b

# Rate analysis for a hybrid quantum repeater

Nadja K. Bernardes1<EMAIL_ADDRESS>Ludmiła Praxmeyer2 Peter van
Loock1<EMAIL_ADDRESS>1Optical Quantum Information Theory Group,
Max Planck Institute for the Science of Light, Günther-Scharowsky-Str. 1/Bau
26, 91058 Erlangen, Germany  Institute of Theoretical Physics I, Universität
Erlangen-Nürnberg, Staudttr. 7/B2, 91058 Erlangen, Germany  2Institute of
Physics, Nicolaus Copernicus University, ul. Grudziadzka 5, 87-100 Torun,
Poland

###### Abstract

We present a detailed rate analysis for a hybrid quantum repeater assuming
perfect memories and using optimal probabilistic entanglement generation and
deterministic swapping routines. The hybrid quantum repeater protocol is based
on atomic qubit-entanglement distribution through optical coherent-state
communication. An exact, analytical formula for the rates of entanglement
generation in quantum repeaters is derived, including a study on the impacts
of entanglement purification and multiplexing strategies. More specifically,
we consider scenarios with as little purification as possible and we show that
for sufficiently low local losses, such purifications are still more powerful
than multiplexing. In a possible experimental scenario, our hybrid system can
create near-maximally entangled ($F=0.98$) pairs over a distance of 1280 km at
rates of the order of 100 Hz.

quantum repeaters

###### pacs:

0000

††preprint: PRE/003

##  I Introduction

In most quantum information processes entanglement plays an essential role. It
enables us not only to teleport quantum information bennett , but also to
achieve perfectly secure quantum communication ekert . Unfortunately, the
quantum channels over which entanglement is distributed are in general noisy.
Owing to fundamental principles, common procedures used in classical
communication, such as amplification or cloning wootters ; dieks , cannot be
applied and therefore the fidelity of transmission will be limited by the
length of the channel. To avoid the exponential decay with the distance and be
able to perform long-distance quantum communication, quantum repeaters were
proposed briegel ; dur . Instead of distributing entanglement over long
distances, entanglement will be generated in smaller segments and a
combination of entanglement swapping zukowski  and entanglement purification
bennett ; deutsch  enables one to extend the entanglement over the entire
channel.

There are various promising proposals for implementing quantum repeaters. The
most prominent of these approaches use some heralding mechanism based on
single-photon detection to generate entangled pairs duan ; childress1 ;
childress2 . In these schemes, typically, high-fidelity entangled pairs are
created, while the success probabilities in the initial entanglement
distribution are very low. Other schemes, employing bright multiphoton
signals, are much more efficient, but have only modest initial fidelities,
since they are more sensitive to photon losses in the communication channel.
As a consequence, these coherent-state-based protocols require more
purification steps PvLa ; Ladd ; PvLb .

Schemes for practically implementing a quantum repeater are not
straightforward, not even for not too long distances such as a few hundred
kilometers. The steps of entanglement distillation and swapping require
advanced local quantum logic, such as two-qubit entangling gates; furthermore,
the typical duration to successfully generate an entangled pair imposes severe
constraints on the quantum memory decoherence times. Depending on how
imperfect these operations are and how short the quantum memory decoherence
time is, the final fidelity may still exponentially decay with the total
distance. Another important issue are the rates at which quantum transmission
succeeds. Assuming that this rate is mainly determined by the channel
transmissivity, the overall transmission rate will also decay exponentially,
unless sufficient quantum memories and/or quantum error detection mechanisms
are available.

In this paper, we will perform a study on the rates for a hybrid quantum
repeater PvLa ; Ladd ; PvLb . In this proposal, entanglement is created
between atomic qubits through an optical coherent state. In order to simplify
the analysis, we shall assume that perfect memories are available, that is,
memories with infinite decoherence times. Moreover, we will assume that
optimal entanglement generation probabilities and deterministic swapping
routines are available. More specifically, the generalized measurements on the
optical mode that lead to the initial, conditional entangled qubit pairs can
be as good and efficient as allowed quantum mechanically. We will derive exact
formulas for the time needed to generate an entangled pair and hence for the
final rates at a given target fidelity. As opposed to previous, mainly
numerical studies PvLa ; Ladd ; vanmeter ; munro , our rate calculations are
fully analytical. For this purpose, we shall first analyze the significance of
nested purifications and multiplexing with the aim of using as little as
possible of these experimentally demanding techniques. Our assumption of
perfect memories makes the use of probabilistic entanglement purification
preferable to alternate techniques based upon deterministic quantum error
correction (QEC) perseguers ; jiang ; fowler ; munro2 . These schemes, using
QEC codes, are experimentally more demanding due to their need of sufficient
spatial memory resources and more complicated quantum gates for encoding and
syndrome identification. In general, there will be a trade-off between the
requirements on the memory decoherence times and those on the quantum error
detection mechanism used to suppress the exponential fidelity decay. As
mentioned, in our case, we shall focus on a scenario where memories are ideal
and quantum error detection as simple as possible. The paper is structured as
follows. In Sec. II, we briefly describe the hybrid quantum repeater,
explaining how to generate, purify, and swap entanglement. In Sec. III, we
calculate the rates to generate an entangled pair over the entire distance for
a hybrid quantum repeater considering different strategies. We conclude in
Sec. IV and give more details on various formulas and derivations in the
appendix.

##  II Hybrid Quantum Repeater

###  II.1 Entanglement generation

In a hybrid quantum repeater scheme, the entanglement distribution mechanism
is based on dispersive light-matter interactions, obtainable from the Jaynes-
Cummings interaction Hamiltonian in the limit of large detuning schleich .
Such a Hamiltonian can be obtained by single electrons trapped in quantum dots
bracker  or by neutral donor impurities in semiconductors strauf . It leads to
a conditional phase-rotation of the field mode,
$\hat{U}_{int}=e^{i\theta\hat{\sigma}_{z}\hat{a}^{\dagger}\hat{a}}$, where
$\theta$ is an effective interaction time, $\hat{\sigma}_{z}$ is the qubit
Pauli-Z operator and $\hat{a}$ ($\hat{a}^{\dagger}$) is the annihilation
(creation) operator of the electromagnetic field mode. Using a coherent state
$|\alpha\rangle$ as the probe beam and an electron-spin system in a cavity
(i.e., a two-level system or a “$\Lambda$-system” as an effective two-level
system), the total output state will be ideally described as

