draft-irtf-qirg-principles-03.txt

Quantum Internet Research Group                             W. Kozlowski
Internet-Draft                                                 S. Wehner
Intended status: Informational                                    QuTech
Expires: September 6, 2020                                  R. Van Meter
                                                         Keio University
                                                              B. Rijsman
                                                              Individual
                                                       A. S. Cacciapuoti
                                                              M. Caleffi
                                        University of Naples Federico II
                                                           March 5, 2019


            Architectural Principles for a Quantum Internet
                     draft-irtf-qirg-principles-03

Abstract

   The vision of a quantum internet is to fundamentally enhance Internet
   technology by enabling quantum communication between any two points
   on Earth.  To achieve this goal, a quantum network stack should be
   built from the ground up as the physical nature of the communication
   is fundamentally different.  The first realisations of quantum
   networks are imminent, but there is no practical proposal for how to
   organise, utilise, and manage such networks.  In this memo, we
   attempt lay down the framework and introduce some basic architectural
   principles for a quantum internet.  This is intended for general
   guidance and general interest, but also to provide a foundation for
   discussion between physicists and network specialists.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
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   This Internet-Draft will expire on September 6, 2020.





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Copyright Notice

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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Quantum information . . . . . . . . . . . . . . . . . . . . .   4
     2.1.  Qubit . . . . . . . . . . . . . . . . . . . . . . . . . .   4
     2.2.  Multiple qubits . . . . . . . . . . . . . . . . . . . . .   5
   3.  Entanglement as the fundamental resource  . . . . . . . . . .   6
   4.  Achieving quantum connectivity  . . . . . . . . . . . . . . .   8
     4.1.  Challenges  . . . . . . . . . . . . . . . . . . . . . . .   8
       4.1.1.  The measurement problem . . . . . . . . . . . . . . .   8
       4.1.2.  No-cloning theorem  . . . . . . . . . . . . . . . . .   8
       4.1.3.  Fidelity  . . . . . . . . . . . . . . . . . . . . . .   9
     4.2.  Direct transmission . . . . . . . . . . . . . . . . . . .   9
     4.3.  Bell pairs  . . . . . . . . . . . . . . . . . . . . . . .  10
     4.4.  Teleportation . . . . . . . . . . . . . . . . . . . . . .  10
     4.5.  The life cycle of entanglement  . . . . . . . . . . . . .  11
       4.5.1.  Elementary link generation  . . . . . . . . . . . . .  11
       4.5.2.  Entanglement swapping . . . . . . . . . . . . . . . .  12
       4.5.3.  Distillation  . . . . . . . . . . . . . . . . . . . .  13
       4.5.4.  Delivery  . . . . . . . . . . . . . . . . . . . . . .  14
   5.  Architecture of a quantum internet  . . . . . . . . . . . . .  14
     5.1.  New challenges  . . . . . . . . . . . . . . . . . . . . .  14
     5.2.  Classical communication . . . . . . . . . . . . . . . . .  15
     5.3.  Abstract model of the network . . . . . . . . . . . . . .  16
       5.3.1.  Elements of a quantum network . . . . . . . . . . . .  16
       5.3.2.  Putting it all together . . . . . . . . . . . . . . .  17
     5.4.  Network boundaries  . . . . . . . . . . . . . . . . . . .  18
       5.4.1.  Boundaries between different physical architectures .  18
       5.4.2.  Boundaries between different administrative regions .  18
     5.5.  Physical constraints  . . . . . . . . . . . . . . . . . .  19
       5.5.1.  Memory lifetimes  . . . . . . . . . . . . . . . . . .  19
       5.5.2.  Rates . . . . . . . . . . . . . . . . . . . . . . . .  19
       5.5.3.  Communication qubits  . . . . . . . . . . . . . . . .  20



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       5.5.4.  Homogeneity . . . . . . . . . . . . . . . . . . . . .  20
   6.  Architectural principles  . . . . . . . . . . . . . . . . . .  20
     6.1.  Goals of a quantum internet . . . . . . . . . . . . . . .  20
     6.2.  The principles of a quantum internet  . . . . . . . . . .  23
   7.  Comparison with classical networks  . . . . . . . . . . . . .  25
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  27
   9.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  27
   10. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  27
   11. Informative References  . . . . . . . . . . . . . . . . . . .  27
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  29

1.  Introduction

   Quantum networks are distributed systems of quantum devices that
   utilise fundamental quantum mechanical phenomena such as
   superposition, entanglement, and quantum measurement to achieve
   capabilities beyond what is possible with classical networks.
   Depending on the stage of a quantum network [5] such devices may be
   simple photonic devices capable of preparing and measuring only one
   quantum bit (qubit) at a time, all the way to large-scale quantum
   computers of the future.  A quantum network is not meant to replace
   classical networks, but rather form an overall hybrid classical
   quantum network supporting new capabilities which are otherwise
   impossible to realise.  This new networking paradigm offers promise
   for a range of new applications such as secure communications [1],
   distributed quantum computation [2], or quantum sensor networks [3].
   The field of quantum communication has been a subject of active
   research for many years and the most well-known application of
   quantum communication, quantum key distribution (QKD) for secure
   communications, has already been deployed at short (roughly 100km)
   distances.

   Fully quantum networks capable of transmitting and managing entangled
   quantum states in order to send, receive, and manipulate distributed
   quantum information are now imminent [4] [5].  Whilst a lot of effort
   has gone into physically realising and connecting such devices, and
   making improvements to their speed and error tolerance there are no
   worked out proposals for how to run these networks.  To draw an
   analogy with a classical network, we are at a stage where we can
   start to physically connect our devices and send data, but all
   sending, receiving, buffer management, connection synchronisation,
   and so on, must be managed by the application itself at what is even
   lower than assembly level where no common interfaces yet exist.
   Furthermore, whilst physical mechanisms for transmitting quantum
   states exist, there are no robust protocols for managing such
   transmissions.





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2.  Quantum information

   In order to understand the framework for quantum networking a basic
   understanding of quantum information is necessary.  The following
   sections aim to introduce the bare minimum necessary to understand
   the principles of operation of a quantum network.  This exposition
   was written with a classical networking audience in mind.  It is
   assumed that the reader has never before been exposed to any quantum
   physics.  We refer to e.g. [10] for an in-depth introduction to
   quantum information.

2.1.  Qubit

   The differences between quantum computation and classical computation
   begin at the bit-level.  A classical computer operates on the binary
   alphabet { 0, 1 }. A quantum bit, a qubit, exists over the same
   binary space, but unlike the classical bit, it can exist in a so-
   called superposition of the two possibilities:

   a |0> + b |1>,

   where |X> denotes a quantum state, here the binary 0 and 1, and the
   coefficients a and b are complex numbers called probability
   amplitudes.  Physically, such a state can be realised using a variety
   of different technologies such as electron spin, photon polarisation,
   atomic energy levels, and so on.

