Schott, Sebastian Starosi-

elec, Sascha R Valentin, Andreas D Wieck,

et al. A gated quantum dot strongly coupled

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 14

to an optical microcavity. Nature, pages 1–1,

2019. DOI: 10.1038/s41586-019-1709-y.

[69] GD Fuchs, Guido Burkard, PV Klimov, and

DD Awschalom. A quantum memory intrin-

sic to single nitrogen–vacancy centres in dia-

mond. Nature Physics, 7(10):789–793, 2011.

DOI: 10.1038/nphys2026.

[70] Dmitry Solenov, Sophia E Economou, and

Thomas L Reinecke. Two-qubit quantum

gates for defect qubits in diamond and

similar systems. Physical Review B, 88

(16):161403, 2013. DOI: 10.1103/Phys-

RevB.88.161403.

[71] ML Goldman, TL Patti, D Levonian,

SF Yelin, and MD Lukin. Optical control of a

single nuclear spin in the solid state. Physical

Review Letters, 124(15):153203, 2020. DOI:

10.1103/PhysRevLett.124.153203.

[72] Xing Rong, Jianpei Geng, Fazhan Shi, Ying

Liu, Kebiao Xu, Wenchao Ma, Fei Kong,

Zhen Jiang, Yang Wu, and Jiangfeng Du.

Experimental fault-tolerant universal quan-

tum gates with solid-state spins under am-

bient conditions. Nature communications,

6(1):1–7, 2015. DOI: 10.1038/ncomms9748

(2015).

[73] Nir Bar-Gill, Linh M Pham, Andrejs

Jarmola, Dmitry Budker, and Ronald L

Walsworth. Solid-state electronic spin co-

herence time approaching one second. Na-

ture communications, 4:1743, 2013. DOI:

10.1038/ncomms2771.

[74] CJ Ballance, TP Harty, NM Linke, MA Se-

piol, and DM Lucas. High-ﬁdelity quan-

tum logic gates using trapped-ion hyper-

ﬁne qubits. Physical review letters, 117

(6):060504, 2016. DOI: 10.1103/Phys-

RevLett.117.060504.

[75] Ye Wang, Mark Um, Junhua Zhang, Shuom-

ing An, Ming Lyu, Jing-Ning Zhang, L-

M Duan, Dahyun Yum, and Kihwan Kim.

Single-qubit quantum memory exceeding

ten-minute coherence time. Nature Photon-

ics, 11(10):646, 2017. DOI: 10.1038/s41566-

017-0007-1.

[76] Severin Daiss, Stephan Welte, Bastian

Hacker, Lin Li, and Gerhard Rempe. Single-

photon distillation via a photonic parity

measurement using cavity qed. Physical re-

view letters, 122(13):133603, 2019. DOI:

10.1103/PhysRevLett.122.133603.

[77] C. Y. Hu, A. Young, J. L. O’Brien, W. J.

Munro, and J. G. Rarity. Giant optical Fara-

day rotation induced by a single-electron

spin in a quantum dot: Applications to

entangling remote spins via a single pho-

ton. Physical Review B, 78(8):085307, Aug

2008. DOI: 10.1103/PhysRevB.78.085307.

URL http://prb.aps.org/abstract/PRB/

v78/i8/e085307.

[78] Kazuki Koshino, Satoshi Ishizaka, and Ya-

sunobu Nakamura. Deterministic photon-

photon swap gate using a λ system. Phys-

ical Review A, 82(1):010301, 2010. DOI:

10.1103/PhysRevA.82.010301.

[79] Serge Rosenblum, Scott Parkins, and Barak

Dayan. Photon routing in cavity QED: Be-

yond the fundamental limit of photon block-

ade. Physical Review A, 84:033854, 2011.

DOI: 10.1103/PhysRevA.84.033854.

[80] Shuo Sun, Hyochul Kim, Glenn S Solomon,

and Edo Waks. A quantum phase switch

between a single solid-state spin and a pho-

ton. Nature Nanotechnology, 11(6):539–544,

2016. DOI: 10.1038/nnano.2015.334.

[81] LM Wells, Sokratis Kalliakos, Bruno Villa,

DJP Ellis, RM Stevenson, AJ Bennett, Ian

Farrer, DA Ritchie, and AJ Shields. Pho-

ton phase shift at the few-photon level and

optical switching by a quantum dot in a mi-

crocavity. Physical Review Applied, 11(6):

061001, 2019. DOI: 10.1103/PhysRevAp-

plied.11.061001.

