# Metropolitan-scale heralded entanglement of solid-state qubits

A.J. Stolk  These authors contributed equally to this work  K.L. van der Enden
These authors contributed equally to this work  M.-C. Slater  I. te Raa-Derckx
P. Botma  J. van Rantwijk  QuTech & Kavli Institute of Nanoscience, Delft
University of Technology, 2628 CJ Delft, The Netherlands  J.J.B. Biemond
R.A.J. Hagen  R.W. Herfst  W.D. Koek  A.J.H. Meskers  R. Vollmer  E.J. van
Zwet  QuTech & Kavli Institute of Nanoscience, Delft University of Technology,
2628 CJ Delft, The Netherlands  Netherlands Organisation for Applied
Scientific Research (TNO), P.O. Box 155, 2600 AD Delft, The Netherlands  M.
Markham  A.M. Edmonds  Element Six Innovation,Fermi Avenue, Harwell Oxford,
Didcot, Oxfordshire OX11 0QR, United Kingdom  J.F. Geus  F. Elsen  Fraunhofer
Institute for Laser Technology ILT, 52074 Aachen, Germany  Chair for Laser
Technology, RWTH-Aachen University, 52074 Aachen, Germany  B. Jungbluth
Fraunhofer Institute for Laser Technology ILT, 52074 Aachen, Germany  C.
Haefner  Fraunhofer Institute for Laser Technology ILT, 52074 Aachen, Germany
Chair for Laser Technology, RWTH-Aachen University, 52074 Aachen, Germany  C.
Tresp  J. Stuhler  S. Ritter  TOPTICA Photonics AG, Lochhamer Schlag 19, 82166
Graefelfing, Germany  R. Hanson  Correspondence to<EMAIL_ADDRESS>QuTech
& Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ
Delft, The Netherlands

###### Abstract

A key challenge towards future quantum internet technology is connecting
quantum processors at metropolitan scale. Here, we report on heralded
entanglement between two independently operated quantum network nodes
separated by 10km. The two nodes hosting diamond spin qubits are linked with a
midpoint station via 25km of deployed optical fiber. We minimize the effects
of fiber photon loss by quantum frequency conversion of the qubit-native
photons to the telecom L-band and by embedding the link in an extensible
phase-stabilized architecture enabling the use of the loss-resilient single-
photon entangling protocol. By capitalizing on the full heralding capabilities
of the network link in combination with real-time feedback logic on the long-
lived qubits, we demonstrate the delivery of a predefined entangled state on
the nodes irrespective of the heralding detection pattern. Addressing key
scaling challenges and being compatible with different qubit systems, our
architecture establishes a generic platform for exploring metropolitan-scale
quantum networks.

00footnotetext: M.-C. S. is currently at: AIT Austrian Institute of Technology
GmbH. Giefinggasse 4, 1210 Vienna, Austria.

Future quantum networks distributing entanglement between distant quantum
processors [1, 2] hold the promise of enabling novel applications in
communication, computing, sensing, and fundamental science [3, 4, 5, 6]. Over
the past decades a range of experiments on different qubit platforms have
demonstrated the rudimentary capabilities of quantum networks at short
distances including photon-mediated entanglement generation [7, 8, 9, 10, 11,
12, 13]. These short-range qubit networks are useful for testing of improved
hardware [14], developing a quantum network control stack [15] and for
exploring quantum network protocols in a lab setting [16, 17, 18].

The next major challenge is to develop quantum network systems capable of
generating, storing and processing quantum information on metropolitan scales.
Such systems face several new requirements. First, the large physical
distance, the consequential significant communication times and need for
scalability demand that the network nodes operate fully independently. Second,
as the optical fibers connecting nodes will extend for tens of kilometers,
photon loss becomes a critical parameter that must be mitigated. Third, as
advanced network applications require the heralded delivery of shared
entangled states ready for further use, the qubit systems must be able to
store quantum information for extended times and the network system must be
capable of applying real-time feedback to the qubits upon successful
entanglement generation.

Recent qubit experiments have shown promising progress towards the latter two
criteria, including the integration with efficient quantum frequency
converters [19, 20, 21, 22, 23, 24], demonstration of long coherence times on
qubit systems that can be extended into multi-qubit registers [25, 26] and
entanglement generation between nearby qubits via tens of kilometers of
optical fiber [27, 28]. In parallel, experiments on ensemble-based quantum
memories have pioneered significant advances on the first two criteria [29,
30, 31, 32].

Here, we report on the realization of a deployed quantum link between two
solid-state qubit nodes separated by $10\text{\,}\mathrm{km}$ matching all
three criteria. The two network nodes are combined with a midpoint heralding
station via $25\text{\,}\mathrm{km}$ of deployed fiber, with all relevant
classic and quantum signals propagating over the same fiber bundle in telecom
bands (see Fig. 1). We implement an extensible architecture that enables the
nodes to operate fully independently at large distance, mitigates the effects
of photon loss on the entangling rate and allows for full heralding of
entanglement generation. Furthermore, the network architecture features
precise polarization and timing control as well as active stabilization of the
relative optical phase between photons emitted from the nodes, enabling the
use of the loss-resilient single-click protocol for efficient entanglement
generation [33, 34]. We benchmark the performance of the architecture by
parameter monitoring and by generating entanglement in post-selection.
Finally, we use the full network capabilities of heralding and real-time
feedback to deliver entangled states shared between the nodes ready for
further use. This demonstration establishes a critical capability for future
applications and scaling and presents a key milestone towards large-scale
quantum networking.

Figure 1: The metropolitan-scale quantum link. Cartographic layout of the
distant quantum link and the route of the deployed fiber bundle, with similar
quantum processor nodes in Delft and The Hague. Fiber length between node
Delft and midpoint is $15\text{\,}\mathrm{km}$, and between node The Hague and
midpoint is $10\text{\,}\mathrm{km}$, with losses on the quantum channels at
$5.6\text{\,}\mathrm{dB}$ and $5.2\text{\,}\mathrm{dB}$, respectively. Inset
to the quantum processor are the used qubit energy levels where the qubit is
encoded in the electronic ground state addressable with microwave pulses (MW),
and the spin-selective optical transition
($\lambda=$637\text{\,}\mathrm{nm}$$) is used for entanglement generation and
state readout.

##  I Deployed quantum network link architecture

In order to meet the challenges of metropolitan-scale entanglement generation,
we designed and implemented the control architecture depicted in Fig. 2A. Each
node contains a CVD grown diamond chip hosting a Nitrogen-Vacancy (NV) center
electronic spin qubit that can be faithfully initialized and read out by
resonant laser light and controlled using microwave pulses. The NV center
optical transition at $637\text{\,}\mathrm{nm}$ is used for generating qubit-
photon entanglement.

Each node is equipped with a stand-alone quantum frequency converter (QFC)
unit that converts the $637\text{\,}\mathrm{nm}$ NV photons down to the
telecom L-band at $1588\text{\,}\mathrm{nm}$ such that photon loss in the
deployed fiber is minimized. The QFCs further serve as a tuning mechanism for
compensating strain-induced offsets between the native emission frequencies
typical for solid-state qubits. Through independent feedback on the frequency
of the individual QFC pump lasers, we achieve conversion to a common target
wavelength despite the few gigahertz difference in qubit emission frequencies
[35]. The QFC in Delft is based on a novel noise-reduced approach (NORA) [36]
that produces two orders of magnitude lower background counts than the
periodically-poled Lithium Niobate (ppLN) with integrated waveguide based QFC
in The Hague [37].

