]

#  $\nu$Oscillation: a software package for computation and simulation of
neutrino propagation and interaction

Seonghyeok Jang  Eunil Won  Kyungmin Lee<EMAIL_ADDRESS>Department of
Physics, Korea University, Seoul 02841, Korea  Eunju Jeon  Young Ju Ko
<EMAIL_ADDRESS>Center for Underground Physics, Institute for Basic Science,
Daejeon, 34047, Korea

([)

###### Abstract

The behavior of neutrinos is the only phenomenon that cannot be explained by
the standard model of particle physics. Because of these mysterious neutrino
interactions observed in nature, at present, there is growing interest in this
field and ongoing or planned neutrino experiments are seeking solutions to
this mystery very actively. The design of neutrino experiments and the
analysis of neutrino data rely on precise computations of neutrino
oscillations and scattering processes in general. Motivated by this, we
developed a software package that calculates neutrino production and
oscillation in nuclear reactors, neutrino-electron scattering of solar
neutrinos, and the oscillation of neutrinos from radioactive isotopes for the
search of sterile neutrinos. This software package is validated by reproducing
the result of calculations and observations in other publications. We also
demonstrate the feasibility of this package by calculating the sensitivity of
a liquid scintillator detector, currently in planning, to the sterile
neutrinos. This work is expected to be used in designs of future neutrino
experiments.

neutrino oscillation, neutrino detector, detector simulation

###### pacs:

14.60.Pq, 13.15.+g, 29.40.Mc

##  I Introduction

The standard model (SM) of particle physics ref:SM ; ref:SM2 ; ref:SM3 ;
ref:SM4 ; ref:QCD ; ref:qcd2  describing the electroweak and strong nuclear
interactions of subatomic particles in microscopic scales has been validated
almost over a half-century extremely successfully, consistent with its
prediction, except for one notable exceptional deviation in the neutrino
sector. The SM predicts that neutrinos are precisely massless, yet experiments
ref:superK_atmos ; ref:sno2001 ; ref:kamland ; ref:SNO2  indicate that
neutrinos have small but finite mass values, allowing quantum oscillations
among eigenstates of the weak force in subatomic particle interaction. In
order to search for solutions to this one of profound fundamental problems in
modern particle physics, tremendous efforts are focused, on ongoing and future
neutrino experiments ref:hyperkamiokande ; ref:dune , not to mention previous
precious experiments. These experimental efforts often require computation of
neutrino production, propagation, and interaction with matter at different
experimental conditions such as nuclear reactors, solar, accelerator, or
radioactive source-based environments.

For the detailed study of the phenomena with neutrinos, there have been number
of software packages developed to calculate the interaction and propagation of
neutrinos ref:n_packages ; ref:n_packages2  and references therein. A large
number of existing packages are theoretically oriented, mathematically very
rigorous, and focused on the calculation of given Hamiltonian, including non-
standard interactions. For feasibility study and design optimization of future
experiments, a software tool that also offers enough precision and flexibility
is desired to carry out detailed studies on future neutrino experiments.

Motivated by these, we developed a unified software package, the
$\nu$Oscillation, which calculates neutrino production and oscillation in
nuclear reactors, neutrino-electron scattering of the neutrinos originating
from the Sun, and the oscillation of SM neutrinos from radioactive isotopes to
hypothetical non-SM sterile neutrinos. This work is expected to be utilized to
identify the physics potential and in the design optimization of future
neutrino experiments.

##  II Implementation of the Software

At present, observed neutrinos appear to be point-like, spin $1/2$ elementary
particles and to have three different kinds, commonly called flavours, in
their weak interactions with matter. Because of aforementioned their non-zero
values of mass, the relation between flavour and mass eigenstates of neutrinos
are different and their relations are commonly described by the lepton mixing
matrix, known as the Pontecorvo–Maki–Nakagawa– Sakata (PMNS) matrix ref:pmns

$\displaystyle|\nu_{\alpha}\rangle=\sum_{i}U^{*}_{\alpha i}|\nu_{i}\rangle,$
(1)

where the flavour index $\alpha$ is $e,\mu$ or $\tau$, the mass index $i$ runs
over $1,2,3$, and $U_{\alpha i}$ are elements of the PMNS matrix. The time-
evolution of Eq. (1), can be described by time-dependent Schrödinger equation,
naturally involve mixing of flavour eigenstates as the given system evolves,
and is referred to as the neutrino oscillation. The probability amplitude for
the neutrino of flavour $\alpha$ to oscillate into a different flavour $\beta$
at the time $t$ is given by $\langle\nu_{\beta}|\nu(t)\rangle$, providing an
oscillation probability

$\displaystyle
P(\nu_{\alpha}\rightarrow\nu_{\beta})=|\langle\nu_{\beta}|\nu_{\alpha}(t)\rangle|^{2}=\Bigg{|}\sum_{i}U_{\beta
i}U^{*}_{\alpha i}e^{-iE_{i}t}\Bigg{|}^{2}$ (2)

where $E_{i}$ is the energy of the $i$-th mass eigenstate. Our package,
$\nu$Oscillation computes Eq. (2) at the ultra-relativistic limit of
neutrinos, and for our computation, we use analytic expressions in Ref.
ref:Zito .

Our implementation of the software, $\nu$Oscillation mainly computes the time
evolution of Eq. (1) in given initial conditions set by users. The
$\nu$Oscillation also contains computation of inverse beta decay (IBD) cross
section, neutrino-electron scattering cross section, and the flux of anti-
neutrinos from nuclear reactors, and neutrinos from the Sun, in order to be
flexible to different experimental scenarios. We also include one type of
fictitious sterile neutrino in the computation of time-evolution of neutrinos
as an option to study the sensitivity of the sterile neutrino search on
proposed experiments.

Our code is written in C++ implemented as C++ header and source codes
utilizing libraries in ROOT ref:root , one of the open-source data analysis
frameworks actively used in the field of particle physics. Note also that the
ROOT is the de facto standard analysis framework for the experimental high
energy physics community. Intended users of $\nu$Oscillation are required to
install ROOT along with $\nu$Oscillation, but nothing more. This allows users
to utilize $\nu$Oscillation on any operating system that supports ROOT. The
source code can be provided upon request.

The $\nu$Oscillation package has the following structure as shown in Fig. 1.
The vOscillation class has functions and variables to compute neutrino
oscillations based on the PMNS matrix, and its subclass, vSterile introduces
sterile neutrinos in the oscillation based on so-called (3+1) model. Note that
the vSterileEE class is a simplified implementation of the (3+1) model
accounting only the electron flavour initial state. Users can freely set the
neutrino oscillation parameters such as $\Delta m^{2}_{ji}$ and $\theta_{ij}$
where $(i,j)$ can be $(1,2),(1,3)$ or $(2,3)$. Here, $\Delta m^{2}_{ji}$ and
$\theta_{ij}$ represent the mass squared difference between two mass
eigenstates of a given neutrino pair and the mixing angle between them,
respectively ref:Zito . Users can also set charge-conjugation and parity
$(CP)$ phase. The matter effect, which is non-negligible in the long-baseline
neutrino experiments, has not been yet implemented in this package.

The calculation of energy spectra of neutrinos from radioactive sources,
nuclear reactors, and the Sun is implemented in the three distinct classes.
The vBetaSpectrum class computes the beta spectrum of radioactive sources,
employing the formula derived by Fermi ref:Fermi_Wilson . Its subclass,
vCe144Spectrum, specifies the mass number, released energy, and atomic number
of the daughter nucleus to compute the beta spectrum from ${}^{144}\text{Ce}$.
In a similar manner, the computation of energy spectra for various radioactive
sources can be implemented. The vReactorSpectrum and vSolarFlux classes are
responsible for computing the neutrino spectra from the reactors and the Sun.
The details of these implementations will be described in the following
sections.

The vIBD class computes the IBD cross section using the formula with first-
order corrections for recoil and weak magnetism ref:Vogel . The vScatteringve
class computes the scattering cross section between neutrino and electron
based on the formula considering radiative and electroweak corrections
ref:Bahcall_1995 . Both classes provide the option to use standard formulas
without corrections for comparison ref:tHooft_1971 ; ref:Bahcall_1987 .

The Monte Carlo simulation of radioactive-source-based experiments is
implemented in vModelGenerator and its subclasses. The simulation generates
the expected spectrum of detected neutrinos at the detector. The detector and
the source are defined with the vDetector class and the vSource class. Both
classes include functions for generating random positions within the detector
or the source, simulating the travel distance of the neutrino. Constants used
in the calculations, such as the masses of proton or neutron, are defined in
the vConstant class.

Figure 1:  A simplified functional block diagram of the $\nu$Oscillation
package, which includes calculations for the neutrino flux from nuclear
reactors, the Sun, or from radioactive sources. It covers reactions such as
IBD, beta decay, and neutrino-electron scattering. The package also implements
neutrino mixing based on the PMNS matrix.

In order to validate the mixing effects implemented in the package
$\nu$Oscillation, we reproduce an illustration of the neutrino oscillation in
Ref. ref:nature_JUNO . It shows the flavour fractions as a function of the
propagation distance when the neutrino energy is 4 MeV. We use the same
oscillation parameters as in the Ref. ref:pars_PDG2014 , and the result is
shown in Fig. 2. This confirms that our result is at least qualitatively
consistent with the reference by comparing the locations of the bumps and the
flavour fraction of the neutrinos.

Figure 2:  The fractions of each flavour as a function of distance computed by
$\nu$Oscillation, are shown when there are $\bar{\nu}_{e}$ only in the initial
state.