[82] Petros Androvitsaneas, Andrew Young,

Joseph Lennon, Christian Schneider, Sebas-

tian Maier, Janna Hinchliﬀ, George Atkin-

son, Edmund Harbord, Martin Kamp, Sven

Höﬂing, et al. An eﬃcient quantum photonic

phase shift in a low q-factor regime. ACS

Photonics, 2019. DOI: 10.1021/acsphoton-

ics.8b01380.

[83] Robert H Hadﬁeld. Single-photon detec-

tors for optical quantum information appli-

cations. Nature photonics, 3(12):696, 2009.

DOI: 10.1038/nphoton.2009.230.

[84] Yuan Zhan and Shuo Sun. Deterministic

generation of loss-tolerant photonic cluster

states with a single quantum emitter. Phys-

ical Review Letters, 125(22):223601, 2020.

DOI: 10.1103/PhysRevLett.125.223601.

[85] C. Y. Hu, A. Young, J. L. O’Brien, W. J.

Munro, and J. G. Rarity. Giant optical Fara-

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 15

day rotation induced by a single-electron

spin in a quantum dot: Applications to en-

tangling remote spins via a single photon.

Physical Review B, 78(8):085307, 2008. DOI:

10.1103/PhysRevB.78.085307.

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 16

Supplementary Materials: Resource requirements for eﬃcient quantum com-

munication using all-photonic graph states generated from a few matter qubits

Bounds for memory-based repeater protocols

As discussed in the main text, the rate of a memory-based quantum repeater based on heralded

entanglement generation is limited by the rate hT

−1

at which a given memory qubit can be re-

used. hT

i is given by the sum of the average time required for generating entanglement hT

ent

i and

the time during which the entanglement is stored hT

store

i. hT

ent

i is given by T

trial

ent

, where T

trial

the time required for a heralded entanglement generation attempt, and P

ent

is the probability that an

attempt is successful. Let’s consider three heralded entanglement procedures [25–27]:

• In Ref. [26], T

trial

= L

/c, and the probability is P

ent

= e

−L

att

/2 (assuming perfect photon

emission and detection probability).

• In Ref. [27], T

trial

= 2L

/c, and the probability is P

ent

= e

−L

att

• In Ref. [25], T

trial

= L

/c, and the probability is P

ent

∝ e

−L

/(2L

att

)

. However, the protocol needs

to be implemented in a regime where the probability to emit a photon is very small to avoid

two-photon emission, which degrades the entanglement ﬁdelity. Consequently, P

ent

 1/2.

Thus in all these cases, if we consider small values of L

 L

att

, we have:

ent

i ≥

att

≥

. (17)

This limitation can be understood from the fact that the photonic qubit and the classical signal need

to travel back and forth either to a measurement node (twice the distance L

/2) or to the next QR

(twice the distance L

). If the photon travels to a quantum repeater, it is possible to deterministically

entangle the QM and the photon, so the entanglement probability is only bounded by the photonic

losses. If the photon travels to an intermediate node, the entanglement generation procedure uses

linear optics and has a success probability that cannot reach more than 1/2 (unless ancillary photonic

qubits are used, but this requires another quantum emitter to produce them [52–54]).

From this limitation, we ﬁnd the upper bound R

(QM)

max

≥ max(R/N

) in the case of QR protocols

with two QMs. In this case, the rate R is limited by hT

−1

≤ hT

ent

−1

= c/(2L

). The number

of matter qubits is given by twice the number of repeaters plus Alice and Bob’s matter qubits, thus

= 2(N

+ 1) = 2L/L

, therefore:

(QM)

max

= c/4L. (18)