To further mitigate photon loss, we employ the single-photon entangling
protocol [33, 34, 38] for which the entangling rate favourably scales with the
square root of the photon transmission probability across the entire link. In
this protocol, each qubit is first prepared in an unbalanced superposition
state $\ket{\psi}=\sqrt{\alpha}\ket{0}+\sqrt{1-\alpha}\ket{1}$. Application of
an optical $\pi$-pulse resonant for qubit state $\ket{0}$ and subsequent
spontaneous emission then results in qubit-photon entanglement, where the
photonic qubit is encoded in the photon number state (0 or 1). Overlap of the
photonic states at the beam splitter at the midpoint removes the which-path
information, followed by single-photon detection by superconducting nanowire
single photon detectors. Upon measurement of one photon after interference,
entanglement of the qubit states is heralded to
$\ket{\Psi^{\pm}}=\left(|01\rangle\pm e^{i\theta}|10\rangle\right)/\sqrt{2}$
with maximum fidelity $1-\alpha$, where the $\pm$ sign is set by which output
arm the photon was detected in. The entangled state phase $\theta$ is
dependent on the optical phase difference between the photonic modes arriving
at the interference beam splitter from Delft and The Hague.

Figure 2: Quantum node components and metropolitan-scale stabilization
performance  (A) Detailed components of the quantum nodes and fiber link
connections. A micro-controller ($\mu$c) orchestrates the experiment, which
together with an arbitrary waveform generator (AWG) shapes laser- and
microwave (MW) pulses. Solid-state qubit entangled photon emission and
stabilization light from each node is converted to the telecom L-band by the
NORA (ppLN) based QFC in node Delft (node The Hague) and sent to a central
midpoint. There, long distance qubit-qubit entanglement is heralded via single
photon measurement (superconducting nanowire single photon detector (SNSPD),
efficiency$\approx 60\%$, darkcount rate $\approx 5s^{-1}$) with detection
outcomes fed back in real-time. The stabilization light is used for phase
locking at the nodes $\left(\theta_{1},\theta_{2}\right)$, at the midpoint
$\left(\theta_{3},\theta_{4},\theta_{5}\right)$ and for phase lock
desaturation to the QFC pump lasers at the nodes $\left(\nu\right)$. The
performance of stabilization over the deployed link over 24hrs is shown for
(B) Time of arrival, (C) Phase and frequency and (D) Polarization. Hardware
providing active feedback (header) keeps these parameter that are drifting
over time (line histogram) stable (shaded histograms) by enabling continuous
feedback faster than the experienced drifts (insets). Vertical lines show the
modeled impact on fidelity and rate.

This entanglement generation critically relies on photon indistinguishability
at the heralding station and therefore all degrees of freedom of the photons
(frequency, arrival time, phase, polarization) must be actively controlled at
the metropolitan scale. A defining feature of our architecture is that
stabilization laser light used for phase locking and polarization
stabilization is time-multiplexed with the single-photon signals used for
entanglement generation and sent over the same fiber from the nodes to the
midpoint. To this end, we operate the link at a pre-defined heartbeat at
$100\text{\,}\mathrm{kHz}$, defining a common time division. During each
heartbeat period of $10\text{\,}\mathrm{\SIUnitSymbolMicro s}$, stabilization
light is sent to the midpoint continuously, except for a
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ period where the entangling photonic
states are sent out. This allows for near-continuous stabilization with high
feedback bandwidth while performing entanglement generation. Below we discuss
the major sources of drift at metropolitan scale affecting the entangled state
generation and our strategy for mitigating them.

First, length drift of the deployed fibers can result in reduced overlap of
the photonic modes on the beam splitter. The timing of the optical $\pi$-pulse
is locally controlled and disciplined by an optically linked distributed clock
that doubles as Ethernet connection (White-Rabbit protocol [39]) over a
dedicated fiber. Fig. 2B shows a histogram of measured offsets in time-of-
arrival for the two deployed-fiber segments over 24 hours, with the inset
displaying the drift speeds ($\sigma$ =
$5\text{\,}\mathrm{ps}\text{\,}{\mathrm{min}}^{-1}$). By using this data to
compensate for the drift via timing adjustments of the control electronics
every $\sim$15 minutes, the offset is kept below $50\text{\,}\mathrm{ps}$,
much smaller than the photon $1/e$ decay time of $12\text{\,}\mathrm{ns}$. The
resulting entangled state infidelity due to length drifts is below 0.1%.

Figure 3: Post-selected entanglement over the deployed link (A) Space-time
diagram depicting the generation of entanglement in post-selection. Horizontal
grey lines indicate the periodic heartbeat of $100\text{\,}\mathrm{kHz}$.
Local qubit control used to generate entanglement and perform state readout
(pop-out) all fit within one heartbeat period of
$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$. A local phase (LP) pulse is
followed by spin-photon entanglement (SPE) generation, an echo and basis
selection microwave pulse. Finally, the state is read out (RO).(B) Outcome of
correlation measurements, with different detector signature for the top and
bottom panels. We show the qubit-qubit readout outcomes per correlators(left),
as well as the resulting values per correlator (right). The calculated state
fidelity is given inside each figure. The number in parenthesis indicates the
amount of events recorded for that correlator. Horizontal grey bars indicate
the theoretical model. (C) Average state fidelity (left vertical axis) and
entanglement generation rate (right vertical axis) for varying photon
acceptance window length. Circling indicates the window used in (B), black
solid line is a model with no free parameters (see Suppl. Mat.). Inset shows
signal-to-noise ratio for the various window lengths. All measurement outcomes
are corrected for tomography errors, errorbars are 1 standard deviation.

Second, as the phase difference of the photonic modes interfering at the beam
splitter is imprinted on the entangled state, this phase must be known in
order for the generated entangled states to be useful. To achieve a known and
constant optical phase setpoint, five individual phase locking loops are
implemented across the total link, see Fig. 2A. At each node, a local phase
lock is closed between reflection of the resonant excitation light off the
qubit device surface and the stabilization laser light via the controllers
$\theta_{1}$ and $\theta_{2}$, stabilizing the in-fiber and free-space
excitation and collection optics. Phase noise from the long fiber and
excitation laser is mitigated by the controllers $\theta_{3}$ and $\theta_{4}$
via interference of light from the midpoint reference telecom laser and
frequency down-converted stabilization light at nanowatt levels. Importantly,
analog phase feedback is performed locally at the midpoint directly on the
incoming light yielding a high stabilization bandwidth exceeding
$200\text{\,}\mathrm{kHz}$. Lastly, interference of telecom stabilization
light from both nodes at the central beam splitter is measured by the single-
photon detectors and input to controller $\theta_{5}$, closing the global
phase lock between the nodes (see Suppl. Mat. for additional details). We note
that the modularity of this architecture directly allows for the connection of
multiple nodes to the same midpoint in a star topology, as the synchronization
of all incoming signals to a central reference and relative phase
stabilization between links can be performed using the control system at the
midpoint.

Fig. 2C displays a histogram of the resulting phase errors during 24 hours of
operation. Stable operation is achieved with a few-percent impact on the
entangled state fidelity per connection. As this architecture yields full
control over the phase difference at the beam splitter, the phase $\theta$ of
the entangled state can be tuned on demand by adjusting the setpoint of the
phase lock (see Suppl. Mat.). In order to maintain the phase lock under
frequency drifts at the nodes, two individual feedback loops
$\left(\nu\right)$ between the nodes and the midpoint adjust the frequency of
the individual QFC pump lasers at an update rate of $500\text{\,}\mathrm{Hz}$.
Importantly, this desaturation allows for a large dynamic range of the phase
feedback, as required to handle the observed frequency drift range of
$>$$50\text{\,}\mathrm{MHz}$, see Fig. 2C inset.

Third, polarization drifts, though significantly slower than phase drift, must
also be mitigated. To this end, the stabilization light is additionally used
for electronic polarization compensation (EPC) at the midpoint. The amplitude
of the error signal input at $\theta_{3}$ and $\theta_{4}$ is dependent on
polarization overlap with an in-fiber polarizer. We use this as input for a
gradient ascent algorithm to feedback on the polarization of the incoming
light at the midpoint. Data on the deployed link (inset Fig. 2D) shows that
polarization drift occurs on second-timescales; our feedback at a few Hz
bandwidth keeps the polarization aligned to within a few percent (Fig. 2D).
Any remaining polarization mismatch is removed by the polarizers at the cost
of a slightly reduced entanglement generation rate.