##  III Nuclear Reactor Neutrinos

The first step in computing the neutrino oscillation from nuclear reactors is
to calculate the flux of anti-neutrinos produced by nuclear fission reactions
within the reactors. For this calculation, we implement two models: the Huber-
Mueller ref:Huber ; ref:Mueller  and the Gutlein model ref:Gutlein  in the
vReactorSpectrum class. In the Gutlein model, the isotopes 235U, 238U, and
239Pu, are included, while the Huber-Mueller model includes 241Pu in addition.
This calculation requires input values of the relative abundance of each
isotope and the thermonuclear power (in Watts we denote
$\textrm{W}_{\textrm{th}}$) of a reactor. Using them as input information, the
averaged differential neutrino flux at a distance $L$ from a nuclear reactor
core is computed as

$\displaystyle\Bigg{\langle}\frac{d\phi}{dE_{\bar{\nu}_{e}}}\Bigg{\rangle}=\frac{W_{\textrm{th}}}{\sum_{i}f_{i}E_{\textrm{re},i}}\frac{1}{4\pi
L^{2}}\sum_{i}f_{i}\frac{d\phi_{i}}{dE_{\bar{\nu}_{e}}}$ (3)

where $W_{\textrm{th}}$ is the thermal power of the reactor, $f_{i}$ is the
fraction of a given fission isotope $i$, $E_{\textrm{re},i}$ is the thermal
energy per fission, and $d\phi_{i}/dE_{\bar{\nu}_{e}}$ is the differential
neutrino flux for the isotope $i$ at the nuclear reactor. For the detection of
anti-neutrinos from a reactor, one has to consider two factors. First is the
oscillation effect due to the neutrino mixing, and the second is the
interaction of the anti-neutrinos in the detector, and for that one has to
compute the event rate using the IBD cross section. Based on these, the
detection probability can be expressed as

$\displaystyle
P^{\textrm{det}}_{\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}}(L)=\frac{\int^{\infty}_{0}\langle\frac{d\phi}{dE_{\bar{\nu}_{e}}}\rangle\sigma(E_{\bar{\nu}_{e}})P_{\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}}(L,E_{\bar{\nu}_{e}})dE_{\bar{\nu}_{e}}}{\int^{\infty}_{0}\langle\frac{d\phi}{dE_{\bar{\nu}_{e}}}\rangle\sigma(E_{\bar{\nu}_{e}})dE_{\bar{\nu}_{e}}}$
(4)

where $\sigma(E_{\bar{\nu}_{e}})$ is the IBD cross section, and
$P_{\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}}(L)$ is the survival probability for
electron anti-neutrino, respectively. Here we validate our computation to the
existing nuclear reactor based experiment by reproducing a result in Ref.
ref:nature_JUNO , and our result is shown in Fig. 3. The expected spectrum of
the detected electron-antineutrinos is calculated by considering the reactor
spectrum, the IBD cross section, and the neutrino oscillation with assuming a
reactor located 53 km away from the detector. The energy resolution of 5% is
assumed.

Figure 3:  The expected spectrum of the detected electron-antineutrinos from a
reactor located 53 km away from the detector is shown. The energy resolution
of 5% is assumed.

For a case study with our $\nu$Oscillation, we choose an underground facility
recently developed in Yemilab ref:Yemilab . This is an underground laboratory
located at $37.2^{\circ}$ N, $128.7^{\circ}$ E in latitude and longitude of
the Republic of Korea. A large hall, capable of hosting a 2 kton detector is
located about 1000 m below the Yemi mountain. There are four nuclear reactors
to be considered for the reactor anti-neutrino flux, and they are 65, 180,
216, and 282 km away with nuclear thermal power (name) of 20.8 (Hanul), 11.8
(Wolsong), 21.3 (Gori), and 16.9 (Hanbit) $\text{GW}_{\text{th}}$,
respectively111International Atomic Energy Agency (IAEA), Power reactor
information system, https://pris.iaea.org/, accessed: 2023/08/22. .

Figure 4:  Top: our computation of the survival probability of anti-electron
neutrino as a function of the distance from a nuclear reactor. The dashed
lines indicate the location of the detector if a given reactor is at 0 km.
Bottom: The expected IBD events from four nuclear reactors.

The distributions of IBD events as a function of distance and neutrino energy
are shown in Fig. 4. Despite of having the lowest detection probability, the
Hanul reactor is the dominant source of the IBD events as it is closest to the
detector site. The estimated event rate of this source is approximately 860
events per year. In principle, with large statistics of data with excellent
energy resolution, one can see the fine structure of a rapid oscillation due
to $\theta_{13}$, but in practice, it will be limited by the volume of the
detector and the energy resolution of the detector that originates from the
photo-electron statistics.

##  IV Solar Neutrinos

The Sun is a natural source of electron neutrinos by nuclear fusion processes
and by now so called the standard solar model ref:Bahcall  describes the flux
of electron neutrinos from various different fusion processes very precisely.
In our study, we use parameters from the B16-GS98 model ref:Vinyoles  to
compute the solar neutrino flux. Solar neutrinos, arriving at detectors on
Earth interact with detectors via neutrino-electron scattering. The
differential yields of the scattering at different recoil kinetic energy ($T$)
of the electron can be expressed as

$\displaystyle\frac{dN_{e}}{dT}$ $\displaystyle=$ $\displaystyle
n_{e}\int^{E_{\nu,\textrm{max}}}_{E_{\nu,\textrm{min}}}dE_{\nu}\frac{d\Phi}{dE_{\nu}}(E_{\nu})$
$\displaystyle\Bigg{(}\frac{d\sigma_{\nu_{e}}}{dT}(E_{\nu},T)P_{ee}+\frac{d\sigma_{\nu_{\mu},\nu_{\tau}}}{dT}(E_{\nu},T)(1-P_{ee})\Bigg{)}$
(5)

where $n_{e}$ is the target electron number, $\frac{d\Phi}{dE_{\nu}}$ is the
differential solar neutrino flux, $P_{ee}$ is the oscillation probability to
the electron neutrino final state, $\frac{d\sigma_{\nu_{e}}}{dT}$ is the
differential cross section of electron neutrino-electron ($\nu_{e}$-$e$)
scattering ref:Bahcall_1995 , and $\frac{d\sigma_{\nu_{\mu},\nu_{\tau}}}{dT}$
is sum of the differential cross sections of $\nu_{\mu}$-$e$ and
$\nu_{\tau}$-$e$ scattering.

Figure 5:  Neutrino-electron elastic scattering cross sections with different
electron recoil energy values.

Figure 5 shows the implemented cross sections of neutrino-electron scatterings
in Eq. (5) at different electron recoil energy values with different flavours
of neutrinos. They are integrated with the solar neutrino flux model mentioned
earlier in order to obtain spectra of scattered electron kinetic energy. Such
spectra for $pp$, $pep$, CNO, ${}^{7}\textrm{Be}$, and ${}^{8}\textrm{B}$
neutrinos are shown in Fig. 6. The yield is normalized to the events per day
per 1 keV per 100 ton of a liquid scintillator detector and the 5% energy
resolution is assumed. The normalization and shapes are in general in
agreement with results in Ref. ref:Borexino2021 . We also include two
irreducible backgrounds. The first one is ${}^{14}\textrm{C}$ and is populated
low end of the electron kinetic energy range. Here we assume the concentration
of ${}^{14}\text{C}/^{12}\text{C}=1.94\times 10^{-18}$ ref:Borexino1998 . The
second one is the so-called cosmogenic ${}^{11}\textrm{C}$ background that is
sensitive to the depth of the site. Here we assume a site with 1000 m
overburden. The ${}^{11}\text{C}$ neutrino spectrum differs in its shape from
the reference spectrum ref:Borexino2021  because our result does not include
detector responses such as the quenching as they are detector specific
information.

Figure 6:  Expected electron kinetic energy distribution from various fusion
processes in the Sun. The yield is normalized to the number per day per 1 keV
per 100 tons of a detector. Also included two major irreducible backgrounds
from ${}^{14}\textrm{C}$ and ${}^{11}\textrm{C}$.

##  V Sterile Neutrinos

Recently, several experiments indicate that the survival probability of
electron neutrino in short range may be smaller than expected ref:LSND_anomaly
; ref:miniboone_anomaly ; ref:stereo_anomaly . This may originate due to the
presence of a different type of neutrino, so-called sterile neutrino. We
implement a (3+1) neutrino model in the $\nu$Oscillation, in which four mass
eigenstates including one sterile neutrino, are mixed to construct three
flavours.

Figure 7:  Energy distributions of neutrinos from $\beta^{-}$ decays of
${}^{144}\textrm{Ce}$ and ${}^{144}\textrm{Pr}$.

In order to estimate the sensitivity to the sterile neutrino search with the
given radioactive source, we assume a radioactive source of 144Ce with
activity of 100 kCi as proposed earlier by other experiments ref:celand ;
ref:dayabay ; ref:sox . The source is assumed to be located at the distance
9.5 m from the center of a cylindrical linear alkylbenzene (LAB) based liquid
scintillator of 2 kton with a radius of 7.5 m. The ${}^{144}\textrm{Ce}$
decays to ${}^{144}\textrm{Pr}$ with the half life of 285 days, and
${}^{144}\textrm{Pr}$ decays to ${}^{144}\textrm{Nd}$ with the half life of 17
minutes, emitting $\bar{\nu}_{e}$. The energy distributions of anti-neutrinos
from radioactive sources are shown in Fig. 7. Note that since the threshold
for IBD is 1.8 MeV, only neutrinos from ${}^{144}\textrm{Pr}$ are sensitive to
IBD. These spectra are implemented in $\nu$Oscillation accordingly.

Our likelihood for the binned data, $\bf n$, and given parameter set,
$\Theta$, is defined as

$\mathcal{L}(\Theta|{\bf
n})=\prod_{i=1}^{N}\frac{m_{i}^{n_{i}}e^{-m_{i}}}{n_{i}!},$ (6)

where $N$ is the number of bins, $n_{i}$ is the number of observed events in
the $i$-th bin, and $m_{i}=m_{i}(\Theta)$ is the expected number of events
from the model. Here, the binned data is in $L/E$, the ratio of the travel
distance ($L$) of a neutrino to its energy ($E$). The model describes the
expected distribution of neutrino events as a function of $L/E$. The parameter
set, $\Theta=\\{\sin^{2}2\theta_{14},\Delta m^{2}_{41}\\}$, including the
oscillation angle and the squared mass difference between the mass eigenstates
1 and 4, is commonly used to describe neutrino oscillation including sterile
neutrino.

