1. Introduction
The international race to build practical quantum computers is heating up. The US Congress recently passed the National Quantum Initiative Act to secure the leading position of U.S. in quantum computing. In the meantime, several grand quantum computing projects, which amount to billions of dollars, have been announced by China and the EU. Advances in quantum computing hardware have been very rapid over the past few years. Google, IBM, and Intel all announced their quantum chips of , , and
superconducting qubits (quantum bits), respectively, in 2018.
Modern computing and communication rest on the digital abstraction of information, measured in bits. A bit has a state either or . Quantum mechanics allows a quantum bit (qubit) to be in a superposition of both states 0 and 1. In addition, the dimension of the state space grows exponentially with the number of qubits. These properties endow a quantum computer the power to achieve tasks that are beyond the capability of classical computers. For example, Grover’s search algorithm (Grover, 1996), for querying an unsorted database on a quantum computer, affords a quadratic speedup when compared to its best classical competitor. Even more impressive is Shor’s factoring algorithm (Shor, 1997), which provides an exponential speedup over the best known classical factoring approach. It is anticipated that “quantum supremacy”– the superiority of quantum computing over classical devices for a welldefined computational problem – will be achieved by NISQ (noisy intermediate scale quantum) devices in the near future (Preskill, 2018). All these phenomena indicate that we are in the transitions from studing quantum theory to engineering quantum information—the second quantum revolution (MacFarlane et al., 2003).
An extraordinary quantum effect is entanglement– a strong quantum correlation between qubits that are even far apart. With preshared quantum entanglement between different parties, communication of quantum information can be done by socalled quantum teleportation (Bennett et al., 1993). Thus establishing entanglement attracts lots of attention (Pirandola and L. Braunstein, 2008). A remarkable work is the system for longdistance and highfidelity qubit teleportation developed by a team of researchers from Massachusetts Institute of Technology and Northwestern University in 2004 (Lloyd et al., 2004). In 2018, deterministic delivery of remote entanglement on a quantum network was realized using diamond spin qubit nodes (Humphreys et al., 2018).
Being able to send qubits from one quantum processor to another allows them to form a quantum computing cluster. These quantum processors can together build an entangled qantum sytem. In this way several less powerful quantum processors are allowed to jointly perform more powerful tasks that are not possible with presentday technologies. This is often referred to as networked quantum computing, or distributed quantum computing, which may provide exceptional savings in communication complexity, compared with classical distributed computation (see, e.g., (Raz, 1999; Regev and Klartag, 2011) and the references therein). Therefore, networked quantum computing offers a path towards scalability for quantum computers, since more and more quantum processors can naturally be added to the network.
With several quantum networks joined together, the quantum internet would play an indubitable role in the development of the second quantum revolution. Besides the potential of exploring the computational power of quantum mechanics, the quantum internet could help to build ultrasharp telescopes (Gottesman et al., 2012), remote quantum computers and secure cloud quantum computing (Broadbent et al., 2009; Mahadev, 2018; Brakerski et al., 2018). Immediately, the quantum internet would provide unparalleled capabilities that are provably impossible by only classical computation. For example, in 1984, Bennett and Brassard introduced the first and the most famous application of quantum internet, quantum key distribution (QKD) (Bennett and Brassard, 1984; Bennett et al., 1992), which enables two remote end nodes to establish provablysecure random keys. To implement quantum key distribution, it is sufficient for the quantum processors to be capable of preparing and measuring only a single qubit at a time. Many research teams have succeeded in building and operating quantum cryptographic devices since last century. In (Elliott et al., 2003), the world’s first network, the DARPA Quantum Network, was built, which delivers end to end network security via highspeed QKD by BBN, Harvard, and Boston University. Building the 1,200mile quantum communication landline between Beijing and Shanghai in 2016 and the quantum communication satellite (known as Micius) in 2017, China has the world’s first spaceground quantum network (Yin et al., 2017).
Rapid experimental progress in recent years has brought first rudimentary quantum networks within reach. Physicists can control and manipulate quantum signals much better than before (Kimble, 2008; Reiserer and Rempe, 2015). In 2018, an idea is illustrated to store quantum information for hours at a time using special diamonds (Astner et al., 2018). This makes the quantum information even more stable than the conventional information stored in the working memory of our computers. A team at Delft has already started to build the first genuine quantum network, which will link four cities in the Netherlands. Chicago Quantum Exchange, a research hub that is building a 30mile quantum teleportation network using telecommunication fibers. It is likely that we will see the birth of the first multinode quantum network in the next few years.
Very recently, the design of the quantum internet received much attention from an engineering communication perspective. Many challenges are formulated in (Caleffi et al., 2018; Cacciapuoti et al., 2018). In a recent seminal work (Wehner et al., 2018), Wehner and coauthors have drawn a road map for the future quantum internet. They proposed stages of development toward a fullblown quantum internet and highlighted experimental and theoretical progress needed to achieve them. Referring to the development history of the classical internet, the next step toward quantum internet is to estabilish a methodology of unified, reliable, ordered, and errorchecked delivery of a stream of qubits between applications running on quantum endnodes. In particular, packet switching is a technique for reliably and efficiently transmitting data over a communication channel. A similar feature for quantum communication is desired. Thus we will imitate this packet switching technique and propose the quantum analogue for the quantum internet.
Moreover, Transmission Control Protocol/Internet Protocol (TCP/IP), developed by Vinton Cerf and Bob Kahn in (Vinton and Robert, 1974), is one of the core components of the Internet that solves the previously mentioned problems in classical internet. TCP/IP provide a systematic classification of tasks that allows the different types of information processing to be accomplished by specific systems using the least amount of digesting of information, and thus enable the widespread use and applications of the classical internet. Almost all operating systems in use today, including all consumertargeted systems, adopt a TCP/IP implementation. Therefore a unified protocol that allows quantum computers on different platforms to interconnect would be of extremely convenient, especially that there are different ideas of building quantum computers: superconducting qubits led by IBM and Google, ion trap led by the University of Maryland, topological qubits led by Microsoft, and so on. Unfortunately, such a network stack for quantum internet has not been presented yet but only some basic elements have been noted (Meter and Touch, 2013).
The frames of classical TCP/IP cannot be directly applied in the quantum network because of the significant difference between classical bits and quantum bits. Retransmission is one of the basic mechanisms used by protocols operating over a packet switched computer network to provide reliable communication (such as that provided by a reliable byte stream, for example TCP). However, retransmission is generally impossible for transmitting qubits due to the nocloning theorem as we mentioned before.
In this paper, we investigate and propose protocols for packet quantum network intercommunication: quantum User Datagram Protocol (qUDP) and quantum Transmission Control Protocol (qTCP). The qTCP provides reliable, ordered, flowcontrolled transmission of packets over the interlinked networks. To protect the fragile quantum information in the quantum internet, qTCP employs techniques of quantum errorcorrecting codes, quantum secret sharing, as well as classical techniques of stack design. In particular, the creation of the logical processtoprocess connections of qTCP is accomplished by a quantum version of the threeway handshake protocol.
This paper is organized as follows. In Section 2, we introduce the basics of quantum mechanics. In Section 3, two models of quantum internet are given: repeaterbased model and plain model. In Section 4, we discuss the main difficulties in designing quantum packet switching for quantum internet. In Section 5, a fourlayer model of quantum network is shown. In Section 6, the four layers for the repeaterbased quantum internet are given in detail, including the qUDP, qTCP protocols with the quantum threeway handshake protocol and similarly the four layers for plain quantum network are given in Section 7. Finally, in Section 8, we conclude with some highlighted research.
2. Preliminaries
Here we present the barebones of quantum mechanics for our purpose. For more details, interested readers are referred to (Nielsen and Chuang, 2011). We start with the postulates of quantum mechanics.
2.1. State space
The state space of a closed quantum system is a complex Hilbert space and a pure
quantum state is an arbitrary unit vector in the Hilbert space. In particular, a qubit system has a twodimensional vector space with an orthonormal basis
. Thus the system can be in an arbitrary superposition of these two states, say , where and are complex numbers satisfying the normalization condition . The dimension of the quantum system grows exponentially with the number of qubits: an qubit system has a dimensional complex linear space with an orthonormal basis and an qubit (pure) state can be an arbitrary superposition of these states. Note thatis the tensor product and will be omitted with no ambiguity.
More generally, a quantum state can be represented by a density operator , which is positive semidefinite and has trace equal to one. For a pure quantum state , its density operator is . Quantum mechanics allows the quantum system in a more complicated state: a mixed quantum state, which is a convex combination of some pure states such that and .
2.2. Quantum evolution
The evolution of a closed quantum system can be described by a unitary transformation. An operator is unitary if , where is the complex conjugate transpose of , and is the identity operator. Suppose initially the system is in and it evovles to after time . Then there exists a unitary such that A basis for the linear operators on a qubit is the Pauli matrices .
A general quantum operation, usually denoted by , is a completely positive linear map, which preserves the trace.
2.3. Quantum measurement
A quantum measurement on a system is described by a collection of measurement operators , which satisfy and the index stands for the measurement outcome. If a quantum system in the state
is measured, the probability that outcome
occurs is and the postmeasurement state is We can also describe the behavior of quantum measurement in the language of density operators. If is measured by , then the probability that outcome occurs is and the postmeasurement state isNext we introduce the effects of quantum teleportation, entanglement swapping, the nocloning theorem, and quantum error correction.
2.4. Entanglement
A twoqubit pure state is entangled if they cannot be described by two independent singlequbit states. This, unlike the behavior of two correlated random bits, is a thoroughly nonclassical behavior. Entanglement leads to nonlocal correlations, which seem to violate causality, so that Einstein, Podolsky, and Rosen mistakenly suggested that quantum mechanics might not be complete (Einstein et al., 1935). The state is called the maximallyentangled state, or simply EPR, where the subscript means that it is shared between and . We usually use to denote .
2.5. Quantum teleportation