##  II Post-selected entanglement generation over a deployed link.

We now turn to the performance of the deployed link in generating entanglement
between the solid-state qubits at the remote nodes. The proper functioning of
all components of our system is first validated in a set of experiments with
all devices of the link in a single lab in Delft, showing successful
entanglement generation at state fidelities exceeding $0.6$ (see Fig. S3).
After connecting and calibrating the equipment at the remote locations, we
first focus on generating entanglement in post-selection. In this protocol the
qubits are measured directly after generating spin-photon entanglement, and
successful photon detection at the midpoint is used in post-processing to
analyze entanglement generation. This scenario is compatible with quantum key
distribution, but does not allow for more advanced protocols as the entangled
state is not available for further use [2].

Our implementation of the post-selected entanglement generation is depicted in
the space-time diagram of Fig. 3A, where each horizontal grey line depicts one
$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$ heartbeat period. Both nodes signal
their start-of-experiment after passing their own charge-resonance check (CR
Check) that ensures that the lasers are on resonance with the relevant optical
transitions [16]. After communicating their readiness they resolve the
earliest heartbeat to start attempting entanglement generation. In the first
20 heartbeat periods both nodes stabilize their local phase, followed by 540
rounds of entanglement generation, with one attempt per heartbeat period.
Every seventh entanglement round also contains optical pulses for maintaining
stability of the local phase. The operations performed at the nodes for each
round are detailed in the pop-out of Fig. 3A. The sequence returns to the CR
Check after completing the preset number of rounds.

We characterize the generated non-local states by measuring qubit-qubit
correlations in different readout bases. In Fig. 3B we plot the outcomes for
the three basis settings split out per detector, showing the expected
(anti-)correlations. Combining the outcome probabilities we calculate the
correlators $\langle ZZ\rangle$, $\langle XX\rangle$ and $\langle YY\rangle$.
Note that as the two detectors herald different Bell states ($\Psi^{+}$ vs
$\Psi^{-}$), the corresponding $\langle XX\rangle$ and $\langle YY\rangle$
correlations have opposite sign. We also plot the values predicted by our
detailed model without any free parameters (grey lines, see Suppl. Mat.), and
observe good agreement with the data. The asymmetry in the amount of events is
caused by a difference in quantum efficiency between the two single-photon
detectors. We find that the measured fidelities
$F(\ket{\Psi^{\pm}})=\frac{1}{4}*\left(1-\langle ZZ\rangle\pm\langle
XX\rangle\pm\langle YY\rangle\right)$ with respect to the ideal Bell states
are significantly above $0.5$, proving the generation of post-selected two-
qubit entanglement (Fig. 3B).

In the above analysis, we included photon emission up to
$10\text{\,}\mathrm{ns}$ after the optical $\pi$-pulse. By varying the
analysis window of the photon detection, we can explore the trade-off between
rate and fidelity (Fig. 3C). We find that the entangled state fidelity slowly
decreases with increasing window length, following the signal-to-noise ratio
of the photon detection at the midpoint (see inset to Fig. 3C). At the same
time, as more photon detection events are accepted with increasing window
length, the success rate increases. The achieved entanglement generation rate
reaches $0.48\text{\,}\mathrm{Hz}$ (success probability per attempt of
$7.2\cdot 10^{-6}$) for a $20\text{\,}\mathrm{ns}$ window.

##  III Fully Heralded entanglement generation over a deployed quantum link

In a final demonstration that highlights the capabilities of the deployed
platform, we generate fully heralded qubit-qubit entanglement. In contrast to
the post-selected entanglement generation described above, “live” entangled
states are now delivered to the nodes that can be further used for quantum
information tasks. Such live entanglement delivery is a fundamental
requirement for many future applications of long-range entangled states [2].

Figure 4: Fully heralded entanglement over the deployed link (A) Space-time
diagram of fully heralded entanglement generation. An attempt is successful
upon registering a heralding signal at the polling time, after which a feed-
forward is applied on the qubit and readout is performed. Absence of a
heralding signal communicates a failed attempt, where we retry for a maximum
of 228 attempts or until success. Pop-out depicts the local qubit control,
basis selection and readout pulses. The time between spin-photon entanglement
(SPE), heralding poll and basis selection is node-dependent. (B) Hahn-echo
experiment on the communication qubit, showing the revivals of the coherence
[40]. Solid vertical line indicates the heralding poll, the dotted line the
time of the basis selection. All times are with respect to echo sequence
start. (C) Correlation measurement for full heralding, showing both detector
outcomes delivering the same $\Psi^{-}$ state. Upper (lower) plot shows events
per detector (combined). Bars indicate data, the number in parenthesis
indicate the amount of events. Horizontal lines indicate the theoretical
model. (D) Average state fidelity (left axis) and entanglement generation rate
(right axis) for varying photon acceptance window length. Circling indicates
the window used in (C), black line is a model with no free parameters (see
Suppl. Mat.). Inset shows signal-to-noise ratio for respective window lengths.
Measurement outcomes are corrected for tomography errors, errorbars are 1
st.dev.

We emphasize that this protocol requires that all relevant heralding signals
(including which detector clicked) are processed at the node before the
entanglement delivery is completed. To this end, we employ the experimental
sequence depicted in Fig. 4A. To preserve the qubit states with high fidelity
while waiting for the heralding signals to return and be processed, the
refocusing echo pulse is applied to the qubits halfway the sequence to
dynamically decouple them from spin bath noise in their solid-state
environment. Fig. 4B shows the resulting qubit preservation as a function of
time. While the qubits are protected at the nodes, the photons travel to the
midpoint in about $52\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and
$73\text{\,}\mathrm{\SIUnitSymbolMicro s}$ from The Hague and Delft,
respectively. An FPGA at the midpoint processes the output of the single-
photon detectors, establishing whether a photon was detected in a
predetermined time window and in which detector. The electronic output of the
FPGA is optically communicated to the nodes, taking another
$52\text{\,}\mathrm{\SIUnitSymbolMicro s}$
($73\text{\,}\mathrm{\SIUnitSymbolMicro s}$). There, the signal is detected
and processed live to choose the next action. The time at which this final
processing is completed per node is indicated by the solid pink lines in Fig.
4B.

We choose to use the first echo revival (orange dotted vertical line in Fig.
4B) after the expected arrival time of the heralding signal to complete the
delivery of the entangled state. For these setpoints, a detailed measurement
of the echo contrast shows that the wait times introduce a $1.3$ ($3.2$)
percent reduction in coherence for node Delft (The Hague). We note that using
more advanced dynamical decoupling sequences the NV qubit coherence time could
be extended towards a second [41].

The entanglement generation runs are automatically repeated by the nodes until
a successful heralding signal is received from the midpoint. Once such a
signal is received, the system jumps to a different control sequence in which
a basis selection gate is applied to each qubit followed by single-shot qubit
readout. This control sequence incorporates the which-detector information
communicated from the midpoint in real time. We exploit this full heralding
capability to apply a phase flip conditioned on which detector clicked,
thereby delivering the same Bell state
$\Psi^{-}=\frac{|01\rangle-|10\rangle}{\sqrt{2}}$ for each of the two possible
heralding signals.

The measured correlations per readout basis are shown in Fig. 4C, showing the
expected anti-correlated outcomes for all three bases. The top bars show the
outcomes divided per detector, displaying now the same $\Psi^{-}$ thus showing
the successful operation of the real-time feedback. We find that the delivered
entangled states have a fidelity of $0.534(15)$. This result establishes the
first demonstration of heralded qubit-qubit entanglement at metropolitan
scale, with all the heralding signals processed in real-time and the entangled
states delivered ready for further use.

