Networked Quantum Services†

The novel contributions of our review article are as follows.

We provide a comprehensive review of networked quantum services, with a detailed overview of the different technologies, platforms, and applications.

The structure of this survey is as follows. Section 2 summarizes the related surveys. In Section 3, we comprehensively review the fundamental methods and recent research on networked quantum services. We study the networked quantum devices, the architecture of networked quantum computers, and networked quantum machine learning. Section 4 focuses on the quantum software of networked quantum services, the quantum programming languages, quantum application programming interfaces (APIs), and standardization approaches. Section 5 reviews the implementation basis. Finally, Section 6 concludes the survey, and outlines open problems and future research directions.

For a comparative analysis, the primary scope of the related works is summarized in Table 1.

The various technologies [10,76,77] that are available for the implementation of quantum computations (such as photonic [78], superconducting [79,80], silicon-spins [81], trapped neutral atom [82], giant atoms [83], trapped ion [84], topological [85], nitrogen-vacancy [86], nuclear magnetic resonance-based [87,88] implementations) require a generalized framework and guidelines for the development of hardware for quantum computation. The aim of the layer architecture is to serve as an abstract hardware-independent standard for scalable fault-tolerant quantum hardware [89].

The layer architecture of a general (gate-model) quantum device is as follows:

Physical Layer : a hardware layer for physical quantum states, physical operations, quantum hardware, and physical environment.

Similar to the traditional OSI (Open Systems Interconnection) layer architecture, a layer architecture can also be defined for distributed quantum computation and networking with the aim of a hardware-independent standardization framework [32–34,90,91].

The general layer architecture of distributed quantum computation consists of the following layers:

Physical Layer : a hardware layer for physical mechanisms to connect distant nodes (quantum entanglement generation and distribution, entanglement swapping, entanglement purification, quantum teleportation, specific operations [91]).

The layer structure of a quantum computer (gate-model quantum device) and the layered model for a distributed quantum computation are shown in Figure 2.

Figure 2.

Layer architectures. (a) Layer architecture of a quantum computer (general gate-model quantum device). (b) A general layer architecture for distributed quantum computation.

While the layer structure can referred to as a general layer architecture, we note that other layer approaches are also available for distributed quantum computation [32,41,45,70,92]. The main attributes of the different approaches to layer architectures are summarized in Table 3.

Quantum computing utilizes the fundamental concepts of quantum mechanics for computing, such as superposition, entanglement, and interference [80,98–102]. As the development of quantum computers evolve [13,80,98,99,101,103–119], the relevance of quantum computations has become more interpretable for problem-solving tasks [80,120–123]. Quantum computers could definitely help solve difficult computational problems [12,98,124–130].

As a first step, Shor’s factorization algorithm [131] has been implemented via a small NMR (nuclear magnetic resonance) device in 2001 [87]. In 2010, the quantum annealing computers have been announced by D-Wave,² with limited application possibilities. The quantum computing research at Google has been started in 2013.³ In 2016, they developed a simulated annealing algorithm [132] and also simulated the energy levels of the hydrogen molecule [133]. In 2019, Google announced a 53-qubit gate-model superconducting quantum computer [80], which, in theory, can solve optimization problems in only 3 min that would take a classical supercomputer 10,000 years. A theoretical framework of the experiments of Google’s 53-qubit quantum computer has been defined in [100]. Quantum computing research at IBM was started in 2016⁴ via the IBM quantum experience program. IBM has made it possible for users to utilize the IBM Quantum Cloud Services⁵ to run algorithms on the IBM quantum processor from the users’ computers. Between 2016 and 2019, IBM developed 5-qubit and 27-qubit quantum processors, while in 2020, they announced the 65-qubit IBM Quantum Hummingbird processor, a 127-qubit processor in 2021, and a 433-qubit processor in 2022. The 1121-qubit processor has been released in 2023.⁶ In 2020, Amazon launched the Amazon Braket Cloud Quantum Computing Service⁷ to provide quantum-based web services.

In the NISQ era [98,130], gate-model quantum computers have particular relevance since these architectures can be implemented in near-term settings [13,80,98–101,103–111,113–119,134–137]. In a gate-model quantum computer, the computational steps are realized via unitary gates. The gates are associated with a gate parameter value, while the computational problem fed into the quantum computer identifies an objective function [101,110–112,136,137]. For objective function examples, see [101,103,106,107,136,137]. The aim of problem solving is to maximize the objective function value via several iteration steps. Each iteration step includes the application of unitary gates, as well as a measurement of the resulting quantum states. From the measurement results, an averaged value can be determined to estimate the actual objective function value [101]. Many hybrid variational quantum algorithms also use gate-model circuits and classical objective functions [138–142], similar to the quantum approximate optimization algorithm (QAOA) [101,110].