$\hat{U}_{int}\left[\frac{(|0\rangle+|1\rangle)}{\sqrt{2}}|\alpha\rangle\right]=\frac{|0\rangle|\alpha\rangle+|1\rangle|\alpha
e^{-i\theta}\rangle}{\sqrt{2}}.$ (1)

This interaction enables one to generate an entangled two-qubit state. First,
we let a bright coherent-state pulse, or “qubus”, which we denote as system
“B”, interact with an atomic qubit superposition state, system “A”, resulting
in the state (1). The coherent state is then sent through a lossy channel and
interacts in a second cavity with system “C”, resulting in an entangled state
between the two qubits (systems “A” and “C”) and the probe beam (system “B”).
This state, after local transformations, is given by PvLb

$\mu_{E}^{2}|\Phi^{+}\rangle\langle\Phi^{+}|+(1-\mu_{E}^{2})|\Phi^{-}\rangle\langle\Phi^{-}|$
(2)

where

$\displaystyle|\Phi^{+}\rangle=\frac{1}{\sqrt{2}}|\sqrt{\eta}\alpha\rangle_{B}|\phi^{+}\rangle_{AC}+\frac{1}{2}e^{-i\eta\xi}|\sqrt{\eta}\alpha
e^{i\theta}\rangle_{B}|10\rangle_{AC}$
$\displaystyle+\frac{1}{2}e^{i\eta\xi}|\sqrt{\eta}\alpha
e^{-i\theta}\rangle_{B}|01\rangle_{AC},$
$\displaystyle|\Phi^{-}\rangle=\frac{1}{\sqrt{2}}|\sqrt{\eta}\alpha\rangle_{B}|\phi^{-}\rangle_{AC}-\frac{1}{2}e^{-i\eta\xi}|\sqrt{\eta}\alpha
e^{i\theta}\rangle_{B}|10\rangle_{AC}$
$\displaystyle+\frac{1}{2}e^{i\eta\xi}|\sqrt{\eta}\alpha
e^{-i\theta}\rangle_{B}|01\rangle_{AC},$

with the maximally entangled Bell states
$|\phi^{\pm}\rangle=(|00\rangle\pm|11\rangle)/\sqrt{2}$. Photon losses in the
channel are described by a beam splitter which transmits, on average, $\eta$
photons, $\xi\equiv\alpha^{2}\sin{\theta}$, and
$\mu_{E}=(1+e^{-(1-\eta)\alpha^{2}(1-\cos{\theta})})^{1/2}/\sqrt{2}$
($\alpha\in\mathbb{R}$). Considering a standard telecom fiber, where photon
loss is assumed to be 0.17 dB per km, the transmission parameter will be
$\eta(L,L_{att})=e^{-L/L_{att}}$, where $L$ is the total distance of the
channel and the attenuation length is assumed to be $L_{att}=25.5$ km.

Measuring the state of the qubus mode permits the preparation of a two-qubit
entangled state. One way to achieve this final step is through homodyne
detection. This is a very efficient and practical way, but the final
fidelities are rather modest (e.g. $F<0.8$ for 10 km). Another slightly less
practical way, but with a considerable improvement in the final fidelities, is
the unambiguous state discrimination (USD) approach. High initial fidelities
can then be achieved at the expense of lower entanglement generation rates. In
general, using the USD measurement, fidelities can be tuned in the whole range
$0.5<F<1$ for any given elementary distance $L_{0}$, with correspondingly
smaller success probabilities for larger fidelities. In any case, the USD
approach gives us ultimate bounds on the performance of the entanglement
generation procedure, when the quantum mechanically optimal USD is considered
PvLb .

In the USD approach, we must be able to distinguish between the state
$|\sqrt{\eta}\alpha\rangle$ and the set of states $\\{|\sqrt{\eta}\alpha
e^{i\theta}\rangle,|\sqrt{\eta}\alpha e^{-i\theta}\rangle\\}$. The fidelity
between these two density operators gives a lower bound to the failure
probability (the probability for obtaining an inconclusive measurement
outcome),

$P_{?}\geq
F=\sqrt{\langle\sqrt{\eta}\alpha|\hat{\rho}|\sqrt{\eta}\alpha\rangle},$ (3)

where

$\hat{\rho}=\frac{1}{2}(|\sqrt{\eta}\alpha
e^{i\theta}\rangle\langle\sqrt{\eta}\alpha e^{i\theta}|+|\sqrt{\eta}\alpha
e^{-i\theta}\rangle\langle\sqrt{\eta}\alpha e^{-i\theta}|).$

This leads to an optimal (minimal) failure probability

$P_{?}^{opt}=e^{-\eta\alpha^{2}(1-\cos{\theta})}.$ (4)

By looking at Eq. (2), it is possible to establish a connection between the
fidelity of the successfully created, entangled pair (a rank-2 mixture of the
$|\phi^{\pm}\rangle$ Bell states for an error-free identification of the state
$|\sqrt{\eta}\alpha\rangle$) and the optimal failure probability. Considering
that $\mu_{E}^{2}\equiv F$, this failure probability will be given by

$P_{?}^{opt}=(2F-1)^{\eta/(1-\eta)}.$ (5)

A practical implementation of a suboptimal USD measurement based upon linear
optics and photon detection can be found in PvLb . A protocol for implementing
the optimal USD is given in azuma .

Assuming that (5) is an optimal bound for the failure probability, the optimal
upper bound for the success probability to generate an entangled pair is
$P_{success}^{opt}=1-P_{?}^{opt}$. We will use this bound for the probability
of success in the following sections.