Fig. 4D displays the rate-fidelity trade-off in analogy to Fig. 3C, showing a
similar trend. The reduction in rate
($$0.022\text{\,}{\mathrm{s}}^{-1}$=$1.3\text{\,}{\mathrm{min}}^{-1}$$ at the
pre-defined $15\text{\,}\mathrm{ns}$ window length) compared to the post-
selected case is mainly due to the added communication delay needed for the
heralding signal to travel to the nodes, making each attempt a factor of
$\approx 20$ slower. The observed fidelity vs. window size is well captured by
our model without any free parameters, with the reduction in fidelity compared
to the post-selected case mainly coming from the additional decoherence and a
reduced phase stability (black line, see Suppl. Mat.). The improved SNR (Fig.
4D inset) compared to Fig. 3C is due to an improved trade-off between
detection efficiency and dark counts following an optimization of the single-
photon detector bias currents.

##  IV Conclusion and outlook

We have realized a deployed quantum link and demonstrated heralded
entanglement delivery between solid-state qubits separated at metropolitan
scale. The architecture and methods presented here are directly applicable to
other qubit platforms [42, 43, 44, 45, 27, 13] that can employ photon-
interference to generate remote entanglement and frequency conversion to
minimize photon losses. Additionally, the ability to phase-lock remote signals
without the need for ultra-stable reference cavities can be of use for
ensemble-based quantum memories [30, 29, 46, 31].

This work can benefit from future developments in the following ways. Real-
time correction of false heralds [17] can be realized by using detection
events of phonon side-band photons on the nodes, upon which the fidelity of
the delivered entangled state improves (see Suppl. Mat.). Near-term
developments can substantially improve the signal-to-noise ratio, which is
currently limiting the entangled state fidelity (about 30% contribution). For
instance, the signal can be significantly boosted by embedding the NV-centre
in an open micro-cavity [47, 48] or by using different color centers that
exhibit a more efficient spin-photon interface [49, 24, 28, 50]. As a
quantitative example, replacing the NV center with a diamond SnV center (which
has a 16 times higher probability of coherent photon emission), employing NORA
QFCs at all nodes [36] (here used only in the Delft node), and fixing a known
imperfection in the local phase stabilization would already increase the
heralded state fidelity to above 80%. Furthermore, improving coherence
protection using XY4 sequences and systematic reduction of the remaining small
error sources could lift the fidelity beyond 90% (see Table S4). The
extensible nature of our architecture opens the door to connecting more than
two qubit nodes to a midpoint, which, in combination with using local memory
qubits [25, 51, 14, 52], would enable the exploration of more advanced
protocols on a metropolitan-scale network [53, 17, 18], as well testing
quantum control stacks [15] on a distributed multi-node quantum network.

## Acknowledgements

We thank Christine Satter, Castro do Amaral, Eftychia Tsapanou-Katranara and
Jacob Dalle for help with characterizing and fabricating the diamond device,
Leon van Dooren for help in developing the local phase stabilization
implementation, Jaco Morits for system engineering support, Theo Lodewijkx for
managing the engineering tasks in the project, Ingrid Romijn for program
management support and Stephanie Wehner for project development in the early
stages of the project, Otmar Ubbens for support on various stabilization
tasks, Sidney Cadot, Henri Ervasti, Ravi Budhrani for software development and
support, Raymond Vermeulen and Raymond Schouten for development of custom
electronics, Siebe Visser and Vinod Narain for technical assistance during the
moving of the equipment, Laurens Feije for assistance with the move of the
midpoint, Marco Gorter and Nico Coesel for providing hardware support with the
OPNT time synchronization devices, and Paul Slootmaker and Ed van de Bovenkamp
for their organizational and on-site support at the KPN locations. Funding: We
acknowledge funding from the Dutch Research Council (NWO) through the project
“QuTech Part II Applied-oriented research” (project number 601.QT.001), the
Spinoza prize 2019 (project number SPI 63-264), and the Zwaartekracht program
Quantum Software Consortium (project no. 024.003.037/3368). We further
acknowledge financial support from the EU Flagship on Quantum Technologies
through the project Quantum Internet Alliance (EU Horizon 2020, grant
agreement no. 820445) and from the Dutch Ministry of Economic Affairs and
Climate Policy (EZK), as part of the Quantum Delta NL programme, and Holland
High Tech through the TKI HTSM (20.0052 PPS) funds. Finally, we acknowledge
funds from Fraunhofer-Gesellschaft zur Förderung der angewandten Forschung
e.V. and Fraunhofer Institute for Laser Technology ILT through the ICON
program. Author contributions: A.J.S., K.L.vd.E., E.v.Z. and R.Hanson devised
the experiment. R.W.H., W.D.K., A.J.H.M., R.Hagen, J.J.B.B. tested and built
experimental hardware, prepared the remote locations and supervised moving.
J.v.R., P.B. and I.t.R-D. developed the software for remote control and
monitoring of equipment, and heralding with input from A.J.S. and K.L.vd.E.
M.M. and A.M.E. grew and prepared the diamond device substrates. R.V.
fabricated one of the diamond devices. J.F.G, F.E., B.J. and C.H. designed,
built and tested the NORA QFC apparatus. C.T., J.S., S.R. designed, built and
tested the low phase noise excitation lasers. A.S. and K.L.vd.E. wrote the
measurement code, carried out the experiments and analyzed the data. A.J.S.
and M.-C.S. developed supporting simulations. A.J.S., K.L.vd.E. and R.Hanson
wrote the main manuscript. A.J.S. and K.L.vd.E. wrote the supplementary
information. R.Hanson conceived the project and supervised the experiments.
All authors commented on the manuscript. Competing interests: The authors
declare no competing interests. Data and materials availability: The data
supporting this manuscript is available at 4TU.ResearchData [54].

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Supplementary Materials

##  S1 Experimental setup

The experimental set-ups used in this work are built on top of the hardware
described in Ref. [35] and they use software based on the Quantum Measurement
Infrastructure, a Python 3 framework for controlling laboratory equipment
[55]. In this section we will discuss the hardware changes made that enabled
compatibility with metropolitan-scale entanglement generation.

###  S1.1 Quantum Node

We exchanged the diamond quantum device of Node The Hague for a newly
fabricated one. All parameters of the substrate (Element Six) and device, such
as purity, crystal orientation and carbon isotopes are the same as previously
reported. To allow for phase stability between physically separated setups,
the TOPTICA DLC DL pro $637\text{\,}\mathrm{nm}$ was exchanged for an upgraded
prototype employing optical feedback from an additional cavity to reduce the
phase noise to $<$40\text{\,}\mathrm{mrad}$$ integrated from
$100\text{\,}\mathrm{kHz}$ to $100\text{\,}\mathrm{MHz}$.

###  S1.2 Quantum Frequency Converters

As mentioned in the main text, we have replaced one of the in-house built QFC
modules with the NORA QFC described in [36]. This design mitigates the amount
of noise photons generated by the frequency conversion due to imperfections in
the waveguide and poling period of periodically poled Lithium Niobate (ppLN)
crystals. Briefly, the novel approach consists of identifying the right bulk
material that can be critically phase-matched to perform efficient conversion,
and embedding the crystal in a single-resonant cavity to enhance the effective
pump power in the conversion process. We compare the NORA QFC to the ppLN QFC
in Table S1, and refer to [36] for more information.

QFC |  Node |  Non-linear medium |  Efficiency (front to end) |  Noise $\left[$\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{GHz}}^{-1}$\right]$ |  Eval. at filter bandwidth
---|---|---|---|---|---
QuTech [23, 35] |  The Hague |  ppLN waveguide |  $50\%$ |  $2104$ |  $50\text{\,}\mathrm{MHz}$
Fraunhofer ILT [36] |  Delft |  bulk KTA |  $48\%$ |  $19$ |  $374\text{\,}\mathrm{GHz}$
Table S1: Comparison of the QFCs used in the experiment.