In the case of multiplexed repeaters, by multiplying the number of memories per repeater by N, the

entangling rate between two repeaters is bounded by Nc/(2L

), but the number of matter qubits is

now N

= 2NL/L

, so that the total rate is still bounded by R

(QM)

max

Ref. [55] also proposes a method to generate heralded entanglement deterministically by detecting

photons. The idea is to excite the two quantum emitters (each consisting of two ground states and one

excited state that couples to only one of the ground states) such that they both emit a photon whose

which-path information is erased using linear optics. When the ﬁrst photon is detected a signal is sent

to each quantum emitter to perform a rotation between the ground states. This rotation needs to be

performed before the second photon is emitted. Therefore, the latency between the photon detection

and the rotation should be much smaller than the lifetime of the quantum emitter’s excited state. A

direct implication of this is that the photon detectors should be placed very close to the quantum

emitters, which is impractical. For example, for quantum emitters with a lifetime under 10 ns (which

is typical for QDs and defects in diamond), the position of the detectors should be much smaller than

1 m. In addition, to obtain high ﬁdelities with this scheme, the two quantum emitters are excited for

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 17

a typical time T

which should be much larger than the classical signaling time L

/c, and thus the

previous analysis should still hold in that case.

In the case of memory-based schemes using heralded entanglement and 2 QMs per QR, it is possible

to ﬁnd a stricter upper bound. To derive this bound, consider two repeater nodes labeled 1 and 2.

Node 1 contains two memory qubits QM1 and QM1

, while node 2 contains QM2 and QM2

. Suppose

that we succeed in entangling QM1 and QM2 via a Bell measurement at an intermediate measurement

node. How long do we have to maintain this entanglement before further entanglement swappings at

neighboring nodes free up QM1 and QM2, allowing them to be re-used for creating the next Bell pair

to be shared between Alice and Bob? The answer to this question is what we called hT

store

i above,

the average amount of time we have to store the entanglement. The upper bound on the rate is then

−1

, where hT

i = hT

ent

i + hT

store

i. We obtained a bound for hT

ent

i above and used this to

derive the bound on the rate given in Eq. (18). There, we neglected the role of hT

store

i. Here, we

reinstate this quantity to obtain a stricter bound.

To determine hT

store

i, we ﬁrst note that the entanglement between QM1 and QM2 needs to be

maintained until QM1

and QM2

become entangled with memory qubits on other nodes. Once this

happens, the entanglement can then be swapped, leaving QM1 and QM2 completely disentangled and

ready for re-use. Of course, QM1

and QM2

could already be entangled with other nodes by the time

QM1 and QM2 become entangled. Thus, there are four equally probable cases to consider:

• If QM1

and QM2

are already connected to other nodes, we immediately swap the entanglement

and thus do not wait any longer. In this case hT

case1

store

i = 0.

• If one QM (either QM1

or QM2

) is already connected to other nodes, we need to wait an average

time of hT

ent

i for the second QM to become entangled as well. In this case, hT

case2

store

i = hT

case3

store

i =

ent

• If neither QM1

or QM2

are connected, we need to wait for both to become connected. In this

case, hT

case4

store

i =

3−2P

ent

2−P

ent

× hT

ent

The expression for hT

case4

store

i above requires some explanation. It comes from the fact that both QM1

and QM2

become successfully entangled with other nodes after n attempts with probability

Pr(n, P

ent

) = [1 − (1 − P

ent

)

]

. (19)

The average storage time in this case then depends on the average number of attempts needed before

entanglement is successfully created and on the time T

trial

each attempt takes:

case4

store

i = T

trial

∞

n=0

n[Pr(n, P

ent

) − Pr(n − 1, P

ent

)] =

3 − 2P

ent

2 − P

ent

i. (20)

This implies that the overall average storage time is

store

i =

case1

store

i + hT

case2

store

i + hT

case3

store

i + hT

case4

store

7/4 − P

ent

2 − P

ent

i. (21)

The upper bound on the rate per matter qubit is therefore

(2QM)

max

(hT

ent

i + hT

store

i)N

2 − P

ent

15 − 8P

ent

, (22)

where we set hT

ent

i → 2L

/c and N

= 2L/L

as appropriate for the memory-based scheme. If we set

ent

→ 1 [85], this becomes

(2QM)

max

for P

ent

= 1. (23)

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 18

Measurement

node

Alice Bob

(a) (b)

(c)

Figure 5: (a) Recursive deﬁnition of the quantum state |RGS

i. (b) Realization of a bridge between two RGSs

|RGS

k,bridge

i. (c) Linear cluster state shared between Alice, Bob and all the measurement nodes.