The NISQ quantum devices are noisy, the decoherence and the imperfections of gates and measurements could easily destroy any possible quantum speedup on these devices. The lack of error-correction prevents scaling. Faulttolerance requires precise physical control over the qubits (two-qubit gates with 99.99% fidelity) [8,98]. Another crucial problem of quantum computing is scalability. As the number of qubits and the number of quantum gates increase, so do the errors and noise. Networked quantum computers allow us to overcome the limitations of the current NISQ quantum hardware. In a networked setting, the quantum computations are realized by clusters of quantum nodes in a distributed manner. The aim of a distributed quantum computer architecture is to use multiple smaller quantum computers (equipped with a small number of qubits and quantum gates) instead of a single large quantum computer to perform the same computational task. The total number of qubits and gates of the networked architecture can exceed the number of qubits and gates of the large quantum computer, allowing efficient scalability with low error rates and noise levels. The interconnection between the quantum nodes requires control mechanisms in the hardware and software layers.

In a networked quantum computer architecture, the remote quantum processing units (QPUs) use classical and quantum channels to communicate with the classical network and quantum network. The quantum channel can be an optical link, free space optical channel, or other physical implementation (see also Section 5). In a large quantum computer, the levels of error and noise are high, and the coherence times are limited. In a distributed architecture, the QPUs are equipped with significantly fewer qubits and quantum gates, which reduces the errors and noise, and overcomes the problem of limited coherence times. It allows using simplified, low-complexity error-correction techniques in the nodes, and decreases the noise that arises from the imperfections of the quantum gates. Both attributes are crucial in an NISQ setting, where the quantum hardware and quantum resources are limited.

We discuss networked quantum computers using the following categorization:

Multichip : an input computational problem is divided into several smaller quantum circuits for parallel processing. The quantum circuits can work independently without the need for direct communication between them. The smaller quantum circuits fit in one QPU, no circuit decomposition is needed. The quantum communication between the quantum circuits is implementation-specific, and classical communication is always available. If only classical communication is available, it is also called embarrassing parallelism [32] (Table 4).

The distributed architecture of networked quantum services allows the upscaling of the computational power of smaller quantum devices to a high computational capacity. In the multichip approach, several smaller quantum chips are used in parallel to realize a scalable quantum system. In the circuit distribution and splitting methods, a large quantum circuit of a large quantum computer is split into several smaller subcircuits and implemented in the distant quantum nodes. The splitting of the QC can be made in different ways: horizontally or vertically, in general. Note, circuit splitting can refer to gate-cutting [143] or wire-cutting [144], both can be shown to be equivalent [145]. The main attributes of the multichip approach with the related works are summarized in Table 4

The details of the circuit distribution method with the related works are given in Table 5

Relevant attributes of the circuit splitting method with the related works are included in Table 6

The multichip, circuit distribution and circuit splitting architectures are depicted in Figure 3

Figure 3.

Distributed quantum computer architectures. (a) Multichip: the n smaller quantum circuits (QC) fit in a quantum processing unit (QPU), quantum communication is implementation-specific, classical communication is always available via Classical Network. (b) Circuit distribution: a large quantum circuit (QC) is distributed among the n QPUs such that each node realizes a smaller subcircuit of it with relation ∑i=1nwiQCi=QCwhere QC_i is the local circuit of the i -th QPU, while w_i is a real weighting coefficient. Quantum communication is available (via Quantum Network), classical communication is available (via Classical Network). (c) Circuit splitting approach: a large quantum circuit (QC) is split among the n QPUs, such that each node realizes a smaller subcircuit with relation ∑i=1nwiQCi=QCno quantum communication is available, classical communication is available.

A distributed CNOT gate between two distant QPUs is shown in Figure 4

Figure 4.

