Principles of quantum mechanics promise a future quantum internet that connects a wide variety of quantum devices together in a coherent and secure fashion. It is well known that due to the size of this quantum internet, quantum repeaters will be a critical part in a similar fashion to the importance of repeaters in today's telecommunications internet. Given the inherent differences between classical and quantum physics, it is essential to establish how a quantum internet will function including how we route information as well as the functionality quantum repeaters will need to provide. Our considerations here go far beyond quantum key distribution and instead focus on a true network of connected quantum devices, including computers and sensors. We show how the efficient operation of such quantum networks relies on the seamless integration of both quantum and classical communication resources.
Recent technological advances over the last decades have made devices based on the principle of quantum mechanics a reality.1–5 We have seen quantum sensor and imaging systems, secure communication systems, and even small-scale quantum simulations and computers demonstrated.6–18 In fact, quantum computational advantage has been achieved in several platforms.19,20 Furthermore, quantum random number generators and point-to-point quantum key distribution (QKD) systems are commercially available.21 However, more general quantum communication systems are much less advanced with a number of key device technologies needing to be developed, especially if the real promise of quantum networked technologies (and a future quantum internet22) is going to be truly implemented. We already know that small-scale quantum computers and sensing devices will need to be networked together in an efficient manner with lossy channels connecting them. Those losses will have a profound effect on the reliability and performance of systems (as they do in our classical world).
In the modern telecommunications industry, loss issues are addressed through the use of both optical amplifiers (OAs) and optical repeaters (ORs), each having a slightly different functionality.23 Optical amplifiers (OAs)23 are simple optoelectronic circuits present within the communications node that amplifies the distorted light signal it receives back to its original intensity (unfortunately amplifying the noise as well) before transmitting it further down the channel to the next node. Each amplification process decreases the signal-to-noise ratio, which gets worse until we reach a point where the signal cannot be distinguished from the noise. Here is where optical repeaters (ORs) come in. Optical repeaters23 amplify and pulse reshape the received signal to significantly reduce the extra unwanted noise that has accumulated. As such, they allow one to create planetary scale telecommunication networks that route information between arbitrary nodes with very little signal degradation. One then needs to consider how information is moved around those networks.
There are a number of different approaches to how telecommunication networks operate,24 but they are based on the transmission of signals (messages) from node to node (direct transmission). The archetypical examples are the circuit switched25 and packet switched networks,26,27 which have fundamentally different modes of operation. Circuit switched networks setup a dedicated path between the two parties (as shown in Fig. 1) before information is transferred between Alice and Bob. That dedicated circuit cannot be used by other network users until it is released. Packet switched networks, on the contrary, operate by grouping the information to be sent into small packets that are then sent over a shared network with multiple users. The path between Alice and Bob may not be known to them in advance and can change. Furthermore, software defined networking has allowed the process of forwarding network packets to be separated from the routing process, making the overall network highly programmable and efficient.28
The natural question that arises now is how we transfer information in the quantum world? Channel losses are, of course, still a serious issue. In fact, they could be even worse, as many quantum states are quite fragile. We will need quantum amplifiers (QAs) and repeaters; however, their functionality may be a little different.
Quantum amplifiers (QAs) are well known,29 and like their optical counterparts, they not only boost the quantum signal back to its original intensity but also add excess noise to that signal.30 This limits the communication distance necessitating the use of repeater-like devices. Alternatively, there are noiseless quantum amplifiers (which as the name indicates do not add excess noise), but they are probabilistic (and heralded) in nature.31–39
Quantum repeaters (QRs), however, cannot be like their classical counterparts due to the no-cloning theorem meaning we cannot remove noise that has already been added.40 So what alternatives do we have and what form do such quantum repeaters take?
II. QUANTUM REPEATERS
The functionality required of quantum repeaters really depends on the task that they are going to be used for. It is quite different to think about general purpose applications than quantum key distribution (QKD) or secret sharing alone.41–43 As such, our focus here will be on the situation where Alice wants to transfer quantum information (maybe even part of a complex entangled quantum state) to Bob through a lossy channel. The typical approach would be to use quantum teleportation, as depicted in Fig. 1, to transfer the message from Alice to Bob.44,45 This requires that a high quality entangled link has already been established between them. The use of the quantum repeaters is then associated with creating high quality entangled links between the parties. Quantum repeaters are characterized by the form of their component operations.46,48–52 These include the following:
Entanglement distribution: It is the process by which entangled links are created between adjacent nodes using heralded entanglement distribution (HED)53,54 or loss-based quantum error correction (LQEC)50,55,56 approaches. The HED approach is technologically simpler and allows for larger separation between nodes (hundreds of kilometers), however, at the expense of performance, as one must wait for the classical heralding messages between nodes before the links can be used. LQEC approaches, on the contrary, are technologically much more difficult, requiring high fidelity local operations and limit the nodal separation to less than 15 km in telecom fiber. They, however, remove the heralding requirements allowing for much better performance.50
Entanglement distillation: It is the process by which the degree of entanglement between two repeater nodes is increased, typically using quantum error detection (QED)57–59 or correction (QEC)47–49,60 techniques. The QED approach can handle higher error rates but is probabilistic and heralded in nature. This necessitates the need for classical signaling between the nodes involved. Purification57,61,62 is a form of QED.