Figure 8:  Exclusion sensitivity of the (3+1) neutrino model at 90%, 95%, and
99% confidence levels. A simulated data set of 34,000 IBD events without
including sterile neutrino is compared with the (3+1) models. The one year of
data taking and 70% detection efficiency are assumed. The best fit of global
analysis of disappearance data and the BEST experiment are also marked, along
with the $1$-$3\sigma$ confidence levels of the result of BEST indicated with
green contours.

We simulate a data set without including the sterile neutrino oscillations. In
total 34,000 neutrino events are generated by assuming the one year of data
taking with 70% detection efficiency. The resolution for energy and position
are not considered in this data. By comparing this data to the expected models
with the sterile neutrino oscillations, we quantify the exclusion sensitivity
at 90%, 95% and 99% confidence levels as shown in Fig. 8. The best fit of the
global analysis of disappearance data ref:globalfit2020  and the result of the
Baksan Experiment on Sterile Transitions (BEST) ref:best2022  are also marked.

##  VI Conclusions and Outlook

We developed a software package, $\nu$Oscillation, that calculates neutrino
production and oscillation in nuclear reactors, neutrino-electron scattering
from neutrinos originating from the Sun, and the oscillation of neutrinos
originating from radioactive isotopes for the search of sterile neutrinos.
Using our package, we discussed applications to the various neutrino
experiments. We validated the package by reproducing the simulations on the
neutrino spectra of reactor, solar and source neutrinos and the detection of
them. We also verified the feasibility of this by calculating the sensitivity
of a liquid scintillator detector to the sterile neutrinos. The current
version of this software does not include the matter effect on the neutrino
oscillation. This feature will be also implemented in the next release. This
work is expected to be used in designs and optimizations of future neutrino
experiments.

###### Acknowledgements.

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (No. NRF-2022R1A2B5B02001535) and
Institute for Basic Science (IBS) under the project code IBS-R016-D1.

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# Resource-efficient fault-tolerant one-way quantum repeater with code
concatenation

Kah Jen Wo<EMAIL_ADDRESS>QuTech, Delft University of Technology, Lorentzweg
1, 2628 CJ Delft, The Netherlands  Guus Avis<EMAIL_ADDRESS>QuTech,
Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
Quantum Computer Science, EEMCS, Delft University of Technology, Lorentzweg 1,
2628 CJ Delft, The Netherlands  Kavli Institute of Nanoscience, Delft
University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands  Filip
Rozpędek  Pritzker School of Molecular Engineering, University of Chicago,
Chicago, IL 60637, USA  Maria Flors Mor-Ruiz  Universität Innsbruck, Institut
für Theoretische Physik, Technikerstraße 21a, 6020 Innsbruck, Austria  Gregor
Pieplow  Department of Physics, Humboldt-Universität zu Berlin, Newtonstraße
15, 12489 Berlin, Germany  Tim Schröder  Department of Physics, Humboldt-
Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany  Liang Jiang
Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL
60637, USA  Anders S. Sørensen  Center for Hybrid Quantum Networks (Hy-Q), The
Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100
Copenhagen Ø, Denmark  Johannes Borregaard<EMAIL_ADDRESS>QuTech,
Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

(June 12, 2023)

###### Abstract

One-way quantum repeaters where loss and operational errors are counteracted
by quantum error correcting codes can ensure fast and reliable qubit
transmission in quantum networks. It is crucial that the resource requirements
of such repeaters, for example, the number of qubits per repeater node and the
complexity of the quantum error correcting operations are kept to a minimum to
allow for near-future implementations. To this end, we propose a one-way
quantum repeater that targets both the loss and operational error rates in a
communication channel in a resource-efficient manner using code concatenation.
Specifically, we consider a tree-cluster code as an inner loss-tolerant code
concatenated with an outer 5-qubit code for protection against Pauli errors.
Adopting flag-based stabilizer measurements, we show that intercontinental
distances of up to 10,000 km can be bridged with a minimal resource overhead
by interspersing repeater nodes that each specializes in suppressing either
loss or operational errors. Our work demonstrates how tailored error-
correcting codes can significantly lower the experimental requirements for
long-distance quantum communication.

##  I Introduction

The ability to faithfully transmit quantum information over long distances
opens up new opportunities for secure communication [1, 2], sensing networks
[3] and distributed quantum computing [4]. There has been impressive progress
towards the realization of quantum networks with the demonstration of long-
distance entanglement distribution through satellites [5], memory-enhanced
quantum communication [6], and a multi-node quantum network [7]. Quantum
communication over intercontinental distances, however, remains a formidable
challenge due to attenuation and degradation of the quantum signal as a result
of loss and operational errors.

There are, in general, two classes of quantum repeater architectures that have
been proposed to overcome these obstacles [8, 9]. Two-way repeaters divide the
total distances into smaller links where heralded entanglement can be created
in a probabilistic manner by direct photon transmission and success is
heralded by two-way communication between the repeater nodes [10, 11, 12].
Such architectures require long-lived multi-mode quantum memories to reach
high communication rates [13]. An alternative approach, which is the focus of
this work, is to encode the quantum information in quantum error-correcting
codes to battle both loss and operational (Pauli) errors as the signal is
being transmitted from one repeater node to the next in a one-way architecture
[14, 15, 16, 17, 18, 19]. Such fault-tolerant one-way repeaters allow bridging
arbitrary distances with high communication rates since the repetition rate is
set by the local processing time of the repeater nodes. However, they often
have daunting requirements in terms of resources needed for the physical
implementation of the repeater [15, 16, 20, 21]. Focusing on the dominant
error of photon loss and abandoning fault tolerance for operational errors can
significantly relax these requirements as recently demonstrated in Ref. [17].
In particular, it was shown that by using only loss-tolerant photonic tree-
cluster encoding, the bottleneck of transmission loss could effectively be
overcome with only three spin qubits per repeater node. That approach is,
however, not fault-tolerant against operational errors, and as a result the
operational error experienced by the qubit in transmission must be very low
($\sim\\!10^{-5}$) in order to bridge intercontinental distances.

Alternatively, the concatenation of two different quantum error correction
codes has been considered to ensure fault-tolerant operation and efficiently
address both loss and operational errors in quantum repeaters. In particular,
the concatenation of a continuous-variable (CV) GKP code with small discrete-
variable (DV) codes has recently been proposed [18, 22, 19, 23], and was shown
to have increased communication performance while reducing the resource cost.

In this article, we propose a novel variant of DV-based one-way quantum-
repeater architecture (see fig. 1) that uses code concatenation to achieve
fault-tolerant operation in a resource-efficient manner. Specifically, we
combine a loss-tolerant tree code as an inner code [17] with a 5-qubit code
operated with a flag qubit as an outer code. The 5-qubit code [24] ensures
efficient correction of operational errors for which the tree code does not
provide protection. By adopting flag-based stabilizer measurements, we show
that communication rates in the kHz range can be achieved over thousands of
kilometers for qubit transmission errors $\sim\\!10^{-3}$.

Our design intersperses two types of repeater nodes that operate differently
on the two codes similarly to the CV-DV architecture of Ref. [18]. Type i
nodes perform error correction solely on the tree code while type ii nodes
perform error correction on both codes. Consequently, type ii nodes are more
complex than type i nodes. We show how the extra cost of type ii nodes can be
included in the design of the repeater architecture to maximize network
performance while minimizing the cost. Finally, we outline how both types of
repeater nodes can be constructed from 5 modular processors containing only 1
quantum emitter and at most 4 memory spins each. This makes our design
suitable for implementation with current quantum-network hardware such as
solid-state defect centers coupled to a small qubit registers of nuclear spins
[25, 26].

The paper is structured as follows. In section II, we introduce the high-level
repeater protocol and its building blocks. In section III, we discuss a
possible implementation with efficient spin-photon interfaces based on cavity-
coupled quantum emitters and in section IV, we quantify the performance of the
repeater, which we optimize for various distances in section V.

Figure 1:  (a) Overview of the hybrid one-way quantum repeater network
containing two types of repeater nodes: type i and type ii. (b) In the start
node (greyed because no error correction is involved), a message qubit is
encoded using the 5-qubit code into five data spin qubits. Then, each of those
data spin qubits is encoded in parallel into a photonic tree-cluster state via
a Bell state measurement (orange dashed box) with the root spin qubit of the
tree. The $[2,2]$ tree is used as an example here. The trees (each encased in
a green box) are then sent in parallel along the repeater network where the
nodes a distance $L_{0}$ apart from each other. Type i nodes consist of five
processing blocks. Each of these blocks decodes the received tree and re-
encodes the decoded qubit into a fresh tree via heralded storage [17]. (c)
Type ii nodes decode the incoming tree at the tree level, then perform
stabilizer operations (i.e., two-qubit gates and syndrome extraction on the
5-qubit code level) between the decoded qubit and the ancilla qubits with
accompanying flag qubits in the nodes. After the stabilizer operations, the
decoded qubit is re-encoded back into a new tree and sent off to the next
node. (d) At the end node, the incoming five trees are received and decoded in
parallel. The 5-qubit code corrections are applied according to the syndromes
obtained along the network and finally they are decoded back into the message
qubit.

##  II Quantum repeater protocol

The repeater architecture incorporates three main steps. First, a message
qubit is encoded in an error-correction code at the starting node. This is
followed by re-encoding and error correction at subsequent repeater nodes
before a final decoding at the end node. A high-level schematic of the entire
repeater network is shown in fig. 1, and we will now address each of the three
steps separately in more detail below.