A distributed CNOT gate realized by two QPUs between distant quantum states |ψ ₁〉 and |ψ ₂〉. Local system |ψ ₁〉 is at QPU 1, while |ψ ₂〉 is at QPU 2. QPU 1 has the quantum circuit QC ₁(dashed line frame) and QPU 2 has QC ₂(dashed line frame). The quantum nodes also share a Bell state |β00〉=12(|00〉+|11〉)The QPUs use their local quantum circuits and classical communication (doubled lines) to realize the CNOT gate between |ψ ₁〉 and |ψ ₂〉 in a distributed manner. The greenshaded box is the cat-entangler sequence [192], and the yellow-shaded box is the cat-disentangler sequence. Related implementations: [209–213].

Recent implementations of multichip and circuit splitting approaches are given in Table 7

In this section, we are focusing on distributed quantum neural networks (QNN). Quantum neural networks are also referred to as parameterized quantum circuits (PQC) or variational quantum circuits (VQC) [130,140,231–237].

The most promising approach for quantum machine learning in the NISQ setting is to use variational quantum algorithms (VQAs) [103,129,238]: this includes classical hardware to overcome the limitations of the current quantum hardware. VQAs are useful for machine learning problems [103,129,238] and combinatorial optimization [239,240]. A VQA consists of a feature map, an architecture (ansatz), loss function, and an optimization method [234,241]. These attributes make them a strong candidate for achieving a quantum advantage in the near term. The feature map encodes the classical input data into quantum states [234,242]. The algorithms use an optimizer on classical hardware to train a PQC, which is used to find the quantum state containing the solution to the problem [234,243]. However, the optimal choice of the optimizer, the structure of the PQC, and the hyperparameters are problem-specific and have a major impact on the performance of a VQA.

A schematic model of a VQA is depicted in Figure 5

Figure 5.

Schematic model of a VQA. The classical data is mapped onto qubits via feature map F (x), and fed into the parameterized quantum circuit U (θ) (PQC), where θ is the parameter of unitary U The output of U (θ) is measured via measurement M in the computational basis, which results in a classical bit string z ∈ {0, 1}ⁿ of length n This string is fed to a classical computer to evaluate a cost function C (z). Depending on the value of C (z), the parameter ¸ is updated via a classical optimizer and propagated back to U (θ). The circuit is re-run multiple times until a convergence or other stopping criterion is fulfilled.

In general, quantum circuits are sequences of unitaries each of which depends on a few parameters. Due to the lack of error correction in the NISQ era, the number of gates that can be applied is limited by the fidelities of the quantum gates, and the tolerable error is low, which is a challenge for the implementation of gate-model quantum neural networks in near-term devices [21,112,129,238]. Depending on whether classical or quantum data is used, and a classical or quantum algorithm, the quantum machine learning architectures can be classified into different groups [35,244,245]. These groups with their main attributes and the related works are summarized in Table 8

Major challenges in the implementation of VQAs are trainability, mainly by the occurrence of the barren plateau [234,262–264] and narrow gorge [265], efficiently calculable cost functions and expectation values, multiple local minima in the loss function [266], high precision measurements [234], and ensuring high accuracy on near-term quantum processors with high levels of noise and errors. The hyperparameters of VQAs are the configuration parameters, such as those for the optimizer or the PQC [234,243,267,268]. It may be assumed that a further quantum advantage can be reached in terms of space efficiency, performance, and expressiveness of learning models in the NISQ era. Translational and exploratory approaches are studied in [269]. Different types of quantum neural networks have been defined in the literature, such as orthogonal, convolutional, feed-forward, self-supervised quantum neural networks, variational depth quantum circuits, quantum Boltzmann machines, along with the applications of linear regressions or quantum amplitude estimation. For a comprehensive review of quantum machine learning applications, see [21].

In a distributed quantum neural network, the training is realized in a parallelized way between distant nodes. The parallelization can be made via data splitting (Table 9) or circuit splitting (Table 6). In data splitting approaches, the dataset is split across the quantum nodes such that the actual model is loaded into each node (term “model” refers to a PQC in this context). In circuit splitting, the model (a PQC) is split across the distant nodes such that each node has access to the full dataset.

The main attributes of data splitting approaches (encoding, data storage, and use of classical side information) in distributed quantum neural networks are summarized in Table 9

The data splitting and circuit splitting methods in a distributed quantum neural network setting are compared in Figure 6

Figure 6.