Entanglement swapping: It is the process where the distance at which repeater nodes are entangled increases.45
These three operations each require some form of classical networking—whether they are deterministic or heralded in nature. All measurements give at least a binary result, which generally needs to be processed at other nodes in the network. A fast and efficient classical network needs to carry such messages—otherwise bottlenecks will arise.
Now with two primary mechanisms for each of the entanglement distribution/distillation operations, we can characterize three distinct repeater designs (generations) as shown in Table I.52 The technical requirements of the devices dramatically increase as we move to the higher generations (the third generation can be viewed as a networked quantum computer). Fortunately, so does their potential performance.50,52 Furthermore, all repeater protocols have a maximum effective distance they can span in reality with reasonable performance.52,63 Further details for the first-generation schemes are presented in Appendix A.
|.||Distribution .||Distillation .||Performance .|
|.||HED .||LQEC .||QED .||QEC .||Time .||Resources .|
|.||Distribution .||Distillation .||Performance .|
|.||HED .||LQEC .||QED .||QEC .||Time .||Resources .|
III. QUANTUM DIRECT TRANSMISSION
The third-generation quantum repeater approaches mentioned above in Table I allow the direct transmission of quantum messages between Alice and Bob through a number of intermediate repeater nodes as shown in Fig. 2. One does not need to create long-range entanglement. Instead, the approach50 begins with Alice encoding her message into a photonic quantum state of the form with being the error correctable logical qubit basis states. This photonic state (potentially in one optical pulse) is then sent to the first repeater node, where loss events are first identified. Then loss and local gate errors are corrected, which is a critical consideration for a quantum repeater. The resultant quantum state, which is very close in form to the original , is then sent to the next repeater, where the same process is repeated. This continues until the message is received by Bob with little added noise. Error syndrome results can be sent along with the quantum photonic signal at the same time meaning Bob does not need to wait to use his received state.50 Further, this approach naturally supports most of the classical techniques for determining how we “route” information between Alice and Bob.
Quantum communication based on the third-generation quantum repeater approaches, however, imposes significant constraints onto the repeaters themselves. Fundamentally, losses between adjacent nodes (including those within the nodes themselves) must be less than 50% to avoid violating the no-cloning principle but realistically should be below 20% ( km for telecom fiber). Furthermore, local gate errors must be below 1% but the exact values for both depend heavily on the form of error correction code used, remembering that there is a trade-off between loss and local gate errors.73 The smaller the local gate errors, the higher the loss can be and vice versa. It is important to explicitly mention that once these error correction codes are implemented fault tolerantly, the infidelity in the transmitted message between Alice and Bob can be made arbitrary small. This in turn means Alice can near deterministically transmit a message to Bob using only per node.
IV. ENTANGLEMENT DISTRIBUTING REPEATERS
Our discussion previously has focused on the transmission of quantum information from Alice to Bob. The traditional use of quantum repeaters has been associated with the generation of long-distance entangled resources, and hence, let us examine that situation now. The previous direct transmission scheme can be used to accomplish this by simply storing half an entangled Bell pair at Alice and sending the other half to Bob. Alice generally, unfortunately, has to store her half of the Bell pair for the entire duration until the second qubit arrives at Bob. This in turn means Alice will need significantly more resources for most quantum information processing tasks (and especially, computational ones) if she is attempting to establish many entangled pairs. Of course, there are applications including secure communication protocols where Alice can measure her qubits straight away, alleviating the need for memories all together. A future quantum internet needs to support all possible use mechanisms including distributed quantum computation where one creates complex entangled states involving many parties.