###  S1.3 Phase stability on the nodes

On both nodes we added the components to define the local interferometers that
lock the excitation laser path to the stabilization path of the respective
set-ups. We use a combination of up- and down-shift Acousto-Optical Modulators
(AOM, Gooch and Housego Fiber-Q), already used for amplitude modulation, to
define a $400\text{\,}\mathrm{MHz}$ frequency offset. The excitation light
reflected off the diamond surface is separated from the zero-phonon-line (ZPL)
path by a polarizing beam splitter (PBS), where it spatially overlaps with the
stabilization light. We guide the light to a set of birefringent crystals that
maximize the complex overlap between the orthogonally polarized reflected beam
and the stabilization light. A polarizer then selects a common polarization
such that both beams can interfere in intensity, after which we focus the
light into a multi-mode fiber (core diameter $\approx 100um$) and measure the
interference beat signal using a fast avalanche photodiode (Menlo Systems
APD210). After amplification and filtering, we extract the phase of the
interference by mixing it with an electronic reference signal generated by a
stable RF source (Anapico APSIN6010). The feedback signal generated is passed
through a Track-And-Hold amplifier (Texas Instruments OPA1S2384) and input to
the Frequency Modulation input of the Excitation light AOM RF-driver (TimeBase
DIM3000), which closes the feedback loop. Because the frequency modulation has
integral feedback on the phase that can not be switched on/off fast, any
bias/offset in the error-signal reduces the free evolution time during which
the local phase remains stable. In a future implementation, this actuator will
be replaced by a linear phase shifter, and an improvement of the local phase-
stability can be realized.

###  S1.4 Midpoint

The previous design of the midpoint used for two-photon quantum interference
[35] suffered from an increased noise floor due to the presence of bright
laser pulses incident on the nanowires every
$200\text{\,}\mathrm{\SIUnitSymbolMicro s}$. To reach an adequate feedback
bandwidth for phase-locking of our excitation lasers to the telecom reference,
we have to reduce the period of bright pulses to
$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$, making this even more challenging.
We have made the following improvements to allow for polarization and fast
phase feedback, to reduce crosstalk between optical channels and lower the
background counts in the SNSPDs:

  * •

Split the high-power reference light into three paths, two used to generate an
optical beat to lock the stabilization light coming from each node to the
reference laser, and one used to generate an error signal to stabilize the
ultra-narrow Fibre-Bragg grating filters.

  * •

Added two low-loss Variable Optical Attenuators (VOAs, Boston Applied
Technologies Nanona VOA) that shield the detectors from bright pulses coming
from the nodes.

  * •

Added two Electronic Polarization Controller (OZOptics
EPC-400-11-1300/1550-9/125-S-3A3A-1-1 ) to allow for full control over the
polarization state to compensate for fiber drifts.

  * •

Added two AOMs to have fast and integral control of the phase of the incoming
light, used for phase stabilization.

  * •

Removed one circulator from the design and splicing multiple connectors where
the single-photons pass through to reduce single-photon loss

All the error signals for the phase- and polarization stabilization, and the
ultra-narrow fiber Bragg-grating (FBG) for one node are generated on the same
balanced photodiode (Thorlabs PDB480C-AC), and subsequently extracted by a
combination of power-splitter, bandpass filters and amplifiers.

###  S1.5 Heralding

To allow for low-latency feedback of photon detections at the midpoint, the
midpoint was upgraded with an FPGA development board (Digilent Arty-A7) with
custom-built PCBs converting the external pin I/O to SMA connectors with the
possibility of $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ matching. Single photon
signals were input on a custom pulse-stretcher, cleaning up the electrical
pulse from the SNSPD to a block-like pulse shape of around
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and input on the I/O of the FPGA.
The complete I/O of the FPGA is shown in Fig. S9. The acceptance window of
sending out a heralding signal is predetermined at the time within a heartbeat
period where we expect spin-photon entangled photons to arrive. The exact
timing of the acceptance window is derived from the centralized clock and
heartbeat signal available at the midpoint. To be able to signal which
detector clicked on a photon count, we use two digital pulses to herald, one
for the success, and one for which detector clicked. Since the delay in time-
of-flight between the nodes is different, the digital pulses need to be
aligned individually to arrive at the nodes at the same time within one
heartbeat. To clarify, this does not mean they arrived at the same global time
at the nodes, but that the digital pulses arrive at the same time modulo the
heartbeat ($10\text{\,}\mathrm{\SIUnitSymbolMicro s}$) at the nodes. Lastly, a
local FPGA code loading server was employed that could upload compiled-VHDL
that was either manually or automatically generated remotely. A change in
timing of sending out pulses could be updated on the underlying VHDL code
automatically based on a custom template code. This allowed fast multi-core
compilation locally and remote upload to significantly reduce debugging time
when determining the required exact timing of the heralding pulses.

##  S2 Phase stabilization in the midpoint

We stabilize the incoming light from each node to the same telecom reference
(fast stabilization). The relative optical phase between the nodes is
stabilized using interference at the central beam splitter in the midpoint,
and measured with the SNSPDs (global stabilization). The optical fields and
respective frequencies are shown in Fig.S1, and the distribution of the fields
in time shown in Fig. S4 and S5.

###  S2.1 Fast stabilization

Part of the stabilization light used in the local phase lock also propagates
via the QFC, through the same deployed fiber, to the midpoint. There it is
split off in frequency from the single-photons via the FBG, subsequently
interfered with a telecom reference laser (NKT Adjustik fiber laser), where a
heterodyne interference beat is measured by the balanced photodiode. A phase-
frequency detector (Analog Devices HMC3716LP4E) processes this signal by
comparing it to a reference signal (Wieserlabs FlexDDS-NG DUAL). Its output
serves as the error-signal for a high-speed servo controller (Newport
LB1005-S) performing PID control. This controller provides feedback on the AOM
to stabilize the incoming light, synchronizing it with the reference at a
bandwidth of over $200\text{\,}\mathrm{kHz}$. To stay within the low-loss
performance of the AOM close to the central frequency, we offload the
accumulation of phase error (frequency drifts) to the QFCs at the nodes. This
can be done much slower ($500\text{\,}\mathrm{Hz}$) and makes the feedback
only limited by the frequency range of the QFCs
($\gg$$1\text{\,}\mathrm{GHz}$). The fast feedback removes all high-frequency
noise from the incoming light, which contains the excitation laser line-width,
as well as phase noise introduced by expansion/contraction and vibrations of
the deployed fiber.

###  S2.2 Global stabilization

To further compensate phase drift that occurs in the midpoint and to set the
relative phase between the incoming optical modes, we employ a control loop
based on the stabilization light from both nodes that interferes at the
central beam splitter. This light leaks through the FBG, and is of sufficient
low power to not blind the SNSPDs. The voltage pulses from the SNSPDs are
amplified, and a difference amplifier (Krohn-Hite Model 7000) generates a
heterodyne beat by subtracting the count-rates from each detector. This beat
is compared to a reference signal (AimTTi) by mixing it and low-pass filtering
($150\text{\,}\mathrm{Hz}$), generating an error-signal. This is again input
to a servo controller (Newport LB1005-S), of which the output is used to
modulate the phase of the reference of one of the fast stabilization
controllers.

###  S2.3 Verifying phase stability

We can verify the performance of the full optical phase stabilization by
interfering the two excitation lasers from the nodes at the central beam
splitter in the midpoint, using the time-resolved counts in the SNSPDs to
generate the error signal. By changing the phase of the reference signal, we
can measure a full visibility fringe of the interference. By correcting the
measured interference for the imbalance in photon flux from each node, we can
calculate the interference contrast, see Fig. S2B. A more in-depth description
and technical analysis of the total phase stabilization will be given in a
separate manuscript [56].

##  S3 Overview communication times and decisions

In this section we discuss the different optical and electrical pulses or
feedback signals that are needed to control this distributed experiment. We
will first discuss the local nodes, then which signals are necessary to be
timed at the midpoint for single photon processing and phase stability.
Lastly, we will discuss how we align the arrival time of the photonic states
at the midpoint telecom detectors.