Quantum teleportation (Bennett et al., 1993) is arguably the most famous quantum communication protocol. With the help of preshared entanglement between the sender and the receiver, quantum information can be transmitted from one location to another using only classical communication.
The teleportation protocol is as follows. Suppose Alice and Bob share an EPR pair and Alice wants to send to Bob an unknown qubit She performs a Bell measurement on her two qubits , where
for . Alice then sends Bob her measurement outcome . Interestingly, this twobit message contains all the information that Bob needs to recover on his side. To see this, observe that
According to Alice’s measurement outcome , Bob’s qubit will be in the state and thus can be recovered after a Pauli correction on Bob’s qubit.
To sum up, with the help of an EPR pair, the transmission of two classical bits is enough to transmit one qubit. Thus, one can transmit qubits by transmitting classical bits using preshared EPR pairs.
2.6. Entanglement swapping
Suppose that Alice and Bob share an EPR pair , and Bob and Charlie share another EPR pair . It is possbile to construct an EPR pair shared between Alice and Charlie by socalled entanglement swapping.
The procedure is as follows: Bob performs a Bell measurement on Bob’s two qubits and sends the twobit outcome to Charlie, who then performs a Pauli correction according to Bob’s measurement outcome.
This can also be regarded as Bob teleporting his particle to Charlie by consuming the ERP pair . In other words, quantum correlations can be teleported.
2.7. Nocloning theorem
A fundamental property of quantum mechanics is that learning an unknown quantum state from a given specimen would disturb its state (Bennett et al., 1994). In particular, the quantum nocloning theorem (Dieks, 1982; Wootters and Zurek, 1982) states that an arbitrary unknown quantum state cannot be cloned. Generally speaking, there is no quantum operation that can transform an unknown quantum state into . Intuitively, this is because a quantum operation is always linear, while the desired mapping is not.
2.8. Quantum error correction
Qubits are maddeningly errorprone. A quantum error correcting code, first proposed by Shor (Shor, 1995), can protect quantum information against decoherence.
A noise channel is modeled as a quantum operation . If there exist encoding and decoding quantum operations and such that
for any input state , we say that and correct the error of . That is, any quantum information encoded by can be perfectly recovered from the noise process by applying the recovering map .
One can observe the very useful fact that quantum errorcorrecting codes can correct errors coherently in the following sense. For encoding, noise and decoding operations on , and can correct the noise for any quantum system .
3. Quantum Internet Models
The Internet is the largest engineered system ever created by mankind. With new quantum computing power, it can be further supplemented to form a more powerful internet—the quantum internet.
Figure 1 illustrates the concept of a quantum network, where several quantum computers are distributed and connected along with a classical network. PCs, workstations, Web servers, quantum computers, etc, are called hosts, or endnodes. Endnodes are connected together by a network of communication links and routers.
Here we discuss two models for the quantum internet: repeaterbased model and plain model, assuming that two neighboring quantum devices share EPR pairs for teleportation or they are directly connected by quantum channels, respectively. These are natural generalizations of the CleveBuhrman model (Cleve and Buhrman, 1997) and the Yao model (Yao, 1993) for twoparty quantum communication, where in the CleveBuhrman model quantum communication is done by teleportation with shared entanglement and classical communication, while in the Yao model qubit channels are used. ( Note that repeater quantum networks have been discussed in the literature but there is no specification about how teleportation is done explicitly. ) We further assume that all the nodes within the quantum internet can communicate classically, for example, over the classical internet, in order to exchange control information (such as the measurement outcome for Pauli correction in teleportation).
We emphasize that both our models features fullduplex data transmission as in the classical internet. In other words, (quantum or classical) data can be transmitted in both directions on a signal carrier at the same time.
In this paper, we only discuss the transmission of quantum information through the internet. Therefore, we will focus on the subnetwork consisting of routers, hosts with the power of quantum information processing, the quantum communication links and its corresponding classical communication links. This can be done by associate onebit index with the IP for which host or router with quantum power.
3.1. Repeaterbased Quantum Internet
A quantum channel between two neighboring quantum devices is required for the transmission of qubits. However, quantum channels are inherently lossy and they are not reliable for longdistance communication. Consequently, the techniques of quantum repeaters are used as intermediate nodes to reach long distances (Jiang et al., 2009; Simon et al., 2007; William et al., 2015; Sangouard et al., 2011).
The idea is to do a sequence of entanglement swapping or teleportation between two consecutive nodes so that quantum communication between two endnodes can be implemented as shown in Fig. 5. Suppose two neighboring nodes and are connected by repeaters , each of which has two (or more) qubits. EPR pairs are constantly created between repeaters and for , and between and , and . Then each repeater performs a Bell measurement on its two qubits and passes the measurement outcomes to for Pauli correction, using classical channels (the gray lines in Fig. 5). This will create an EPR pair between and and thus fullduplex quantum communication can be implemented.
One may establish endtoend EPR pairs at the beginning and then only classical communication remains to be done for quantum communication. But, in this way, the quantum buffer of the receiver will be occupied during stage of the classical communication. Thus we keep the flexibility that the entanglement swapping or teleportation is done when necessary. Another issue is the time cost. In order to transmit one qubit information from endnodes to using an endtoend EPR pair, the time cost is roughly the time cost of establishing an EPR pair between and , given EPR pairs between neighboring nodes are available, plus the time cost of transmitting two classical bits from to to accomplish the teleportation. If the path consists of repeaters, the time cost of establish an endtoend EPR pair is roughly the transmission time for transmitting two bits for Pauli correction through edges. This is basically equal to the time cost of classical communication for teleportation. Thus the time cost would be halved if we do sequential teleportation or entanglement swapping.
3.1.1. StoredecodeandForward Transmission
A delicate step of a repeaterbased quantum channel is that the Pauli correction for teleportation can be deferred. For example, If a qubit is sent from node to via a router . Originally, sends the measurement outcome and then does a Pauli correction according to . Instead, can perform a Bell measurement on his qubits and send the outcome , together with , to for Pauli correction. In fact, having is enough for to recover the transmitted qubit. The only quantum operation that has to do is a Bell measurement. This manner is called storedecodeandforward transmission, that is, an intermediate node stores the message for Pauli correction from the previous node, updates the message for Pauli correction according to his Bell measurement, and forwards this message to the next node.
Notice that, router must wait for the message of about the positions of the qubits to do the Bell measurements since a router may have many qubits. If a Bell measurement is applied to wrong target qubits, quantum information will be destroyed.
3.2. Plain model of Quantum internet
In the plain model of quantum internet, we assume that a fullduplex quantum channel exists between two neighboring quantum devices, such as two hosts, two routers, a host and a router, and they do not preshare any quantum entanglement. Also these quantum channels are paired with fullduplex classical channels.
When some data are to be sent, they will be segmented and attached with certain header bytes. The resulting packages of information, socalled packets, are then sent through the network to the destination, at which they are reassembled into the original data. Routers that are assigned quantum computing power can take quantum packets as well as classical packets. In particular, any quantum communication between two neighboring quantum devices is accomplished by sending packets of quantum messages over the quantum channel connecting them, together with classical packets over a classical channel connecting them. (For simplicity, the quantum packet and its corresponding classical packet are called a quantumclassical packet.) Moreover, we assume that synchronization can be done for the plain model of quantum internet: we assume that a packet of quantum messages and its corresponding packet of classical messages always arrive at the next node simultaneously.
Although the distribution of quantum states over long distances is not possible with current technology, it will be improved in the near future. Comparing with the repeaterbased model, the plain model is more straightforward, and much more efficient in the following sense. In order to transmit one qubit information to a neighboring router, the previous model requires onequbit transmission in the entanglement establishment plus two bits classical information transformation.
3.2.1. StoreandForward Transmission
A router takes a packet arriving on one of its incoming communication links and forwards that packet on one of its outgoing communication links. In the plain model, a router must receive the entire quantumclassical packet before it can transmit the first qubit and the first bit of the packet onto the outbound link. This is called storeandforward transmission.
4. Difficulties and Challenges: Quantum Packet Loss
As mentioned in the previous section, in classical network, the technique of packet switching is used so that messages are split into small packets, which are then sent independently through the network. Packets from different messages can be interspersed to give greater responsiveness for interactive computing; and individual packets can be resent if necessary, rather than entire messages. Whenever one party sends something to the other party, it retains a copy of the data it sent until the recipient has acknowledged that it received it. In a variety of circumstances, the sender automatically retransmits the lost packet using the retained copy.
Within each network, quantum communication may be disrupted due to unrecoverable mutation of the data or missing data. The reasons include quantum decoherence, imperfect operations, the network packet loss, and others.
Endtoend restoration procedures are desirable to allow complete recovery from these conditions in quantum network, we will imitate the packet switching technique and propose the quantum analogue for the quantum internet. In order to ensure that all quantum data are eventually transferred from source to destination, we are facing many difficulties. The nocloning theorem is a vital ingredient in quantum cryptography, as it forbids eavesdroppers from creating copies of a transmitted quantum cryptographic key. In contrast, it prevents us from using classical techniques in quantum computing. In particular, this affects our problem of transmitting qubits over the internet in two fundamental ways. First, any logical quantum data leak into the environment because the noisy channel cannot be recovered by the communicating parties. Second, the parties hold a joint quantum state that evolves with the protocol, but they cannot make copies of the joint state without corrupting it.
We will employ the techniques of quantum error correcting code together with the classical techniques of packet design to solve the problem of quantum packet loss, see Section 6 for the repeater quantum network, and Section 7 for the plain quantum network.
5. Layer model
The classical internet has a layered structure, which has conceptual and operational advantages. To introduce a similar structure in the design of quantum internet protocols, quantum network hardwares and softwares are required to operate in the following four layers:
Layered model 