It is straightforward to modify the direct transmission scheme to the entanglement distribution task50 as shown in Fig. 3. The butterfly approach begins in a central repeater node with the creation of a logically encoded Bell pair. A direct transmission is then used to send each of the encoded qubits to Alice and Bob, respectively. It is critical that the classical signaling associated with the error correction of the encoded qubits is sent along with the quantum signal at the same time to avoid performance issues. This can be seen in the third-generation repeater designs using a parallel configuration like that depicted in Fig. 3 (inset), where the entanglement between all the adjacent nodes occurs at the same time. The classical signals follow on later, and typically, Alice and Bob must wait a time before they can use the resulting Bell pair. This compares with for the butterfly design. For the parallel arrangement to maintain the same performance as that of the butterfly design, Alice and Bob need times the number of resources. For communication only between them, the intermediate nodes do not need those extra resources. However, in a multiuser environment, it is likely these intermediate nodes (especially if they have users associated with them) will also need those extra resources.
V. REPEATERS OR RELAYS?
It is clear that quantum repeaters require the three key operations mentioned previously to distribute high quality entanglement over arbitrary long distances. However, to distribute entanglement (or quantum states) over mid-range distances, the entanglement distillation (purification) step may not be necessary giving rise to devices known as quantum relays. The quality of that entanglement will decrease the more relays it passes through, until one reaches one point at which entanglement cannot be distributed or the entanglement that is distributed is not of the quality required.
The form of the quantum relay is quite interesting depending on the fundamental building blocks used in the entanglement distribution and swapping operations. They can be passive or active in nature:
A passive quantum relay only performs predefined operations—not conditioned on information associated with the arriving entangled pairs. It cannot, for instance, use a quantum memory to store one of the incoming photons while waiting for another to arrive before some further operation occurs dependent on both photons being present.
An active quantum relay, on the other hand, can perform any non-distillation operation, which can be conditioned on information associated with the arriving entangled pairs. Arriving information can be stored in quantum memories awaiting the arrival of photons from other nodes.
This second case is particularly interesting because an active quantum relay may utilize many of the same building blocks used in quantum repeaters. Active techniques, including loss-based error correction and quantum memories, can be used to mitigate the effect of channel loss without such devices necessarily becoming quantum repeaters. Local gate errors will accumulate exponentially with the number of nodes used in the network. The quantum nodes are thus acting like active quantum relays, returning a distorted version of the original signal with the same number of photons present.
The difference between quantum repeaters and relays is then obvious in terms of how the signal-to-noise ratio scales with the number of nodes—as long as one accepts that local gate errors (and other channel imperfections) are present. Typically, one would expect the fidelity of the resultant state created between Alice and Bob to scale as Fn (or worse), where F is the fidelity of the initially created encoded state and n is the number of relay nodes. On the contrary, quantum repeaters could, in principle, scale as F or less. It is not to say, however, that quantum amplifiers/relays are not useful. They can, in principle, allow messaging over quite long distances as long as the local gate errors are small. In the QKD scenario, even if the fidelity of the shared states is very low, a key rate might be extracted, showing that a quantum relay can represent a valid alternative for a quantum repeater at specific requirements.
VI. COMPLEX QUANTUM NETWORKS
The direct transmission approach outlined above provides an ideal starting point to consider the transfer of a single quantum bit of information in a complex network as illustrated in Fig. 1. That information would be encoded into a quantum packet also containing the sending and receiving addresses, for instance. The question is how we choose the path between the sender Alice and the receiver Bob. This, of course, is where routing comes in.
A. Quantum routing
Routing is the process where one establishes the path that data packets must take to reach their destination. It is not about how the packet is forwarded itself. In telecommunications there are two main approaches:
Static routing: where every node has a manually configured routing table, which is important for full connectivity. Those routes are fixed and do not automatically change if the network changes.
Dynamic (adaptive) routing: where the node can select a different route based on the current conditions of the overall network. Dynamic routing allows as many routes as possible to remain valid if the network changes.
In a quantum network, the same approaches can obviously be used, with the direct transmission model naturally supporting it. There are, however, other options available due to our quantum principles. We can introduce the concept of Quantum routing—where the node can select a number of different routes to act coherently together. Quantum routing is not essential for quantum networking but does add a new capability which can be useful to mitigate path failures and/or limited resources with the nodes.
B. Quantum networking
Given that we have techniques to determine how the routing of quantum information could work, we now need to explore how the networking could be achieved in a little more detail. In the telecommunications arena, we have two traditional networking approaches:25,26,28
Circuit switched networking,25 where the nodes in the chosen route between Alice and Bob are determined, setup in advance and dedicated to those two users for the duration of their communication (corresponding to the solid lines in Fig. 1).
Packet switched networking,26 where packets of data are routed through the network based on the destination address (thin lines in Fig. 1). Different packets can be routed different ways and so no dedicated path is established.
These approaches can be used to control the transmission of quantum information in a straightforward manner. Packet switched networking seems ideally suited for the second- and third-generation quantum repeaters, while circuit switched networking seems the only viable method for the first-generation ones.