###  II.2 Entanglement purification and swapping

Using the same interactions as presented in the preceding section, we are able
to perform entanglement swapping and entanglement purification. In either
case, local two-qubit gates are needed. Following Ref. PvLc , a measurement-
free, deterministic controlled-phase gate can be achieved with a sequence of
four conditional displacements of a coherent-state probe interacting with the
two spins. More specifically, by appropriate choice of $\beta_{1}$ and
$\beta_{2}$ ($\beta_{1}\beta_{2}=\pi/8$), up to a global phase and local
unitaries, the total unitary operator representing the controlled-phase gate
will be

$\hat{D}(i\beta_{2}\hat{\sigma}_{z2})\hat{D}(\beta_{1}\hat{\sigma}_{z1})\hat{D}(-i\beta_{2}\hat{\sigma}_{z2})\hat{D}(-\beta_{1}\hat{\sigma}_{z1}).$
(6)

Here the operator $\hat{D}(\beta)=e^{\beta\hat{a}^{\dagger}-\beta^{*}\hat{a}}$
describes a phase-space displacement of the probe by $\beta$. In fact, it can
be shown PvLc  that the sequence of (6) can be achieved through uncontrolled
displacements and controlled rotations of the probe via the same Jaynes-
Cummings-type interaction as used for the entanglement generation. Another
scheme to obtain a two-qubit gate is proposed in Refs. PvLa ; Ladd . Although
in this scheme less conditional operations are needed, making it less
sensitive to losses, even without losses, a small amount of decoherence is
introduced due to the remaining entanglement between the probe and the spins
after the gate operation. This decoherence effect depends on $\theta$ and on
the initial probe state, and scales with $\alpha\theta^{2}$ (for an initial
coherent-state probe with amplitude $\alpha$), thus becoming negligible for
sufficiently small $\theta$ and $\alpha\theta\sim 1$.

Single-qubit rotations, measurements, and this controlled-phase gate are
sufficient resources to implement the standard purification protocol
introduced in Ref. deutsch . From Eq. (2), it is possible to see that, after
the USD measurement has taken place, the output state will be a rank-2 state
(a mixture of two Bell states). The probability of success of the entanglement
purification is

$P_{purification}=F^{2}+(1-F)^{2},$ (7)

and the final fidelity becomes

$F_{purification}=\frac{F^{2}}{F^{2}+(1-F)^{2}}.$ (8)

As was shown in Ref. dur , the purification protocol introduced by Deutsch et
al. deutsch  is particularly efficient for these classes of states (i.e., it
is more efficient than for full-rank mixtures).

The same kind of operations as for purification are sufficient to implement
the entanglement swapping. For a state given as in Eq. (2), the swapping will
be deterministic ($P_{swapping}\equiv 1$) and its final fidelity is given by

$F_{swapping}={F^{2}+(1-F)^{2}}.$ (9)

In a more realistic approach, considering errors in the gates caused by local
losses, Eqs. (7-9) will no longer be valid. According to the analysis
presented in Ref. Louis , dissipation will be introduced in the probe mode
between and during each interaction with the spins. A coherent-state matrix
element $|\gamma\rangle\langle\beta|$ after dissipation changes to
$\langle\beta|\gamma\rangle^{1-T}|\gamma\sqrt{T}\rangle\langle\beta\sqrt{T}|$,
where $T$ is the transmission parameter walls . The interaction sequence (6)
will then change to terms that cause single-qubit dephasing, but also one term
which causes a dephasing on both qubits; this effect scales with
$(1-T)\alpha^{2}\sin{\theta}^{2}$ (for an initial coherent-state probe with
amplitude $\alpha$). For further details see Appendix A. The resulting gate is
extremely sensitive to losses. However, the aim of our work is a better
understanding of the building blocks in the hybrid quantum repeater, and hence
we shall minimize the effect of local losses by choosing in most parts of our
analysis a sufficiently high local transmission parameter. In those figures
where the parameter T is not explicitly given, it is assumed that the error is
at least as small as $1-T=0.001\%$. The main influence of photon losses then
occurs in the communication channel and therefore depends on the communication
distance.

##  III Rate analysis

In this section, the rates of entanglement generation over the entire distance
for a hybrid quantum repeater will be calculated. The memories are considered
ideal, and the entanglement connection deterministic, such that the primary
source of errors will be photon losses in the channel.

Figure 1: (color online). Idealized quantum repeater. A total distance L is
divided in $2^{n}$ segments with length $L_{0}=L/2^{n}$. Initially,
entanglement is generated between neighboring repeater stations. The qubits at
the intermediate stations are then connected. Finally, entanglement over the
entire distance L is obtained.

Let us first consider a general quantum repeater as illustrated in Fig. (1). A
total distance L is divided in $2^{n}$ segments, each of length
$L_{0}=L/2^{n}$. First, entanglement is generated between the adjacent nodes,
which is accomplished with probability $P_{0}$. Then these segments are
connected, extending the entanglement from $L_{0}$ to $2L_{0}$. This step is
performed many times, until the terminal nodes, separated by $L=2^{n}L_{0}$,
are entangled. The initial distribution of entanglement over small segments
does not prevent the final fidelity from decaying exponentially. Even if the
connecting operations are perfect, we must take into account that the initial
entangled pairs are not perfectly entangled pairs, and this will cause a decay
of the final fidelity. As a consequence, purification and quantum memories are
essential for the full quantum repeater, but their implementations introduce a
series of experimental difficulties. There are various ways to optimize the
combinations of swapping and purification (e.g., considering residual
entanglement razavi1 ; razavi2 , multiplexing collins , blind connecting
measurements razavi2 ). However, these approaches are more significant for the
case of imperfect memories and probabilistic swappings, whereas our model uses
perfect memories and deterministic swappings.

In the following sections, we will focus on the effects of spatial
multiplexing compared to a parallel scheme for the hybrid quantum repeater. In
a parallel repeater, the $i$th memory pair in one segment interacts only with
the $i$th pair in neighboring segments, however, for the multiplexing scheme,
resources can be dynamically allocated. Moreover, it will be discussed what
the impact of purification on the rates is and how the two methods of
multiplexing and purification compare 111Note that there are different ways to
do and use multiplexing. The more conventional application is spatial
multiplexing, as typically used in discrete-variable single-photon-based
repeaters collins . In this case, the probability to successfully generate an
entangled pair, $P_{0}<1$, is fairly small and multiplexing means that
neighboring segments even from different, parallel repeater chains can be
connected immediately after the corresponding pairs have been created.
Another type of multiplexing is temporal munro2 . In this case, the initial
entanglement is generated almost instantly by transmitting in parallel
sufficiently many probe pulses (after their interactions with a corresponding
number of spins) to a receiving spin that interacts with these pulses almost
simultaneously to guarantee that at least one entangled pair is created with
an effective, near-unit success probability, $P_{0}^{\rm eff}\to 1$.
Experimentally, to make this kind of temporal multiplexing efficient,
sufficiently fast local interactions and gates are needed. As a result, when
also swapping and error detection are deterministic (by replacing purification
by error correction), the full repeater protocol becomes near-deterministic,
as opposed to our scheme which keeps both the initial entanglement generation
and the entanglement purification probabilistic.
In principle, our rate analysis could be directly combined with the ideas of
Ref. munro2  by simply substituting our $P_{0}$ by an effective probability
of, for instance, $P_{0}^{\rm eff}\equiv(1-(1-P_{0})^{n})$, where $n$ is the
number of cavities transmitting the probe pulses and $P_{0}$ in this formula
is still the same function of fidelity, losses, etc., as described in the main
text for our USD-based entanglement distribution. Then, $P_{0}^{\rm eff}\to 1$
can be achieved for sufficiently large $n$ even when the original $P_{0}$ is
as small as 10-20 %. However, as our goal is to keep the number of spatial
resources as small and the scheme as simple as possible, we will not consider
this type of temporal multiplexing here. Our setting contains perfect
memories, but modest spatial resources, such that $P_{0}^{\rm eff}\equiv
P_{0}$ remains of the order of 1-10 %..