The Application Layer creates user quantum data and communicate this data to other applications on another host. The Transport Layer performs hosttohost quantum communications. The Network Layer exchanges quantum datagrams across network boundaries. The Network Access Layer provides the means for the system to deliver quantum data to the other devices on a directly attached network.
In the following, we will take a topdown approach to explain the four layers, first covering the Application Layer and then proceeding downwards, for the quantum repeater network and plain quantum network, respectively.
6. Quantum repeater network layers
The quantum repeater network is a teleportationbased network. EPR pairs have to be created between neighboring hosts and routers. Other than that, the data that are actually transmitted in the network are classical bits. Therefore, this model has the following difference from classical network model. To obtain the data to be transmitted, Bell measurements must be jointly applied on the quantum data of the Transport Layer and the particles of the shared EPRs in the Network Access Layer as the initial step of the teleportation. The classical data, measurement outcomes, are packeted in the Transport Layer then passed into the Network Layer.
6.1. Application Layer
The Application Layer is where quantum network applications and their applicationlayer protocols reside. In a quantum network application, endnodes exchange messages with each other.
In addition to a classical storage, Host holds two quantum registers and as send buffer and receive buffer, respectively, and also a local working register . At each stage of a general quantum application protocol, Host would send to another Host , receive quantum packet from the network at , and then apply a local operation on its local registers and . (Note that both the quantum repeater network and plain quantum network can be recast in this framework.)
In the quantum network, the global state of the system can be arbitrary inputs that are allowed, and it is possibly unknown (totally or partially) to hosts.
To send quantum information between two endnodes, messages are divided into smaller packets, which are then sent through communication links and routers. In this scenario, a packet error or loss can destroy the global state, and hence the followup tasks. Such a problem will be handled in the next layer–Transport Layer.
6.2. Transport Layer
The Transport Layer of the quantum network transports Application Layer messages between application endpoints. Before providing the two transport protocols, qUDP (Quantum User Datagram Protocol) and qTCP (Quantum Transmission Control Protocol), we first introduce some tools from quantum error detection.
6.2.1. Quantum error detection
The widely used error detection method, including the parity bit, Cyclic Redundancy Check, and the Checksum, can be characterized by a check function in the following encoding procedure: a given bit string is encoded as of bits. This has been generalized in quantum computing.
Suppose a host wants to send an qubit register through the internet, using a check function . Let be a purifed state of for a reference system . The encoding is done by firstly appending ancilla qubits in to , and the state of is of the form
Then, the host applies a controlled unitary on to obtain
where is defined by and is called a check unitary. Upon receiving , the receiver simply applies the decoding unitary to . If the data has not been changed, the decoded state would be
Then measuring in the computational basis will give us the outcome . Otherwise, if the measurement outcome is nonzero, the receiver knows that at least one qubit error has occurred. It could be the case that some error occurs but the measurement outcome is , which will lead to a decoding error. This situation occurs with a small chance if the check function is chosen appropriately.
6.2.2. Quantum User Datagram Protocol
Just like its classical counterpart, UDP, the quantum UDP protocol uses a simple connectionless communication model with a minimum of protocol mechanism.
Two communicating quantum processes, says Alice and Bob, use classical UDP sockets to interact. After establishing sockets, Alice, the sender at this round, firstly applies the quantum checksum as the quantum analog of the checksum of the classical UDP protocol using the idea we mentioned in the beginning of this subsection. This is for quantum error detection, and her qubits are now called the quantum segments, says qubits.
In order to apply for teleportation, Alice has to apply the joint measurements on her segments and the particles of the EPRs. Which EPRs she uses would directly correspond to the next router the segments will send to. Therefore, she asked the Network Layer for proper EPRs by sending the Network Layer the destination, which can be done by the adjusted IP protocol discussed in the next subsection.
After that, she applies the joint measurement on her quantum segments and the particles of EPRs, obtains a bit string . She uses the classical checksum on and sends the resulting classical bits by the classical UDP protocol. Now Alice can generate the qUDP packet for quantum repeater network, using these data in the structure:
Classical UDP header  Indicator  Data 
where the Indicator is used to indicate that this is qUDP packet of quantum repeater network. Because the action of the routers and receiver is different from the UDP packet of classical internet. Besides the correction of Pauli measurement outcomes, the data part also contains the positions of the corresponding EPRs between two nodes that just been consumed. One can choose the size of this qUDP packet to fit the current UDP structure.
The behavior of the sender is as in Protocol 1.
In step 4 the qUDP will split the qubits into several parts, where each part will be teleported to a corresponding router. This is because we want (1) the size of each packet to fit the Maximum Transmission Unit of the Network Layer, and (2) the data been transmitted through different routing paths.
The behaviour of the reciever is as in Protocol 2. Note that the receiver must wait for all pieces of before he can decode the message. The receiver maintains a timeout setting. If some is not received in time, he drops all the received and release the corresponding quantum registers .
If there is no packet loss or corrupted during the networks, the state will be successfully transferred to the registers of the receiver.
The qUDP provides no recovery procedure for lost packets, and it favors reduced latency over reliability like the classical case. Applications that use qUDP can flexibly define their own mechanisms for handling packet loss.
6.2.3. Quantum Transmission Control Protocol
The quantum Transmission Control Protocol (qTCP) introduced here will provide a connectionoriented, reliable, ordered, and errorchecking delivery of a quantum data stream between hosts. In the transmission of applicationlayer messages, a long message will be divided into shorter segments by qTCP, just like TCP in the classical network. This service includes guaranteed delivery of applicationlayer messages to the destination and flow control (that is, sender/receiver speed matching).
Quantum information is fragile through the transmission over the internet. To guarantee datagram delivery, a quantum version of information retransmission is needed. However, the nocloning theorem prevents quantum information from being copied. Herein we show how information retransmission can be achieved using the techniques of quantum secret sharing (Cleve et al., 1999). This guarantees that the quantum data stream transmitted through qTCP will have exactly the same quantum information and correlation as the original stream.
The qTCP packet is designed as follows:
Classical TCP header 