###  II.1 Encoding

The message qubit at the start node is first encoded into the [[5,1,3]] code,
otherwise known as the 5-qubit code (see fig. 1b). Here, [[n, k, d]] refers to
a code that encodes $k$ logical qubits using $n$ physical qubits with a code
distance $d$. We have chosen the 5-qubit code as the outer code because it is
the smallest quantum error correcting code that can correct single arbitrary
Pauli errors [24], an attractive trait in minimizing physical resource
requirements. We note that the encoding of the logical state can be performed
fault-tolerantly using a scheme recently demonstrated with a diamond Nitrogen-
Vacancy platform in Ref. [27]. This involves heralding the desired encoded
logical state via repeated stabilizer measurements in conjunction with a flag
qubit.

After encoding the message qubit into the outer 5-qubit code, each of the five
data qubits of the code are then further encoded into a photonic tree-cluster
state (inner code) that provides loss tolerance via information redundancy
[28]. The tree-cluster code can be described by its branching vector
$\vec{t}=[b_{0},b_{1},\ldots,b_{d}]$ which determines the degree of branching
from each level in the tree beginning from the root qubit (see fig. 2). Every
node in the tree cluster state described by the branching vector represents a
qubit in state $|+\rangle=(\ket{0}+\ket{1})/\sqrt{2}$ and each of the edges
represents CPHASE gate. This resulting tree cluster state is then the unique
state that is stabilized, i.e., it has eigenvalue $+1$ for each of the
operators $K_{\nu}=X_{\nu}\prod_{b\in\mathscr{N}_{\nu}}Z_{b}$, where $\nu$
labels the qubits of the tree state and $\mathscr{N}_{\nu}$ denotes the set of
qubits connected to the $\nu^{\mathrm{th}}$ qubit. To encode a data qubit in
the tree-cluster code, a single Bell state measurement of the root qubit of
the tree-cluster state and the respective data qubit is sufficient as detailed
in Ref. [17] and shown in fig. 1b.

Figure 2: A $[3,2]$ tree-cluster state is shown as an example to illustrate
the different levels of a tree. Each vertex/circle represents a qubit
initially prepared in the state $\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$. The
edges connecting the vertices correspond to a CPHASE gate being applied
between the two qubits.

At this point, the message qubit has been encoded into 5 photonic tree-cluster
states, which are sent down the repeater network in parallel (see fig. 1a).
Note that we are assuming a generation scheme of the photonic tree-cluster
states where each photon of the tree is generated sequentially from a single
emitter featuring a multi-level electronic ground state manifold [29], hence
the photons are sent down the optical fibers using time-bin encoding. For
details on this scheme, we refer to Ref. [17]. Consequently, the 5 trees can
be transmitted using 5 single-mode fibers.

###  II.2 Re-encoding and error correction

In the repeater network, the incoming photonic qubits from the previous node
will be received by either a type i or type ii repeater node, each
specializing in dealing with types of errors. The type i nodes only operate on
the inner tree code to correct transmission loss, and we envision that each
type i node consists of 5 parallel repeater subnodes similar to the nodes
considered in Ref. [17]. Each of these 5 subnodes receives and operates on 1
of the 5 tree-encoded qubits of the outer 5-qubit code. Their basic operation
is to decode each incoming photonic tree-encoded data qubit into a spin-qubit
and then re-encode it into a new photonic tree encoding. In this process, loss
will be corrected but not logical errors on the outer 5-qubit code level.

The re-encoding of the data qubit into the new tree-cluster state is achieved
with a Bell state measurement between any one of the first-level photonic
qubits in the incoming tree and the root qubit of the new tree along with the
measurement of the remaining qubits in the incoming tree in appropriate
single-qubit bases according to the stabilizer generators of the tree (see
section II.1). The whole procedure can be performed in a loss-tolerant way
with only two memory spins and a single cavity-coupled emitter as outlined in
Ref. [17]. Note that we assume that decoherence errors in all memory spins
considered in this article is negligible compared to other gate errors because
the information is only stored short-term in the memory spins in our network
protocol.

Importantly, the Bell state measurement between a first-level photonic qubit
of the incoming tree and the stationary root qubit of the new tree is not
protected against faulty gate operations, which introduces re-encoding error
$\epsilon_{\mathrm{r}}$ on our encoded qubit on the tree code level. In
addition, we assume that each qubit in the trees is subjected to a single
qubit depolarizing error of $\epsilon_{0}$, which comes from the inherent
operational error in the underlying hardware. The presence of this re-encoding
error is why the scheme in Ref. [17] is not fault-tolerant and requires very
low re-encoding error rates for long-distance communication. Here, we solve
this issue with the outer 5-qubit code, which provides an extra layer of
protection for the message qubit in the type ii repeater nodes.

The type ii repeater nodes are designed to primarily correct for the
accumulated re-encoding errors in the network by operating on both the inner
tree encoding and the outer 5-qubit encoding. We refer to this as the
accumulated re-encoding error between type ii nodes as the transmission error
$\epsilon_{\mathrm{trans}}$. This transmission error experienced by the
encoded qubit is modeled via the single qubit depolarizing channel discussed
in Methods VII.2. The first step to correct for $\epsilon_{\mathrm{trans}}$ is
to decode the tree encoding using the same procedure as the type i nodes
thereby correcting loss errors. Once the 5 incoming trees are decoded into 5
single spin qubits, the syndrome extraction of the 5-qubit code is performed
in a fault-tolerant manner using a flag qubit to correct for
$\epsilon_{\mathrm{trans}}$. For the fault-tolerant error correction protocol,
refer to Methods VII.1. After the syndrome extraction, the decoded qubits are
re-encoded back into newly generated trees and are sent down the network. Note
that since a type ii node also performs the same decoding and encoding
procedure as a type i node, we also take into account the errors generated by
these steps in the transmission error. Refer to Methods VII.2 for how these
are taken into account.

Additionally, we consider noisy two-qubit gates in type ii nodes, which we
model via a two-qubit depolarizing channel with an error rate of
$\epsilon_{0}$ discussed in Methods VII.2. Note that, as we will discuss in
section III, there are two-qubit gates in each of the type ii nodes that have
to be performed in a teleported manner, which have an error rate of
$3\epsilon_{0}$ because each of them is essentially comprised of 3 two-qubit
gates.

In section V, we discuss that the inherent operational error is related to the
re-encoding error via the relation $\epsilon_{\mathrm{r}}\approx
3\epsilon_{0}$ found through our numerical simulation (see Methods VII.2).
This means with the increase of the re-encoding error, the two-qubit gates in
type ii nodes become noisier and that placements of type ii nodes in the
network would become more sparse. Note that we assume errors introduced by
single qubit gates are negligible since they are typically much smaller than
errors induced by two-qubit gates in an architecture based on diamond defects
[30]. We also assume that the dominant errors in the qubit readout operations
enter through the noisy two-qubit gates because the qubit readout operations
considered in this paper are always associated with stabilizer operations of
the 5-qubit code. Therefore, the qubit readouts themselves are considered
error-free. In summary, the faulty re-encoding procedure in both type i and
type ii nodes, and the noisy two-qubit gates in the type ii nodes are the only
source of operational errors in our model.

Besides correcting for transmission errors, the 5-qubit code can also correct
for failed decoding attempts of the photonic tree cluster states due to loss
of too many photons. In general, any quantum error-correcting code capable of
correcting $t$ arbitrary Pauli errors could also be used to correct
$2t$-erasure errors [31]. Consequently, the 5-qubit code can correct for up to
two failed decoding attempts (erasure errors) of the incoming photonic tree-
states. While the probability of 2-erasure errors occurring within the same
type ii node is very small due to the loss correction of the type i nodes, we
find that correcting for a single lost tree, i.e., 1-erasure errors, at the
type ii nodes can significantly boost the communication rate (see supplemental
material). We therefore apply the erasure-error correction in this work in
section IV, for more details please refer to Methods VII.1.2.

###  II.3 Decoding

The extracted syndromes of the error correction, which provide information
about whether a 1-erasure error has occurred or not, are simply stored in a
classical register. This register is classically communicated to the end node
where it is interpreted and only when required single-qubit gates are applied
to correct the errors according to the interpreted syndromes. Thus, it is not
necessary to perform any correction on the quantum level at the repeater
nodes. This allows to both avoid additional decoherence due to latency
introduced by applying the error correction at every type ii node and allows
for performing corrections based on the joint information of all error
syndromes from the repeater network. This approach is known as Pauli frame
updating [32].

The end node is a type ii node, where the tree-cluster states are first
decoded into the 5 data spin qubits. Error correction is then performed on the
5 data spin qubits according to the syndromes extracted throughout the
network, after which they are then further decoded back into the original
message qubit and measured. Note that since the encoding and decoding
procedure on the 5-qubit code level is only done at the start and end node, we
assume that errors introduced at these steps are negligible.

##  III Implementation

We will now discuss a specific modular design of the repeater nodes that
allows for an implementation with small qubit processors containing 1 cavity-
coupled quantum emitter and at most 4 memory spins. Our choice of a modular
design is motivated by current hardware capabilities with e.g. solid state
defect centers where efficient spin-photon interfaces can be achieved through
coupling to nanophotonic resonators [26, 33, 25]. By coupling to a few near-by
nuclear spins such small processors can be envisioned. Coupling between the
processors (needed for type ii nodes) can be achieved through photon-mediated
interactions between the emitters as we outline below.

As described above, a type i node consists of 5 parallel repeater subnodes
similar to the node described in Ref. [17], with each subnode requiring only 2
memory spins and 1 quantum emitter for the generation of depth-3 photonic
tree-cluster states to perform the loss-tolerant re-encoding operation.

The photonic tree generation follows the scheme of Ref. [29], where the
photons of the tree are sequentially emitted from the emitter with
intermediate entangling operations between the emitter and the 2 memory spins.
In this way the branches of the tree are generated one by one starting from
the bottom. We note that this generation scheme requires a subsequent re-
ordering of the photons using optical switching and a delay line as detailed
in Ref. [17] such that the first-level photons lead when arriving at the next
repeater node. This is necessary since the absence/presence of a first-level
photon determines the measurement bases of the remaining photons of the
branch.