###  III.1 Entanglement generation and swapping

First, let us calculate the rate for generating an entangled pair for a
quantum repeater in parallel without any purification or dynamical allocation
of resources. For the ideal memory case, we calculate the simple case of
entanglement-length doubling ($n=1$) with a single memory per half node. Since
the $n$-level quantum repeater is the entanglement-length doubling of two
$(n-1)$-level systems, it is essential to understand the basic process with
$n=1$.

Even simpler, let us start with just one segment, $n=0$. The average time
necessary to generate an entangled pair at distance $L$ (in this case
$L=L_{0}$) is given by

$\left\langle T\right\rangle_{0}=\frac{T_{0}}{P_{0}},$ (10)

where $T_{0}=2L_{0}/c$ is the minimum time to successfully generate
entanglement over $L_{0}$, assuming that this is the time spent on classical
communication to verify the success of the entanglement generation over
$L_{0}$ (including the initial transmission of the probe beam) and $c$ is the
speed of light in an optical fiber ($2\times 10^{8}m/s$).

For two pairs, $n=1$, the average time necessary to generate an entangled pair
at distance $L$ is then given by collins

$\left\langle
T\right\rangle_{1}=\frac{T_{0}}{P_{0}}\frac{(3-2P_{0})}{(2-P_{0})}.$ (11)

Recall that our memories are assumed to be ideal such that one successfully
created pair can be kept until a second pair is created in the neighboring
segment. The result in Eq. (11) is equivalent to the problem of a
geometrically distributed random variable with success probability $P_{0}$.

For $2^{n}$ pairs next to each other in $2^{n}$ segments, the average time
needed to generate an entangled pair at distance $L$ is given by

$\left\langle T\right\rangle_{n}=T_{0}Z_{n}(P_{0}),$ (12)

where the average number of steps to successfully generate entanglement in all
$2^{n}$ pairs, $Z_{n}(P)$, is

$Z_{n}(P)=\sum^{2^{n}}_{j=1}\binom{2^{n}}{j}\frac{(-1)^{j+1}}{1-(1-P)^{j}},$
(13)

and $P$ is the probability of success. Detailed calculations are presented in
Appendix B.

The rate to successfully generate entanglement in all of $2^{n}$ pairs over L
and to eventually obtain one L-distant pair can now be written as

$R_{n}=\frac{1}{\left\langle T\right\rangle_{n}}=\frac{1}{T_{0}Z_{n}(P_{0})}.$
(14)

Figure 2: (color online). Rates for a hybrid quantum repeater over a total
distance $L=1280$ km with $L_{0}=20$ km without purification and without
multiplexing, but including perfect memories. Comparison between the exact
formula to generate entangled pairs (green line) as given by Eqs. (12-14) and
the approximated one (black dashed line) corresponding to
$(\frac{2}{3})^{n}\frac{P_{0}}{T_{0}}$.

Commonly in the literature sangouard , for small $P_{0}$, these rates are
approximated by $\frac{P_{0}}{T_{0}}\left(\frac{2}{3}\right)^{n}$. However, as
illustrated in Fig. (2) for a total distance of $L=1280$ km and $L_{0}=20$ km,
the approximate formula is underestimating the rates in some regimes by more
than $50\%$ for our case of the USD-based hybrid repeater.

Figure 3: (color online). Different strategies to connect memory elements
situated in different columns. The parallel architecture a) connects only
elements in the same chain. Contrary to it, multiplexing b) connects any
available entangled pairs.

###  III.2 Multiplexing versus parallelization

How would the rates be affected, if multiplexing is introduced in the system?
In the multiplexed scheme, where more than one memory per half node exist, as
shown in Fig. (3), entangled pairs are connected not only with pairs in the
same chain (as in the parallel scheme). Instead, as soon as one entangled pair
is successfully generated in one of the columns, it can be connected to a
neighboring pair, no matter in which chain they are positioned. The rate to
generate one entangled pair over the total distance for $n=1$ and $r$ memories
per half node is given by

$R_{mult,1,r}=\frac{1}{T_{0}Z_{mult,1,r}(P_{0})}=\frac{1}{T_{0}}\frac{1-(1-P_{0})^{2r}}{1+2(1-P_{0})^{r}}.$
(15)

For further details see Appendix C. Plotting this rate for a total distance of
$L=40$ km and four memory pairs per segment, $r=4$, see Fig. (4), it is
possible to confirm that the multiplexed scheme performs better than the
parallel one. However, an experimental implementation of this type of spatial
multiplexing would require fairly demanding feedforward techniques on the
pulses that enact the gates for entanglement swapping. It is therefore more
practical and efficient to operate the repeater in parallel, provided perfect
memories are available, as it is the case in our scenario. In the following
sections, the influence of purification will be studied. By comparing
purification with multiplexing, we will justify why we shall not consider
multiplexing in the further analysis.

Figure 4: (color online). Rates for a total distance $L=40$ km, $L_{0}=20$ km.
Comparison between the parallel (red, dashed line) and the multiplexed scheme
(black, solid line) for 2 segments ($n=1$) and 4 memories per half node
($r=4$).

