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- Bell Pairs:
- A special type of quantum state that is two qubits. The two qubits show a correlation that cannot be observed in classical information theory. We refer to such correlation as quantum entanglement. Bell pairs exhibit the maximal quantum entanglement. One example of a Bell pair is (|00>+|11>)/(Sqrt(2)). The Bell pairs are a fundamental resource for quantum communication.
- Bit:
- Binary digit (i.e., fundamental unit of information in classical communications and classical computing). Bit is used in the Classical Internet where the state of a bit is deterministic. In contrast, qubit is used in the Quantum Internet where the state of a qubit is uncertain before it is measured.
- Classical Internet:
- The existing, deployed Internet (circa 2020) where bits are transmitted in packets between nodes to convey information. The Classical Internet supports applications that may be enhanced by the Quantum Internet. For example, the end-to-end security of a Classical Internet application may be improved by a secure communication setup using a quantum application. Classical Internet is a network of classical network nodes that do not support quantum information technology. In contrast, Quantum Internet consists of quantum nodes based on quantum information technology.
- Entanglement Swapping:
- It is a process of sharing an entanglement between two distant parties via some intermediate nodes. For example, suppose that there are three parties (A, B, and C) and that each of the parties (A, B) and (B, C) share Bell pairs. B can use the qubits it shares with A and C to perform entanglement-swapping operations, and as a result, A and C share Bell pairs. Entanglement swapping essentially realizes entanglement distribution (i.e., two nodes separated in distance can share a Bell pair).
- Fast Byzantine Negotiation:
- A quantum-based method for
fast agreement in Byzantine negotiations
. - Local Operations and Classical Communication (LOCC):
- A method where nodes communicate in rounds, in which (1) they can send any classical information to each other, (2) they can perform local quantum operations individually, and (3) the actions performed in each round can depend on the results from previous rounds.
- Noisy Intermediate-Scale Quantum (NISQ):
- NISQ was
defined in
to represent a near-term era in quantum technology. According to this definition, NISQ computers have two salient features: (1) the size of NISQ computers range from 50 to a few hundred physical qubits (i.e., intermediate-scale) and (2) qubits in NISQ computers have inherent errors and the control over them is imperfect (i.e., noisy). - Packet:
- A self-identified message with in-band addresses or other information that can be used for forwarding the message. The message contains an ordered set of bits of determinate number. The bits contained in a packet are classical bits.
- Prepare and Measure:
- A set of Quantum Internet scenarios where
quantum nodes only support simple quantum functionalities (i.e.,
prepare qubits and measure qubits). For example, BB84
is a prepare-and-measure quantum key distribution protocol. - Quantum Computer (QC):
- A quantum end node that also has quantum memory and quantum computing capabilities is regarded as a full-fledged quantum computer.
- Quantum End Node:
- An end node that hosts user applications and interfaces with the rest of the Internet. Typically, an end node may serve in a client, server, or peer-to-peer role as part of the application. A quantum end node must also be able to interface to the Classical Internet for control purposes and thus be able to receive, process, and transmit classical bits and/or packets.
- Quantum Internet:
- A network of quantum networks. The Quantum Internet is expected to be merged into the Classical Internet. The Quantum Internet may either improve classical applications or enable new quantum applications.
- Quantum Key Distribution (QKD):
- A method that leverages quantum mechanics such as a no-cloning theorem to let two parties create the same arbitrary classical key.
- Quantum Network:
- A new type of network enabled by quantum information technology where quantum resources, such as qubits and entanglement, are transferred and utilized between quantum nodes. The quantum network will use both quantum channels and classical channels provided by the Classical Internet, referred to as a "hybrid implementation".
- Quantum Teleportation:
- A technique for transferring quantum information via Local Operations and Classical Communication (LOCC). If two parties share a Bell pair, then by using quantum teleportation, a sender can transfer a quantum data bit to a receiver without sending it physically via a quantum channel.
- Qubit:
- Quantum bit (i.e., fundamental unit of information in quantum communication and quantum computing). It is similar to a classic bit in that the state of a qubit is either "0" or "1" after it is measured and denotes its basis state vector as |0> or |1> using Dirac's ket notation. However, the qubit is different than a classic bit in that the qubit can be in a linear combination of both states before it is measured and termed to be in superposition. Any of several Degrees of Freedom (DOF) of a photon (e.g., polarization, time bib, and/or frequency) or an electron (e.g., spin) can be used to encode a qubit.
- Teleport a Qubit:
- An operation on two or more carriers in succession to move a qubit from a sender to a receiver using quantum teleportation.
- Transfer a Qubit:
- An operation to move a qubit from a sender to a receiver without specifying the means of moving the qubit, which could be "transmit" or "teleport".
- Transmit a Qubit:
- An operation to encode a qubit into a mobile carrier (i.e., typically photon) and pass it through a quantum channel from a sender (a transmitter) to a receiver.

