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) [35], 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 [2527]. 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 [2834] 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
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arXiv:2005.07198v4 [quant-ph] 11 Feb 2021
L
0
L
0
Alice Bob
QR node
Measurement
node
(a) 2-quantum-memory protocol
L
0
Classical channel
Photon
Entanglement
Swapping
HEG
Multiplexed QR
(b) Multiplexed QR
2N
=
b
0
=3
b
1
=2
m=3
(c) RGS
(d) RGS protocol
Measurement
node
RGS node
L
0
L
0
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
6
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 [4447]. 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-
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ice and Bob is divided into smaller distances L
0
by N
QR
= L/L
0
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
i
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
i
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 [4954]), 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
0
/2, this means that
the overall heralded entanglement generation at-
tempt takes total time T
trial
L
0
/c. This in-
cludes the time L
0
/(2c) for the single-photon
transfer from the repeater to the measurement
Notation Definition
L Total distance between Alice and Bob.
L
0
Distance between adjacent nodes.
N
QR
Number of repeater nodes.
2m Number of arms of the RGS.
~
b Branching vector of an error-correction
tree (
~
b = (b
0
, b
1
, . . . , b
n1
)).
T
CZ
CZ gate time.
η
t
(l) Transmission of a fiber of length l (η
t
(l) =
exp(l/L
att
)).
η
c
, η
d
In-fiber collection and detection efficien-
cies of photons.
L
att
, c Attenuation distance of the fiber and
speed of light in a fiber. (L
att
20km)
t
att
Average time of flight of photons in the
fiber (t
att
= L
att
/c).
T
RGS
Generation time of an RGS.
R, R
m
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
0
/(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
i
1
= P
ent
/T
trial
< c/(2L
0
).
We derived this result for a specific heralded en-
tanglement generation protocol but it also holds
for all known protocols [2527] (see Supplemen-
tary Materials).
From these results, we can show that the rate
per matter qubit (where the number of matter
qubits is N
m
= 2(N
QR
+ 1) = 2L/L
0
) has a the-
oretical upper bound, R
(QM)
max
:
R
(QM)
N
m
hT
ent
i
1
N
m
c
4L
= R
(QM)
max
. (1)
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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 =
Y
(i,j)E
CZ
ij
|+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
0
by N
QR
= L/L
0
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
AB
much higher than direct
fiber transmission: P
AB
η
t
(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
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RGS:
R
(RGS)
=
P
RGSRGS
L/L
0
T
RGS
, (3)
with P
RGSRGS
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:
P
RGSRGS
= (1 (1 P
Bell
)
m
)
× Pr(M
X,`
)
2
Pr(M
Z,`
)
2m2
,
(4)
where P
Bell
= P
ph
2
/2 is the probability of a
successful Bell measurement. This depends on
P
ph
= η
c
η
d
η
t
(L
0
/2), which is the probability
that a single photon is emitted and collected
into the fiber (η
c
), is transmitted to the mea-
surement node (η
t
(L
0
/2)) and detected (η
d
).
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
X
)
2
Pr(M
Z
)
2m2
= P
ph
2m
η
t
(L
0
), 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
ph
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
ph
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
0
, b
1
, ..., b
n1
) (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
ph
) or, if
it fails (with probability 1 P
ph
), perform an in-
direct measurement (with probability r
k
). Thus,
the overall success probability of a Z measure-
ment at level k is:
Pr(M
Z,k
) = P
ph
+ (1 P
ph
)r
k
. (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
i
, 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
B
jN(B)
Z
j
|Gi
= X
B
Z
A
i∈{1,b
k+1
}
Z
C
i
|Gi ,
(6)
so we have:
Z
A
|Gi = X
B
i∈{1,b
k+1
}
Z
C
i
|Gi . (7)
A single indirect measurement has a success
probability s
k
. Note, however, that the tree
structure allows b
k
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
r
k
= 1 (1 s
k
)
b
k
, (8)
with
s
k
= P
ph
Pr(M
Z,k+2
)
b
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
ph
. Logical measurements
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A
B
C
2
C
1
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
0
,
Pr(M
Z,`
) = (P
ph
+ (1 P
ph
)r
1
)
b
0
= Pr(M
Z,1
)
b
0
.