###  III.3 Entanglement purification

Taking into account that purification is an expensive task, either in terms of
spatial or in terms of temporal resources, the most efficient and simplest
extension beyond those schemes described in the preceding sections would
employ just one round of purification at the first nesting level. Intuitively,
considering the purification procedure probabilistic and entanglement swapping
deterministic, performing the purification at the beginning will make better
use of the memories than by doing it at the end or somewhere in between; for a
more quantitative justification, see Appendix D. Starting again with $n=0$,
already initially there is a need to generate at least two pairs, such that
purification is possible. In this way, the average time needed to generate a
purified pair will be

$\left\langle
T\right\rangle_{purif,0}=\frac{T_{0}}{P_{0}P_{1}}\frac{(3-2P_{0})}{(2-P_{0})}.$
(16)

Here we employ the same purifying protocol as the one introduced in zeilinger
; in this case the probability of one round of purification to succeed,
$P_{1}$, is given by Eq. (7).

For $n>0$, the calculation of the rates is not straightforward. In this
regime, we have found an upper and a lower bound for the time needed to
generate $2^{n}$ purified pairs. The former times, which give us the lower
bounds for the rates, correspond to those cases when purification will start
only after all the pairs are successfully generated, even if two pairs in the
same column are already present after a shorter time. In this slowest case,
the average time needed to generate $2^{n}$ one-round purified pairs is given
by

$\left\langle T\right\rangle_{purif,upper,n}=T_{0}Z_{n+1}(P_{0})Z_{n}(P_{1}).$
(17)

For the most optimistic case, corresponding to the fastest possible way to
achieve purification, we imagine that if purification fails, it is not
necessary to start from the beginning, trying to generate new pairs again, but
we assume that the two pairs necessary for purification are still available,
imagining that they have not been destroyed. The average time needed to
generate $2^{n}$ one-round purified pairs is then given by

$\left\langle
T\right\rangle_{purif,lower,n}=T_{0}(Z_{n+1}(P_{0})+Z_{n}(P_{1})).$ (18)

Aiming to find a compromise between the upper and lower bounds for the rates,
the rates can be calculated in an approximate fashion, combining the ideas
from the recurrence and the exact formula. The average time needed to generate
one purified pair will be

$\left\langle T\right\rangle_{purif,approx,0}=\frac{T_{0}}{P_{L_{0}}},$ (19)

where $P_{L_{0}}$ is an effective probability, from (16),
$P_{L_{0}}={P_{0}P_{1}}\frac{(2-P_{0})}{(3-2P_{0})}$. From this, it follows
that

$\left\langle T\right\rangle_{purif,approx,n}=T_{0}Z_{n}(P_{L_{0}}).$ (20)

Figure 5: (color online). Rates for a hybrid quantum repeater over a total
distance $L=1280$ km and $L_{0}=20$ km. Comparison between the lower bound
(green, solid line), upper bound (red, dot dashed line) and the approximate
formula (blue, dashed line).

How are the upper bound, the lower bound, and the approximate times, and their
corresponding rates, related to each other? Fig. (5) shows for a total
distance $L=1280$ km, the rates in these three cases. The approximate formula
gives a result which always stays in the middle between the upper and the
lower bounds. The rates are then given by

$\displaystyle R_{purif,n}$ $\displaystyle=$
$\displaystyle\frac{1}{\left\langle T\right\rangle_{purif,approx,n}}$
$\displaystyle=$ $\displaystyle\frac{1}{T_{0}Z_{n}(P_{L_{0}})}.$

Figure 6: (color online). Rates for total distance $L=320$ km (below), $L=640$
km (center), $L=1280$ km (above) with $L_{0}=20$ km. Comparison between the
rates for a hybrid quantum repeater with one round of purification (black,
dashed line), two rounds of purification (red, solid line), and three rounds
of purification (blue, dot dashed line) at the first nesting level.

There is one remaining question: do more rounds of purification at the first
nesting level always increase the rates for the hybrid quantum repeater?
Utilizing (III.3) we plotted in Fig. (6) the rates to generate an entangled
purified pair over different total distances applying one, two, or three
rounds of purification 222$P_{L_{0}}$ varies depending on the number of rounds
of purification. For one round,
$P_{L_{0},1purif}={P_{0}P_{1}}\frac{(2-P_{0})}{(3-2P_{0})}$. For two rounds,
$P_{L_{0},2purif}={P_{L_{0},1purif}P_{1}}\frac{(2-P_{L_{0},1purif})}{(3-2P_{L_{0},1purif})}$.
And for three rounds,
$P_{L_{0},3purif}={P_{L_{0},2purif}P_{1}}\frac{(2-P_{L_{0},2purif})}{(3-2P_{L_{0},2purif})}$..
We see that not always does the increase of rounds of purification result in
an increase of the rates. More specifically, for this scheme and these
distances, three rounds of purification only increase the rates for very high
final fidelities. It should be pointed out here that the number of initial
resources 333These initial resources can be either spatial or temporal, where
in the latter case, for every new round of purification, a fresh initial pair
is created (so-called entanglement pumping dur ). Such an approach requires a
minimum of spatial resources childress1 ; childress2 , however, a maximum of
time. This means one needs extremely good memories and, nonetheless, the total
rates remain fairly low. Even though in our setting we do assume perfect
memories, throughout we shall stick to the faster standard purification
methods at the first nesting level using a sufficient initial supply of
spatial resources, similar to Ref. PvLa . increases also with the number of
rounds of purification, and so, in a more appropriate analysis, keeping the
numbers of initial resources the same, the rates for multiple-rounds of
purification should even perform worse.

###  III.4 Multiplexing versus purification

Figure 7: (color online). Rates to generate one entangled pair over a total
distance of $L=40$ km (below), $L=640$ km (center) and $L=1280$ km (above)
with $L_{0}=20$ km. Comparison between multiplexing for 2 memories per half
node (red solid line) and one round of purification in the first nesting level
(blue dashed line).

After analyzing the effects of multiplexing and purification independently, we
would like to compare both strategies. Although in Eq. (15) the rates for
$n=1$ and $r$ memories per half node have been calculated, for $n>1$, there is
not such an analytical formula. In this case, the rates were calculated
through an effective probability according to Eq. (III.3) with different
$P_{L_{0}}$ in each case. For the scheme with purification, we have
$P_{L_{0}}=P_{0}P_{1}\frac{(2-P_{0})}{(3-2P_{0})}$ and for multiplexing,
$P_{L_{0}}=\frac{1-(1-P_{0})^{2r}}{1+2(1-P_{0})^{r}}$. We kept the number of
initial resources equal in both schemes (two memories per half node). As it is
possible to observe in Fig. (7), even for a total distance as small as $L=40$
km, for sufficiently high fidelities, the scheme with just one round of
purification is providing higher rates. For longer distances, $L=640$ km and
$L=1280$ km, one round of purification is clearly more powerful than
multiplexing without purification.