Classical signaling does not limit the repetition rate of the RGS protocol

In this section, we show that classical signaling does not limit the repetition rate of the RGS protocol.

This is important to consider because some of the X and Z measurements that are performed on the

RGS will, depending on the measurement outcomes, leave the remaining photons in a state that is

only locally equivalent to a graph state. In this case, additional single-qubit gates must be applied

to bring the state to a proper graph state before the next measurements are performed. Otherwise,

the bases for these subsequent measurements will be eﬀectively rotated, and the measurements will

not modify the state in the intended way. In the following, we show that the extra single-qubit gates

only need to be applied on qubits located at the same measurement node where the measurements

on which they depend occur. Thus, classical communication between nodes is not needed. The only

exception to this occurs at the very end of the protocol, when we are left with a linear cluster state

in which the two end qubits are those of Alice and Bob. The ﬁnal step is to turn this state into a

Bell pair shared by Alice and Bob by performing X measurements on the inner qubits of the linear

cluster state. Afterward, Alice and Bob’s qubits form a Bell pair whose Pauli frame depends on the

X measurement outcomes. This information therefore needs to be transferred to them using classical

signaling. However, the protocol can be repeated before this classical signal is received and thus, the

classical signaling does not limit the repetition rate. The latter is only limited by the generation time

of RGSs.

Recursive construction of a repeater graph state.

We denote the RGS with k arms by |RGS

i. For the purposes of this argument, we allow k to be

either odd or even. We want to deﬁne this quantum state recursively. As illustrated in Fig. 5(a), we

can make this state by starting with an RGS with k − 1 arms and then adding one new 1

-leaf qubit

and one new 2

-leaf qubit by applying CZ gates between these two new qubits and between the new

-leaf qubit and all the existing 1

-leaf qubits in |RGS

k−1

i. In other words, we can start from a

2-qubit cluster state and |RGS

k−1

i in a product state: (|0

, +

i + |1

, −

i) ⊗ |RGS

k−1

i. Here, k

and k

denote the 1

-leaf qubit and the 2

-leaf qubit from the k

arm. We ignore the normalization

factors troughout this section. The remaining step is to apply CZ gates between k

and all the 1

-leaf

qubits in |RGS

k−1

i, which yields

|RGS

i = |0

, +

i ⊗ |RGS

k−1

i + |1

, −

i ⊗



RGS

k−1

, (24)

where we have introduced the notation



RGS

≡

i=1

|RGS

i . (25)

Here, Z

is the Pauli Z operator applied to the j

-leaf qubit of the i

arm of the RGS.

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 19

Note that if the RGS has a logical tree encoding, then instead of attaching a 2-qubit cluster state to

|RGS

k−1

i, we need to attach a star graph with b

arms (in the case of a depth-two tree), where b

is the

ﬁrst branching parameter of the tree. To create |RGS

i, each arm of the star graph needs to connect

to every 1

-leaf qubit of |RGS

k−1

i; this requires a total of b

(k − 1) CZ gates. The resulting RGS

can still be decomposed as in Eq. (24), but we need to replace |0

i by a sum over all the even-parity

bit strings on b

qubits, and |1

i needs to be replaced by all the odd-parity bit strings on b

qubits.

In what follows, we will continue to use Eq. (24) with the understanding that the k

qubit states need

to be replaced in this manner if a logical tree encoding is included. This does not otherwise alter the

analysis below.

Bell state measurement

Let’s now consider the eﬀect of a Bell measurement on 2

-leaf qubits from two diﬀerent RGSs. We

suppose that one of these qubits resides in arm k of one RGS, and the other in arm k

of the other RGS.