Data splitting and circuit splitting in distributed quantum neural networks. (a) Data splitting. The full dataset (Dataset) is split into parts, denoted by |ψ ₁〉,…, |Dⁿ 〉, across the n quantum nodes (QPUs), while satisfying ∑i−1n|Di〉=DatasetThe model (PQC) is loaded into each QPU. The QPUs process and train in parallel, each QPU has access to a classical network (Classical Network). (b) Circuit splitting. The input PQC is split into n parts, PQC ₁…,PQC_n and loaded into the n QPUs along with the full quantum dataset (Dataset); ∑i=1nciPQCi=PQCPQC_i where is the local PQC of the i -th QPU, while c_i is a real weighting coefficient. The QPUs process and train in parallel; each QPU has access only to a classical network in general.

While data splitting is a general strategy for data parallelization in quantum neural networks, other methods have also been investigated. Data reduction (or coreset technique) is a different method that can also be used in quantum neural networks to optimize the amount of data shared between the quantum nodes. In data reduction approaches, an approximation of the dataset is distributed between the nodes to improve the parallel processing and to decrease the size of the dataset. In parameter distribution, classical side information is distributed between the nodes in a parallelized way for optimization purposes. Table 10 summarizes the main attributes of data reduction and data distribution in quantum neural networks.

In [290], a modified distributed learning scheme called quantum distributed optimization (QUDIO) has been proposed for variational quantum algorithms. The model consists of a distributed optimization scheme whose back ends support both quantum devices and various quantum simulators to further improve the capabilities of the VQAs. The proposed method collaborates with multiple quantum machines or simulators to complete learning tasks. The model is depicted in Figure 7

Figure 7.

A QUDIO model for VQAs. The nodes are trained in parallel using a PQC or a quantum simulator (QSIM). A classical central server communicates with all local quantum nodes to synchronize the trainable parameters of the nodes.

In blind quantum computations [291] multiple users (untrusted devices) perform computations and analysis on the joint data without sharing or revealing their own data with each other. The calculations are hidden from the users and the encrypted input algorithm is received by a server from a trusted client. The data of the users are fed into local quantum devices and a distributed quantum computation is established between them via a particular distributed quantum algorithm [33,91]. Classical communication is generally allowed between the quantum devices. A particular user has no knowledge about the other users’ data, nor about the actual distributed quantum algorithm; however, some part of a hidden algorithm and the joint state can be learned by the parties, and the server can also have partial information about some parameters of the calculations. Due to these security issues, another approach for the realization of quantum blind computation is the use of a trusted secure quantum cloud [36,292,293]. If the users are connected via a secure quantum cloud, their local quantum devices cannot communicate with each other directly, only via the secure quantum cloud. On the experimental realization of blind quantum computing, see [49–51].

Quantum cloud services allow the users to run quantum algorithms on quantum computers. These cloud services can be accessed via classical computers allowing the users to have access to quantum devices and resources. Several different cloud-based platforms have been developed with different resource capabilities, quantum computing models, and supported software development kits [36,37,294–296]. In general, the different quantum cloud platforms provide commercial quantum computing as a main service. For a comprehensive review of quantum cloud computing, see [36].

The main attributes of recent quantum cloud platforms with the related works are summarized in Table 11 Superconducting technology: IBM Quantum⁸ Google Quantum⁹ Rigetti¹⁰ OCO: Oxford Quantum Circuits¹¹ QuTech¹² Trapped ion technology: IonQ¹³ Quantinuum¹⁴ Neutral atom technology: Pasqal¹⁵ QuEra¹⁶ Quantum annealing technology: D-Wave¹⁷ Solid-state spins technology: QuTech.

Remote quantum sensing uses the fundamentals of quantum mechanics for highly sensitive measurements in a distributed manner via an entangled network of distributed quantum sensors [65,66]. Applications of remote sensing include atomic quantum clocks, quantum sensors for geology, biomedical applications [317] and navigation, nanoscale vortex imaging using a cryogenic quantum magnetometer [318], quantum-enhanced accelerometers and gyroscopes, magnetic field detection, and gravitational wave detection.

Quantum interferometry can be also realized in a distributed way using multiple local quantum interferometers at distant quantum computers [33,319]. The local quantum computers process the results of the interferometers and use a distributed quantum algorithm to extract information to improve the accuracy and to decrease the noise level.

On the use of QAOA in the NISQ setting, see [101,110,320–322], on variational quantum eigensolvers, see [323–326], and on pseudorandom quantum circuit-based solutions for quantum advantage, see [327–329]. The complexity of NISQ has been derived in [330], while the structure of shallow quantum circuits has been studied in [75,331]. For details on the methods of loading classical information into quantum computers, see [332].