One may argue that the effects of multiplexing become more significant in a
configuration with more memories per half node, i.e., bigger $r$. However, as
can be seen from Fig. (8), the rate to generate an entangled pair after one
round of purification (only two memories per half node) is higher than the
rate to generate an entangled pair for a scheme using multiplexing with 32
available memory pairs per segment, $r=32$, for fidelities larger than
$~{}0.95$. For multiplexing schemes with $r=16$, the purification is
performing better already for a final fidelity $F_{final}>0.84$. Taking into
account all these results and the fact that multiplexing is difficult to
implement, multiplexing will not be considered in our final analysis. It has
to be pointed out here that this argument is not valid anymore if the losses
considered in the controlled- phase gates are bigger than $0.01\%$ (see Sec.
II.B and App. A).

Figure 8: (color online). Rates to generate one entangled pair for a total
distance of $L=1280$ km with $L_{0}=20$ km. Comparison between one round of
purification in the first nesting level (blue dashed line) and multiplexing
(red solid line) for 32 memories per half node (above) and for 16 elements
memories per half node (below).

###  III.5 Results

Figure 9: (color online). Rates for total distance $L=80$ km (below), $L=160$
km (center), $L=320$ km (above) with $L_{0}=20$ km. Comparison between direct
transmission with one round of purification (blue, thick line), direct
transmission without purification (red, dashed line), quantum relay (black,
dotted line), hybrid quantum repeater with one round of purification (purple,
thick line), and hybrid quantum repeater without purification (black, dot
dashed line).

In Fig. (9) the rates for total distances $L=80$ km, $160$ km, $320$ km are
shown, with $L_{0}$ always equal to $20$ km. We have compared the case where
one round of purification takes place at the beginning before the connecting
steps (solid lines) to the case with no purification (dashed/dotted lines). We
have further compared the rates for the hybrid quantum repeater to the rate
upper bounds for transmitting the state directly (i.e., without swapping) and
to the rates of a quantum relay scheme, where all entangled pairs have to be
successfully generated in each segment at the same time (i.e., without using
memories). Even for a relatively small total distance, $L=80$ km, the hybrid
quantum repeater is performing better than the quantum relay or direct
transmission. For larger distances, $L=160$ km and $320$ km, the difference
between the rates of those schemes is even bigger, and the rates for the
quantum relay and direct transmission are effectively zero.

Aiming to analyze the length dependence of the rates, in Fig. (10), we plotted
the rates for the hybrid quantum repeater scheme with perfect memories and two
rounds of purification in the first nesting level for a total distance of
$L=1280$ km, $2560$ km, $5120$ km and $10240$ km. We observe that the rates
only decrease inverse-linearly with the total distance. For ideal memories and
using purification, we show that it is possible to avoid the exponential decay
of the rates with the distance.

In Fig. (11) we can show the rates of the hybrid quantum repeater for $L=1280$
km with different losses in the local gates, T. We show that the protocol can
run with local gate errors, however, depending on T, the final fidelity and
the generation rates can decrease very fast. For $1-T=0.001\%$ the rates are
practically the same as for $T=1$. For $1-T=0.01\%$ and a final fidelity of
0.95, rates of near 16 pairs per second can be achieved. On the other hand,
for $1-T=0.1\%$ rates of the order of only 10 pairs per second with a final
fidelity below 0.84 can be achieved.

Figure 10: (color online). Rates for the hybrid quantum repeater with two
rounds of purification for total distance $L=1280$ km (blue, thick line),
$L=2560$ km (red, dashed line), $L=$ 5120 km (purple, dotted line), $L=$ 10240
km (black, thin line). Figure 11: (color online). Rates for the hybrid quantum
repeater with two rounds of purification for total distance $L=1280$ km for
different parameters of local losses in the controlled-phase gate, $1-T=0$
(blue, thick line), $1-T=0.001\%$ (red, dashed line), $1-T=0.01\%$ (pink,
dotted line), $1-T=0.1\%$ (black, thin line).

##  IV Conclusion

In this work, we calculated analytically the rates for the so-called hybrid
quantum repeater. Comparing these rates with those obtainable through direct
entanglement distribution (without using swapping, purification, or memories)
and through quantum relay schemes (using swapping, but no purifications and no
memories), for sufficiently long distances, the hybrid quantum repeater scheme
leads to significantly better rates. In fact, only the hybrid quantum repeater
scales sub-exponentially with the channel length, while both direct
distribution and quantum relay keep the exponential decay with distance. We
found the rates for a system where the subroutines (generation, swapping, and
purification) can all be implemented through weak dispersive light-matter
interaction, as proposed in Ref. PvLa .

In the hybrid repeater scheme with homodyne measurement from Ref. PvLa , a
rate of 15 pairs per second with final $F=0.98$ was achieved using 16 qubits
per half node for a total distance of $L=1280$ km, segment lengths of
$L_{0}=10$ km, and local gate errors of $0.1\%$. In our case, by employing
generalized measurements (instead of homodyne projection measurements) for an
optimal unambiguous discrimination of coherent states (USD) as proposed in
Ref. PvLb , for just 4 qubits per half node, two rounds of purification in the
first nesting level, $L=1280$ km, $L_{0}=20$ km, and gate errors of $0.001\%$,
we achieved rates of the order of 100 pairs per second with $F=0.98$. Note
that in the scheme from Ref. PvLa  the rates were calculated performing Monte
Carlo simulations where not only more rounds of purification in different
nesting levels are allowed, but also multiplexing. Compared to the banded
scheme from Ref. vanmeter  for the same distances and final fidelities, we
achieve rates with similar order of magnitude, however, it should be pointed
out that Ref. vanmeter  uses a scheme with initially 50 qubits per half node,
and correspondingly more rounds of purification in different nesting levels.