Indicator 
Pseudo acknowledgement number 
Pseudo Window 
Data 
The indicator implies that this is a qTCP packet for quantum repeater network. Besides the correction of Pauli measurement outcomes, the data part also contains the positions of the corresponding EPRs between two nodes that just been consumed. Others are just the same as classical TCP.
To reach reliable transmission, a packet of quantum information is not transmitted in one step in our qTCP, but in at least two stages. Only if the transmission of both parties is successful, the quantum information is successfully transmitted. The Pseudo acknowledgement number and Pseudo Window are used to record the status of the transmission. We will explain the Pseudo acknowledgement number in the following Data transfer part.
The qTCP protocol operations may be divided into three phases. The logical processtoprocess connections of qTCP is established by a quantum version of the threeway handshake protocol before entering the data transfer phase. After data transmission is completed, the connection termination closes established virtual circuits and releases all allocated resources.
Connection establishment–quantum threeway handshake.
To establish a connection, we propose a quantum version of the threeway handshake protocol.
Just like its classical counterpart, Host B establishes a passive open, and then Host A initiate an active open. To establish a quantum connection, the quantum threeway handshake protocol operates as follows:

SYN: Host A establishes local EPR pairs . Host A sends SYN to Host B, together with the quantum information of (stored as by Host B) by a qTCP packet.