###  S3.1 Pulses and timing at node

Since the central clock in the midpoint is distributed to the nodes by the
White Rabbit enabled switches, it provides a common reference frame to align
the time-of-arrival for each of the entanglement generation attempts, see
Section S3.3. A full timing overview of a single heartbeat length when we
perform a full entanglement attempt without heralding is shown in Fig. S4A and
with heralding in Fig. S4B. The entanglement generation attempt with heralding
contains exactly the same pulses as in Fig. S4A, except for that the time
between the spin superposition state generation and basis-selection pulse is
at a revival of the spin coherence, as previously shown in the pop-out of Fig.
4A of the main text.

###  S3.2 Pulses and timing at midpoint

The laser, electrical and feedback pulses used at the midpoint by design
operate agnostically of the experiment that is being run. For about
$\sim$$8\text{\,}\mathrm{\SIUnitSymbolMicro s}$ we can perform phase,
frequency and FBG stability actions using the interference from the reference
light and the incoming stabilization light from the node (see Fig. S5). In the
remaining $\sim$$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ the local reference
light is shut off, the phase Track-and-Hold amplifier is set to ‘hold’ and the
VOAs opened fully to allow for single photons to pass through to the SNSPDs.

###  S3.3 Time-of-arrival alignment

One of the requirements for indistinguishability of the photons is the time
aligned photonic mode overlap at the beam splitter in the midpoint. The timing
of the emission of the photonic state determines, together with the time of
flight (ToF), the time at which the photonic state arrives from the node at
the midpoint. This is especially crucial in the metropolitan distance case,
where fiber distance varies with tens of picoseconds per hour (see Table S2).
Since we have unbalanced fiber connection lengths between the nodes-midpoint,
Node Delft will be allowed to send its photon as soon as possible. Node The
Hague has to hold its pulse sequence start for 3 heartbeats. Both nodes
require sub-heartbeat timing alignment to align the arrival time of the solid-
state entangled photons in the last
$\sim$$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ of the midpoint scheme (Fig.
5). This last alignment step is achieved with the arbitrary waveform generator
(AWG, Zurich Instruments HDAWG) sequencer and the built in skewing parameter
of the AWG outputs.

Due to the constantly changing ToF, the alignment in time is performed in two
steps. A first time alignment is required to be performed manually where laser
pulse timing is recorded at the midpoint timetagger (PicoQuant Multiharp,
$80\text{\,}\mathrm{ps}$ resolution), after which the time difference between
arrival times of the pulses from the two nodes can be calculated and adjusted.
This value is recorded as calibration parameter. Secondly, the reported ToF
from the time synchronization system is used to automatically adjust the
skewing parameter every $\sim$15 minutes with the difference in fiber ToF with
respect to the initial calibration.

##  S4 Other midpoint stabilization

###  S4.1 Polarization stabilization

We continuously stabilize the polarization of the incoming stabilization light
using the same beat generated on the balanced photodiode as mentioned in the
phase stability section (Section S2) . Because of our in-fiber polarizer, the
amplitude of the beat will depend on the overlap of the incoming light with
the polarizer. Maximizing this amplitude will guarantee the same polarization
between the nodes, as needed for interference. Using a phase-gain detector
(Analog Devices EVAL-AD8302) we can monitor the amplitude of the beat by
comparing it to an in-phase RF tone (generated by Wieserlabs FlexDDS-NG DUAL)
over multiple orders of magnitude. We sample the output of this phase-gain
detector with an Analog Discovery 2 (Digilent), and process the error signal
in a python process running a Gradient Ascent algorithm. This in turn sends
commands to the EPC that adapts the polarization state and completes the
control loop. The whole process can be monitored and adjusted via a Graphical
User Interface.

###  S4.2 Ultra-narrow FBG stabilization

We adapted the control loop for the temperature stabilization of the ultra-
narrow FBGs (UNF) to allow for a significant reduction of the optical power
reflected back to the SNSPDs. We achieved this by using a heterodyne beat
measurement that is sensitive to the transmission through the UNF, instead of
a simple power measurement at DC. This allows for multiple orders higher
sensitivity, and allows us to stabilize the filters with only $\approx$
$10\text{\,}\mathrm{fW}$ of light traveling through the filter in the
backwards direction.

##  S5 Deployed fiber characterization and stability

We use a fiber bundle with four fibers to perform all the communication
between the locations. These are provided to us by KPN, a major Dutch telecom
provider, and are part of their telecom infrastructure. The reality of using
deployed fibers means that it is built up of sections of continuous fiber,
which are spliced or connected together at various locations along the path.
Therefore the losses per kilometer of deployed fiber are significantly higher
than the ideal losses of
$\approx$$0.2\text{\,}\mathrm{dB}\text{\,}{\mathrm{km}}^{-1}$ @
$1580\text{\,}\mathrm{nm}$, and can differ from fiber to fiber. During the
period of measuring over the deployed link in 2023, several parts of the fiber
were adapted and re-routed as part of restructuring of the network by KPN,
changing both their length ($\pm$ $100\text{\,}\mathrm{m}$) and loss ($\pm$
$1\text{\,}\mathrm{dB}$) multiple times over the span of a few weeks. Table S2
shows typical loss values measured during this period. Thanks to the modular
approach of our system, we have complete freedom in changing the arrival time
of the photon in the midpoint, allowing for easy re-alignment of the
propagation delay, both for the fiber restructuring as well as the daily
expansion and contraction.

Table S2: Parameters of Quantum Link. Link |  Loss local |  Loss deployed fiber |  Total |  Typ. Round-trip time, drift
---|---|---|---|---
The Hague to midpoint |  $3\text{\,}\mathrm{dB}$ QFC, $4.9\text{\,}\mathrm{dB}$ filters |  $5.2\text{\,}\mathrm{dB}$, $0.51\text{\,}\mathrm{dB}\text{\,}{\mathrm{km}}^{-1}$ |  $13.1\text{\,}\mathrm{dB}$ |  $103.16\text{\,}\mathrm{\SIUnitSymbolMicro s}$ $\pm$$40\text{\,}\mathrm{ps}\text{\,}{\mathrm{h}}^{-1}$
Delft to midpoint |  $4\text{\,}\mathrm{dB}$ QFC, $4.5\text{\,}\mathrm{dB}$ filters |  $5.6\text{\,}\mathrm{dB}$, $0.39\text{\,}\mathrm{dB}\text{\,}{\mathrm{km}}^{-1}$ |  $13.9\text{\,}\mathrm{dB}$ |  $145.31\text{\,}\mathrm{\SIUnitSymbolMicro s}$ $\pm$$80\text{\,}\mathrm{ps}\text{\,}{\mathrm{h}}^{-1}$

##  S6 Calibrations

###  S6.1 Entanglement generation

We employ a method where we interleave the generation of entanglement with
three calibrations that measure and re-calibrate critical parameters and
settings. These calibrations are an adapted version of the entanglement
generation sequence that measure specific key elements of the entanglement
generation: the signal-to-noise ratio (SNR), optical phase stability (PHASE)
and the entangled state phase (XsweepX). Typical results of these three
calibrations are shown in Fig. S2. When these calibrations have passed their
threshold, we proceed to measure the correlators used to calculate the
entangled state fidelity, FID for short.

The first calibration is the measurement of the single-photon signal from both
nodes and the background, which includes all the losses that are present in
the system. We prepare the communication qubit in the $\ket{0}$ to maximize
the single photon signal. The measurement is the same experimental sequence as
shown in Fig. S4A, where now the state generation $\alpha$ pulse is removed to
measure the signal, and replaced by a $\pi$-rotation to measure the noise.
Furthermore, the arrival time of the photons is shifted by
$100\text{\,}\mathrm{ns}$ to measure the brightness of both nodes
individually.

Second is determining the contrast and phase of the relative optical phase
between the nodes. We realize this by interfering a bright pulse
($1\text{\,}\mathrm{\SIUnitSymbolMicro s}$) of the excitation lasers which we
reflect off the diamond surface, and follows the exact path as the NV-center
emission from there. By varying the setpoint of the phase in the stabilization
scheme, we can measure a full interference fringe at the SNSPDs as seen in
Fig. S2B and characterize the residual phase noise of the system at the
specific time window where single photons interfere.