   Upon measurement, the qubit loses its superposition and irreversibly
   collapses into one of the two basis states, either |0> or |1>.  Which
   of the two states it ends up in is not deterministic, but it can be
   determined from the readout of the measurement, a classical bit, 0 or
   1 respectively.  The probability of measuring the state in the |0>
   state is |a|^2 and similarly the probability of measuring the state
   in the |1> state is |b|^2, where |a|^2 + |b|^2 = 1.  This randomness
   is not due to our ignorance of the underlying mechanisms, but rather
   it is a fundamental feature of a quantum mechanical system [6].

   The superposition property plays an important role in fundamental
   gate operations on qubits.  Since a qubit can exist in a
   superposition of its basis states, the elementary quantum gates are
   able to act on all states of the superposition at the same time.  For
   example, consider the NOT gate:

   NOT (a |0> + b |1>) -> a |1> + b |0>.







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2.2.  Multiple qubits

   When multiple qubits are combined in a single quantum state the space
   of possible states grows exponentially and all these states can
   coexist in a superposition.  For example, the general form of a two-
   qubit register is

   a |00> + b |01> + c |10> + d |11>

   where the coefficients have the same probability amplitude
   interpretation as for the single qubit state.  Each state represents
   a possible outcome of a measurement of the two-qubit register.  For
   example, |01>, denotes a state in which the first qubit is in the
   state |0> and the second is in the state |1>.

   Performing single qubit gates affects the relevant qubit in each of
   the superposition states.  Similarly, two-qubit gates also act on all
   the relevant superposition states, but their outcome is far more
   interesting.

   Consider a two-qubit register where the first qubit is in the
   superposed state (|0> + |1>)/sqrt(2) and the other is in the
   state |0>.  This combined state can be written as:

   (|0> + |1>)/sqrt(2) x |0> = (|00> + |10>)/sqrt(2),

   where x denotes a tensor product (the mathematical mechanism for
   combining quantum states together).  Let us now consider the two-
   qubit CNOT gate.  The CNOT gate takes as input two qubits, a control
   and target, and applies the NOT gate to the target if the control
   qubit is set.  The truth table looks like

                               +----+-----+
                               | IN | OUT |
                               +----+-----+
                               | 00 |  00 |
                               | 01 |  01 |
                               | 10 |  11 |
                               | 11 |  10 |
                               +----+-----+

   Now, consider performing a CNOT gate on the ensemble with the first
   qubit being the control.  We apply a two-qubit gate on all the
   superposition states:

   CNOT (|00> + |10>)/sqrt(2) -> (|00> + |11>)/sqrt(2).





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   What is so interesting about this two-qubit gate operation?  The
   final state is *entangled*. There is no possible way of representing
   that quantum state as a product of two individual qubits, they are no
   longer independent and their behaviour cannot be fully described
   without accounting for the other qubit.  The states of the two
   individual qubits are now correlated beyond what is possible to
   achieve classically.  Neither qubit is in a definite |0> or |1>
   state, but if we perform a measurement on either one, the outcome of
   the partner qubit will *always* yield the exact same outcome.  The
   final state, whether it's |00> or |11>, is fundamentally random as
   before, but the states of the two qubits following a measurement will
   always be identical.

   Once a measurement is performed, the two qubits are once again
   independent.  The final state is either |00> or |11> and both of
   these states can be trivially decomposed into a product of two
   individual qubits.  The entanglement has been consumed and if the
   same measurement is to be repeated, the entangled state must be
   prepared again.

3.  Entanglement as the fundamental resource

   Entanglement is the fundamental building block of quantum networks.
   To see this, consider the state from the previous section:

   (|00> + |11>)/sqrt(2).

   Neither of the two qubits is in a definite |0> or |1> state and we
   need to know the state of the entire register to be able to fully
   describe the behaviour of the two qubits.

   Entangled qubits have interesting non-local properties.  Consider
   sending one of the qubits to another device.  This device could in
   principle be anywhere: on the other side of the room, in a different
   country, or even on a different planet.  Provided negligible noise
   has been introduced, the two qubits will forever remain in the
   entangled state until a measurement is performed.  The physical
   distance does not matter at all for entanglement.

   This lies at the heart of quantum networking, because it is possible
   to leverage the non-classical correlations provided by entanglement
   in order to design completely new types of application protocols that
   are not possible to achieve with just classical communication.
   Examples of such applications are quantum cryptography, blind quantum
   computation, or distributed quantum computation.






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   Entanglement has two very special features from which one can derive
   some intuition about the types of applications enabled by a quantum
   network.

   The first stems from the fact that entanglement enables stronger than
   classical correlations, leading to opportunities for tasks that
   require coordination.  As a trivial example consider the problem of
   consensus between two nodes who want to agree on the value of a
   single bit.  They can use the quantum network to prepare the state
   (|00> + |11>)/sqrt(2) with each node holding one of the two qubits.
   Once any of the two nodes performs a measurement the state of the two
   qubits collapses to either |00> or |11> so whilst the outcome is
   random and does not exist before measurement, the two nodes will
   always measure the same value.  We can also build the more general
   multi-qubit state (|00...> + |11...>)/sqrt(2) and perform the same
   algorithm between an arbitrary number of nodes.  These stronger than
   classical correlations generalise to more complicated measurement
   schemes as well.

   The second feature of entanglement is that it cannot be shared, in
   the sense that if two qubits are maximally entangled with each other,
   than it is physically impossible for any other system to have any
   share of this entanglement.  Hence, entanglement forms a sort of
   private and inherently untappable connection between two nodes once
   established.

   Entanglement is created through local interactions between two qubits
   or as a product of the way the qubits were created (e.g. entangled
   photon pairs).  To create a distributed entangled state one can then
   physically send one of the qubits to a remote node.  It is also
   possible to directly entangle qubits that are physically separated,
   but this still requires local interactions between some other qubits
   that the separated qubits are initially entangled with.  Therefore,
   it is the transmission of qubits that draws the line between a
   genuine quantum network and a collection of quantum computers
   connected over a classical network.

   A quantum network is defined as a collection of nodes that is able to
   exchange qubits and distribute entangled states amongst themselves.
   A quantum node that is able only to communicate classically with
   another quantum node is not a member of a quantum network.

   More complex services and applications can be built on top of
   entangled states distributed by the network, see e.g. [5]>







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4.  Achieving quantum connectivity

   This section explains the meaning of quantum connectivity and the
   necessary physical processes at an abstract level.

4.1.  Challenges

   A quantum network cannot be built by simply extrapolating all the
   classical models to their quantum analogues.  One cannot just send
   qubits like one can send bits over a wire.  There are several
   technological as well as fundamental challenges that make classical
   approaches unsuitable in a quantum context.