SYNACK: Host B receives the qTCP packet. Firstly, he applies the Pauli correction to , and then sends SYN+1 to Host A, together with the quantum information of (stored as by ). Also, Host B establishes local EPR pairs . Then Host B sends ACK to Host A, together with the quantum information of (stored as by ) by a qTCP packet. After verifying SYN+1, Host A performs a multiqubit Bell measurement on and checks whether the measurement outcome is .

ACK: Host A sends ACK+1, and transfers the quantum information of to of Host B. After verifying ACK+1, Host B performs a multiqubit Bell measurement on and check whether the measurement outcome is .
At this point, both Hosts have received an acknowledgment of the connection. One can observe that the quantum channel between two hosts is noiseless if and only if the distribution of EPRs is noiseless. To see the efficiency of detecting the channel noise, we consider the extreme case that the entanglement is destroyed either during the teleportation from to or from to . Assume the state of at the end of step 2, , is not entangled (or socalled separable
). In other words, there exist probability distribution
and quantum states and such that . When we perform a multiqubit Bell measurement on it, the probability of obtaining satisfiesThe steps 1, 2 establish the classical and quantum connections for one direction and it is acknowledged. The steps 2, 3 establish the quantum connection for the other direction and it is acknowledged. With these, a fullduplex quantum communication is established.
Data transfer
A reason that the classical TCP works well is because classical information can be correctly read and copied. When one party sends something to the other, it will keep a copy until it gets acknowledged by the recipient. In a variety of circumstances, a sender may automatically retransmit the data using the retained copy.
To handle quantum retransmission, we start with a simple question:
How to achieve reliable transmission of a onequbit state from Host A to Host B through a noisy quantum channel?
We consider a multiround protocol, which can always be modelled as follows:

Host A encodes into , and sends register to Host B.

Host B sends Host A the acknowledgement whether the transmission is successful.

If unsuccessful, the hosts will do other actions.

Otherwise, Host A sends to Host B.