(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
ph
, the Hadamard gate H,
measurements in the Pauli bases M
X
, M
Y
, M
Z
and the CZ gate. The generation of an RGS with
2m arms and a tree graph encoding with branch-
ing vector
~
b = (b
0
, b
1
, ..., b
n1
) requires n+1 mat-
ter qubits Q
1
, ..., Q
n+1
, and is given by the se-
quence (see also Fig. 2(b))
M
Y,Q
1
(M
X,Q
3
M
X,Q
2
CZ
Q
1
,Q
3
CZ
Q
2
,Q
3
E
ph,Q
3
G
3
b
0
)
2m
with G
k
= M
Z,Q
k
H
Q
k
E
ph,Q
k
CZ
Q
k1
,Q
k
G
k+1
b
k2
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
T
RGS
= 2m
1 + f(
~
b, n 1)
T
E
ph
+ T
M
+ 2m
2 + f(
~
b, n 2)
(T
M
+ T
CZ
)
+ 2mf(
~
b, n 2)T
H
,
(12)
with f (
~
b, k) =
P
k
i=0
Q
i
j=0
b
j
and with T
E
ph
, T
M
,
T
H
and T
CZ
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
CZ
is much longer
than the durations of the other operations, and so
we set T
H
= T
M
= T
E
ph
= 0 for simplicity. With
this assumption, the generation time T
RGS
only
depends on the number of CZ gates and their
duration:
T
RGS
= 2m
2 +
n2
X
k=0
k
Y
j=0
b
j
T
CZ
. (13)
This is for an RGS with 2m arms and a log-
ical tree encoding with branching vector
~
b =
(b
0
, ..., b
n1
). In the following, we will assume
a depth-two tree graph state (
~
b = (b
0
, b
1
)), 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
N
m
= (L/L
0
1)(n + 1) = 3(L/L
0
1).
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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
R
(RGS)
m
T
CZ
(here normalized by the parameter T
CZ
1
)
and optimal node separation L
0
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
0
, b
1
, m. (a) Three-dimensional plot showing optimal
rate and node separation as a function of b
0
, b
1
and
m. For each point in the plot, the value of L
0
is op-
timized to maximize R
(RGS)
m
T
CZ
. Point sizes represent
maximized R
(RGS)
m
T
CZ
values while point colors repre-
sent the optimal values of L
0
(indicated with the color
scale). The RGS parameters that achieve the largest
value of R
(RGS)
m
T
CZ
in this case are indicated with
dashed lines. The corresponding maximal R
(RGS)
max
T
CZ
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)
m
, 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
c
η
d
= 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
R
(RGS)
m
of the RGS protocol for a total distance
L is inversely proportional to the CZ gate time
T
CZ
, but it depends non-linearly on the separa-
L/L
att
L
0
/L
att
m b
0
b
1
R
(RGS)
max
T
CZ
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
0
, b
1
and node
separation L
0
for several different total network dis-
tances L. Here, L
att
is the attenuation length of optical
fibers.
tion distance, L
0
, between two RGS nodes and
the RGS shape (number of arms 2m and the tree
branching vector
~
b = (b
0
, b
1
)). 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
0
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
0
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
0
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
0
, 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
0
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
R
(RGS)
max
, as the latter is inversely proportional to
the CZ gate time, T
CZ
. If T
CZ
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
CZ
T
O
. Decreasing T
CZ
ini-
tially increases the rate, but if T
CZ
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
CZ
below 6 × 10
4
t
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
0
ranges
from 0.15 0.19L
att
3 3.8 km. As an
illustration of the RGS performance, for L =
1000 km and T
CZ
= 10 ns, the total rate is
R = 220 kHz for N
m
= 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
η
c
η
d
6= 1. Fig. 4(c) shows the optimized rate per
matter qubit for different values of η
c
η
d
. 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
CZ
and
η
c
η
d
for a total distance L = 50L
att
. These re-
sults show that the photon losses must be below
15% (η
c
η
d
> 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
AB
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
AB
[59, 60]:
R
skr
= R(1 2h(F
AB
)), (14)
where h is the binary entropy function:
h(F ) = F log
2
(F ) (1 F ) log
2
(1 F ). (15)
Accepted in Quantum 2021-02-03, click title to verify. Published under CC-BY 4.0. 8