4.1.1.  The measurement problem

   In classical computers and networks we can read out the bits stored
   in memory at any time.  This is helpful for a variety of purposes
   such as copying, error detection and correction, and so on.  This is
   not possible with qubits.

   A measurement of a qubit's state will destroy its superposition and
   with it any entanglement it may have been part of.  Once a qubit is
   being processed, it cannot be read out until a suitable point in the
   computation, determined by the protocol handling the qubit, has been
   reached.  Therefore, we cannot use the same methods known from
   classical computing for the purposes of error detection and
   correction.

4.1.2.  No-cloning theorem

   Since directly reading the state of a qubit is not possible, one
   could ask the question if we can simply copy a qubit without looking
   at it.  Unfortunately, this is fundamentally not possible in quantum
   mechanics.

   The no-cloning theorem states that it is impossible to create an
   identical copy of an arbitrary unknown quantum state.  Therefore, it
   is also impossible to use the same mechanisms that worked for
   classical networks for signal amplification, retransmission, and so
   on as they all rely on the ability to copy the underlying data.
   Since any physical channel will always be lossy, connecting nodes
   within a quantum network is a challenging endeavour and its
   architecture must at its core address this very issue.








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4.1.3.  Fidelity

   In general, it is expected that a classical packet arrives at its
   destination without any errors introduced by hardware noise along the
   way.  This is verified at various levels through a variety of
   checksums.  Since we cannot read or copy a quantum state a similar
   approach is out of question for quantum networks.

   To describe the quality of a quantum state a physical quantity called
   fidelity is used.  Fidelity takes a value between 0 and 1 -- higher
   is better, and less than 0.5 means the state is unusable.  It
   measures how close a quantum state is to the state we desire it to be
   in.  It expresses the probability that one state will pass a test to
   identify as the other.  Fidelity is an important property of a
   quantum system that allows us to quantify how much a particular state
   has been affected by noise from various sources (gate errors, channel
   losses, environment noise).

   Interestingly, quantum applications do not need perfect fidelity to
   be able to execute -- as long as it is above some application-
   specific threshold, they will simply operate at lower rates.
   Therefore, rather than trying to ensure that we always deliver
   perfect states (a technologically challenging task) applications will
   specify a minimum threshold for the fidelity and the network will try
   its best to deliver it.

4.2.  Direct transmission

   Conceptually, the most straightforward way to distribute an entangled
   state is to simply transmit one of the qubits directly to the other
   end across a series of nodes while performing sufficient forward
   quantum error correction to bring losses down to an acceptable level.
   Despite the no-cloning theorem and the inability to directly measure
   a quantum state, error-correcting mechanisms for quantum
   communication exist [7].  However, quantum error correction makes
   very high demands on both resources (physical qubits needed) and
   their initial fidelity.  Implementation is very challenging and
   quantum error correction is not expected to be used until later
   generations of quantum networks.

   An alternative relies on the observation that we do not need to be
   able to distribute any arbitrary entangled quantum state.  We only
   need to be able to distribute any one of what are known as the Bell
   pair states[12].







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4.3.  Bell pairs

   Bell pair states are the entangled two-qubit states:

   |00> + |11>,
   |00> - |11>,
   |01> + |10>,
   |01> - |10>,

   where the constant 1/sqrt(2) normalisation factor has been ignored
   for clarity.  Any of the four Bell pair states above will do, as it
   is possible to transform any Bell pair into another Bell pair with
   local operations performed on only one of the qubits.  That is,
   either of the nodes that hold the two qubits of the Bell pair can
   apply a series of single qubit gates to just their qubit in order to
   transform the ensemble between the different variants.

   Distributing a Bell pair between two nodes is much easier than
   transmitting an arbitrary quantum state over a network.  Since the
   state is known handling errors becomes easier and small-scale error-
   correction (such as entanglement distillation discussed in a later
   section) combined with reattempts becomes a valid strategy.

   The reason for using Bell pairs specifically as opposed to any other
   two-qubit state, is that they are the maximally entangled two-qubit
   set of basis states.  Maximal entanglement means that these states
   have the strongest non-classical correlations of all possible two-
   qubit states.  Furthermore, since single-qubit local operations can
   never increase entanglement, less entangled states would impose some
   constraints on distributed quantum algorithms.  This makes Bell pairs
   particularly useful as a generic building block for distributed
   quantum applications.

4.4.  Teleportation

   The observation that we only need to be able to distribute Bell pairs
   relies on the fact that this enables the distribution of any other
   arbitrary entangled state.  This can be achieved via quantum state
   teleportation.  Quantum state teleportation consumes an unknown
   quantum state that we want to transmit and recreates it at the
   desired destination.  This does not violate the no-cloning theorem as
   the original state is destroyed in the process.

   To achieve this, an entangled pair needs to be distributed between
   the source and destination before teleportation commences.  The
   source then entangles the transmission qubit with its end of the pair
   and performs a read out on the two qubits (the sum of these
   operations is called a Bell state measurement).  This consumes the



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   Bell pair's entanglement turning the source and destination qubits
   into independent states.  The measurements yields two classical bits
   which the source sends to the destination over a classical channel.
   Based on the value of the received two classical bits, the
   destination performs one of four possible corrections (called the
   Pauli corrections) on its end of the pair which turns it into the
   unknown quantum state that we wanted to transmit.

   The unknown quantum state that was transmitted was never fed into the
   network itself.  Therefore, the network needs to only be able to
   reliably produce Bell pairs between any two nodes in the network.

4.5.  The life cycle of entanglement

   Reducing the problem of quantum connectivity to one of generating a
   Bell pair has facilitated the problem, but it has not solved it.  In
   this section we discuss, how these entangled pairs are generated in
   the first place, and how its two qubits are delivered to the end-
   points.

4.5.1.  Elementary link generation

   In a quantum network, entanglement is always first generated locally
   (at a node or an auxiliary element) followed by a movement of one or
   both of the entangled qubits across the link through quantum
   channels.  In this context, photons (particles of light) are the
   natural candidate for entanglement carriers, the so-called flying
   qubits.  The rationale for this choice is related to the advantages
   provided by photons such as moderate interaction with the environment
   leading to moderate decoherence, convenient control with standard
   optical components, and high-speed low-loss transmissions.  However,
   since photons cannot be stored, a transducer must transfer the flying
   qubit's state to a qubit suitable for information processing and/or
   storage (often referred to as a matter qubit).

   Since this process may fail, in order to generate and store
   entanglement efficiently, we must be able to distinguish successful
   attempts from failures.  Entanglement generation schemes that are
   able to announce successful generation are called heralded
   entanglement generation schemes.

