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