A key element for the re-encoding operation is a cavity-mediated CPHASE gate
between an incoming photon and the emitter [34, 35, 36, 37]. The basic
principle of this gate is that if only one of the ground states of the emitter
is coupled to the cavity mode, an incoming photon will be reflected
with/without a $\pi$-phase shift if the emitter is in the uncoupled/coupled
state. This operation allows for transferring the quantum state of a photon to
the emitter, heralded by subsequent detection of the photon. As shown in Ref.
[17], this makes the re-encoding operation robust to transmission losses by
first transferring the state of a first-level photon of the incoming tree to
the emitter in a heralded way followed by a Bell state measurement between the
emitter and a memory spin. The memory spin constitutes the root qubit of a
fresh tree cluster state emitted prior to the reception of the incoming tree.

The requirements of type ii nodes are different from the type i nodes since
they perform error correction on the outer 5-qubit code in addition to the
loss correction on the inner tree-cluster code. Here we outline how a modular
architecture can enable this increased functionality with a very modest
increase in resources compared to type i nodes. Similar to type i nodes, we
consider an implementation with 5 cavity-coupled emitters that each can
receive and generate their tree-cluster states using 2 additional memory spins
per emitter (see fig. 3). To allow for the syndrome extraction of the 5-qubit
code, we only need one of these emitters to be coupled to 2 extra memory
spins, which act as ancilla and flag qubits of the 5-qubit code. Consequently,
we require a sequential extraction of the syndromes, which increases the
duration of the error-correction procedure. A faster parallel extraction would
be possible with the added expense of more ancilla and flag qubits, but here
we choose to focus on the sequential operations to minimize the number of
qubits per repeater node.

We assume direct coupling between an emitter and its nearby memory spins
based, for example, on spin-spin interactions, which allows for the
implementation of multi-qubit gate operations. However, as shown in fig. 4,
the syndrome extraction of the 5-qubit code requires two-qubit gates between
the ancilla qubits belonging to different emitter-cavity systems and the data
spin qubit. To implement such non-local operations, teleported gates can be
used.

Figure 3:  The physical resources of the type ii repeater nodes: 5 sets of
single-sided cavities, each with a quantum emitter
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/red_spin.pdf}}}\mkern-1.6mu_{E_{j}}$
with
$j\mkern-3.0mu\in\mkern-3.0mu\\{\mkern-1.0mu1,\mkern-1.2mu2,\mkern-1.2mu3,\mkern-1.2mu4,\mkern-1.2mu5\mkern-1.0mu\\}$
and accompanying memory spins. The 5 emitter spins host the decoded qubits
from the 5 trees. They are equipped with optical switches that allow for
performing teleported CNOT gates between the desired decoded qubits that are
physically far apart. In all the sets, 2 extra qubits, i.e.,
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/red_spin.pdf}}}\mkern-1.6mu_{M_{j}}$
and
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/red_spin.pdf}}}\mkern-1.6mu_{R_{j}}$,
are needed for tree generation and teleported two-qubit gates. In one of the
sets, another 2 qubits are needed for the ancilla and flag qubits, which are
shown as
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/purple_spin.pdf}}}\mkern-1.6mu_{A}$
and
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/green_spin.pdf}}}\mkern-1.6mu_{F}$,
respectively.  Figure 4: The circuit that a type ii node performs to fault-
tolerantly correct for operational errors using the 5-qubit code. For the
error correction protocol, refer to Methods VII.1.

The teleported CNOT gate involves heralding a Bell pair between two emitters
via the transmission and subsequent reflection of a photon [38, 39, 40, 41]
(see fig. 5). The creation of such a Bell pair can be performed using the same
operations and same spins, which are shown enclosed in dashed gray triangles
in fig. 3, to decode an incoming tree-cluster and generate a new tree-cluster.
Note that we can only perform the Bell pair creation once the fresh tree has
been generated with the memory spin $R$ hosting the root qubit and the emitter
spin is “freed”, i.e., reinitialized. Specifically, for creating a Bell pair,
a photon entangled with one of the emitters is generated by emission and then
scattered off another cavity-spin system followed by a heralding measurement
of the photon. This will prepare the two emitter spins in a Bell state, which
can be used to mediate a gate between two remote spin qubits through gate
teleportation. To perform teleported CNOT gates between two qubits of
interest, local CNOT gates between the qubits of interest and the emitter
spins are performed followed by single-qubit measurements of the emitter
spins. Note that the emitter spins in the repeater nodes hold the decoded data
qubits upon the reception of trees, therefore we need to first free them up by
transferring the decoded data qubits onto the auxiliary memory spins as shown
in fig. 5 before performing the CNOT gate. This will then amount to a CNOT
operation between the two distant spin qubits up to a single qubit correction
dictated by the measurement outcome. In order to connect the pairs of cavity-
emitter systems dictated by the error-syndrome-extraction circuit (see fig.
4), we imagine that fast optical switching [42] is employed to ensure the
generation of Bell pairs between the respective emitter spins.

Figure 5:  Procedure for performing a teleported CNOT gate between an ancilla
qubit (control) and a data qubit (target) by heralding a Bell pair between two
emitters and performing a measurement on the Bell pair. For simplicity, not
all spins coupled to each cavity are shown (fig. 3). We use the realization of
a teleported CNOT gate between an ancilla qubit
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/purple_spin.pdf}}}\mkern-1.6mu_{A}$
and the decoded qubit in
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/red_spin.pdf}}}\mkern-1.6mu_{E_{2}}$
as an example. The steps are as follows: (1) The decoded qubits in the emitter
spins
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/red_spin.pdf}}}\mkern-1.6mu_{E_{j}}$
are transferred to the auxiliary memory spins
$\vbox{\hbox{\includegraphics[width=7.79999pt]{assets/red_spin.pdf}}}\mkern-1.6mu_{M_{j}}$.
(2) A Bell pair is heralded between the two emitter spins. (3) Local spin-spin
CNOT gates are performed between spins of interest. (4) The qubits
corresponding to the Bell pair heralded in step (2) are measured to perform a
teleported CNOT gate between desired control and target qubits up to some
Pauli gate(s) correction according to the measurement outcome.

##  IV Repeater performance

In this section, we benchmark the performance of the repeater network by
looking at the secret key rates that are achieved when executing a Quantum Key
Distribution (QKD) protocol. This quantity encapsulates both the fidelity of
the final message qubit and the raw bit rate of the network. Therefore, the
determination of the secret key rate provides an excellent means of assessing
the general performance of the network. Since our repeater network is based on
the one-way quantum repeater protocol, we consider the prepare-and-measure-
based six-state protocol [43] as the most suitable QKD protocol, whose secret
key fraction is dependent on the effective error rate of the message qubit at
the end node. Details about this secret key fraction is given in Methods
VII.3.

Furthermore, we boost the secret key rate by leveraging the 5-qubit code’s
ability to correct for erasure errors as explained in section II.2. Note that
our secret key rate depends on 2 facts: (1) the effective error rate at the
end node depends on how many erasure errors occurred in the network, and (2)
the presence of erasure errors results in a different successful transmission
probability of the message qubit through the network (see Methods VII.3).
Recall that we only consider correcting for 1-erasure errors and not 2-erasure
errors at the 5-qubit code level for reasons explained in section II.2. These
dependencies are accounted for in the secret key rate calculation by
performing a weighted sum over the possible number of 1-erasure error
occurrences in the network, effectively resulting in an average secret key
rate

$S\\!K\\!R=\tau_{\mathrm{tot}}^{-1}\sum^{m_{\mathrm{II}}}_{i=0}\Big{(}\begin{matrix}m_{\mathrm{II}}\\\
i\end{matrix}\Big{)}f_{m_{\mathrm{II}},i}\,p_{\mathrm{trans}}(m_{\mathrm{II}},i),$
(1)

where $m_{\mathrm{I}}$ ($m_{\mathrm{II}}$) is the number of type i (type ii)
nodes, $f_{m_{\mathrm{II}},i}$ is the secret key fraction,
$p_{\mathrm{trans}}(m_{\mathrm{II}},i)$ is the probability of successful
transmission with 1-erasure errors in $i$ distinct type ii nodes (see Methods
VII.3), and

$\tau_{\mathrm{tot}}=\tau_{\mathrm{tree}}+14\tau_{\mathrm{ss}}+26\tau_{\mathrm{tele}}+8\tau_{\mathrm{meas}},$
(2)

is the total processing time of each type ii node for 1 logical qubit. Here,
$\tau_{\mathrm{ss}}$ is the local spin-spin two-qubit gate time,
$\tau_{\mathrm{tele}}$ is the teleported two-qubit gate time, and
$\tau_{\mathrm{meas}}$ is the spin readout time. We arrived at eq. 2 by
considering the longest possible time taken by the fault-tolerant error
correction protocol since that will become the processing bottleneck of the
network (see supplemental material). We assume that the readout of the ancilla
and flag qubit can be done simultaneously. Finally, a new tree-cluster has to
be generated in the type ii node with tree vector
$\vec{t}=[b_{0},b_{1},\ldots,b_{d}]$ which takes time $\tau_{\mathrm{tree}}$
and is estimated as [17]

$\displaystyle\tau_{\mathrm{tree}}$ $\displaystyle\approx
b_{0}\Big{[}100+b_{1}(1+b_{2}(1+\cdots
b_{d-1}(1+b_{d})\cdots))\Big{]}\tau_{\mathrm{ph}}$
$\displaystyle+b_{0}\Big{[}3+b_{1}(1+b_{2}(1+\cdots
b_{d-2}(1+b_{d-1})\cdots))\Big{]}\tau_{\mathrm{ss}},$ (3)

where $\tau_{\mathrm{ph}}$ is the emission time of a single photonic qubit and
the emission time of the first-level photons is assumed to be
$100\tau_{\mathrm{ph}}$. The longer emission time of the photons means a
narrower frequency bandwidth. This is necessary since the first-level photons
may participate in a cavity-mediated CPHASE gate at the next repeater station.
If the frequency bandwidth of the incoming photon is not narrow compared to
the cavity (Purcell) enhanced linewidth of the emitter, the gate operation
will be imperfect. Assuming an emission time of $100\tau_{\mathrm{ph}}$ will
ensure that these imperfections lead to gate errors $\lesssim 10^{-4}$ [17].