The reliability requires at least the following fact: once a transmission failed, the hosts are able to recover the original state from the remaining qubits.
Suppose the first step of transmitting to Host B fails, the left system is enough to recover the original state . Then the information of is not enough to recover . Otherwise, this provides a cloning procedure, contradicts to the nocloning theorem. This implies that alone contains no information of the original state . In the stage of (2b), to make sure that the information of the original state is not lost even when is lost, must contain enough information to recover the original state . Moreover, we want that if the transmission of successes, Host B can recover the original state from .
In other words, we want an encoding scheme from to such that any two of can reconstruct the unknown . This can actually be done by the threshold scheme of quantum secret sharing (Cleve et al., 1999), using the theory of quantum errorcorrecting codes. An threshold scheme is such that any shares, but not fewer, 1 can jointly recover the secret.
Now we outline our quantum retransmission procedure. First Host A sends to Host B after generating from . If is not received, then Host A reconstructs from , and runs the procedure again. Otherwise if is received, then Host A sends to Host B. If it is received, Host B can reconstruct from . If is not received, Host A repeats the procedure on the remaing . The procedure can be repeated recursively until Host B has enough information to recover or the procedure aborts after a number of rounds. More explicitly, the transmission succeeds if two consecutive messages are correctly received by Host B.
In a less noisy channel, one can make sure that this scheme always succeeds within finite steps. However, a potential problem here is that the storage of Host B for this single message is not unbounded. Although Host B may need an unbounded classical data structure to maintain the status of the transmission, he has only four types of possible status:

B is waiting for some since the previous is not received or valid;

is received, and B is waiting for ;

is received, but the corresponding is not valid;

and are both validly received.
The Pseudo acknowledgement number and Pseudo Window of the qTCP packet are used for the acknowledgement of the status of Host B.
Now we give the construction of our qTCP packet as follows. For any register with qubits, Host A first applies a check unitary to and ends with qubits as previously discussed in the qUDP part.

Use the threshold scheme to obtain qubits .

Ask the Network Layer for EPRs, perform Bell measurements, and record the outcome .

Construct a classical TCP packet for and send it.
By carefully choosing and , we can ensure that can be put in one TCP packet.
The sender just combines this with the Quantum retransmission Protocol 3.
The router’s action for any packet is illustrated in Protocol 4 in qUDP.
Host B’s actions for any packet are

Verify classical checksum;

Apply Pauli correction;

Store it.
Once all the pieces are received, Host B

Decodes the threshold scheme;

Performs and Bell measurements to verify the quantum checksum;

Acknowledges Host A;

Uses the qubits and releases the buffer.
It would not be surprising that most techniques in classical TCP, including Doubling the Timeout Interval, Fast Retransmit, selective acknowledgment, for congestion control are applicable to this quantum repeater network by using this qTCP data structure.
Let be a register that is about to be sent. Host A maintains statuses of his data for some and , “Sent and Acknowledged”; “Send But Not Yet Acknowledged”, “Not Sent, Recipient Ready to Receive”; “Not Sent, Recipient Not Ready to Receive”. Host A always holds some corresponding . Similarly, Host B maintains a classification of his data for some and , “Received and ACK Not Send to Process”, “Received Not ACK”, “Not Received”.
The “Pseudo acknowledgement number” and “Pseudo Window” are used to record these statuses. Together with “Acknowledgement number” and window, sliding window protocol can be implemented.
Connection termination
To terminate the connection, we just use the fourway handshake as in classical TCP, where each side of the connection is terminated and each buffer is released independently.
6.3. Network Layer
After receiving a transportlayer segment and a destination address from the Transport Layer protocol (qUDP or qTCP) in a source host, the Network Layer then provides the service of delivering the segment to the Transport Layer in the destination host.
We just use the celebrated Internet Protocol (IP). The routing protocol can be slightly modified such that quantum packets must be transmitted to a router with quantum power. This can be done by the associate onebit index of the IP. In particular, each host and router maintains a dynamic table with the information about the numbers of the EPRs that he shares with his neighbor. The neighbor shared more EPRs has priority in the routing protocol.
Now we describe the action of the router for qUDP packet or qTCP packet as follows.
The behaviour of the router is very different from the behaviour of router in classical networks. The data of quantum repeater network are just the Pauli measurement outcomes. It can only be used in completing the teleportation, which would accomplish the transformation of quantum states. The classical checksum is used to check the data integrity. In order to transmit the “received” quantum state to the next node, the router has to implement teleportation, which would generate new data. This data has no correlation with the received classical data, generally. The router then replaces the data and checksum by the newly generated data in the qUDP packet or qTCP packet to obtain a new one.
6.4. Network Access Layer
The Network Access Layer in the quantum repeater network provides the services that a classical Network Access Layer does, including the CRC checking for the classical data of qTCP. It provides the distribution of entanglement, in particular EPRs, as an additional service. The entanglement distribution is the lifeline of the quantum repeater network since it determines the connectivity of the quantum network due to the irreplaceable role of EPRs in quantum teleportation.
Many entanglement distribution protocols can be used to establish EPRs between node and quantum repeaters. Here, we use the idea of quantum GilbertVarshamov bound (Keqin and Zhi, 2004) for low noise quantum channel. We have the following robust entanglement distribution protocol, where is the binary entropy function and is a noise parameter.
After obtaining these shortrange EPRs, teleportation is used to establish the EPRs between neighbor nodes. Hosts and routers would need to record their positions of the EPRs and the destinations that the EPRs connected.
The entanglement is a perishable resource in the sense that the entanglement among entangled parties is progressively lost over time. Nodes of the internet need to refresh their EPRs constantly. To do so, the time that the EPRs are created is also needed to be recorded.
It would be convenient if each node in the network is quantum connected to all his neighbors, each with a considerable number of EPRs enough for transmitting a UDP packet or qTCP packet, in its free time. This is just in case of potential request quantum communication to some neighbor.
When a node is working on qUDP or qTCP transformations, he would need to dynamically change its quantum connections by updating the information of EPRs.
7. Plain quantum network layers
In this section, we outline the quantum network protocols for the plain quantum Network Layers. In this model, the communication links between hosts and routers are quantum channels. The transmitted information flow through the network is quantum.
As the progress of quantum information transformation technology, the quantum state distribution will be more and more accurate between neighboring nodes of the network. This would be more suitable for modeling such quantum network. This model is more clear as its quantum communication links are fixed. Comparing with quantum repeater network, the behaviour of the layers in this model is closer to that of the classical internet layers.
The Application Layer is the same as that of the quantum repeater network as discussed in Subsubsection 6.1.
7.1. Transport Layer
The qUDP and qTCP for plain quantum network is different from those of quantum repeater network in the following sense.
In this model, the data packet is a classical packet head along with several qubits as quantum packet. As the two components of a quantum packet, they always arrive at the next node simultaneously. The classical packet head of qUDP (qTCP) is exactly the same as that of UDP (TCP), where the length is given by the number of qubits. The data part contains the qubits.