   There exist three basic schemes for heralded entanglement generation
   on a link through coordinated action of the two nodes at the two ends
   of the link [13]:

   o  "At mid-point" scheme: the key idea is that an entangled pair
      source sends an entangled photon through a quantum channel to each
      of the nodes, where transducers are invoked to transfer the



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      entanglement from the flying qubits to the matter qubits.  In this
      scheme, the transducers know if the transfers succeeded and are
      able to herald successful entanglement generation via a message
      exchange over the classical channel.

   o  "At source" scheme: the key idea is that one of the two nodes
      sends a flying qubit that is entangled with one of its matter
      qubits.  A transducer at the other end of the link will transfer
      the entanglement from the flying qubit to one of its matter
      qubits.  Also in this scheme, the transducer knows if its transfer
      succeeded and is able to herald successful entanglement generation
      with a classical message sent to the other node.

   o  "At both end-points" scheme: the key idea is that both nodes send
      a flying qubit that is entangled with one of their matter qubits.
      A detector somewhere in between the nodes performs a joint
      measurement on the two qubits, which stochastically projects the
      remote matter qubits into an entangled quantum state.  In this
      scheme, the detector knows if the entanglement succeeded and is
      able to herald successful entanglement generation by sending a
      message to each node over the classical channel.

   The "mid-point" scheme is more robust to photon loss, but in the
   other schemes the nodes retain greater control over the entangled
   pair generation.

4.5.2.  Entanglement swapping

   The problem with generating entangled pairs directly across a link is
   that its efficiency decreases with its length.  Beyond a few 10s of
   kms in optical fibre or 1000 kms in free space (via satellite) the
   rate is effectively zero and due to the no-cloning theorem we cannot
   simply amplify the signal.  The solution is entanglement swapping.

   A Bell pair between any two nodes in the network can be constructed
   by combining the pairs generated along each individual link on the
   path between the two end-points.  Each node along the path can
   consume the two pairs on the two links that it is connected to in
   order to produce a new entangled Pair between the two remote ends.
   This process is known as entanglement swapping.  Pictorially it can
   be represented as follows:

   +---------+      +---------+      +---------+
   |    A    |      |    B    |      |    C    |
   |         |------|         |------|         |
   |      X1~~~~~~~~~~X2   Y1~~~~~~~~~~Y2      |
   +---------+      +---------+      +---------+




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   where X1 and X2 are the qubits of the entangled pair X and Y1 and Y2
   are the qubits of entangled pair Y.  The entanglement is denoted with
   ~~. In the diagram above nodes A and B share the pair X and nodes B
   and C share the pair Y, but we want entanglement between A and C.

   To achieve this goal we simply teleport the qubit X2 using the pair
   Y.  This requires node B to performs a Bell state measurement on the
   qubits X2 and Y1 which result in the destruction of the entanglement
   between Y1 and Y2.  However, X2 is transmitted and recreated in Y2's
   place carrying with it its entanglement with X1.  The end-result is
   shown below:

   +---------+      +---------+      +---------+
   |    A    |      |    B    |      |    C    |
   |         |------|         |------|         |
   |      X1~~~~~~~~~~~~~~~~~~~~~~~~~~~X2      |
   +---------+      +---------+      +---------+

   Depending on the needs of the network and/or application a final
   Pauli correction at the recipient node may not be necessary since the
   result of this operation is also a Bell pair.  However, the two
   classical bits that form the read out from the measurement at node B
   must still be communicated, because they carry information about
   which of the four Bell pairs was actually produced.  If a correction
   is not performed, the recipient must be informed which Bell pair was
   received.

   This process of teleporting Bell pairs using other entangled pairs is
   called entanglement swapping.  Quantum nodes that create long-
   distance entangled pairs via entanglement swapping are called quantum
   repeaters in academic literature [12] and we will use the same
   terminology in this memo.

4.5.3.  Distillation

   Neither the generation of Bell pairs nor the swapping operations are
   noiseless operations.  Therefore, with each link and each swap the
   fidelity of the state degrades.  However, it is possible to create
   higher fidelity Bell pair states from two or more lower fidelity
   pairs through a process called distillation (sometimes also referred
   to as purification).

   To distil a quantum state, a second (and sometimes third) quantum
   state is used as a "test tool" to test a proposition about the first
   state, e.g., "the parity of the first state is even."  When the test
   succeeds, confidence in the state is improved, and thus the fidelity
   is improved.  The test tool states are destroyed in the process, so
   resource demands increase substantially when distillation is used.



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   When the test fails, the tested state must also be discarded.
   Distillation makes low demands on fidelity and resources, but
   distributed protocols incur round-trip delays [11].

4.5.4.  Delivery

   Eventually the Bell pairs must be delivered to an application (or
   higher layer protocol) at the two end-nodes.  A detailed list of such
   requirements is beyond the scope of this memo.  At minimum, the end-
   nodes require information to map a particular Bell pair to the qubit
   in their local memory that is part of this entangled pair.

5.  Architecture of a quantum internet

   It is evident from the previous sections that the fundamental service
   provided by a quantum network significantly differs from that of a
   classical network.  Therefore, it is not surprising that the
   architecture of a quantum internet will itself be very different from
   that of the classical Internet.

5.1.  New challenges

   This subsection covers the major fundamental challenges building
   quantum networks.  Here, we only describe the fundamental
   differences, technological limitations are described later.

   1.  Bell pairs are not equivalent to payload carrying packets.

       In most classical networks, including Ethernet, Internet Protocol
       (IP), and Multi-Protocol Label Switching (MPLS) networks, user
       data is grouped into packets.  In addition to the user data each
       packet also contains a series of headers which contain the
       control information that lets routers and switches forward it
       towards its destination.  Packets are the fundamental unit in a
       classical network.

       In a quantum network the entangled pairs of qubits are the basic
       unit of networking.  These pairs are handled individually -- they
       are not grouped into packets and they do not carry any headers.
       Therefore, quantum networks will have to send all control
       information via separate classical channels which the repeaters
       will have to correlate with the qubits stored in their memory.

   2.  An entangled pair is only useful if the locations of both qubits
       are known.

       A classical network packet logically exists only at one location
       at any point in time.  If a packet is modified in some way,



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       headers or payload, this information does not need to be conveyed
       to anybody else in the network.  The packet can be simply
       forwarded as before.

       In contrast, entanglement is a phenomenon in which two or more
       qubits exist in a physically distributed state.  Operations on
       one of the qubits change the mutual state of the pair.  Since the
       owner of a particular qubit cannot just read out its state, it
       must coordinate all its actions with the owner of the pair's
       other qubit.  Therefore, the owner of any qubit that is part of
       an entangled pair must know the location of its counterpart.
       Location, in this context, need not be the explicit spatial
       location.  A relevant pair identifier, a means of communication
       between the pair owners, and an association between the pair ID
       and the individual qubits is sufficient.