To analyze this measurement, we expand the two-RGS product state using the recursive deﬁnition of

an RGS shown above:

|RGS

i ⊗



RGS



= |RGS

k−1

i ⊗ |0

, +

, 0

i ⊗



RGS

k−1



+ |RGS

k−1

i ⊗ |0

, +

, −

, 1

i ⊗



RGS

k−1



RGS

k−1

⊗ |1

, −

, +

, 0

i ⊗



RGS

k−1





RGS

k−1

⊗ |1

, −

, 1

i ⊗



RGS

k−1

(26)

We want to perform a Bell state measurement on qubits k

and k

. After the Bell measurement, we

want to obtain the state:

|RGS

k,bridge

i = |RGS

k−1

i ⊗ |0

, 0

i ⊗



RGS

k−1



+ |RGS

k−1

i ⊗ |0

, 1

i ⊗



RGS

k−1



RGS

k−1

⊗ |1

, 0

i ⊗



RGS

k−1



−



RGS

k−1

⊗ |1

, 1

i ⊗



RGS

k−1

(27)

which corresponds to the state represented in Fig. 5(b) where an edge replaces a pair of 2

-leaf qubits

shared by two RGSs.

The Bell state measurements are realized in the basis:





= |0, +i ± |1, −i ,





= |1, +i ± |0, −i .

(28)

These are not the standard Bell states but are related to them by a Hadamard gate on the second

qubit. By applying such a Hadamard gate, a standard Bell state analyzer can be used. Using this

basis for the 2

-leaf qubits k

and k

, we can rewrite the state before the Bell measurement as

|RGS

i ⊗



RGS



= |RGS

k−1

i ⊗ |0i

⊗





−





−





⊗ |0i

⊗



RGS

k−1



+ |RGS

k−1

i ⊗ |0i

⊗





−



−





−



−





⊗ |1i

⊗



RGS

k−1



RGS

k−1

⊗ |1i

⊗





−



−



−



−





⊗ |0i

⊗



RGS

k−1





RGS

k−1

⊗ |1i

⊗



−



−





−



−





⊗ |1i

⊗



RGS

k−1

(29)

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 20

Suppose that we can resolve only the two states |ψ

i with a linear optics Bell state analyzer.

Depending on the measurement outcomes, the state becomes:



−



→ Z

|RGS

k,bridge

i ,



→ Z

|RGS

k,bridge

i .

(30)

Thus, we ﬁnd that if the outcome is





(respectively |ψ

−

i) we need to apply a Z rotation on qubit

(on qubits k

and k

) to recover the desired state. Note that this qubit is at the measurement node

where we perform the Bell measurement.

Z measurements

If the measurement fails or if one Bell measurement has already succeeded, we need to perform Z

measurements on 1

-leaf qubits to disconnect them and their associated 2

-leaf qubits from the

graph. Depending on the outcomes of these measurements, the resulting state is one of four possible

states:

|RGS

k−1

i ⊗



RGS

k−1





RGS

k−1

⊗



RGS

k−1



|RGS

k−1

i ⊗



RGS

k−1



RGS

k−1

⊗



RGS

k−1

(31)

Note that if a Bell measurement has already succeeded, then each of the above outcomes needs to be

acted upon by a CZ gate on the two 1

-leaf qubits associated with the successful measurement. We

leave this possible CZ gate implicit in the following. Ideally, we want to recover |RGS

k−1

i⊗



RGS

k−1

Therefore, we need to be able to convert a state



RGS

k−1

into |RGS

k−1

i by performing measurements

only on qubits present in the measurement node. Let’s consider any other arm l of the RGS that is

present at the same measurement node and rewrite the state as

|RGS

k−1

i = |0

, +

i ⊗ |RGS

k−2

i + |1

, −

i ⊗



RGS

k−2

. (32)

By applying Z gates on all the 1

-leaf qubits, we also have:



RGS

k−1

= |0

, +

i ⊗



RGS

k−2

− |1

, −

i) ⊗ |RGS

k−2

i . (33)

A comparison of these two results leads to the identity:



RGS

k−1

= i |RGS

k−1

i . (34)

By applying local rotations of qubits situated at the same measurement node, it is therefore possible to

recover the desired state. The last two Z measurements on the two qubits denoted k

and k

requires

special care as the only two other qubits, l

and l

, positioned at this measurement node, are the

one which connect the two RGSs (if a Bell measurement has succeeded) and should undergo an X

measurement. In that case, it is still possible to use l

and l

to recover the desired graph state by

performing single-qubit rotations that depends on the measurement outcomes of k

and k



, 0

→ I

⊗ I



, 1

→ −iZ

⊗ Y



, 0

→ −iY

⊗ Z



, 1

→ −X

⊗ X

(35)

where I

denotes the identity on qubit i.