- Quantum cryptography applications:
- Refer to the use of
quantum information technology for cryptographic tasks (e.g.,
Quantum Key Distribution
). - Quantum sensor applications:
- Refer to the use of
quantum information technology for supporting distributed sensors
(e.g., clock synchronization
). - Quantum computing applications:
- Refer to the use of
quantum information technology for supporting remote quantum
computing facilities (e.g., distributed quantum computing
).

- Secure communication setup:
- Refers to secure
cryptographic key distribution between two or more end nodes. The
most well-known method is referred to as "Quantum Key Distribution (QKD)"
. - Fast Byzantine negotiation:
- Refers to a quantum-based
method for fast agreement in Byzantine negotiations
, for example, to reduce the number of expected communication rounds and, in turn, to achieve faster agreement, in contrast to classical Byzantine negotiations. A quantum-aided Byzantine agreement on quantum repeater networks as proposed in includes optimization techniques to greatly reduce the quantum circuit depth and the number of qubits in each node. Quantum-based methods for fast agreement in Byzantine negotiations can be used for improving consensus protocols such as practical Byzantine Fault Tolerance (pBFT) as well as other distributed computing features that use Byzantine negotiations. - Quantum money:
- Refers to the main security requirement
of money is unforgeability. A quantum money scheme aims to exploit
the no-cloning property of the unknown quantum states. Though the
original idea of quantum money dates back to 1970, these early
protocols allow only the issuing bank to verify a quantum
banknote. However, the recent protocols such as public key quantum
money
allow anyone to verify the banknotes locally.

- Network clock synchronization:
- Refers to a world wide
set of high-precision clocks connected by the Quantum Internet to
achieve an ultra precise clock signal
with fundamental precision limits set by quantum theory. - High-sensitivity sensing:
- Refers to applications that
leverage quantum phenomena to achieve reliable nanoscale sensing of
physical magnitudes. For example,
uses an entangled quantum network for measuring the average phase shift among multiple distributed nodes. - Interferometric telescopes using quantum information:

Refers to interferometric techniques that are used to combine signals from two or more telescopes to obtain measurements with higher resolution than what could be obtained with either telescope individually. It can make measurements of very small astronomical objects if the telescopes are spread out over a wide area. However, the phase fluctuations and photon loss introduced by the communication channel between the telescopes put a limitation on the baseline lengths of the optical interferometers. This limitation can potentially be avoided using quantum teleportation. In general, by sharing Einstein-Podolsky-Rosen pairs using quantum repeaters, the optical interferometers can communicate photons over long distances, providing arbitrarily long baselines.

- Distributed quantum computing:
- Refers to a collection
of small-capacity, remote quantum computers (i.e., each supporting
a relatively small number of qubits) that are connected and work
together in a coordinated fashion so as to simulate a virtual
large capacity quantum computer
. - Blind quantum computing:
- Refers to private, or blind,
quantum computation, which provides a way for a client to delegate
a computation task to one or more remote quantum computers without
disclosing the source data to be computed
.

- The quantum node A encodes classical bits to qubits. Basically, the node A generates two random classical bit strings X and Y. Among them, it uses the bit string X to choose the basis and uses Y to choose the state corresponding to the chosen basis. For example, if X=0, then in case of the BB84 protocol, Alice prepares the state in {|0>, |1>}-basis; otherwise, she prepares the state in {|+>, |->}-basis. Similarly, if Y=0, then Alice prepares the qubit as either |0> or |+> (depending on the value of X); and if Y =1, then Alice prepares the qubit as either |1> or |->.
- The quantum node A sends qubits to the quantum node B via a quantum channel.
- The quantum node B receives qubits and measures each of them in one of the two bases at random.
- The quantum node B informs the quantum node A of its choice of bases for each qubit.
- The quantum node A informs the quantum node B which random quantum basis is correct.
- Both nodes discard any measurement bit under different quantum
bases, and the remaining bits could be used as the secret key.
Before generating the final secret key, there is a post-processing
procedure over authenticated classical channels. The classical
post-processing part can be subdivided into three steps, namely
parameter estimation, error correction, and privacy
amplification. In the parameter estimation phase, both Alice and Bob
use some of the bits to estimate the channel error. If it is larger
than some threshold value, they abort the protocol or otherwise move to
the error-correction phase. Basically, if an eavesdropper tries to
intercept and read qubits sent from node A to node B, the
eavesdropper will be detected due to the entropic uncertainty
relation property theorem of quantum mechanics. As a part of the
post-processing procedure, both nodes usually also perform
information reconciliation
for efficient error correction and/or conduct privacy amplification for generating the final information-theoretical secure keys. - The post-processing procedure needs to be performed over an
authenticated classical channel. In other words, the quantum node A
and the quantum node B need to authenticate the classical channel to
make sure there is no eavesdroppers or on-path attacks,
according to certain authentication protocols such as that described in
. In , the authenticity of the classical channel is checked at the very end of the post-processing procedure instead of doing it for each classical message exchanged between the quantum node A and the quantum node B.