For the fundamental details of parameterized quantum circuits, we suggest [130,140,231–237]. On the interoperability of quantum neural networks and quantum computations, and on the attributes of the different training procedures of quantum neural networks, we suggest [149,251,333–336]. A quantum-classical hybrid algorithm for determining eigenstate energies in quantum chemistry is proposed in [337]. On resource-efficient quantum algorithms for quantum chemistry, see [338]. For an enhanced feature encoding and classification on distributed quantum hardware, see [286]. Machine learning algorithms for near-term quantum computers can also be found in [242,339–342]. For the different learning architectures of quantum neural networks, see [235,343–348]. For further works on the barren plateau phenomenon in quantum neural networks, see [233,263,264,349–351]. The implementation aspects of quantum neural networks on current quantum hardware architectures for different problems are studied in [231,352–359].

A quantum API is a software interface that allows interaction with quantum computers in applications. These APIs also connect the different programming languages and the quantum hardware. The quantum computing vendors provide various APIs (IBM Quantum Experience API, Amazon Braket API, D-Wave API, etc.), therefore a standardization of access to quantum services is an important problem regarding quantum APIs [37,399–401]. A solution to the standardization problem and for the optimization of accessing quantum services could be the use of quantum API gateways [399,402]. Quantum API gateways integrate the multiple quantum vendors and offer the best quantum hardware available for a particular task based on the input parameters received from the users’ applications.

A solution to the problem of service accessing the different platforms is an extension of the quantum API gateway, application of code generators, and a workflow to automate the development and deployment of the services [369,403]. It allows to deploy quantum code without specific knowledge of the quantum platforms, which can simplify the development of quantum applications and workflow integration.

An aim of the utilization of programming tools and middleware units is to allow access to quantum services using a standardized format for all quantum providers. A standard format for requesting services and handling service responses also allows developers to manage the communication workflow in an abstract-level, independent from the actual physical quantum hardware and the constraint parameters of the different quantum providers. An approach to standardize the process of defining quantum services using the OpenAPI specification has been proposed in [37]. The method can generate source code of quantum services from an API specification and a quantum circuit. For a detailed discussion on the subject of standardization, see the works of Moguel et al. [24,37].

The different standardization approaches of service access, service development, and programming are summarized in Table 14

Any two communicating nodes in the system are connected by a quantum channel, a communication medium that is influenced by nature in the form of noise [417]. The theoretical modeling of quantum channels has a long history. Many articles have related the theoretical maximum of the transmission capacity of the channels to their capacity; for understandable reasons, the capacity for classical information has primarily been considered, since in the initial point-to-point systems, the end users were human beings who could only interpret this type of information. However, even in this case, a very diverse range of questions arise; for example, how much classic information can be safely transferred over the channel, and by how much is the capacity increased if we share entangled pairs between the nodes before communication (i.e., when investigating entanglement-assisted capacity)?

The situation becomes completely different when networked quantum devices communicate with each other. This case involves the transmission of quantum information, which is much more sensitive to noise effects, meaning that error correction codes that enable noise control play an even more prominent role. Discussion of this topic is further complicated by the fact that the definitions of the various quantum capacities are not comprehensive, and in some cases there is even a lack of consensus on the quantity describing the information to be maximized.

The capacity of the quantum channel is a guiding quantity, as it maximizes the amount of information that can be transferred. In a practical implementation, error correction codes/algorithms compete for the best approach to capacity [418].

In a similar way to the planning and deployment of classical communication networks, it is also necessary in the quantum case to refine the theoretical channel models based on measurements, to bring them closer to reality.

According to current trends in quantum communication, there are two parallel directions for future development. Optical fiber-based systems will use existing classical optical communication links that are widely installed worldwide; however, these have a different effect on classical photon packets and the information encoded in the polarization of photons. In a real-world implementation, additional challenges arise if classical and quantum communication travels on the same optical fiber, as artificial interference also inevitably appears in addition to the noise caused by nature.

Quantum free space communication primarily involves the exchange of information with GEO/LEO (geostationary equatorial orbit/low Earth orbit) satellites. As the Earth’s atmosphere is relatively thin, its effect on photons is significantly weaker than that of optical fibers, meaning that longer distances can be bridged. This type of communication is also asymmetric, meaning that it is primarily suitable for communication in the satellite-to-Earth direction.