In principle, type i nodes could operate faster than type ii nodes since they
are only performing loss correction at the tree-code level. However, type i
nodes would have to wait for the type ii nodes to complete their operations
before transmitting the tree-encoded qubits. Thus, the operation time of the
type ii nodes becomes the bottleneck that sets the repetition rate of the
repeater network and faster type ii nodes are not required. As a consequence,
the hardware architecture at type i and ii nodes can be quite similar,
facilitating the large scale building of such nodes.

##  V Results

$10^{2}$$10^{3}$$10^{4}$$10^{5}$$S\\!K\\!R$ [Hz]()$\epsilon_{\mathrm{r}}$$0.1\text{\textperthousand}$$0.3\text{\textperthousand}$$0.5\text{\textperthousand}$$0.1\%$$0.2\%$$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{6}$$10^{7}$$10^{8}$$L_{\mathrm{tot}}$ [km]Cost() Figure 6:  (a) The secret key rate $S\\!K\\!R$ corresponding to (b) the minimized cost function $C_{\mathrm{min}}$ as a function of the distance $L_{\mathrm{tot}}$ for various re-encoding error probabilities $\epsilon_{\mathrm{r}}$ with fixed repeater relative weight $\kappa=1$. For comparison, the secret key rates and the costs of the homogeneous repeater network from Ref. [17] multiplied by a factor of 5 are included as dashed lines as well. The factor of 5 is included to compensate for the fact that the concatenated repeater is similar to 5 parallel homogeneous repeater networks. The cost function of the homogeneous repeater is the same as eq. 4 with $m_{\mathrm{II}}=0$, and was minimized independently.  Quantity |  | Value
---|---|---
Spin-spin gate time, | $\tau_{\mathrm{ss}}$ |  $\scriptscriptstyle\mathbin{\Diamond}$ |  $100$ ns
Single photon emission time, | $\tau_{\mathrm{ph}}$ |  $\scriptscriptstyle\mathbin{\Diamond}$ |  $1$ ns
Spin readout time, | $\tau_{\mathrm{meas}}$ |  |  $1$ $\upmu$s
Teleported two-qubit gate time, | $\tau_{\mathrm{tele}}$ |  |  $1$ $\upmu$s
Optical fiber’s attenuation length, | $L_{\mathrm{att}}$ |  $\scriptscriptstyle\mathbin{\Diamond}$ |  $20$ km
Effective photon-detection efficiency, | $\eta_{\mathrm{d}}$ |  $\scriptscriptstyle\mathbin{\Diamond}$ | $0.95$
Table 1: Values of the quantities used in the minimization of the cost
function shown in eq. 4. The symbol $\scriptscriptstyle\mathbin{\Diamond}$
denotes that the values were also used in Ref. [17]. The effective photon-
detection efficiency, $\eta_{\mathrm{d}}$ is the combined efficiency of in/out
coupling of the photon, frequency conversion and the efficiency of the photon
detectors.

To find the optimal configuration of the repeater network, that is to maximize
the secret key rate while using as little physical resources as possible, we
perform a numerical minimization of a dimensionless cost function for a
specific total distance

$C=S\\!K\\!R^{-1}\frac{L_{\mathrm{att}}}{\tau_{\mathrm{ph}}L_{\mathrm{tot}}}(m_{\mathrm{I}}+\kappa
m_{\mathrm{II}}),$ (4)

over the parameters $L_{0}$, $m_{\mathrm{II}}$, and $\vec{t}$ with constraints
discussed in Methods VII.4. Note that for simplicity, we assume a uniform
interspersing of type i and type ii nodes in the repeater network. For more
details on how the repeaters are interspersed, refer to the supplemental
material. The coefficient $\kappa$ is the relative cost of a type ii to a type
i node. In a repeater network with multiple types of nodes, one type of node
may require more functionality (as is the case here) and thus be more
expensive than another. For instance, if $\kappa=1$, then the cost of a type i
node is the same as that of a type ii node.

To quantify the cost per unit length, we divide the number of nodes in the
cost function with the total distance between the start and end node
$L_{\mathrm{tot}}$. This is expressed in units of the attenuation length
$L_{\mathrm{att}}$ to make the cost function dimensionless. We also quantify
the cost per unit time by including the inverse of the secret key rate
($S\\!K\\!R$) which is in unit Hertz. Note that the secret key rate depends on
the photon emission time $\tau_{\mathrm{ph}}$, therefore its presence in the
denominator in eq. 4 serves to make the cost function dimensionless. It also
means that we are expressing the cost per unit time in units of the photon
emission time. The values for the constants used are shown in table 1.

The results of the optimizations for different error rates are shown in fig. 6
for $\kappa=1$. For short distances, the homogeneous repeater scheme from Ref.
[17] (dashed lines) is superior since it does not possess the time overhead of
error correction that comes with type ii nodes and thus enables a higher
repetition rate. For longer total distances, however, the secret key rate of
the concatenated repeater protocol (solid lines with markers) greatly
surpasses that of the homogeneous repeater protocol of Ref. [17] due to the
added protection from the outer 5-qubit code. For instance, for
$\epsilon_{\mathrm{r}}=0.1\%$ (purple line), the secret key rate for the
concatenated repeater is $\sim\\!\\!5.5$ kHz at $10^{3}$ km, while for the
homogeneous counterpart (dashed line), the rate at the same total distance is
less than 1 Hz. Furthermore, despite the higher secret key rates of the
homogeneous repeater scheme, we see that from fig. 6b that its cost is
significantly higher than that of the concatenated repeater protocol. This
means that the secret key rate itself is not the only relevant indicator of
overall performance, but the cost associated with building the proposed
network needs to be considered as well. Note that even if the cost function is
ignored and if the repeater network is configured by purely maximizing the
secret key rate, the gain in the secret key rate would be minimal with the
tradeoff being a significantly increased cost (see supplemental material).

The accompanying results in fig. 7a reveal the optimal inter-repeater
distances required to achieve the best secret key rates whilst maintaining
minimum cost. When considering higher re-encoding error probabilities
$\epsilon_{\mathrm{r}}$ for fixed total distances, the optimum inter-repeater
distance $L_{0}$ decreases to minimize the loss error to counter the
increasing re-encoding error rate. From fig. 7b, we see that the configuration
with the highest re-encoding error has the lowest proportion of type ii nodes
in the network, even though it has the lowest inter-repeater distance. This is
because the re-encoding error rate scales with the error rate of the two-qubit
gates in the type ii nodes as explained in section II.2. Therefore, adding
more type ii nodes no longer entails better error suppression, but instead
introduces more error in the network. Hence, we find an asymmetry between the
number of type i and type ii nodes as the ratio of $m_{\mathrm{I}}$ to
$m_{\mathrm{II}}$ deviates from unity. This result demonstrates that even when
we consider the cost of type i and type ii nodes to be equal, i.e.,
$\kappa=1$, the asymmetry between loss and operation errors in the repeater
network is still present. Note that it is in principle possible for $\kappa$
to have values below unity. In our calculation, however, because of the
constraint on the $m_{\mathrm{II}}$ set, the minimum value that the ratio of
$m_{\mathrm{I}}$ to $m_{\mathrm{II}}$ can attain is unity.

To demonstrate how the value of $\kappa>1$ influences the cost and secret key
rate when considering type ii nodes that are more expensive than type i nodes,
we optimize with $\kappa\in\\{1,2,10\\}$ for fixed re-encoding error rate
$\epsilon_{\mathrm{r}}=0.1\%$ in fig. 8. As the value of $\kappa$ increases,
we see that the secret key rate is only minimally affected as shown in fig. 8a
whilst the number of type ii nodes in the network is significantly decreased
as shown in fig. 8b. From fig. 8c, we infer that this is followed by an
increase in the number of type i nodes in the network. This is expected as
type ii nodes are more costly. Additionally, fig. 8c shows how the ratio of
type i to type ii nodes changes as function of the distance. For high-cost
type ii nodes ($\kappa=10$) the ratio decreases with distance. This can be
understood from the accumulation of re-encoding errors in the network. For
small distances, the accumulated error is relatively small, and it is
therefore advantageous to employ few type ii nodes. For larger distances, the
accumulated error necessitates more type ii nodes despite their higher cost.
On the other hand, we observe that the ratio of type i to type ii nodes
increases with distance even for a low cost of type ii nodes ($\kappa=1$). We
attribute this to the fact that placing too many type ii nodes in the network
introduces more operational errors than it can suppress.