- There are many enhanced QKD protocols based on
. For example, a series of loopholes have been identified due to the imperfections of measurement devices; there are several solutions to take into account concerning these attacks such as measurement-device-independent QKD . These enhanced QKD protocols can work differently than the steps of BB84 protocol . - For large-scale QKD, QKD Networks (QKDNs) are required, which can
be regarded as a subset of a Quantum Internet. A QKDN may consist of
a QKD application layer, a QKD network layer, and a QKD link layer
. One or multiple trusted QKD relays may exist between the quantum node A and the quantum node B, which are connected by a QKDN. Alternatively, a QKDN may rely on entanglement distribution and entanglement-based QKD protocols; as a result, quantum repeaters and/or routers instead of trusted QKD relays are needed for large-scale QKD. Entanglement swapping can be leveraged to realize entanglement distribution. - QKD provides an information-theoretical way to share secret keys
between two parties (i.e., a transmitter and a receiver) in the
presence of an eavesdropper. However, this is true in theory, and
there is a significant gap between theory and practice. By exploiting
the imperfection of the detectors, Eve can gain information about the
shared key
. To avoid such side-channel attacks in , the researchers provide a QKD protocol called "Measurement Device-Independent (MDI)" QKD that allows two users (a transmitter "Alice" and a receiver "Bob") to communicate with perfect security, even if the (measurement) hardware they are using has been tampered with (e.g., by an eavesdropper) and thus is not trusted. It is achieved by measuring correlations between signals from Alice and Bob, rather than the actual signals themselves. - QKD protocols based on Continuous Variable QKD (CV-QKD) have recently
seen plenty of interest as they only require telecommunications
equipment that is readily available and is also in common use
industry-wide. This kind of technology is a potentially
high-performance technique for secure key distribution over limited
distances. The recent demonstration of CV-QKD shows compatibility
with classical coherent detection schemes that are widely used for
high-bandwidth classical communication systems
. Note that we still do not have a quantum repeater for the continuous variable systems; hence, these kinds of QKD technologies can be used for the short distance communications or trusted relay-based QKD networks. - Secret sharing can be used to distribute a secret key among multiple nodes by letting each node know a share or a part of the secret key, while no single node can know the entire secret key. The secret key can only be reconstructed via collaboration from a sufficient number of nodes. Quantum Secret Sharing (QSS) typically refers to the following scenario: the secret key to be shared is based on quantum states instead of classical bits. QSS enables splitting and sharing such quantum states among multiple nodes.
- There are some entanglement-based QKD protocols, such as that described in
, , and , which work differently than the above steps. The entanglement-based schemes, where entangled states are prepared externally to the quantum node A and the quantum node B, are not normally considered "prepare and measure" as defined in . Other entanglement-based schemes, where entanglement is generated within the source quantum node, can still be considered "prepare and measure". Send-and-return schemes can still be "prepare and measure" if the information content, from which keys will be derived, is prepared within the quantum node A before being sent to the quantum node B for measurement.

- A client node with source data delegates the computation of the source data to a remote computation node (i.e., a server).
- Furthermore, the client node does not want to disclose any source data to the remote computation node, which preserves the source data privacy.
- Note that there is no assumption or guarantee that the remote computation node is a trusted entity from the source data privacy perspective.

- The client delegates a computation function to the server.
- The client does not send original qubits to the server but does send transformed qubits to the server.
- The computation function is performed at the server on the transformed qubits to generate temporary result qubits, which could be quantum-circuit-based computation or measurement-based quantum computation. The server sends the temporary result qubits to the client.
- The client receives the temporary result qubits and transforms them to the final result qubits.

- The BQC protocol in
is a circuit-based BQC model, where the client only performs simple quantum circuit for qubit transformation, while the server performs a sequence of quantum logic gates. Qubits are transmitted back and forth between the client and the server. Universal BQC (UBQC) in is a measurement-based BQC model, which is based on measurement-based quantum computing leveraging entangled states. The principle in UBQC is based on the fact that the quantum teleportation plus a rotated Bell measurement realize a quantum computation, which can be repeated multiple times to realize a sequence of quantum computation. In this approach, the client first prepares transformed qubits and sends them to the server, and the server needs to first prepare entangled states from all received qubits. Then, multiple interaction and measurement rounds happen between the client and the server. For each round: - the client computes and sends new measurement instructions or measurement adaptations to the server;
- the server performs the measurement according to the received measurement instructions to generate measurement results (in qubits or classic bits); and
- then the client receives the measurement results and transforms them to the final results.