Furthermore, this calibration is used to set the phase of the optical
interference at any point along this fringe, which can be used to optimize the
entangled state fidelity [38]. We show the capability of changing the optical
phase to two different setpoints of $0\text{\,}\mathrm{\SIUnitSymbolDegree}$
and $180\text{\,}\mathrm{\SIUnitSymbolDegree}$.

The third and last calibration is the measurement of the entangled state
phase, which has an offset with respect to the optical phase setpoint. Using
the single-click protocol we create a correlated spin-spin state that we can
use to probe this phase. We do this by measuring correlations in the rotated
basis $X$ on Node Delft, while varying the readout basis in the Bloch sphere
equator plane for Node The Hague. We can extract this entangled state phase by
fitting the resulting oscillation with a single cosine, see Fig. 2C. We feed-
forward this phase in the entanglement measurements to always generate the
$\Psi^{\pm}=\frac{1}{\sqrt{2}}\left(\ket{01}\pm\ket{10}\right)$ during the
entanglement generation.

###  S6.2 Experimental setup performance

We maintain similar performance parameters of the experimental setup by
performing calibrations of many different settings at different timescales. On
a daily timescale, performance parameters of the communication qubit are re-
calibrated:

  1. 1.

Laser power (see Fig. S10)

  2. 2.

Single-Shot Readout

  3. 3.

Microwave qubit control ($\pi$, $\pi/2$, $\alpha$)

  4. 4.

Optical excitation (optical $\pi$ pulse) for entanglement generation attempt

  5. 5.

QFC efficiency

Roughly every 20 minutes, a set of shorter timescale calibrations are run.
Some are dependent on performance parameters that are available live during or
right after a measurement set. See Fig. S10 for the flowchart showing the
calibrations and decisions made. As discussed in the previous section, we
require several types of measurements to calibrate the phase of the entangled
state. If in any of those measurements we are outside of the bound of
acceptance, we restart the sequence. Additionally, the FID measurement
sequence is cut into smaller blocks of 10k (30k) successful CR-checks for the
heralded (delayed-choice) metropolitan distance experiments. After each block,
the CR-check passing rate is checked against the average CR-check rate during
the SNR sequence (per node). If the passing rate is below this value, we exit
the FID sequence and continue. We do this because the CR-check passing rate is
a value that includes many derived underlying performance parameters of the
setup as a whole, e.g. laser wavelengths, laser power, objective position and
others. If any of those values changed significantly the CR-check passing rate
is negatively impacted, which we use as a signal that an underlying parameter
of the experiment is drifting off ideal and subsequently we break out of FID
data taking. As can be seen in the flowchart, afterwards all calibrations are
restarted, and we start from a calibrated setup with the SNR sequence, which
is when we expect the highest CR check passing rate, and as such, use that to
define our threshold for the rest of the experiment.

##  S7 Modeling the generated entangled state

To assist in setting up the experiment and simulating the outcomes, we employ
a Monte-Carlo simulation based on the model described in [38]. This is an
extensive description of generating entanglement between remote nodes using
the single-click protocol. It considers the spin-spin density matrix, based on
the detection events where at most two photons reach the central detectors. By
calculating the probability of each detection pattern and its corresponding
spin-state, it arrives at the average density matrix given non photon-number
resolving detectors. It takes into account many physical parameters that have
an impact on the entangled state fidelity, such as photon loss, photon
indistinguishably, and the photon wavepacket shape. It also includes several
experimental parameters such as the effects of residual phase noise, dephasing
noise and darkcount probability.

In the experiments presented in the main text we do not apply a post-
processing correction using the Charge-Resonance check after the readout of
the spin-state. This correction can be done to identify sequences where one of
the nodes is in the NV0 state, and no longer optically responsive to the
excitation laser, but only in a post-processed fashion. To accurately model
the effect of including events when the NV-center is ionized on the fidelity
with the ideal state, we extend the model by including the probability of one
of the nodes being ionized. We assume that, when an NV photon from a
particular node in the midpoint is detected, this node can not be in the
ionized state, but the other node can be with a probability $p_{ion}$. If the
event in the midpoint is due to a noise photon, either of the nodes can be in
the ionized state. This probability is measured by saving the CR-outcomes
after the readout sequence. We include this ionized state as a third level of
our qubit, that is outside the space addressed by our microwave control. When
applying a readout pulse however, it will be read out as the $\ket{1}$
($=$dark) state. The single shot readout (SSRO) fidelity of the solid-state NV
qubits are given in Table S3. State readout correction is performed according
to iterative Bayesian unfolding as described in [57].

An overview of all the simulation parameters used and their measured/used
value for the simulation is given in Table S3. We have also simulated near-
term performance of the same link based on SnV centers (keeping the collection
efficiency the same), which have 16 times higher coherent photon emission
probability per optical excitation, full use of the improved QFC technology
and improved phase stability, shown in Table S4.

Table S3: Error budget of deployed link post-selected and heralded entangled states. Parameters noted with * are averages of data measured between the fidelity measurements, other are calibrated in separate experiments. The probability to detect a photon is given for $\alpha=1$. If only one value is mentioned it is considered equal for both nodes. The infidelity contribution is given as if it was the only error present, to make direct comparison easier. Note that due to the large contribution of noise counts, the infidelity of the final simulated state with all the contributions taken into account does not follow the sum of the individual contributions. Simulation parameter | Delayed-choice |  | $Fullyheralded$ |
---|---|---|---|---
| Value | Error con. | $Value$ | Error con.
Det. prob.* (Delft, DH)[] | 10.6e-6, 8.4e-6 | 32.7% | $10.0e-6,7.1e-6$ | 28.5%
$\alpha$* [] | 0.25 and 0.25 |  | $0.25and0.25$ |
Backgr. counts* (Det1, Det2) [Hz] | 40.3, 42.8 |  | $23.8,22.0$ |
Double excitation probability [] | 0.04, 0.1 | 3.7% | $0.04,0.1$ | 3.8%
Res. phase noise std* [∘] | $35$ | 9.8% | 45.3 | 14.3%
Dephasing probability [] | 0.01, 0.01 | 2.3% | $0.02,0.04$ | 4.5%
Spectral diffusion FWHM [MHz] | 13 | 4.5% | $13$ | 4.8%
Beam splitter imperfections [] | 0.95 |  | $0.95$ |
Detection window length [ns] | 10 | - | $15$ | -
Optical Excitation Rabi angle [∘] | 150, 150 | - | $150,150$ | -
Ionization probability* [] | 0.035, 0.045% | 5.4% | $0.046$ | 6.2%
PSB collection efficiency [] | 0.1, 0.1 | - | $0.0,0.0$ | -
Simulated error (all params) | - | 43.1%, 42.2% | $-$ | 45.6%
Simulated Fidelity | 0.568, 0.576 | - | $0.544$ | -
Measured error | - | 43.2%, 42.4% | $-$ | 46.6%
Avg. SSRO fidelity | 95.14%, 94.49% | - | $94.13\%,94.98\%$ |
Table S4: Error budget of near-term and future improvements. For the near-term performance, we simulated an improved ZPL fraction of the SnV, the implementation of the improved QFC on both nodes, and an improvement of the local phase-stabilization. For future improvements on the system with we have added improvements in qubit coherence via dynamical decoupling, reduction of double excitation via shorter optical excitation,reduced ionization by better control of the charge state of the emitter and improved mode overlap at the central beamsplitter.  Simulation parameter | Near-term |  | $Future$ |
---|---|---|---|---
| Value | Error con. | $Value$ | Error con.
Det. prob. (Delft, DH)[] | 16x10e-6, 16x10e-6 | 4.6% | $16x10e-6,16x10e-6$ | 4.6%
$\alpha$ [] | 0.05 and 0.05 |  | $0.05and0.05$ |
Backgr. counts (Det1, Det2) [Hz] | 5.0, 5.0 |  | $5.0,5.0$ |
Double excitation probability [] | 0.04, 0.04 | 1.3% | $0.01,0.01$ | 0.3%
Res. phase noise std [∘] | 15 | 1.7% | $15$ | 1.7%
Dephasing probability [] | 0.02, 0.04 | 3.0% | $0.01,0.01$ | 1.0%
Spectral diffusion FWHM [MHz] | 13 | 3.3% | $13$ | 1.3%
Beam splitter imperfections [] | 0.95 |  | $0.99$ |
Detection window length [ns] | 15 | - | $15$ | -
Optical Excitation Rabi angle [∘] | 150, 150 | - | $150,150$ | -
Ionization probability [] | 0.046 | 4.7% | $0.01$ | 1.0%
PSB collection efficiency [] | 0.1, 0.1 | - | $0.1,0.1$ | -
Simulated error (all params) | - | 17.2% | $-$ | 10.0%
Simulated Fidelity | 0.828 | - | $0.90$ | -