The quantum information transformation is done by direct transfer via a qUDP or qTCP packet rather than transmitting classical Pauli correction bits together with EPRs, including the threeway quantum handshake. Therefore this qUDP or qTCP header has no information about the data.

In the quantum repeater network, there are one round quantum checksum and one round of classical checksum. The classical checksum is verified by router and renewed. The quantum checksum would only be verified by the receiver.
In the plain quantum network, there are two rounds quantum checksum, which would be only checked by the receiver, not the router.
7.2. Network Layer
The Network Layer is simpler than the quantum repeater network case. Internet Protocol (IP) can be employed for addressing host interfaces based on the classical head information of the qUDP or qTCP packet. As we mentioned at the end of Section 3, the routers in this model use storeandforward transmission, but not checking for the validity of data packet.
7.3. Network Access Layer
The Network Access Layer of plain quantum network provides the services of quantum CRC checking, as we introduced in the quantum error detection of Section 6.
8. Conclusions and Futurework
We have discussed the interconnection of packet quantum network intercommunication for quantum repeater network and plain quantum network. In particular, we have described quantum User Datagram Protocols which allow connectionless communication model with a minimum of protocol mechanism and quantum Transmission Control Protocols which provide reliable quantum packet communication, respectively.
The next important step is to studying techniques for congestion avoidance and control of quantum network protocols. In classical network, packet switching introduces new complexities, since the packets must be reordered and reassembled at the destination. Also, since there are no dedicated circuits, the network links can become congested, potentially resulting in lost packets. Quantum effects bring new challenges in developing congestion control algorithms for qTCP.
Another interesting project is to produce a detailed specification of these protocols and implement a prototype so that some initial simulation, and in the future some experiments, with it can be performed.
9. Acknowledgement
This work does not raise any ethical issues.
References
 (1)
 Astner et al. (2018) Thomas Astner, Johannes Gugler, Andreas Angerer, Sebastian Wald, Stefan Putz, Norbert Mauser, Michael Trupke, Hitoshi Sumiya, Shinobu Onoda, Junichi Isoya, Jorg Schmiedmayer, Peter Mohn, and Johannes Majer. 2018. Solidstate Electron Spin Lifetime Limited by Phononic Vacuum Modes. Nature Materials 17 (2018), 313–317.
 Bennett and Brassard (1984) Charles Bennett and Gilles Brassard. 1984. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing. 175.
 Bennett et al. (1992) Charles H. Bennett, François Bessette, Gilles Brassard, Louis Salvail, and John Smolin. 1992. Experimental Quantum Cryptography. Journal of Cryptology 5, 1 (01 Jan 1992), 3–28.
 Bennett et al. (1993) Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. 1993. Teleporting an unknown Quantum State via dual Classical and EinsteinPodolskyRosen Channels. Physical Review Letters 70 (Mar 1993), 1895–1899. Issue 13.
 Bennett et al. (1994) Charles H Bennett, Gilles Brassard, Richard Jozsa, Dominic Mayers, Asher Peres, Benjamin Schumacher, and William K Wootters. 1994. Reduction of Quantum Entropy by Reversible Extraction of Classical Information. Journal of Modern Optics 41, 12 (1994), 2307–2314.
 Brakerski et al. (2018) Zvika Brakerski, Paul Christiano, Urmila Mahadev, Umesh V. Vazirani, and Thomas Vidick. 2018. A Cryptographic Test of Quantumness and Certifiable Randomness from a Single Quantum Device. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 79, 2018 (FOCS ’18). 320–331.
 Broadbent et al. (2009) A. Broadbent, J. Fitzsimons, and E. Kashefi. 2009. Universal Blind Quantum Computation. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS ’09). 517–526.
 Cacciapuoti et al. (2018) Angela Sara Cacciapuoti, Marcello Caleffi, Francesco Tafuri, Francesco Saverio Cataliotti, Stefano Gherardini, and Giuseppe Bianchi. 2018. Quantum Internet: Networking Challenges in Distributed Quantum Computing. arXiv:1810.08421y (2018).
 Caleffi et al. (2018) Marcello Caleffi, Angela Sara Cacciapuoti, and Giuseppe Bianchi. 2018. Quantum Internet: From Communication to Distributed Computing!. In Proceedings of the 5th ACM International Conference on Nanoscale Computing and Communication (NANOCOM ’18). Article 3, 4 pages.
 Cleve and Buhrman (1997) Richard Cleve and Harry Buhrman. 1997. Substituting Quantum Entanglement for Communication. Physical Review A 56 (Aug 1997), 1201–1204. Issue 2.
 Cleve et al. (1999) Richard Cleve, Daniel Gottesman, and HoiKwong Lo. 1999. How to Share a Quantum Secret. Physical Review Letters 83 (Jul 1999), 648–651. Issue 3.
 Dieks (1982) D. Dieks. 1982. Communication by EPR devices. Physics Letters A 92, 6 (1982), 271 – 272.
 Einstein et al. (1935) A. Einstein, B. Podolsky, and N. Rosen. 1935. Can QuantumMechanical Description of Physical Reality Be Considered Complete? Physical Review 47 (May 1935), 777–780. Issue 10.
 Elliott et al. (2003) Chip Elliott, David Pearson, and Gregory Troxel. 2003. Quantum Cryptography in Practice. In Proceedings of the 2003 Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications (SIGCOMM ’03). 227–238.
 Gottesman et al. (2012) Daniel Gottesman, Thomas Jennewein, and Sarah Croke. 2012. LongerBaseline Telescopes Using Quantum Repeaters. Physical Review Letters 109 (Aug 2012), 070503. Issue 7.