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 21

X measurements and classical signaling

If at least one Bell measurement succeeds at each measurement node and if all the required Z mea-

surements succeed, we are left with a linear cluster state with two 1

-leaf qubits at each measurement

node (see Fig. 5(c)). The next step is to perform X measurements on these qubits. Performing X

measurements on two connected qubits of a graph state has the eﬀect of connecting each qubit in the

neighborhood of the ﬁrst qubit to all the qubits in the neighborhood of the second one [35]. Therefore,

after the X measurements of all the qubits at each measurement node, the two remaining qubits shared

by Alice and Bob are in a maximally entangled state. More precisely, this state is equivalent to a Bell

pair up to local qubit rotations. It is important to note that all the X measurements can be performed

independently of each other. Although the state can deviate from a proper graph state depending

on the measurement outcomes, it is straightforward to show that the state only diﬀers by at most

local Z gates. This can eﬀectively convert some of the X measurements to Y measurements, but both

types of measurements have a similar eﬀect on the state, and a Bell pair is still obtained once all the

measurements are completed. However, the precise state depends on all the measurement outcomes of

each X measurement. This information needs to be sent via a classical channel to Alice and Bob to

recover the desired state.

This classical signal does not limit the repetition rate of the protocol, which can either be used for

quantum teleportation of quantum states or for QKD. Indeed, for quantum teleportation, two quantum

memories situated at Alice’s and Bob’s nodes can be used to store the Bell pair while waiting for the

classical signal to arrive. In that case, the protocol can still be repeated at a high rate as long as

there are enough quantum memories at Alice’s and Bob’s nodes to store the states that are already

transferred. It is still the case that the protocol does not require quantum memories at the nodes

inside the network, which is a clear resource improvement compared to the memory-based approach.

For QKD, Alice’s and Bob’s qubits can be measured in a random basis upon arrival at their nodes,

and they don’t need to be stored in quantum memories until the classical signaling is received. The

classical signaling is still required to recover the Pauli frame of the measured qubits and to check if

Alice’s and Bob’s qubits were measured in compatible basis for QKD.

Error correction using tree graph states

In the main text, we have shown that logical encoding with tree graph states allows measuring a qubit

with a probability Pr(M

Z,`

) or Pr(M

X,`

) that is higher than the single-photon measurement probability

. We now show that the logical encoding based on tree graphs also increases the tolerance of the

RGS protocol to single-qubit errors. Indeed, if m

indirect measurements are performed on a qubit

at level k in the tree, it is possible to use a majority vote of the measurements to reduce the eﬀect

of errors. Suppose that all b

of the possible indirect measurements are performed, where b

is the

branching parameter of the tree. The probability that exactly m

of these measurements succeed

(perhaps with errors) is

) =

(1 − s

)

−m

, (36)

where s

is the probability that a single indirect measurement succeeds.

The average error after b

indirect measurement attempts at level k is:

, (37)

where e

is the average error in the case of m

successful measurements, and r

is the probability

that at least one indirect measurement succeeds. For m

indirect measurements, an error still occurs

in the majority vote if more than half of the indirect measurements (m

/2) are faulty. Thus, if m

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 22

odd:

= Pr(N

errors

> m

/2)

j=dm

/2e

)

(1 − e

)

−j

(38)

where e

is the average error of a single indirect measurement. If m

is even, we have:

−1

j=dm

/2e

− 1

)

(1 − e

)

−1−j

. (39)

The sum goes only up to m

−1 because we can randomly remove one result to return to the odd-number

case and re-use the above formula for the probability.