We achieve our rates (of nearly 100 Hz) in a fairly practical setting where
the spatial resources are reduced (e.g. only four cavities per half node),
quantum error detection is close-to-minimal (i.e., only two rounds of
purification at the very beginning), and without spatial multiplexing.
However, these results are only obtained under the basic assumptions of very
low local losses, optimal USD measurements for entanglement distribution,
deterministic entanglement swapping, and perfect quantum memories. In future
work, we aim to relax one or more of these requirements. In fact, our
analytically obtained rates can be certainly further improved, if we include
multiplexing as well as purification in different nesting levels, possibly
leading to an improvement of two orders of magnitude compared to the scheme
from Ref. PvLa , and similar to the results presented in Ref. munro .
Moreover, further improvements in the initial entanglement distribution can be
achieved, if instead of coherent states and USD measurements, squeezed states
and homodyne measurements (together with extra local phase-space
displacements) are utilized, as was recently shown in Ref. ludmila . Besides
its quantitative significance, the work presented here represents a first step
towards a better understanding of the importance of each of the building
blocks in the architecture of a hybrid quantum repeater.

### Acknowledgments

Support from the Emmy Noether Program of the Deutsche Forschungsgemeinschaft
is gratefully acknowledged. L.P. gratefully acknowledges financial support for
the Future and Emerging Technologies (FET) programme within the Seventh
Framework Programme for Research of the European Commission, under the FET-
Open grant agreement CORNER no. FP7-ICT-213681.

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## Appendix A

Purification and entanglement swapping can be performed utilizing controlled-
phase (CZ) gates, Hadamard operations, and measurements on the qubits.
Assuming that the CZ-gate is the only operation that leads to errors in those
schemes, we are able to calculate the final fidelity for the purification and
swapping, as well as the probability of purification, using the imperfect CZ-
gate obtained in Ref. Louis . These are as follows:

$\displaystyle
P_{purification}^{imp}=\frac{1}{2}+e^{\frac{\pi(T-1)}{\sqrt{T}}}(F-1)F+$
$\displaystyle\left(e^{\frac{\pi(T-1)\left(2-i\text{sech}\left(\frac{\log(T)}{2}\right)\right)}{2\sqrt{T}}}+e^{\frac{\pi(T-1)\left(2+i\text{sech}\left(\frac{\log(T)}{2}\right)\right)}{2\sqrt{T}}}\right)\times$
$\displaystyle\frac{(2(F-1)F+1)}{4},$ (22)

$\displaystyle
F_{purification}^{imp}=\frac{F+Fe^{-\frac{\pi\left(2-2T\right)}{\sqrt{T}}}}{4P_{purification}^{imp}}+$
$\displaystyle\frac{2Fe^{\frac{\pi(T-1)}{\sqrt{T}}}\left(\sin\left(\pi\left(\frac{3}{2}-\frac{2}{T+1}\right)\right)F+F-1\right)}{4P_{purification}^{imp}},$
(23)

$\displaystyle
F_{swapping}^{imp}=\frac{1}{4}+\frac{1}{4}e^{2\pi\sinh\left(\frac{\log(T)}{2}\right)}(1-2F)^{2}+$
$\displaystyle\frac{e^{\pi\sinh\left(\frac{\log(T)}{2}\right)}}{2}(2(F-1)F+1)\cos\left(\frac{\pi}{2}\tanh\left(\frac{\log(T)}{2}\right)\right).$
(24)

## Appendix B

Consider a coin, which, in one toss, gives “tail” as an outcome with
probability $p$ and “head” with probability $q=1-p$. Imagine we are just
interested in the tail’s outcomes. We will here calculate the number of times
we have to toss one or more coins to obtain one, two, three or more tails.
Note that the problem of calculating the average time necessary to have one
success event in this case is analogous to our problem of calculating the
average time to successfully generate one (or many) entangled pair(s) in
different segments of the repeater.

#### 1) one success

Imagine that a coin is flipped repeatedly until the first tail outcome. How
many times on average do we have to toss the coin until we get a tail? The
average time necessary to obtain one tail is:

$\sum_{k=1}^{\infty}kq^{k-1}p=\frac{1}{p}=:\mathcal{Z}_{1}(p).$ (25)

Note that the character $\mathcal{Z}$ is used here in Appendix B for the
average time to have tail as an outcome; when average times are calculated to
generate entanglement over repeater segments, the character $Z$ is used
instead in the main text and Appendix C.

#### 2) two successes

Now imagine we have two identical coins and we flip both coins at once until
we get a “tail and tail” outcome. If two tails are obtained in the same trial,
game is over. If “tail and head” result is obtained, we keep the tail coin
(simulating the situation of memory qubits in the repeater) and flip only the
other coin until the second tail is obtained. In this case, the average time
needed to obtain two tails is:

$\displaystyle\sum_{k=1}^{\infty}k(q^{2})^{k-1}p^{2}+\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}(k+l)((q^{2})^{k-1}2pq)(q^{l-1}p)=$
$\displaystyle\frac{1}{p}\frac{(3-2p)}{(2-p)}=:\mathcal{Z}_{2}(p).\,$ (26)

The first sum describes the cases when we had the head outcomes in both coins
$(k-1)$ times and then 2 tails were obtained at once. The second sum counts
the cases when at $k$-th trial a tail in one coin was obtained and stored and
then after another $l$ (single) trials we got the second success.

#### 3) three and more successes

For 3 successes (in this case 3 tails) in 3 identical coins, the average time
needed will be:

$\displaystyle\\!\\!\\!\\!\\!\\!\sum_{k=1}^{\infty}k(q^{3})^{k-1}p^{3}+\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}(k+l)((q^{3})^{k-1}3p^{2}q)(q^{l-1}p)$
$\displaystyle+\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}(k+l)((q^{3})^{k-1}3pq^{2})((q^{2})^{l-1}p^{2})$
$\displaystyle\\!\\!\\!\\!\\!\\!+\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}\sum_{m=1}^{\infty}(k+l+m)((q^{3})^{k-1}3pq^{2})((q^{2})^{l-1}2pq)(q^{m-1}p)$
$\displaystyle=\\!\\!\frac{p(19+3p(p-4))-11}{p(p-2)(p(p-3)+3)}=:\mathcal{Z}_{3}(p).\;\;$
(27)

In a similar way, we calculate average waiting times for $N$ successes (tails)
in $N$ coins and obtain the following recurrence formula:

$\mathcal{Z}_{N}(p)=\frac{1}{1-q^{N}}\left(1+\sum_{j=1}^{N-1}{N\choose
j}q^{j}p^{N-j}\mathcal{Z}_{j}(p)\right),$ (28)

where $\mathcal{{Z}}_{1}(p)=\frac{1}{p}$.