   3.  Generating entanglement requires temporary state.

       Packet forwarding in a classical network is largely a stateless
       operation.  When a packet is received, the router looks up its
       forwarding table and sends the packet out of the appropriate
       output.  There is no need to keep any memory of the packet any
       more.

       A quantum node must be able to make decisions about qubits that
       it receives and is holding in its memory.  Since qubits do not
       carry headers, the receipt of an entangled pair conveys no
       control information based on which the repeater can make a
       decision.  The relevant control information will arrive
       separately over a classical channel.  This implies that a
       repeater must store temporary state as the control information
       and the qubit it pertains to will, in general, not arrive at the
       same time.

5.2.  Classical communication

   In this memo we have already covered two different roles that
   classical communication must perform:

   o  communicate classical bits of information as part of distributed
      protocols such as entanglement swapping and teleportation,

   o  communicate control information within a network - this includes
      both background protocols such as routing as well as signalling
      protocols to set up end-to-end entanglement generation.

   Classical communication is a crucial building block of any quantum
   network.  All nodes in a quantum network are assumed to have



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   classical connectivity with each other (within typical administrative
   domain limts).  Therefore, quantum routers will need to manage two
   data planes in parallel, a classical one and a quantum one.
   Additionally, it must be able to correlate information between them
   so that the control information received on a classical channel can
   be applied to the qubits managed by the quantum data plane.

5.3.  Abstract model of the network

5.3.1.  Elements of a quantum network

   We have identified quantum repeaters as the core building block of a
   quantum network.  However, a quantum repeater will have to do more
   than just entanglement swapping in a functional quantum network.  Its
   key responsibilities will include:

   1.  Creating link-local entanglement between neighbouring nodes.

   2.  Extending entanglement from link-local pairs to long-range pairs
       through entanglement swapping.

   3.  Performing distillation to manage the fidelity of the produced
       pairs

   4.  Participate in the management of the network (routing etc.).

   Not all quantum repeaters in the network will be the same, here we
   break them down further:

   o  Quantum routers (controllable quantum nodes) - A quantum router is
      a quantum repeater with a control plane that participates in the
      management of the network and will make decisions about which
      qubits to swap to generate the requested end-to-end pairs.

   o  Automated quantum nodes - An automated quantum node is a data
      plane only quantum repeater that does not participate in network
      management.  Since the no-cloning theorem precludes the use of
      amplification long-range links will be established by chaining
      multiple such automated nodes together.

   o  End-nodes - End-nodes in a quantum network must be able to receive
      and handle an entangled pair, but they do not need to be able to
      perform an entanglement swap (and thus are not necessarily quantum
      repeaters).  End-nodes are also not required to have any quantum
      memory as certain quantum applications can be realised by having
      the end-node measure its qubit as soon as it is received.





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   o  Non-quantum nodes - Not all nodes in a quantum network need to
      have a quantum data plane.  A non-quantum node is any device that
      can handle classical network traffic.

   Additionally, we need to identify two kinds of links that will be
   used in a quantum network:

   o  Quantum links - A quantum link is a link which can be used to
      generate an entangled pair between two directly connected quantum
      repeaters.  It may include a dedicated classical channel that is
      to be used solely for the purpose of coordinating the entanglement
      generation on this quantum link.

   o  Classical links - A classical link is a link between any node in
      the network that is capable of carrying classical network traffic.

5.3.2.  Putting it all together

   A two-hop path in a generic quantum network can be represented as:

   | App |-------------------CC-------------------| App |
      ||                                            ||
    ------                 ------                 ------
   |  EN  |----QC & CC----|  QR  |----QC & CC----|  EN  |
    ------                 ------                 ------

   App - user-level application
   QR - quantum repeater
   EN - end-node
   QC - quantum channel
   CC - classical channel

   An application running on two end-nodes attached to a network will at
   some point need the network to generate entangled pairs for its use.
   This will require negotiation between the end-nodes, because they
   must both open a communication end-point (a quantum socket) which the
   network can use to identify the two ends of the connection.  The two
   end-nodes use the classical connectivity available in the network to
   achieve this goal.

   When the network receives a request to generate end-to-end entangled
   pairs it uses the classical communication channels to coordinate and
   claim the resources necessary to fulfil this request.  This may be
   some combination of prior control information (e.g. routing tables)
   and signalling protocols, but the details of how this is achieved are
   an active research question and thus beyond the scope of this memo.





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   During or after the control information is distributed the network
   performs the necessary quantum operations such as generating
   entangled over individual links, performing entanglement swaps, and
   further signalling to transmit the swap outcomes and other control
   information.  Since none of the entangled pairs carry any user data,
   some of these operations can be performed before the request is
   received in anticipation of the demand.

   The entangled pair is delivered to the application once it is ready,
   together with the relevant pair identifier.  However, being ready
   does not necessarily mean once all link pairs and entanglement swaps
   are complete as some applications can start executing on an
   incomplete pair.  In this case the remaining entanglement swaps will
   propagate the actions across the network to the other end.

5.4.  Network boundaries

   Just like classical network, there will various boundaries will exist
   in quantum networks.

5.4.1.  Boundaries between different physical architectures

   There are many different physical architectures for implementing
   quantum repeater technology.  The different technologies differ in
   how they store and manipulate qubits in memory and how they generate
   entanglement across a link with their neighbours.  Different
   architectures come with different trade-offs and thus a functional
   network will likely consist of a mixture of different types of
   quantum repeaters.

   For example, architectures based on optical elements and atomic
   ensembles are very efficient at generating entanglement, but provide
   little control over the qubits once the pair is generated.  On the
   other hand nitrogen-vacancy architectures offer a much greater degree
   of control over qubits, but have a harder time generating the
   entanglement across a link.

   It is an open research question where exactly the boundary will lie.
   It could be that a single quantum repeater node provides some
   backplane connection between the architectures, but it also could be
   that special quantum links delineate the boundary.

5.4.2.  Boundaries between different administrative regions

   Just like in classical networks, multiple quantum networks will
   connect into a global quantum internet.  This necessarily implies the
   existence of borders between different administrative regions.  How




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   these boundaries will be handled is also an open question and thus
   beyond the scope of this memo.

5.5.  Physical constraints

   The model above has effectively abstracted away the particulars of
   the hardware implementation.  However, certain physical constraints
   need to be considered in order to build a practical network.  Some of
   these are fundamental constraints and no matter how much the
   technology improves, they will always need to be addressed.  Others
   are artefacts of the early stages of a new technology.  Here, we
   consider a highly abstract scenario and refer to [5] for pointers to
   the physics literature.