$1$$2$$3$$4$$L_{0}$
[km]()$\epsilon_{\mathrm{r}}$$0.1\text{\textperthousand}$$0.3\text{\textperthousand}$$0.5\text{\textperthousand}$$0.1\%$$0.2\%$$10^{2}$$10^{3}$$10^{4}$$0$$2$$4$$6$$8$$L_{\mathrm{tot}}$
[km]$m_{\mathrm{I}}:m_{\mathrm{II}}$() Figure 7:  (a) The inter-repeater
distance $L_{0}$, and (b) the ratio of number of type i to type ii nodes
$m_{\mathrm{I}}:m_{\mathrm{II}}$ as a function of total distance corresponding
to fig. 6.  $10^{2}$$10^{3}$$10^{4}$$10^{5}$$S\\!K\\!R$
[Hz]()$\kappa$$1$$2$$10$$0$$100$$200$$300$$m_{\mathrm{II}}$()$10^{2}$$10^{3}$$10^{4}$$0$$10$$20$$30$$40$$L_{\mathrm{tot}}$
[km]$m_{\mathrm{I}}:m_{\mathrm{II}}$() Figure 8:  (a) The secret key rate
$S\\!K\\!R$ corresponding to the minimized cost function $C_{\mathrm{min}}$ as
a function of the distance $L_{\mathrm{tot}}$ for various relative node
weights $\kappa$ with fixed re-encoding error probability
$\epsilon_{\mathrm{r}}=0.1\%$. For comparison, the secret key rate of the
homogeneous repeater network multiplied by a factor of 5 from Ref. [17] is
included as dashed line optimized with respect to the cost function shown in
eq. 4 with fixed $m_{\mathrm{II}}=0$. (b) The corresponding number of type ii
nodes in the network and (c) the corresponding ratio of number of type i to
type ii nodes as a function of total distance.

##  VI Discussion

In conclusion, we have outlined how a resource-efficient DV one-way quantum
repeater can be constructed by concatenating a flag-based 5-qubit code with a
loss-tolerant tree-cluster code. The high loss tolerance and conceptually
straightforward encoding/decoding of the tree-cluster code makes it suitable
as an inner code that protects against photon loss in transmission between
repeater nodes. An extra layer of protection is provided by the outer 5-qubit
code which effectively suppresses operational errors that accumulate due to
the non-fault-tolerant nature of the tree-cluster code against Pauli errors.
As a result, the code-concatenated repeater is able to bridge distances of
several thousand kilometers for re-encoding error probabilities as high as
$\sim\\!0.1\%$.

The code concatenation allows for an architecture tailored to the asymmetry
between loss and operational errors that is characteristic for quantum
communication. Specifically, we have shown how the optimizations can include
relative cost of the different repeater nodes to arrive at an optimized
architecture with many relatively cheap type i nodes that only correct errors
due to transmission loss, but far fewer expensive type ii nodes that also
correct operational errors. In particular, we found that such an optimization
could be done with a minimal effect on the optimal secret key rate of the
repeater network.

We have outlined a possible modular implementation of the repeater based on
few-qubit processors with an efficient spin-photon interface through a cavity-
coupled quantum emitter. Solid-state systems such as group-IV defect centers
in diamond [26, 6, 44] are promising hardware candidates having demonstrated
both efficient coupling to nanophotonic cavities [6] and access to small-scale
qubit registers through coupling to nearby nuclear spins [26]. To implement
both type i and type ii nodes, only 5 cavity-emitter systems are required.
Importantly, it is sufficient for each of these systems to have just 1 cavity-
coupled emitter spin with direct, e.g., magnetic, coupling to 4 nearby nuclear
spins. Operations between the processors, which are needed for the type ii
nodes can be achieved through teleported gates facilitated by photon-mediated
interactions between the quantum-emitter spins.

##  VII Methods

###  VII.1 Error correction

####  VII.1.1 Fault-tolerance

In a type ii node, an additional flag qubit in a type ii node is used to
perform error correction, with a circuit shown in fig. 4. This additional
qubit aids in suppressing the otherwise undetectable physical errors
propagated by faulty gates in the stabilizer operations [45]. In fig. 9, we
consider a subsystem of fig. 4 as an example and illustrate how an error that
occurred right before a two-qubit gate during the stabilizer operation could
result in logical errors on the data qubits due to the error being propagated
by the two-qubit gate onto the data qubits. Then, we consider additional
errors induced by the noisy two-qubit gate on the target qubit. With one
additional flag qubit prepared in state $\ket{0}$, the propagated error on the
data qubits would then be detected because the flag qubit would be measured as
being in state $\ket{1}$. When the flag qubit is detected as being in state
$\ket{1}$, we switch over to the unflagged circuit and apply the Pauli
operations according to the syndromes measured during this unflagged circuit
to revert the propagated error. An example is shown in fig. 9b-c. For the
complete fault-tolerant error-correction protocol, see the supplemental
material.

Figure 9: (a) Circuit diagram of the error propagation in a subsection of the
5-qubit code circuit. (b) The errors induce by gate $a$ if it was triggered by
an error on the ancilla qubit and (c) similarly for gate $b$. Figure adapted
from Fig. 1 in Ref. [45].

####  VII.1.2 Erasure-error correction

In the event of a 1-erasure error being heralded at a type ii node, we
initialize the spin in the node corresponding to the lost tree to state
$|{0}\rangle$, while the rest of the intact trees are decoded into spins in
the other nodes as usual. The 5-qubit code stabilizer operations are then
performed on the 5 qubits, projecting the mixed state back into the logical
codespace of the 5-qubit code, up to some Pauli corrections on the 5 qubits.
For example, if the 5th qubit that was part of the encoded 5-qubit logical
state $|\psi_{L}\rangle$ was lost, then the original encoded state would be
restored according to the Pauli corrections in table 2. The procedure is
similar for correcting a 2-erasure error, however, we have not included
corrections of 2-erasure errors in our optimizations since they occur very
rarely compared to 1-erasure errors.

To complete the error correction look-up table, we considered the cases in
which additional Pauli errors also occurred on other non-lost qubits. The
resulting look-up table is shown in table 3. Note that this look-up table is
not unique, i.e., multiple distinct errors could lead to the same syndrome.
For instance, if the $1{\mathrm{st}}$ qubit is lost, the syndrome
$|\\!-\\!+\\!+\\!+\rangle$ corresponds to the cases where either a $X_{2}$,
$Z_{3}$, $Z_{4}$, or $X_{5}$ error also occurred. Since they are
indistinguishable due to the shared syndrome, we choose to only correct for
$X_{2}$. Randomly choosing one out of the four possible errors to correct does
not affect the suppression of error because we are considering the
depolarizing noise model where every Pauli error occurs with the same
probability. However, if one is considering a biased noise model, it is wise
to choose which case to correct in order to maximize the error suppression.

Note that for erasure correction, we do not use the error correction protocol
with the flag qubit since we found that it does not improve the fidelity of
the error corrected state compared to not using the flag qubit. This is due to
the fact that the effectiveness of the flag qubit protocol relies on the
initial 5-qubit logical state having at most a Pauli error on 1 physical
qubit, while an erasure error effectively induces a correlated error, i.e., it
is not merely a single qubit Pauli error.

Syndrome | Projected state
---|---
$|\\!+\\!+\\!+\\!+\rangle$ | $|\psi_{L}\rangle$
$|\\!+\\!+\\!-\\!-\rangle$ | $X_{5}|\psi_{L}\rangle$
$|\\!+\\!-\\!+\\!+\rangle$ | $Z_{5}|\psi_{L}\rangle$
$|\\!+\\!-\\!-\\!-\rangle$ | $X_{5}Z_{5}|\psi_{L}\rangle$
Table 2: The syndrome observed from reading out the ancilla qubits and the
corresponding restored encoded logical states assuming the 5th qubit was lost.
Table 3: Corrections to the 5 data qubits to project them back into the
logical codespace of the 5-qubit code depending on the ancilla outcomes and
which data qubit was lost.

###  VII.2 Error model

We model the noise in the two-qubit gates between two local spins using the
two-qubit depolarizing channel

$\Lambda_{2}(\epsilon)=(1-\epsilon)\varrho+\frac{\epsilon}{15}\sum_{P\in\\{I,X,Y,Z\\}^{\otimes
2}\setminus{I^{\otimes 2}}}P\varrho P.$ (5)

where $\epsilon$ is a general error rate and $\varrho$ is a two-qubit density
matrix. In our model, a non-teleported two-qubit gate is followed by
$\Lambda_{2}(\epsilon_{0})$. Conversely, a teleported two-qubit gate, as shown
in fig. 5, involves 3 two-qubit gates that each has an error of
$\epsilon_{0}$. Thus, we consider that an error of
$\Lambda_{2}(3\epsilon_{0})$ follows immediately after a teleported two-qubit
gate since $\epsilon_{0}\ll 1$.

Another noise channel which we consider is the single qubit depolarizing
channel, which is given by

$\Lambda(\epsilon)=(1-\epsilon)\rho+\frac{\epsilon}{3}\sum_{P\in\\{X,Y,Z\\}}P\rho
P,$ (6)

where $\rho$ is a single qubit density matrix. As previously noted in section
II, each of the 5 logical qubits at the tree code level in transmission is
subjected to such error channel with an error rate of
$\epsilon_{\mathrm{trans}}$, i.e., $\Lambda(\epsilon_{\mathrm{trans}})$. The
expression for the transmission error is given by

$\epsilon_{\mathrm{trans}}=1-(1-\epsilon_{\mathrm{r}})^{n}(1-\epsilon_{0}).$
(7)

where $\epsilon_{\mathrm{r}}$ is the re-encoding error and $n$ is the number
of links between consecutive type ii nodes as illustrated in fig. 10. Note
that we have parameterized the number of type i nodes as
$m_{\mathrm{I}}=m_{\mathrm{II}}(n-1)$. We consider the error induced by both
the tree decoding and encoding step in type ii nodes. The tree encoding step
in type ii nodes introduce an error of $\epsilon_{0}$. Conversely, the
decoding step at a type i and type ii node is similar, hence it introduces an
error of $\epsilon_{\mathrm{r}}$. However instead of decoding the tree upon
reception into a fresh tree, type ii nodes would perform a SWAP gate, which we
assume has an error of $\epsilon_{0}$, between the decoded qubit from a tree
and an auxiliary memory spin in preparation for a teleported two-qubit gate
for the 5-qubit code stabilizer operations as detailed in section III.

Figure 10: Illustration of the error propagated by the repeater operations
between two consecutive type ii nodes with $n$ (parallel) links in between.
Only 1 out of 5 of the parallel links is shown for simplicity.