One has to remember, however, that the quantum resources needed between nodes are likely to be quite large, and therefore, on occasions there may be insufficient resources available directly between those two nodes. In such a case, quantum information could be sent coherently over two or more channels and recombined at the node where it arrives. Quantum memories could then be used to synchronize such operations with different channel lengths. This uniquely quantum phenomena allows us to introduce the concept of
Quantum network aggregation—where packets of quantum data are coherently sent simultaneously through different network nodes, being recombined at certain immediate nodes where necessary.
What our quantum routing and network aggregation78 show are new capabilities that become available once we move into the quantum regime.
C. Quantum network aggregation
It is useful to look at the network aggregation78 in a little more detail given its importance. In Fig. 4, we illustrate its concept, where Alice and Bob are connected by three independent channels which follow different physical paths. Each channel has the potential to simultaneously establish a number of links and may pass through a number of intermediate nodes. Now the aggregation technique works by Alice encoding her qubit of information into a loss tolerant error correction code. The code is then divided up and coherently send over those different paths (each of which has different transmission probabilities). At Bob's node, the code is recombined and quantum error correction is performed to remove the effects of both channel losses as well as local gate errors.
VII. HYBRID APPROACHES
A critical question that we have not addressed so far is whether the different forms of quantum repeaters can be combined between the direct transmission and entanglement distribution approaches. Can we use already established long-range entangled links in combination with direct transmission? The obvious answer seems to be yes, but we need to be careful about how classical signaling works. Let us explore the functionality of a network composed of a quantum backbone with access networks connected to it like the one we depict in Fig. 5. For the hybrid network, there are two logical configurations. The first is Direct transmission for the edge of the network with shared entangled resources for the network backbone. Here, resource factories can establish in advance logical Bell states anywhere along the backbone, using error correction to maintain their quality. Direct transmission techniques can be used on the edge of the network to move the quantum information to the nearest backbone node—where it could be teleported to another backbone node close to the recipient. Classical information must be forwarded to that other backbone node where the other half of the Bell state resides. Limited by the speed of light, this means resources in that particular node will be unavailable for further tasks until that classical information arrives. This is likely to cause significant bottlenecks to arise in the overall network—even if that network is well provisioned.
The second configuration is Shared entangled resources at the network edge with direct transmission for the backbone. In this situation, the edge nodes in the network maintain entangled links to their closest backbone nodes. Pre-established entanglement between the edge nodes and their closest backbone node can be used to teleport the desired state/information to that backbone node. However, before proceeding, we need to wait for the classical teleportation information to arrive—noting that the distance between the edge node and the backbone is small (compared to the overall network size). Once that classical information arrives, we can use direct transmission techniques to move it rapidly across the backbone, with the quantum and classical information propagating at the same time.
In large-scale networks, these hybrid approaches can be quite inefficient compared to solely using the direct transmission approach. This is due to the nature of how quantum teleportation works and the requirement to send the measurement results from the Bell state measurement to the other node. One has temporally separated when the quantum and classical operations are done. It is an important consideration to keep in mind.
When one designs large-scale complex quantum networks or even our future quantum internet, it is important to think about the system as a whole (and not only as parts). There are a number of design aspects that need to be considered:
First and foremost is that a complex quantum network must be supported by an efficient telecommunications network which is able to signal between remote nodes on time scale near the speed of light. If such speeds are not possible, then the performance of the quantum network is likely to be limited by the telecommunications network.
Second is that it is unlikely that the two parties will actually know the path joining them and that path may change over time. One should be looking at packet switched networks when multiple users are on the network. Furthermore, many of the current routing protocols can be used.
Third, the first-generation repeater schemes are inherently slow and require significant classical signaling between nodes due to the probabilistic nature of many of the operations that make them up. In turn, this means one must carefully allocate resources within nodes and introduce cutoff timings to optimize their performance. Their use is probably limited to small-scale networks.
Fourth, the higher generation quantum repeater approaches, in principle, allow both the direct transmission of quantum messages and entanglement generation over very large networks. This allows the usual telecommunication tools to determine how such information may move across the network. Network aggregation techniques can be used to overcome resource limitations within nodes.
Additionally, it is important to note that all quantum repeaters have a maximum distance they can span in the presence of imperfect elements (including memories). We should also remember that the tasks or applications one wants to run on your quantum network can modify these design considerations. Certain applications like QKD and quantum secret sharing allow certain users to measure their qubits straight away, removing the necessity of quantum memories and to delay the transmission of classical messages until later.