5.5.1.  Memory lifetimes

   In addition to discrete operations being imperfect, storing a qubit
   in memory is also highly non-trivial.  The main difficulty in
   achieving persistent storage is that it is extremely challenging to
   isolate a quantum system from the environment.  The environment
   introduces an uncontrollable source of noise into the system which
   affects the fidelity of the state.  This process is known as
   decoherence.  Eventually, the state has to be discarded once its
   fidelity degrades too much.

   The memory lifetime depends on the particular physical setup, but the
   highest achievable values currently are on the order of seconds.
   These values have increased tremendously over the lifetime of the
   different technologies and are bound to keep increasing.  However, if
   quantum networks are to be realised in the near future, they need to
   be able to handle short memory lifetimes, for example by reducing
   latency on critical paths.

5.5.2.  Rates

   Entanglement generation on a link between two connected nodes is not
   a very efficient process and it requires many attempts to succeed.  A
   fast repetition rate for Bell pair generation is achievable, but only
   a small fraction will succeed.  Currently, the highest achievable
   rates of success between nodes capable of storing the resulting
   qubits are of the order of 10 Hz.  Combined with short memory
   lifetimes this leads to very tight timing windows to build up
   network-wide connectivity.








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5.5.3.  Communication qubits

   Most physical architectures capable of storing qubits are only able
   to generate entanglement using only a subset of its available qubits
   called communication qubits.  Once a Bell pair has been generated
   using a communication qubit, its state can be transferred into
   memory.  This may impose additional limitations on the network.  In
   particular if a given node has only one communication qubit it cannot
   simultaneously generate Bell Pairs over two links.  It must generate
   entanglement over the links one at a time.

5.5.4.  Homogeneity

   Currently all hardware implementations are homogeneous and they do
   not interface with each other.  In general, it is very challenging to
   combine different quantum information processing technologies at
   present.  Coupling different technologies with each other is of great
   interest as it may help overcome the weaknesses of the different
   implementations, but this may take a long time to be realised with
   high reliability and thus is not a near-term goal.

6.  Architectural principles

   Given that the most practical way of realising quantum network
   connectivity is using Bell Pair and entanglement swapping repeater
   technology what sort of principles should guide us in assembling such
   networks such that they are functional, robust, efficient, and most
   importantly: they work.  Furthermore, how do we design networks so
   that they work under the constraints imposed by the hardware
   available today, but do not impose unnecessary burden on future
   technology.

   As this is a completely new technology that is likely to see many
   iterations over its lifetime, this memo must not serve as a
   definitive set of rules, but merely as a general set of recommended
   guidelines for the first generations of quantum networks based on
   principles and observations made by the community.  The benefit of
   having a community built document at this early stage is that
   expertise in both quantum information and network architecture is
   needed in order to successfully build a quantum internet.

6.1.  Goals of a quantum internet

   When outlining any set of principles we must ask ourselves what goals
   do we want to achieve as inevitably trade-offs must be made.  So what
   sort of goals should drive a quantum network architecture?  The
   following list has been inspired by the history of the classical
   Internet, but it will inevitably evolve with time and the needs of



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   its users.  The goals are listed in order of priority which in itself
   may also evolve as the community learns more about the technology.

   1.  Support distributed quantum applications

       This goal seems trivially obvious, but makes a subtle, but
       important point which highlights a key difference between quantum
       and classical networks.  Ultimately, quantum data transmission is
       not the goal of a quantum network - it is only one possible
       component of more advanced quantum application protocols.  Whilst
       transmission certainly could be used as a building block for all
       quantum applications, it is certainly not the most basic one
       possible.  For example, QKD, the most well known quantum
       application protocol only relies on the stronger than classical
       correlations and inherent secrecy of entangled Bell pairs and
       does not transmit arbitrary quantum states.

       The primary purpose of a quantum internet is to support
       distributed quantum application protocols and it is of utmost
       importance that they can run well and efficiently.  Thus, it is
       important to develop performance metrics meaningful to
       application to drive the development of quantum network
       protocols.  For example, the Bell pair generation rate is
       meaningless if one does not also consider their fidelity.  It is
       generally, much easier to generate pairs of lower fidelity, but
       quantum applications may have to make multiple re-attempts or
       even abort if the fidelity is too low.  A review of the
       requirements for different known quantum applications can be
       found in [5].

   2.  Support tomorrow's distributed quantum applications

       Early-stage quantum networks will be very limited in terms of
       their capabilities and will only be able to run a limited set of
       applications.  As quantum repeater technology becomes more
       advanced and acquire more sophisticated capabilities, new
       applications will become possible.  The different stages of this
       development are described in [5].

       Therefore, it is important that any proposed architecture for
       general purpose quantum repeater networks should not constrain
       their potential capabilities for the benefit of being able to run
       early-stage applications more efficiently.  For example, it is
       already possible to run QKD efficiently on metropolitan scales
       and such networks are already commercially available.  However,
       they are not based on quantum repeaters and thus will not be able
       to easily transition more sophisticated applications.




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   3.  Support hardware heterogeneity

       There are multiple proposals for realising practical quantum
       repeater hardware and they all have their advantages and
       disadvantages.  Some may offer higher Bell pair generation rates
       on individual links at the cost of more difficult entanglement
       swap operations.  Other platforms may be good all around, but are
       more difficult to build.

       Whilst conceptually they are all capable of the same tasks the
       optimal network configuration will likely leverage the advantages
       of multiple platforms to optimise the provided service.
       Therefore, it is an explicit goal to incorporate varied hardware
       support from the beginning.

   4.  Be flexible with regards to hardware capabilities and limitations

       This goal encompasses two important points.  First, the
       architecture should be able to function under the physical
       constraints imposed by the current generation hardware.  Second,
       it should not make it difficult to run the network over any
       hardware that may come along in the future.  The physical
       capabilities of repeaters will improve and redeploying a
       technology is extremely challenging.

   5.  Ensure security at the network level

       Whilst the priority for the first quantum networks should be to
       simply work, we cannot forget that ultimately they have to also
       be secure.  This is particularly important for quantum networks
       given that one of the key driving factors for the technology is
       the enhance security offered by quantum entanglement.  This has
       implications for the physical realisations (do they satisfy the
       idealised theoretical models) and also the design of the control
       stack.

       It is actually difficult to guarantee security at the network
       level and even if the network did provide such guarantees, the
       application would still need to perform its own verification
       similarly to how one ensures end-to-end security in classical
       networks.

       It turns out that as long as the underlying implementation
       corresponds to (or sufficiently approximates) theoretical models
       of quantum cryptography, quantum cryptographic protocols do not
       need the network to provide any guarantees about the
       authenticity, confidentiality, or integrity of the transmitted
       qubits or the generated entanglement.  Instead, applications,



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       such as QKD, establish such guarantees using the classical
       network in conjunction with he quantum one.  This is much easier
       than demanding that the network deliver secure entanglement in
       the first place.