Regarding the tree generation scheme from Ref. [17] which we consider in our
work, we note that, in principle, it could lead to correlated errors in the
tree cluster states since it involves repeated photon emission from the same
emitter with error rate of $\epsilon_{0}$. This error on the emitter thus
might propagate to more than a single photon in the tree. It is however,
outside the scope of this work to consider the effect of such correlated
errors. We note that they could potentially be kept small by e.g.,
intermediate error detection steps in the generation procedure [25]. We assume
that we are able to suppress the correlated errors, then we can say that each
photon in our trees is merely subjected to $\Lambda(\epsilon_{0})$. From our
numerical simulation of the tree re-encoding procedure, we find that our model
results in a re-encoding error rate of $\epsilon_{\mathrm{r}}\approx
3\epsilon_{0}$ at each type i node for the tree sizes considered in this work
(see supplemental material). This reflects the fact that despite the tree-
clusters consisting of hundreds of photons each subjected to
$\Lambda(\epsilon_{0})$, the dominant error comes from errors on the first-
level photon and spin qubit involved in the re-encoding step. Errors on the
remaining photons, which are measured out, are efficiently suppressed by
employing a majority-voting correction strategy.

###  VII.3 Repeater Performance

The key performance metrics of the repeater network that we consider are the
end-to-end transmission probability of the message qubit and the quality of
the transmitted qubit. To characterize the end-to-end transmission probability
of the message qubit, we start by considering the transmission probability,
$\eta_{\mathrm{e}}$, of a single tree-cluster state with branching vector
$\vec{t}=[b_{0},b_{1},\ldots,b_{d}]$ between consecutive repeater nodes. From
Ref. [28], we have that

$\eta_{\mathrm{e}}=[(1-\mu+\mu R_{1})^{b_{0}}-(\mu R_{1})^{b_{0}}](1-\mu+\mu
R_{2})^{b_{1}},$ (8)

where $R_{k}=1-[1-(1-\mu)(1-\mu+\mu R_{k+2})^{b_{k+1}}]^{b_{k}}$ with
$R_{d+1}=0$, $b_{d+1}=0$, and $\mu=1-\eta\eta_{\mathrm{d}}$. Here,
$\eta=\exp(-L_{0}/L_{\mathrm{att}})$ is the transmission probability of a
single photon between repeater nodes with $L_{0}$ being the inter-node
distance and $L_{\mathrm{att}}$ being the attenuation length of the optical
fiber. Furthermore, $\eta_{\mathrm{d}}$ is the combined efficiency of in/out
coupling of the photon, frequency conversion and the photon detection. We
assume that efficient frequency conversion to the telecom band is possible
such that $L_{\mathrm{att}}=20$ km.

Without erasure-error correction on the 5-qubit code level, the end-to-end
transmission probability of the message qubit would be given by
$\eta_{\mathrm{e}}^{5m_{\mathrm{tot}}}$ where
$m_{\mathrm{tot}}=m_{\mathrm{I}}+m_{\mathrm{II}}$ is the total number of nodes
in the repeater excluding the start node with $m_{\mathrm{I}}$
($m_{\mathrm{II}}$) being the number of type i (type ii) nodes in the network.
Note that we exclude the start node but include the end node when counting
$m_{\mathrm{II}}$. Nonetheless, it is advantageous to leverage the outer
5-qubit code to also correct for erasure errors. Thus, we need to consider how
this affects the end-to-end transmission probability of the message qubit.

We choose to treat cases with more than one erasure error, i.e., more than one
failed tree encoding, at the same type ii node as a failed transmission for
simplicity as mentioned in section VII.1.2. The total transmission probability
in this case can then be expressed as

$\sum^{m_{\mathrm{II}}}_{i=0}\Big{(}\begin{matrix}m_{\mathrm{II}}\\\
i\end{matrix}\Big{)}p_{\mathrm{trans}}(m_{\mathrm{II}},i),$ (9)

with

$p_{\mathrm{trans}}(m_{\mathrm{II}},i)=[\eta_{\mathrm{e}}^{5n}]^{m_{\mathrm{II}}-i}[5\eta_{\mathrm{e}}^{4n}(1-\eta_{\mathrm{e}}^{n})]^{i},\quad
m_{\mathrm{II}}\geq i,$ (10)

being the probability of successful transmission with 1-erasure errors in $i$
distinct type ii nodes. Besides the transmission probability we also need to
assess the quality of the transmitted qubits. To this end, we consider a
scenario where the transmitted qubits are used to distill a secret key. In
particular, we assume that the six-state protocol [43] is employed, which
allows us to quantify the quality of the transmitted qubits using its
asymptotic secret key fraction. This is given by [46]

$f_{m_{\mathrm{II}},i}=\max\\!\Big{(}(1-Q)\Big{[}1-h\Big{(}\frac{1-3Q/2}{1-Q}\Big{)}\Big{]}-h(Q),0\Big{)},$
(11)

where $h(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$ is the binary entropy and

$Q=2\epsilon_{\mathrm{eff}}(m_{\mathrm{II}},i)/3,$ (12)

denotes the QBER (QuBit Error Rate) of the fully decoded qubit at the end
node. The quantity $\epsilon_{\mathrm{eff}}(m_{\mathrm{II}},i)$ is the
effective error rate of the received qubit at the end node given there were
$i$ erasure-error occurrences and $m_{\mathrm{II}}$ type ii nodes.

With our error model, we determine
$\epsilon_{\mathrm{eff}}(m_{\mathrm{II}},i)$ from numerical simulations of the
5-qubit code throughout the repeater network. The details of these simulations
can be found in the supplemental material where we also provide a semi-
analytical approximation to $\epsilon_{\mathrm{eff}}(m_{\mathrm{II}},i)$,
which matches the numerical results to great precision for the relevant range
of effective error rates, i.e., error rate that is less than the threshold
QBER of the associated QKD protocol.

###  VII.4 Numerical minimization

We minimize the cost function $C$ in eq. 4 with respect to $L_{0}$,
$m_{\mathrm{II}}$, and $\vec{t}$ with the following constraints: minimum
inter-repeater distance $L_{0}\geq 1$ km, the maximum number of type ii nodes
never exceed half of the total number of nodes, i.e.,
$m_{\mathrm{II}}\leq\lfloor m_{\mathrm{tot}}/2\rfloor$, the maximum photon
number $N_{\mathrm{max}}=300$ with the total photon number
$N=\sum^{d}_{i=0}\prod^{i}_{j=0}b_{j}$ and a fixed depth of the tree $d=2$.
Then, the minimization of $C$ can be written as

$\displaystyle C_{\mathrm{min}}$
$\displaystyle=\min_{L_{0},m_{\mathrm{II}},\vec{t}}C,\;\;\text{subject to: }$
$\displaystyle\phantom{{}==={}}\text{$L_{0}\geq 1$ km},$
$\displaystyle\phantom{{}==={}}\text{$1\leq m_{\mathrm{II}}\leq\lfloor
m_{\mathrm{tot}}/2\rfloor$},$
$\displaystyle\phantom{{}==={}}\text{$\vec{t}\in\\!\\{[b_{0},b_{1},b_{2}]\,|\,\forall{j},b_{j}\\!\\!\in\\!\mathbb{Z}_{>0}\text{
and }N\\!\leq\\!N_{\mathrm{max}}\\}$},$ (13)

where $\mathbb{Z}_{>0}$ is the set of all positive integers. We chose the
quantity $\lfloor m_{\mathrm{tot}}/2\rfloor$ as the maximum number of type ii
nodes in our optimization since the approximations that make eq. 1 accurate
starts to break down beyond this value. This is because we can no longer
properly place the type ii nodes such that they are evenly spaced beyond this
value (see supplemental material). The rest of the parameters are fixed and
their values are shown in table 1. To ensure that the true optimum does not
lie in regime where only type ii nodes are permitted in the network, we
independently optimized such a configuration with respect to the cost function
and found that both the resulting secret key rate and cost are significantly
worse than for the hybrid configuration (see supplemental material).

## Data availability

The generated data in this study can be found at
https://doi.org/10.4121/b9c7327e-97b2-4ea2-9b74-18c51f265027.v1.

## Code availability

The code used for obtaining the presented numerical results is available at
https://github.com/bernwo/code-concatenated-quantum-repeater.

## Acknowledgments

We thank Stephanie Wehner for illuminating discussions. K.J.W., J.B., and G.A.
acknowledge funding from the NWO Gravitation Program Quantum Software
Consortium (Project QSC No. 024.003.037). G.P. and T.S. were supported by the
Federal Ministry of Education and Research (BMBF, Project QR.X with Subproject
No. KIS6QK4001, Project QPIS No. 16KISQ032K) and the European Research Council
(ERC Starting Grant “QUREP”). M.F.M.R. was supported by the Austrian Science
Fund (FWF) through projects No. P36009-N and No. P36010-N and Finanziert von
der Europäischen Union - NextGenerationEU. A.S.S. acknowledge funding from the
Danish Nation Research Foundation (Center of Excellence “Hy-Q,” Grant No.
DNRF139). F.R. and L.J. acknowledge support from the ARO(W911NF-23-1-0077),
ARO MURI (W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399, FA9550-21-1-0209),
AFRL (FA8649-21-P-0781), NSF (OMA-1936118, ERC-1941583, OMA-2137642), NTT
Research, Packard Foundation (2020-71479), and the Marshall and Arlene Bennett
Family Research Program.

## Competing Interests

The authors declare no competing interests.

## Author Contributions

J.B. conceived the quantum-repeater-network architecture together with A.S.S.
J.B. supervised the project. K.J.W. developed theory and code for performing
the numerical optimization of repeater network with input from A.S.S., F.R.,
and L.J., while G.A. and M.F.M.R developed the code for finding the effective
error suppression provided by specific tree branching vectors. All authors
contributed to the discussion of the results and writing of the manuscript.

## Additional information

Correspondence and requests for materials should be addressed to K.J.W., G.A.,
or J.B.

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