Finally, the role of quantum channel simulators should be emphasized. These are sophisticated tools that are designed to model and analyze the behavior of quantum channels while accounting for various physical effects such as noise, decoherence, and loss. Quantum channel simulators play a crucial role in understanding how quantum states evolve as they are transmitted through the channels, and can help in the development of quantum communication protocols, error correction techniques, and other quantum technologies.

When bridging large distances in classical networks, devices may be needed between the endpoints to mitigate losses and distortions in the communication channel. This task is performed by devices called repeaters, which amplify incoming weak signals and send them on. Amplification involves a kind of duplicating (copying) of the incoming weak signal. However, in quantum networks, the No-cloning theorem makes it impossible to copy any incoming arbitrary quantum state without error.

Quantum repeaters are essential components of quantum communication networks and are designed to extend the range of quantum communication by overcoming the limitations of direct transmission of quantum states [419]. They are used to facilitate long-distance quantum communication by addressing issues such as loss and decoherence, which occur in optical fibers or free-space communication.

Quantum long-distance communication uses a combination of two protocols in order to circumvent the No-cloning theorem. Firstly, using entanglement swapping, entangled pairs are created between endpoints located at great distances from each other. Following this, they undergo teleportation, in which quantum states can be transmitted between the parties by communication over a classical channel. The quantum repeaters play an active role in the first phase, as they create entangled pairs between themselves and then perform an operation (Bell measurement) in a synchronized manner, which creates an entangled pair between the endpoints. From the point of view of implementation, the repeater function relies on an extremely precise classical synchronization mechanism between the endpoints. In addition, the resulting entangled quantum states must be preserved for a sufficient time so that they are available for teleportation; in other words, the decoherence time of the quantum memory function must also be sufficiently long.

Although quantum processors, or computers that have been proven to obey quantum phenomena, have been appearing regularly since 2018 and their physical numbers of qubits are increasing exponentially (2ⁿ) every year, meaning that the state space of manageable problems is growing super-exponentially (2²ⁿ), there is still competition between different technologies.

Quantum computers can be categorized into several types based on their underlying technology and operational principles, as follows:

Superconducting Qubit Computers: These quantum computers use superconducting circuits to create qubits. Superconducting qubits are among the most widely researched and developed types, with companies like IBM and Google leading the way. They operate at extremely low temperatures to maintain quantum coherence.

The wide variety of quantum computers that will be available in the future may limit their networked operation, since only machines in which there is an interface between the qubits performing the calculations and the photons carrying the information will be able to be connected to the network. Another requirement is that quantum computers must be able to preserve quantum stored information for at least as long as the period over which communication between the machines takes place. That is, the coherence time of quantum memories also restricts the range of machines that are capable of cooperation. Here, we draw attention to the possibility that the information storage qubits of quantum memories may be based on a different physical medium. In other words, a convention may be necessary between the qubits that perform calculations, store information, and deliver information; these processes suffer from noise, making the role of the error correction codes even more valuable.

Based on their important roles in quantum communication, we can identify the most promising physical implementations of quantum memory as follows [420]:

Atomic Ensembles: These systems use collections of atoms or ions to store quantum information. The quantum states of the atoms can be manipulated using light fields, which allows for the storage and retrieval of quantum states. Techniques such as electromagnetically induced transparency (EIT) are often employed in these systems.

In summary, it can be concluded that all quantum devices intended for use in network operation must meet three requirements: precise synchronization efficient error-correction coding and memory with adequate coherence time These functions do not necessarily require different physical implementations, and combining them can increase efficiency.

In networked quantum services, a particular quantum state can be affected by noise from various sources such as quantum gate errors, channel losses, and environment noise. The fidelity of the state, 0 ≤ F ≤ 1, can be used to quantify how much a particular state has been affected by these noise sources [70,91]. The generation of Bell pairs, or the process of entanglement swapping are both noisy operations, and as a corollary it degrades the fidelity of a quantum state. Higher-fidelity Bell states can be generated from lower-fidelity pairs via the process of entanglement purification (or entanglement distillation). Quantum error correction (QEC) can also be utilized in entanglement distillation with lower communication costs [421], but with higher demands on the fidelity of the input quantum state. The resulting output fidelity, F ^′ of the entanglement purification at an input state fidelity F ₀ is F′=F02/pS with relation F ′ > F ₀ if F ₀ > 0.5, where p_S is the success probability of entanglement purification, pS=F02+(1−F0)2 The utilization of Bell pairs in networked quantum services depends on a F _thr fidelity threshold, below which the entangled states cannot be used in practice. Different applications can have different requirements on F _thr In networked quantum services, a Quality of Service (QoS) parameter can be the end-to-end Bell pairs per second (BPPS) and the required fidelity of the Bell pairs [46,91,422].