       Nevertheless, control protocols themselves should be security
       aware in order to protect the operation of the network itself and
       limit disruption.  This will primarily involve securing the
       classical control and management traffic by means of
       authentication and possibly encryption.

   6.  Make them easy to manage and monitor

       The fundamental unit of quantum information, the qubit, cannot be
       actively monitored as any readout irreversibly destroys its
       contents.  Furthermore, given one end of an entangled pair, it is
       impossible to tell where the other qubit is without any
       additional information.  Therefore, monitoring quantum networks
       will be more challenging and more important if any meaningful
       network management solution is to be developed.

   7.  Ensure availability and resilience

       A practical and usable network is able to continue to operate
       despite losses and failures, and will be robust to malicious
       actors trying to disable connectivity.  These may be simply
       considered different aspects of security, but it is worthwhile to
       address them explicitly at the architectural level already.  The
       coexistence of two separate channels, a quantum and a classical
       one, will likely prove to be challenging.

6.2.  The principles of a quantum internet

   The principles support the goals, but are not goals themselves.  The
   goals define what we want to build and the principles provide a
   guideline in how we might achieve this.  The goals will also be the
   foundation for defining any metric of success for a network
   architecture, whereas the principles in themselves do not distinguish
   between success and failure.  For more information about design
   considerations for quantum networks see [8] [9] .

   1.  Bell Pairs are the fundamental building block

       The key service that a quantum network provides is the
       distribution of entanglement between the nodes in a network.
       This point additionally specifies that the entanglement is
       primarily distributed in the form of the entangled Bell Pair




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       states which should be used as a building block in providing
       other services, including more complex entangled states.

   2.  Bell Pairs are indistinguishable

       Any two Bell Pairs between the same two nodes are
       indistinguishable for the purposes of an application provided
       they both satisfy its required fidelity threshold.  This point is
       crucial in enabling the reuse of resources of a network and for
       the purposes of provisioning resources to meet application
       demand.  However, the qubits that make up the pair themselves are
       not indistinguishable and the two nodes operating on a pair must
       coordinate to make sure they are operating on qubits that belong
       to the same Bell Pair.

   3.  Fidelity is part of the service

       In addition to being able to deliver Bell Pairs to the
       communication end-points, the Bell Pairs must be of sufficient
       fidelity.  Unlike in classical networks where errors should
       essentially be eliminated for most application protocols, many
       quantum applications only need imperfect entanglement to
       function.  However, different applications will have different
       requirements for what fidelity they can work with.  It is the
       network's responsibility to balance the resource usage with
       respect to the application's requirements.  It may be that it is
       cheaper for the network to provide lower fidelity pairs that are
       just above the threshold required by the application than it is
       to guarantee high fidelity pairs to all applications regardless
       of their requirements.

   4.  Time is part of the service

       With the current technology, time is the most expensive resource.
       It is not the only resource that is in short supply (memory, and
       communication qubits are as well), but ultimately it is the
       lifetime of quantum memories that imposes the most difficult
       conditions for operating an extended network of quantum nodes.
       Current hardware has low rates of Bell Pair generation, short
       memory lifetimes, and access to a limited number of communication
       qubits.  All these factors combined mean that even a short
       waiting queue at some node could be enough for the Bell Pairs to
       decohere.








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7.  Comparison with classical networks

   Creating end-to-end Bell pairs between remote end-points is a
   stateful distributed task that requires a lot of a-priori
   coordination.  Therefore, a connection-oriented approach seems the
   most natural for quantum networks.  In this section, we discuss a
   plausible quantum network architecture inspired by MPLS.  This is not
   an architecture proposal, but a thought experiment to give the reader
   an idea of what components are necessary for a functional quantum
   network.  We use classical MPLS as a basis as it is well known and
   understood in the networking community.

   In connection-oriented quantum networks, when two quantum application
   end-points wish to start creating end-to-end Bell pairs, they must
   first create a quantum virtual circuit (QVC).  As an analogy, in MPLS
   networks end-points must establish a label switched path (LSP) before
   exchanging traffic.  Connection-oriented quantum networks may also
   support virtual circuits with multiple end-points for creating
   multipartite entanglement.  As an analogy, MPLS networks have the
   concept of multi-point LSPs for multicast.

   When a quantum application creates a quantum virtual circuit, it can
   indicate quality of service (QoS) parameters such as the required
   capacity in end-to-end Bell pairs per second (BPPS) and the required
   fidelity of the Bell pairs.  As an analogy, in MPLS networks
   applications specify the required bandwidth in bits per second (BPS)
   and other constraints when they create a new LSP.

   Quantum networks need a routing function to compute the optimal path
   (i.e. the best sequence of routers and links) for each new quantum
   virtual circuit.  The routing function may be centralized or
   distributed.  In the latter case, the quantum network needs a
   distributed routing protocol.  As an analogy, classical networks use
   routing protocols such as open shortest path first (OSPF) and
   intermediate-system to intermediate system (ISIS).

   Given the very scarce availability of resources in early quantum
   networks, a traffic engineering function is likely to be beneficial.
   Without traffic engineering, quantum virtual circuits always use the
   shortest path.  In this case, the quantum network cannot guarantee
   that each quantum end-point will get its Bell pairs at the required
   rate or fidelity.  This is analogous to "best effort" service in
   classical networks.

   With traffic engineering, quantum virtual circuits choose a path that
   is guaranteed to have the requested resources (e.g. bandwidth in
   BPPS) available, taking into account the capacity of the routers and
   links and taking into account the resources already consumed by other



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   virtual circuits.  As an analogy, both OSPF and ISIS have traffic
   engineering (TE) extensions to keep track of used and available
   resources, and can use constrained shortest path first (CSPF) to take
   resource availability and other constraints into account when
   computing the optimal path.

   The use of traffic engineering implies the use of call admission
   control (CAC): the network denies any virtual circuits for which it
   cannot guarantee the requested quality of service a-priori.  Or
   alternatively, the network pre-empts lower priority circuits to make
   room for the new one.

   Quantum networks need a signaling function: once the path for a
   quantum virtual circuit has been computed, signaling is used to
   install the "forwarding rules" into the data plane of each quantum
   router on the path.  The signaling may be distributed, analogous to
   the resource reservation protocol (RSVP) in MPLS.  Or the signaling
   may be centralized, similar to OpenFlow.

   Quantum networks need an abstraction of the hardware for specifying
   the forwarding rules.  This allows us to de-couple the control plane
   (routing and signaling) from the data plane (actual creation of Bell
   pairs).  The forwarding rules are specified using abstract building
   blocks such as "creating local Bell pairs", "swapping Bell pairs",
   "distillation of Bell pairs".  As an analogy, classical networks use
   abstractions that as based on match conditions (e.g. looking up
   header fields in tables) and actions (e.g. modifying fields or
   forwarding a packet to a specific interface).  The data-plane
   abstractions in quantum networks will be very different from those in
   classical networks due to the fundamental differences in technology
   and the stateful nature of quantum networks.  In fact, choosing the
   right abstractions will be one of the biggest challenges when
   designing interoperable quantum network protocols.

