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HE Q nology UANTUM L EAPS ]. [9]. ese per ful 6]. bit m rs al e e e its own strengths and weaknesses [24]. In general, the qubits B. Quantum Networks based on the trapped ion are stable with slow operation, the superconducting qubits work fast with quick decoherence, the The quantum Internet [18] is envisioned to enhance the diamond qubits can operate at room temperature with the classical Internet by enabling quantum communication among difficulty of entanglement generation, and the photonic qubits arbitrary network users [19]. There are three elemental quan- are useful for long-distance communication with probabilistic tum hardware components that make up the quantum Inter- teleportation [25]. However, these characteristics are not defi- net [19], i.e., the quantum channel, quantum repeater, and nite, for example, a new diamond defect is found to have both end nodes. Specifically, the quantum channel is the physical long coherence time and excellent optical properties [26]. medium to transmit the qubits, the quantum repeater is used to In mathematics, topology concerns the properties of spaces enable the quantum networking of arbitrary distances, and the that are invariant under smooth continuous deformations [27]. end nodes are used for information processing, e.g., quantum For half a century, topology has been broadly applied to computers. The quantum repeater establishes the connectivity physics, e.g., the topological matter [28]. In topological matter, through entanglement swapping [44] and then the information a collective particles can behave like an elementary particle, is transmitted through quantum teleportation [45], which is i.e., the quasiparticle, and the quantum states of multiple quasi- envisioned to be the quantum feature of the future Internet particles form the topological qubit. The topological qubits [45]. A review of quantum teleportation is available in [25], are resilient to the outside interference due to the topological e.g., the deterministic and probabilistic teleportation, the active degeneracy of the quasiparticles that are neither bosons nor and passive teleportation. In addition, the quantum networks fermions but are anyons [29]. Specifically, it can be achieved need the quantum memory [46] for the storage and process by either electron fractionalization or ground state degeneracy, of the quantum information [47]. It is depicted that there i.e., storing the quantum information at two distinct places. will be six implementation stages of quantum Internet [19], For example, the topological qubits are realized based on ranging from the trusted repeater networks, which is the the fractional quantum Hall states [30] and the Majorana current implementation status, through the quantum memory zero modes [31] [32]. The topological qubits are coming networks, to the ultimate quantum computing networks. into limelight with the potential for fault-tolerant quantum The optical light has the advantage of long-distance com- computing [33] [29] [34] [23]. munication [47] and photonic channels are usually used to In addition, an interesting realization of the qubits is based establish the quantum links between the quantum repeaters on the Schr¨odinger’s cat states [35]. The cat state means and between the end nodes [19], e.g., the free-space channels the quantum superposition of the classically distinct states and the fiber-based channels. The quantum information can [36], i.e., the Schr¨odinger’s cat is alive and dead at the same be encoded by two types of observables [48], i.e., the dis- time [37]. In an experiment [38], a wave packet of light crete variables, e.g., the polarization state, and the continuous composed of hundreds of particles is placed in two microwave variables, e.g., the intensity and phase of the electric field of cavities bridged by a superconducting artificial atom, which the electromagnetic wave. Contrast to the light-based quantum can be seen either as a cat state with two spatial modes communication, the quantum memory and the quantum com- or as an entangled pair of cat states, and it demonstrates puter are usually matter based, e.g., the solid-state quantum that the Schr¨odinger’s cat lives or dies in the two cavities memories, and a quantum interface is required to convert simultaneously, which is a manifestation of mesoscopic su- the light-based quantum states to the matter-based quantum perposition and entanglement constructed from quasiclassical states and vice versa [18] [47]. It is proposed that the hybrid states and can be used to store the quantum information with approaches are needed to combine the features of both the redundancy [38] [39]. In another experiment, it shows that the discrete-variable and the continuous-variable technologies and Schr¨odinger’s cat states can be deterministically created by to integrate the light-based communication and the matter- using a single trapped atom in a cavity to control the quantum based storage and processing for a scalable quantum Internet states of the reflected light pulse [40] and it is envisioned that [47]. In addition, the integration of the quantum networks and the deterministic cat states can be useful for short- and mid- the classical networks is discussed in [49]. distance quantum communication due to the characteristics of The quantum computers, which are not mandatory for some loss correction [41]. quantum network protocols [19], are the essential components The qubit is the basic element in quantum information of the quantum computing networks, where the quantum processing [42], which encompasses quantum computation, computers can arbitrarily exchange information through the quantum cryptography, quantum communication, quantum quantum network. On the other hand, it is an intriguing way simulation, and so on. For sake of stability, fault tolerance, and to build the quantum computers by connecting a few quantum scalability, quantum error correction [23] is proposed based on systems together [50]. the logical qubits, each of which is emulated by a number of 1) Quantum Computers: There are three types of quantum physical qubits. It is depicted that the development of quantum computers [23], i.e., the analog quantum computers, the digital information processing includes seven stages [43], from the noisy intermediate-scale quantum computers, and the fully operation on physical qubits through the operation on logical error-corrected quantum computers, where the first type of qubits to the fault-tolerant quantum computing. quantum computers are based on the analog methods, e.g., quantum annealing and quantum simulation, and the last two A. Beyond the Analysis of Classical Networks types of quantum computers are based on quantum logic The classical methodology for network performance anal- gates. Since the analog quantum computers can be built ysis is queueing theory, which was initiated for teletraffic by controlling the Hamiltonian of the quantum system in a analysis in telecommunication networks by Agner Krarup straightforward way without full error-correction, it has an Erlang in 1909 [67] and was revived to analyze computer advantage with respect to the digital quantum computers to networks by Leonard Kleinrock in the 1960s [68]. We regard solve practical problems in the near term and is deemed to the queueing analysis as the canonical approach because of its become obsolete due to the control difficulty and be