Grover (1996)
Lov K. Grover.
1996.
A Fast Quantum Mechanical Algorithm for Database
Search. In
Proceedings of the Twentyeighth Annual ACM Symposium on Theory of Computing
(STOC ’96). 212–219.  Humphreys et al. (2018) Peter Humphreys, Norbert Kalb, Jaco Morits, Raymond Schouten, Raymond Vermeulen, Daniel Twitchen, Matthew Markham, and Ronald Hanson. 2018. Deterministic Delivery of Remote Entanglement on a Quantum Network. Nature 299 (2018), 268–273.
 Jiang et al. (2009) Liang Jiang, J. M. Taylor, Kae Nemoto, W. J. Munro, Rodney Van Meter, and M. D. Lukin. 2009. Quantum Repeater with Encoding. Physical Review A 79 (Mar 2009), 032325. Issue 3.
 Keqin and Zhi (2004) Feng Keqin and Ma Zhi. 2004. A finite GilbertVarshamov Bound for Pure Stabilizer Quantum Codes. IEEE Transactions on Information Theory 50, 12 (2004), 3323–3325.
 Kimble (2008) H. Jeff Kimble. 2008. The Quantum Internet. Nature 453 (2008), 1023–1030.
 Lloyd et al. (2004) Seth Lloyd, Jeffrey H. Shapiro, Franco N. C. Wong, Prem Kumar, Selim M. Shahriar, and Horace P. Yuen. 2004. Infrastructure for the Quantum Internet. ACM SIGCOMM Computer Communication Review 34, 5 (Oct. 2004), 9–20.
 MacFarlane et al. (2003) Alistair MacFarlane, Jonathan P. Dowling, and Gerard J. Milburn. 2003. Quantum Technology: the Second Quantum Revolution. Philosophical Transactions of the Royal Society A 361 (2003). Issue 1809.
 Mahadev (2018) Urmila Mahadev. 2018. Classical Verification of Quantum Computations. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 79, 2018 (FOCS ’18). 259–267.
 Meter and Touch (2013) R. V. Meter and J. Touch. 2013. Designing Quantum Repeater Networks. IEEE Communications Magazine 51, 8 (2013), 64–71.
 Nielsen and Chuang (2011) Michael A. Nielsen and Isaac L. Chuang. 2011. Quantum Computation and Quantum Information: 10th Anniversary Edition (10th ed.). Cambridge University Press, New York, NY, USA.
 Pirandola and L. Braunstein (2008) Stefano Pirandola and Samuel L. Braunstein. 2008. Physics: Unite to build a quantum Internet. Nature 453 (2008), 1023–1030. Issue 3.
 Preskill (2018) John Preskill. 2018. Quantum Computing in the NISQ era and beyond. Quantum 2 (2018), 79.
 Raz (1999) Ran Raz. 1999. Exponential Separation of Quantum and Classical Communication Complexity. In Proceedings of the 1999 ACM 31st Annual Symposium on Theory of Computing (STOC ’99). ACM, 358–367.
 Regev and Klartag (2011) Oded Regev and Boáz Klartag. 2011. Quantum oneway Communication can be Exponentially Stronger than Classical Communication. In Proceedings of the 2011 ACM 43rd Annual Symposium on Theory of Computing (STOC ’11). ACM, 31–40.
 Reiserer and Rempe (2015) Andreas Reiserer and Gerhard Rempe. 2015. Cavitybased Quantum Networks with Single Atoms and Optical Photons. Reviews of Modern Physics 87 (Dec 2015), 1379–1418. Issue 4.
 Sangouard et al. (2011) Nicolas Sangouard, Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin. 2011. Quantum Repeaters based on Atomic Ensembles and Linear Optics. Reviews of Modern Physics 83 (Mar 2011), 33–80. Issue 1.
 Shor (1995) Peter W. Shor. 1995. Scheme for Reducing Decoherence in Quantum Computer Memory. Physical Review A 52 (Oct 1995), R2493–R2496. Issue 4.
 Shor (1997) Peter W. Shor. 1997. PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM J. Comput. 26, 5 (Oct. 1997), 1484–1509.
 Simon et al. (2007) Christoph Simon, Hugues de Riedmatten, Mikael Afzelius, Nicolas Sangouard, Hugo Zbinden, and Nicolas Gisin. 2007. Quantum Repeaters with Photon Pair Sources and Multimode Memories. Physical Review Letters 98 (May 2007), 190503. Issue 19.
 Vinton and Robert (1974) Cerf Vinton and Kahn Robert. 1974. A Protocol for Packet Network Intercommunication. IEEE Transactions on Communications 22, 5 (1974), 637–648.
 Wehner et al. (2018) Stephanie Wehner, David Elkouss, and Ronald Hanson. 2018. Quantum Internet: A Vision for the Road ahead. Science 362, 6412 (2018).
 William et al. (2015) Munro William, Azuma Koji, Tamaki Kiyoshi, and Nemoto Kae. 2015. Inside Quantum Repeaters. IEEE Journal of Selected Topics in Quantum Electronics 21, 3 (2015), 78–90.
 Wootters and Zurek (1982) William Wootters and Wojciech Zurek. 1982. A Single Quantum Cannot be Cloned. Nature 299 (1982), 802–803.
 Yao (1993) Andrew ChiChih Yao. 1993. Quantum Circuit Complexity. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science (FOCS ’93). 352–361.
 Yin et al. (2017) Juan Yin, Yuan Cao, YuHuai Li, ShengKai Liao, Liang Zhang, JiGang Ren, WenQi Cai, WeiYue Liu, Bo Li, Hui Dai, GuangBing Li, QiMing Lu, YunHong Gong, Yu Xu, ShuangLin Li, FengZhi Li, YaYun Yin, ZiQing Jiang, Ming Li, JianJun Jia, Ge Ren, Dong He, YiLin Zhou, XiaoXiang Zhang, Na Wang, Xiang Chang, ZhenCai Zhu, NaiLe Liu, YuAo Chen, ChaoYang Lu, Rong Shu, ChengZhi Peng, JianYu Wang, and JianWei Pan. 2017. Satellitebased Entanglement Distribution over 1200 Kilometers. Science 356, 6343 (2017), 1140–1144. https://doi.org/10.1126/science.aan3211
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