Figure S1: Frequency of relevant optical fields for the phase lock in the
midpoint. The grey Lorentzian shape is the ultra-narrow fibre-bragg grating
filter (FBG). It is stabilized using a $160\text{\,}\mathrm{MHz}$ beat
generated by the reference laser and a frequency-shifted tap-off that is back-
propagated through the flank of the FBG ($25\text{\,}\mathrm{MHz}$ from the
transmission peak). The stabilization light is located far from the filter
resonance, detuned by $400\text{\,}\mathrm{MHz}$ from the filter peak and
excitation laser/single-photons, which is locked by the local phase lock
controllers $\theta_{1}$ and $\theta_{2}$. This stabilization light is
reflected off the FBG and interferes with the reference laser generating a
$\approx$215\text{\,}\mathrm{MHz}$$ beat, which is stabilized by the fast
phase lock controller $\theta_{3}$ and $\theta_{4}$. The slight offset between
the nodes of $1.5\text{\,}\mathrm{kHz}$ is measured by the SNSPDs and input to
controller $\theta_{5}$, to close the phase lock between Delft and The Hague.
Figure S2: Typical calibration outcomes of: (A) signal strength (SNR) where
the dotted lines show the used ‘signal’ window for both nodes respectively.
The same windows are used to measure the ’noise’ when the NV qubit state is
rotated to the $\ket{1}$ (=dark) state.(B) Optical phase stability (PHASE),
with fitted contrast (black lines). The vertical lines project to the top
horizontal axis to determine the interference phase for the following
measurements.(C) The entangled state phase (XsweepX) for an interference phase
of $0\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$, showing the capabilities of the
stabilization system to set a specific phase setpoint. The fitted phases
(black solid lines) are 219(6) and 14(6) for the interference phase
$0\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$, respectively. Figure S3: Bayesian
estimation of density matrix for both phase setpoints, using the post-selected
entanglement generation scheme, with both nodes in the same lab in Delft. We
measure all two-spin correlators $\langle M_{i}M_{j}\rangle$ with
$M_{i,j}\in\left[X,Y,Z\right]$ and use Bayesian estimation for tomography [58,
59] to find the most likely density matrix. The calibration measurements in
Fig. S2 allow us to generate the same entangled state, at two different
optical phase setpoints. The overlap with the ideal state for the most likely
density matrix is given as numerical value, that is above $0.6$ for both phase
setpoints and detector outcomes. Figure S4: (A) Local node pulse scheme for
the duration of one heartbeat when performing entanglement generation without
heralding. Red and yellow pulses are laser, orange are microwave pulses, blue-
green is electrical feedback. The NV center is (re-)initialized in the correct
spin and charge state via pumping resonant with $\ket{1}$
($637\text{\,}\mathrm{nm}$) and the ZPL of NV0($575\text{\,}\mathrm{nm}$),
respectively. We limit resonant optical excitation during local phase
stability by preparing the qubit state in the opposite spin state a using
microwave $\pi$-pulse. After the local phase stability, we initialize the spin
into the $\ket{0}$ state, create the unequal superposition with a microwave
pulse with rotation angle $\alpha$, and optically excite using a short optical
excitation, completing the spin-photon entangled state generation. A microwave
$\pi$-pulse echos the spin state, and a $\frac{\pi}{2}$-pulse selects the
readout basis for the resonant optical readout that follows.  (B) Node pulse
scheme during entanglement generation. This sequence now takes 20 heartbeat
periods $\left($200\text{\,}\mathrm{\SIUnitSymbolMicro s}$\right)$. All pulses
required to perform the local phase stability and spin-photon entanglement are
repeated exactly as in (A). However, now the rephasing echo pulse is played at
its calibrated value of $82\text{\,}\mathrm{\SIUnitSymbolMicro
s}$($68\text{\,}\mathrm{\SIUnitSymbolMicro s}$) for Node Delft (The Hague),
while the stabilization light still is periodically sent to the midpoint such
that phase stability feedback can be applied there. The total duration of
$200\text{\,}\mathrm{\SIUnitSymbolMicro s}$ is for one whole entanglement
generation attempt where wait time padding is added node specifically (not
shown) to allow for a common repetition period when dealing with the different
node-midpoint communication delays. Figure S5: Midpoint pulse scheme for the
duration of one heartbeat. This sequence is agnostic to the experiment is run.
Purple colors are telecom wavelength laser light, blue-green is electrical
signal. Stabilization light is converted light coming from the nodes. The
reference and FBG stability light are both originated from the telecom laser
at the midpoint. The error signal for the fast lock is processed by a Track
and Hold (T&H) amplifier before being processed by the controllers. Variable
optical attenuators are used to shield the detectors during the times where
stabilization/reference light is propagating in the system, and they are only
opened for $2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ where single photons from
the nodes can arrive. The global control task is not modulated, as the
feedback bandwidth is much slower than the low-pass filtering of the error
signal ($150\text{\,}\mathrm{Hz}$). For all above pulse schemes holds that
absolute block sizes are not to scale, the denoted time duration is leading.
Figure S6: (A) Correlation measurement for full heralding with correction on
discarding events accompanied by a measured PSB photon, independent of
detector outcome. Upper plot shows events per detector, lower for both
detectors combined. Bars indicate measured data, the number in parenthesis
indicate the amount of events. Horizontal lines indicate the theoretical
model. (B) State fidelity and entanglement generation rate for varying photon
acceptance window length. Inset shows signal-to-noise ratio for the same
window lengths. All measurement outcomes are corrected for tomography errors,
errorbars are 1 standard deviation. Figure S7: Continuous data of important
parameters over deployed fiber link. From top to bottom: the frequency drift,
phase noise, polarization and time of flight as measured by the individual
systems and logged into a central database. For the frequency, phase and
polarization we have conditioned the data on the respective feedback system
being active. The pause in the system overnight was due to one of the feedback
systems being out of lock, which got reset the next morning. Figure S8:
Temperature logging of the three locations during the same time as the
continuous data logging of Fig. S7. Figure S9: Single photon detection signals
are processed by custom FPGA-code on the Digilent Arty A7. If photons arrive
within the externally derived acceptance window from the heartbeat (HB), a
heralding signal is sent to both locations through an Electrical-to-Optical-
Converter (EOC). The heralding signal is also recorded on the timetaggers on
all the three locations. Figure S10: Flowchart for calibrations and performing
the sequence of two-node experiments for the purpose of measuring the
entanglement fidelity of the entangled state that is aimed to be generated.
Between the different steps of the measurement sequence thresholds are checked
of measured parameters that decide continuation of the sequence. Calibrations
are started with a signal-to-noise (SNR) measurement, where the counts-per-
shot of telecom signal from the nodes $\left(\mathrm{CPS_{tel}}\right)$ is
checked. We continue with PHASE and XsweepX and check their respective
parameters. If all thresholds are satisfied, we continue with the fidelity
measurements (FID) until the Charge Resonance passing rate
$\left(\mathrm{CR_{rate}}\right)$ drops below the calibrated value it had
during the SNR $\left(\mathrm{CR_{rate,SNR}}\right)$. Finally, we do another
PHASE check before we restart the entire cycle.