Scalability of networked services requires the extension of quantum systems over long-distances while preserving quantum coherence. The current hardware limitations of quantum devices such as gate fidelities and error rates make the problem of scaling even more difficult [58,423–425]. The handling of external disturbances, unwanted interactions, and decoherence in quantum systems are the main challenges in scaling networked quantum services. To achieve scalability over long distances in networked quantum services, the quantum repeater architecture of the quantum internet can be utilized with advanced quantum networking protocols [45]. An integration of physical layer developments (entanglement swapping, entanglement purification, multipartite entanglement, quantum error correction, quantum memories, controlling) with higher layer advancements (entanglement routing, delay minimization techniques, dynamic path selection, network intelligence, error management, connectivity management, etc.) provides an effective solution for the scalability of practical networked quantum services.

The aim of quantum error correction is to preserve quantum information decoherence and errors. By using quantum error correction codes (such as surface codes, error syndromes, code concatenation) the accuracy and reliability of quantum information processing can be enhanced significantly. The tools of quantum error correction allow us to realize fault-tolerant quantum computations in experiments. According to the threshold theorem [426] (or quantum fault-tolerance theorem), a quantum computer with a physical error rate below a certain threshold can suppress the logical error rate to arbitrarily low levels via the application of quantum error correction. This means that quantum computers can be made fault-tolerant.

Due to the efficient error correction techniques, the percentage of error of quantum gates is decreasing over the years [13,427,428]. The current error rate (%) of quantum gates is in the range of ε ≈ 0.01, which represents approximately two orders of magnitude improvement in the last few years. Table 15 summarizes the gate error rates of quantum processors of different quantum vendors.

An advantage of networked quantum services is the enhanced fault-tolerance and improved scalability due to the distributed architecture [31]. For a detailed discussion on quantum error correction, fault tolerance, and quantum memory limitations, we suggest references [71–74].

In networked quantum services, quantum error correction can be implemented with a high cost due to the conditions on physical resources (high number of physical qubits with high fidelity). According to a current roadmap [70] of quantum networks, quantum error correction will be used only in later generations of quantum networks, while current networked quantum services can use entanglement swapping and quantum teleportation. Using Bell states with small-scale error correction such as entanglement distillation represents an implementable alternative of high-cost quantum error correction schemes in practical scenarios.

Quantum networks can be categorized into Types I-III, based on their loss and error tolerance schemes [70,429], as it is summarized in Table 16

Type I quantum networks use heralded entanglement generation, the entangled states are probabilistically generated and the distribution range is extended via entanglement swapping. The term “heralded” refers to that the entanglement generation process runs until the two non-neighboring stations do not receive an ACK (acknowledge) message from the intermediary station, which confirms that the entanglement swapping operation was successful. High-fidelity entanglement is produced via entanglement purification. These networks use two-way classical communications which reduces significantly the achievable communication rates.

From Type II quantum networks, the entanglement purification procedure can be replaced by QEC. It improves the rate of communication since it can reduce the required amount of classical communications. However, QEC also requires higher state fidelities and additional quantum circuits that make the scheme more complex.

Type III quantum networks use QEC to handle the loss and operational errors without classical communications. These networks can use direct unidirectional communication, but with an improved complexity due to the use of QEC for loss and error tolerance. Type III networks can directly transmit QEC-encoded qubits to adjacent quantum nodes, and the Bell pairs can be created without heralding or the use of classical side-information [70]; however, they require high-fidelity quantum gates and longer coherence times.

Table 17 summarizes the applications of different QEC codes [427] in Type II-III networks.

Regarding quantum memories, a critical problem in achieving persistent storage is the isolation of a quantum system from the environment. Due to the decoherence, the environment adds an uncontrollable noise into the system, which also degrades the fidelity of the state. The lifetime (coherence time) of a quantum memory is implementationspecific; however, the highest achievable values currently in a quantum network hardware are in the range of few seconds to few minutes (see Table18). The coherence time values of quantum memories are continuously increasing as the different technologies evolve. Networked quantum services also have to be able to manage short coherence times. A possible solution to this problem is the reduction of the latency on critical paths of the network [70]. Details of recent quantum memory implementations with their highest coherence time values are given in Table 18