surpassed pervasiveness and irreplaceability in networking. The queueing by the digital quantum computers in the long term [51] [23]. theory deals with the accumulation of the interarrival time and The practical utility, quantum supremacy, and fault-tolerant service time in time unit [69] or the accumulation of arrival computing can be seen as the milestones of the quantum quantity and service quantity in bit unit [66], based on which computer development, and the accumulation of the physical the performance measures are defined, e.g., backlog as the and logical qubits in the quantum systems can serve as the accumulated workload in the queue or the whole system, delay basis to measure the development progress [23], e.g., the as the waiting time or sojourn time with respect to different quantum volume [52]. scheduling schemes, and throughput as the traffic amount pass- It is anticipated that the future quantum computer would ing through the queueing system. A communication network is have a hybrid architecture [24], with the superconducting seen as a network of queues, due to the irregular randomness qubits running algorithms, the trapped ion qubits forming of the queue output, it is difficult to analyze the queueing memory, and the photonic qubits communicating signals. The networks and the queueing analysis based on the accumulation topological methods have the advantage of achieving the of bits usually has an advantage over the analysis based on the logical qubits with far less physical qubits [23]. In addition, accumulation of time. the classical computers are needed to control the quantum Since the system theory is a mathematical tool to facilitate operations and to implement the computations for the quantum the network design and deployment in real world, the funda- error correction [53] [23]. To enable transparent application mental challenge of formulating a system theory lies in what development and network management, the quantum networks form the network is and will be of, or what form of network the need the quantum software to allow the quantum algorithms theory indicates to build with a better performance in terms of and quantum protocols to connect to the quantum hardware some performance measures. If we build the quantum internet [54] [55] [19]. The quantum software are layers of software as an alternative or augmentation to the classical internet tools, for example, the quantum programming languages and architecture, i.e., the quantum internet serves to connect the compilers for quantum application development [56], and quantum computing devices here and there for communication, the quantum error-correcting code for fault tolerant quantum which is exactly the initial motivation of the classical internet computation [57]. It is depicted that the quantum algorithms [70], a few research issues should be addressed, considering will feature hybrid approaches [54] to combine both the clas- the quantum network uniqueness. sical and the quantum processors for the noisy intermediate- scale quantum computers [51], which are different from the B. Quantum Network Characteristics quantum algorithms for the noiseless and large-scale quantum computers [58], and the quantum network software stack will The quantum Internet is not only a hybrid architecture in be a synergy of the quantum computing stack and the classical terms of the diverse implementations, but also a hybrid system network stack [19]. by the complex nature of the quantum physics. Albeit the passion in quantum computer [59] [60] [61], there 1) Quantum diversity. Compared to the classical communi- are pessimistic voices against it, arguing that a quantum com- cation, the quantum communication has additional physical puter needs to control an astronomical amount of continuous resources, i.e., the superposition and entanglement, and the parameters with high precision that is impossible [62] and that quantum concepts usually have a manifold characteristic. From the quantum systems are inherently noisy [63], and there are the information-theoretic perspective, the quantum channel ca- no definite answers on when the useful quantum computers pacity has many concepts [22] [71], i.e., the classical capacity, appear [23] [64] [65]. quantum capacity, private capacity, entanglement-assisted ca- pacity, etc. In addition, it shows that [72] the ability to combine III. OPPORTUNITIES IN PERFORMANCE ANALYSIS quantum channels in a superposition of orders can boost the Technically, the development of quantum Internet requires rate of communication beyond the limits of conventional quan- a hybrid of technologies, which combine the features of both tum Shannon theory, which is due to the quantum causality discrete variable systems and continuous variable systems [47]. with indefinite causal order [73] [74] [75]. These different Theoretically, it is necessary to formulate a system theory capacity concepts describe different facets or capabilities of [66] for the dimension of the network dynamics, i.e., backlog, the quantum internet, which means that the quantum internet delay, and throughput, to deal with the quality-of-service can be used for different purposes with different protocols requirements of the network applications and to help design or technology supports, e.g., the transformation of classical and deploy the quantum networks. information, quantum information, or private information. On the other hand, the information can be either classical bits IV. CONCLUSION or quantum bits, for example, the information coming from The quantum physics is a growing theory still without the classical source can be classical, the information coming a definite answer to the nature of reality and the quan- from the quantum source can be quantum. If such information tum engineering is an expanding field with more and more is to be transmitted through the quantum internet, through advanced quantum technologies. No mater how much the quantum states, the classical information should be coded into quantum Internet mimics the classical Internet in terms of the quantum states, while the quantum information may need the architectures and applications at present, it is hard to to be converted from one type of quantum system state to deny the possibility that the future quantum Internet will be another quantum system state for transmission, e.g., the light- fundamentally different from the present design. Considering matter transformation [18] [47]. The diversity of the quantum the diverse realizations of the quantum bits, quantum devices, concepts implies the diversity of the quantum networks, e.g., quantum networks, and quantum software, we envision an the networking analysis should consider the multiplexing of inclusive framework for the performance analysis of quantum the classical information and the quantum information in a networks with a varied choice of methodologies to address the heterogeneous network of diverse quantum channels, e.g., the different implementations. communication links and storage. 2) Information additivity and causality. The performance anal- REFERENCES ysis considers the information quantity, i.e., the amount of [1] D. Kleppner and R. Jackiw, “One hundred years of quantum physics,” storage space [76], rather than the information content. The Science, vol. 289, no. 5481, pp. 893–898, 2000. information transmission follows the causality principle [77] [2] A. 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