The single indirect Z measurement error e

of a qubit A at level k depends on the error of the

X measurement of qubit B at level k + 1 (which we take to be a constant error probability ) and on

the errors of the b

k+1

measurements on the qubits at level k + 2, denoted C

in Fig. 2(a). These can

either be direct (with error ) or indirect (with error e

k+2

). Let n

be the number of qubits that are

only measured directly (with error ) while b

k+1

− n

qubits are measured indirectly (with error e

k+2

is thus given by:

k+1

Pr(I

Z,k+2

)

k+1

−n

(1 − Pr(I

Z,k+2

))

, (40)

where e

is the error in the case of n

direct qubit measurements, and

Pr(I

Z,k

) =

Pr(I

Z,k

)Pr(M

Z,k

)

Pr(M

Z,k

)

Pr(M

Z,k

)

. (41)

Here, I

Z,k

denotes a successful indirect Z measurement of a photon at level k and thus Pr(I

Z,k

) is

what is called r

in the main text. Therefore, Pr(I

Z,k

) is the probability that a photon at level k

was successfully measured indirectly (I

Z,k

) given that it has been successfully measured (M

Z,k

). Note

that a photon can be both successfully directly and indirectly measured. In that case, we keep the

indirect measurement outcome which should have a smaller error e

compared to the intrinsic single

qubit error , thanks to the error correction.

The only remaining quantity to determine is the error e

. The X measurement on qubit B and the

direct Z measurements on the qubits C

are performed with error , while the remaining b

k+1

− n

indirect Z measurements are performed with error e

k+2

. Note that the indirect measurement outcome

is given by the net parity of all the X and Z measurement outcomes. Thus, the indirect measurement

can still yield the correct outcome even if there are errors in the X and Z measurements, provided

there are an even number of errors so that the parity remains unchanged. Only an odd number of

errors causes the indirect measurement to fail. Thus:

i=0







+ 1



(1 − )

+1−i

k+1

−n

j=0,

i+j=1[2]

k+1

− n

k+2

(1 − e

k+2

)

k+1

−n

−j







(42)

In this equation, we take into account the fact that only odd numbers of measurement errors con-

tribute to e

by restricting the second sum to only values of i + j = 1[2]. This is diﬀerent than what

was obtained in Ref. [31], perhaps due to additional approximations in that reference.

As shown in Fig. 6(a), the logical Z measurement M

Z,`

succeeds if all the 1

-level qubits are measured

directly or indirectly in the Z basis. The logical X measurement M

X,`

succeeds if any of the 1

level

qubits is measured in the X basis and all its corresponding 2

level qubits are successfully measured

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 23

(a) (b)

Figure 6: (a) Logical Z measurement M

Z,`

at the physical level. (b) Logical X measurement M

X,`

at the physical

level. The parity of the Z measurements on the qubits in the green area is the same as the logical M

X,`

measure-

ment on the logical qubit (purple) and is the same as the parity measurement on the two light blue areas (where

measurement bases are indicated next to each qubit).

in the Z basis either directly or indirectly, as shown in Ref. [31]. To understand this, let’s denote by

L the logical qubit that is connected to a set A = {A

, A

, ...} of other qubits, represented by the

green area in Fig. 6(b). Because X

= X

⊗

is a stabilizer of the RGS at the logical level, the

logical X measurement of qubit L, M

X,`

, has the same outcome as Z

, i.e. the parity of the product

of all Z measurements on the qubits in A. At the physical level, the logical qubit is encoded with

a tree graph state where we consider a ﬁrst-level qubit B and its set of neighbor qubits in the tree

C(B). X

C(B)

is a stabilizer of the RGS at the physical level and thus X

C(B)

(represented by

either of the light blue areas in Fig. 6(b)) has the same parity as Z

and therefore constitutes an X

measurement of the logical qubit L.

The logical X and Z errors are thus given by:

¯e

= e

¯e

n=0

Pr(I

)

−n

(1 − Pr(I

))

with e

i=0









(1 − )

n−i

−n

j=0,

i+j=1[2]

− n

(1 − e

)

−n−j







(43)

The derivation of ¯e

follows the same idea as e

(for "k = −1") except that no X measurements are

needed here.

Fidelity of the full protocol

We have used the results already derived in the supplementary materials of Ref. [31] to calculate the

global ﬁdelity of the generated entangled pair of photons shared by Alice and Bob. The total ﬁdelity

is given by

= 1 − (E

+ E

), (44)

with

= E



1 − (1 − 2)

2(N

+1)

(1 − 2¯e

)





1 + (1 − 2)

2(N

+1)

(1 − 2¯e

)

− 2(1 − 2)

2(N

+1)

(1 − 2¯e

)

(1 − 2¯e

)

(2m−2)N



(45)

Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 24

Comments

Products

Project