   In quantum networks, control plane traffic (routing and signaling
   messages) is exchanged over a classical channel, whereas data plane
   traffic (the actual Bell pair qubits) is exchanged over a separate
   quantum channel.  This is in contrast to most classical networks,
   where control plane traffic and data plane traffic share the same
   channel and where a single packet contains both user fields and
   header fields.  There is, however, a classical analogy to the way
   quantum networks work.  Generalized MPLS (GMPLS) networks use
   separate channels for control plane traffic and data plane traffic.
   Furthermore, GMPLS networks support data planes where there is no
   such thing as data plane headers (e.g.  DWDM or TDM networks).






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8.  Security Considerations

   Even though no user data enters a quantum network security is listed
   as an explicit goal for the architecture and this issue is addressed
   in the section on goals.  Even though user data doesn't enter the
   network, it is still possible to attack the control protocols and
   violate the authenticity, confidentiality, and integrity of
   communication.  However, as this is an informational memo it does not
   propose any concrete mechanisms to achieve these goals.

   In summary:

   As long as the underlying implementation corresponds to (or
   sufficiently approximates) theoretical models of quantum
   cryptography, quantum cryptographic protocols do not need the network
   to provide any guarantees about the authenticity, confidentiality, or
   integrity of the transmitted qubits or the generated entanglement.
   Instead, applications such as QKD establish such guarantees using the
   classical network in conjunction with he quantum one.  This is much
   easier than demanding that the network deliver secure entanglement.

9.  IANA Considerations

   This memo includes no request to IANA.

10.  Acknowledgements

   The authors of this memo acknowledge funding received from the EU
   Flagship on Quantum Technologies through Quantum Internet Alliance
   project.

   The authors would further like to acknowledge Carlo Delle Donne,
   Matthew Skrzypczyk, and Axel Dahlberg for useful discussions on this
   topic prior to the submission of this memo.

11.  Informative References

   [1]        Bennett, C. and G. Brassard, "Quantum cryptography: Public
              key distribution and coin tossing", Theoretical Computer
              Science 560, 7-11, 2014,
              <http://www.sciepub.com/reference/53249>.

   [2]        Crepeau, C., Gottesman, D., and A. Smith, "Secure multi-
              party quantum computation. Proceedings of Symposium on
              Theory of Computing", Proceedings of Symposium on Theory
              of Computing , 2002,
              <https://arxiv.org/abs/quant-ph/0206138>.




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   [3]        Giovanetti, V., Lloyd, S., and L. Maccone, "Quantum-
              enhanced measurements: beating the standard quantum
              limit", Science 306(5700), 1330-1336, 2004,
              <https://arxiv.org/abs/quant-ph/0412078>.

   [4]        Castelvecchi, D., "The Quantum Internet has arrived (and
              it hasn't)", Nature 554, 289-292, 2018,
              <https://www.nature.com/articles/d41586-018-01835-3>.

   [5]        Wehner, S., Elkouss, D., and R. Hanson, "Quantum internet:
              A vision for the road ahead", Science 362, 6412, 2018,
              <http://science.sciencemag.org/content/362/6412/
              eaam9288.full>.

   [6]        Aspect, A., Grangier, P., and G. Roger, "Experimental
              Tests of Realistic Local Theories via Bell's Theorem",
              Phys. Rev. Lett. 47 (7): 460-463, 1981,
              <https://journals.aps.org/prl/abstract/10.1103/
              PhysRevLett.47.460>.

   [7]        Muralidharan, S., Kim, J., Lutkenhaus, N., Lukin, M., and
              L. Jiang, "Ultrafast and Fault-Tolerant Quantum
              Communication across Long Distances", Phys. Rev. Lett. 112
              (25-27), 250501, 2014, <https://arxiv.org/abs/1310.5291>.

   [8]        Van Meter, R. and J. Touch, "Designing quantum repeater
              networks", IEEE Communications Magazine 51, 64-71, 2013,
              <https://ieeexplore.ieee.org/document/6576340>.

   [9]        Dahlberg, A., Skrzypczyk, M., Coopmans, T., Wubben, L.,
              Rozpedek, F., Pompili, M., Stolk, A., Pawelczak, P.,
              Knegjens, R., de Oliveira Filho, J., Hanson, R., and S.
              Wehner, "A Link Layer Protocol for Quantum Networks",
              arXiv 1903.09778, 2019,
              <https://arxiv.org/abs/1903.09778>.

   [10]       Nielsen, M. and I. Chuang, "Quantum Computation and
              Quantum Information", Cambridge University Press , 2011.

   [11]       Bennett, C., DiVincenzo, D., Smolin, J., and W. Wootters,
              "Mixed State Entanglement and Quantum Error Correction",
              Phys. Rev. A Vol. 54, Iss. 5, 1996,
              <https://arxiv.org/abs/quant-ph/9604024>.

   [12]       Briegel, H., Dur, W., Cirac, J., and P. Zoller, "Quantum
              Repeaters: The Role of Imperfect Local Operations in
              Quantum Communication", Phys. Rev. Lett. Vol. 81, Num. 26,
              1998, <https://arxiv.org/abs/quant-ph/9803056>.



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   [13]       Cacciapuoti, A., Caleffi, M., Van Meter, R., and L. Hanzo,
              "When Entanglement meets Classical Communications: Quantum
              Teleportation for the Quantum Internet",   , 2019,
              <https://arxiv.org/abs/1907.06197>.

Authors' Addresses

   Wojciech Kozlowski
   QuTech
   Building 22
   Lorentzweg 1
   Delft  2628 CJ
   Netherlands

   Email: w.kozlowski@tudelft.nl


   Stephanie Wehner
   QuTech
   Building 22
   Lorentzweg 1
   Delft  2628 CJ
   Netherlands

   Email: s.d.c.wehner@tudelft.nl


   Rodney Van Meter
   Keio University
   5322 Endo
   Fujisawa, Kanagawa  252-0882
   Japan

   Email: rdv@sfc.wide.ad.jp


   Bruno Rijsman
   Individual

   Email: brunorijsman@gmail.com











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   Angela Sara Cacciapuoti
   University of Naples Federico II
   Department of Electrical Engineering and Information Technologies
   Claudio 21
   Naples  80125
   Italy

   Email: angelasara.cacciapuoti@unina.it


   Marcello Caleffi
   University of Naples Federico II
   Department of Electrical Engineering and Information Technologies
   Claudio 21
   Naples  80125
   Italy

   Email: marcello.caleffi@unina.it

































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