- A hybrid UBQC is proposed in
, where the server performs both quantum circuits like that demonstrated in and quantum measurements like that demonstrated in to reduce the number of required entangled states in . Also, the client is much simpler than the client in . This hybrid BQC is a combination of a circuit-based BQC model and a measurement-based BQC model. - It is ideal if the client in BQC is a purely classical
client, which only needs to interact with the server using classical
channels and communications.
demonstrates such an approach where a classical client leverages two entangled servers to perform BQC with the assumption that both servers cannot communicate with each other; otherwise, the blindness or privacy of the client cannot be guaranteed. The scenario as demonstrated in is essentially an example of BQC with multiple servers. - How to verify that the server will perform what the client
requests or expects is an important issue in many BQC protocols,
referred to as "verifiable BQC".
discusses this issue and compares it in various BQC protocols.

- Leverage quantum mechanics to enhance classical distributed
computing. For example, entangled quantum states can be exploited to
improve leader election in classical distributed computing by
simply measuring the entangled quantum states at each party (e.g., a
node or a device) without introducing any classical communications
among distributed parties
. Normally, pre-shared entanglement first needs to be established among distributed parties, followed by LOCC operations at each party. And it generally does not need to transfer qubits among distributed parties. Distribute quantum computing functions to distributed quantum computers. A quantum computing task or function (e.g., quantum gates) is split and distributed to multiple physically separate quantum computers. And it may or may not need to transmit qubits (either inputs or outputs) among those distributed quantum computers. Entangled states will be needed and actually consumed to support such distributed quantum computing tasks. It is worth noting that: - Entangled states can be created beforehand and stored or buffered;
- The rate of entanglement creation will limit the performance of practical Quantum Internet applications including distributed quantum computing, although entangled states could be buffered.

For example, and have demonstrated that a Controlled NOT (CNOT) gate can be realized jointly by and distributed to multiple quantum computers. The rest of this section focuses on this type of distributed quantum computing.

- The quantum computer A locally generates some sensitive data qubits to be teleported to the quantum computer B.
- A shared entanglement is established between the quantum computer A and the quantum computer B (i.e., there are two entangled qubits: q1 at A and q2 at B). For example, the quantum computer A can generate two entangled qubits (i.e., q1 and q2) and send q2 to the quantum computer B via quantum communications.
- Then, the quantum computer A performs a Bell measurement of the entangled qubit q1 and the sensitive data qubit.
- The result from this Bell measurement will be encoded in two classical bits, which will be physically transmitted via a classical channel to the quantum computer B.
- Based on the received two classical bits, the quantum computer B modifies the state of the entangled qubit q2 in the way to generate a new qubit identical to the sensitive data qubit at the quantum computer A.

- Trusted repeater networks (Stage-1)
- Prepare-and-measure networks (Stage-2)
- Entanglement distribution networks (Stage-3)
- Quantum memory networks (Stage-4)
- Fault-tolerant few qubit networks (Stage-5)
- Quantum computing networks (Stage-6)

- In Stage-1, basic QKD is possible and can be leveraged to support secure communication setup, but trusted nodes are required to provide end-to-end security. The primary requirement is the trusted nodes.
- In Stage-2, the end users can prepare and measure the qubits. In this stage, the users can verify classical passwords without revealing them.
- In Stage-3, end-to-end security can be enabled based on quantum repeaters and entanglement distribution to support the same secure communication setup application. The primary requirement is entanglement distribution to enable long-distance QKD.
- In Stage-4, the quantum repeaters gain the capability of storing and manipulating entangled qubits in the quantum memories. Using these kinds of quantum networks, one can run sophisticated applications like blind quantum computing, leader election, and quantum secret sharing.
- In Stage-5, quantum repeaters can perform error correction; hence, they can perform fault-tolerant quantum computations on the received data. With the help of these repeaters, it is possible to run distributed quantum computing and quantum sensor applications over a smaller number of qubits.
- Finally, in Stage-6, distributed quantum computing relying on more qubits can be supported.

Quantum Internet Stage | Example Quantum Internet Use Cases | Characteristic |
---|---|---|

Stage-1 | Secure communication setup using basic QKD | Trusted nodes |

Stage-2 | Secure communication setup using the QKD with end-to-end security | Prepare-and-measure capability |

Stage-3 | Secure communication setup using entanglement-enabled QKD | Entanglement distribution |

Stage-4 | Blind quantum computing | Quantum memory |

Stage-5 | Higher-accuracy clock synchronization | Fault tolerance |

Stage-6 | Distributed quantum computing | More qubits |