Recurrence formula (28) can be solved, and its solution is given by

$\displaystyle\mathcal{Z}_{N}(p)=\sum_{k=1}^{N}{N\choose
k}\frac{(-1)^{k+1}}{1-(1-p)^{k}}\;.$ (29)

By substituting $N$ by $2^{n}$ in Eq. (29), we can calculate the average
number of steps necessary to successfully generate entanglement in $2^{n}$
pairs, as in Eq. (13).

## APPENDIX C

In order to compare the multiplexing with the parallelization strategy, we
have to calculate the average times needed to successfully generate entangled
pairs over the corresponding segments. We will start our calculation here with
the simplest case (two columns and two rows, similar to Fig. 3 with two
memories per half node).

#### 1) two columns ($n=1$), two rows ($r=2$)

Imagine now that we have 4 coins to toss; coins that can be distinguished by
their positions, arranged into two columns, two rows each. We toss the coins
all at once in the first step, and in the next steps, we toss only those coins
that had a head outcome in the previous trial. The average number of steps
needed for at least one success (tail) in every column will obviously depend
on whether these successes are required to appear in the same row or whether
they are allowed to appear in two different rows. The first case corresponds
to parallelization, the second case to multiplexing. Probabilities and average
numbers of steps are calculated in a similar manner to Appendix B, taking into
account that successes have to be appropriately distributed among columns and
rows. In the following, we will not refer to coins anymore, but to repeater
segments and the probabilities to generate entanglement over these segments.

The probability that at least one entangled pair is created in every column in
the case of multiplexing, i.e., when it does not matter in which row the
entanglement was created, is given by:

$\displaystyle
P_{mult,1,2}=\sum_{k=1}^{\infty}(q^{4})^{k-1}(p^{4}+4p^{3}q+4p^{2}q^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}2p^{2}q^{2}\sum_{l=1}^{\infty}(q^{2})^{l-1}(p^{2}+2pq)$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}(p^{3}+3p^{2}q+2pq^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}pq^{2}\sum_{m=1}^{\infty}(q^{2})^{m-1}(p^{2}+2pq)\;.$

$P_{mult,n,r}$ is the probability to create at least one entangled pair in
every column from the $2^{n}$ columns with $r$ rows, in the case of
multiplexing. The average number of steps needed for at least one success in
every column in this case can be calculated as follows:

$\displaystyle
Z_{mult,1,2}(q)=\sum_{k=1}^{\infty}(q^{4})^{k-1}k(p^{4}+4p^{3}q+4p^{2}q^{2})$
(31) $\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}2p^{2}q^{2}\sum_{l=1}^{\infty}(k+l)(q^{2})^{l-1}(p^{2}+2pq)+$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4pq^{3}\sum_{l=1}^{\infty}(k+l)(q^{3})^{l-1}(p^{3}+3p^{2}q+2pq^{2})+$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}pq^{2}\times$
$\displaystyle\sum_{m=1}^{\infty}(q^{2})^{m-1}(k+l+m)(p^{2}+2pq)$
$\displaystyle=$ $\displaystyle\frac{1+2q^{2}}{1-q^{4}}\;.$

The probability that at least two “parallel” entangled pairs are created is:

$\displaystyle
P_{parallel,1,2}=\sum_{k=1}^{\infty}(q^{4})^{k-1}(p^{4}+4p^{3}q+2p^{2}q^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4p^{2}q^{2}\sum_{l=1}^{\infty}(q^{2})^{l-1}(p^{2}+2pq)$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}(p^{3}+3p^{2}q+pq^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}2pq^{2}\sum_{m=1}^{\infty}(q^{2})^{m-1}(p^{2}+2pq)\,.$

The corresponding average number of steps needed for parallel successes in two
columns ($n=1$) is given by:

$\displaystyle
Z_{parallel,1,2}(q)=\sum_{k=1}^{\infty}(q^{4})^{k-1}k(p^{4}+4p^{3}q+2p^{2}q^{2})$
(33) $\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4p^{2}q^{2}\sum_{l=1}^{\infty}(k+l)(q^{2})^{l-1}(p^{2}+2pq)$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4pq^{3}\sum_{l=1}^{\infty}(k+l)(q^{3})^{l-1}(p^{3}+3p^{2}q+pq^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}2pq^{2}\times$
$\displaystyle\sum_{m=1}^{\infty}(q^{2})^{m-1}(k+l+m)(p^{2}+2pq)$
$\displaystyle=$
$\displaystyle\frac{1+q+5q^{2}+4q^{4}}{1+q+q^{2}-q^{4}-q^{5}-q^{6}}\,.$

#### 2) two columns ($n=1$), $r>2$ rows

Calculations similar to those from Eq. (31) show that for 3, 4,…, r rows (see
Fig. 3 with $r$ memories per half node) the average number of steps needed for
at least one success in every column in the case of multiplexing is given by:

$\displaystyle Z_{mult,1,3}(q)=\frac{1+2q^{3}}{1-q^{6}},$ (34)

$\displaystyle Z_{mult,1,4}(q)=\frac{1+2q^{4}}{1-q^{8}},$ (35)

and

$\displaystyle Z_{mult,1,r}(q)=\frac{1+2q^{r}}{1-q^{2r}},$ (36)

respectively.

In the main text, Eq. (15) is, except for $T_{0}$, exactly the inverse of Eq.
(36) with $q=1-P_{0}$

## Appendix D

The choice of purifying the entangled pairs at the very beginning can be
intuitively justified through the probabilistic character of the purification
procedure and the deterministic entanglement swapping. Taking as much
advantage from the perfect memories as possible, it is reasonable to think
that the earlier the purification starts, the smaller the necessary times to
generate and purify an entangled pair will be. For a more quantitative
justification, let us compare the rates to generate a purified entangled pair
in two extreme cases. For purification occurring at the first nesting level,
the rate will be calculated as in Eq. (19). On the other hand, for
purification at the end (i.e., at the last nesting level), the rates will be
given by $R_{purif,n,end}=\frac{P_{1}}{T_{0}Z_{2n}(P_{0})}$. Recall that both
$P_{0}$ and $P_{1}$ are functions of the fidelity and that we are comparing
the rates for the same final fidelity. Hence these quantities will have
different values in each case. In Fig. (12) we illustrate that a scheme where
purification occurs in the first level performs better than one with
purification at the end.

Figure 12: (color online). Rates for the hybrid quantum repeater with one
round of purification at the first nesting level (blue dashed line) and at the
last nesting level (red line) for total distance $L=1280$ km and $L_{0}=20$
km.