Next, there has been a great interest in systems where the use of quantum repeaters is an advantage in terms of performance (see Appendix C). One needs to be extremely careful in addressing this. Is it that quantum repeaters have better performance than quantum relays or the direct transmission of signals? Such considerations must include both the quantum and classical information costs. When one considers the performance of such devices, there are three important quantities to consider:
the total time to transmit the information (both quantum and classical) over the network (which has to be least and more generally ),
the repetition time between attempts (which can be as low as the time to perform local gate operations, but only if the necessary accompanying classical information is handled appropriately), and
the fidelity of the quantum information, that is, achieved by transmitting it across the network.
One needs to be careful as the performance metric should correspond to the tasks that one wants to accomplish. For instance, in the teleportation of a quantum signal from Alice to Bob, one needs to consider both the time to generate the required entangled link and also the time cost associated with the transmission of the classical bits from the Bell state measurement to Bob. The fastest that could be done is which is slower than direct transmission.
Finally, we also need to consider security when sending quantum information. The entanglement-based QR approaches are not transmitting quantum information from repeater node to repeater node and so naturally have security built in to some degree. On the contrary, the direct transmission approaches are transmitting the user information from node to node in the network—where it could be intercepted and read. As such, how to secure such approaches needs to be carefully considered.
To summarize, the design of large-scale quantum networks greatly depends on the task one wants to perform. A holistic approach that includes different applications such as quantum computation and quantum key distribution is a challenging achievement since the requirements (in terms of entanglement distribution time, efficiencies of the devices used, and local error gates) can be numerous. In this work, we give a perspective on the main issues quantum networks and tomorrow's quantum internet will have to address to be competitive with their classical counterparts.
The authors thank Koji Azuma and Marta P. Estarellas for useful discussions. This work was supported by the JSPS KAKENHI Grant No. 21H04880.
Conflict of Interest
The authors have no conflicts to disclose.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX A: FIRST-GENERATION QR IN NETWORKS
The first-generation schemes, while conceptually simpler, are much more problematic in a large-scale complex quantum network setting. The necessity of long-range purification inherent in the first-generation schemes means we are likely to require a circuit switched approach, where one can simultaneously generate entangled links between adjacent nodes (which in turn reduces the coherence time requirements of the quantum memories within those nodes). Furthermore, we will need to adopt time-efficient purification strategies, which minimize the number of rounds of purification used, as well as use optimal strategies for sharing entangled states across a quantum network, including timing issues.74,75 These tasks can be accomplished by the use of a quantum decision process,76 in which arbitrary quantum policies are applied to optimize figure-of-merit of a quantum network, such as the expected fidelity and the probability that a quantum link is active.76 A specific quantum policy then based on the cutoff time of quantum memories75,77 allows one to determine important requirements for a practical implementation of near-term quantum networks, such as lower and upper bounds of the coherence times of quantum memories, shortest lengths of repeater chains, and device efficiencies for large-scale networks. Quantum error detection code techniques can be used to enable a one round solution.78 However, even with these advanced purification schemes, such networks are unlikely to be robust on the large scale.79
APPENDIX B: CONTINUOUS VARIABLE QUANTUM REPEATERS
The protocols mentioned in the main text use a discrete encoding of quantum information, that is, where quantum information is encoded in finite dimensional bases, such as qubits or qudits. However, there is another paradigm of quantum information that utilizes information encoded in infinite dimensional bases, for example, continuous spectra like the amplitude and phase of light.80 The former is referred to as a discrete variable (DV), while the latter is referred to as a continuous variable (CV). Each paradigm offers its own unique advantages and disadvantages. In principle, CV encodings offer easier state generation, manipulation and detection.81 Another attractive advantage of CV protocols is that they can offer better compatibility with existing telecommunications infrastructure,82 which is of crucial importance to the design of the future quantum internet. However, the field of DV quantum communication is more advanced and to date has been demonstrated over longer distances [605 km (Ref. 83) versus 202 km (Ref. 84) in terrestrial fiber channels].
For the long-distance quantum communication, CV states are plagued by loss in different ways than DV states. As such, while the first repeater protocol was conceptualized in 199853 and has been through three aforementioned distinct iterations of improvement, the first CV repeater protocols did not come about until recent years.64–66 An early protocol for a CV repeater scheme64 used noiseless linear amplification31 via a single quantum scissor85 for entanglement purification and entanglement swapping via Gaussian continuous variable teleportation;86 however, the scheme was limited in usefulness to the very high loss regime. The scheme65 uses non-Gaussian entanglement distillation protocols,87,88 heralded non-Gaussian entanglement swapping, and Gaussificiation. Alternatively, the protocol66 combined non-Gaussian entanglement swapping and noiseless linear amplification. Recently, the proposal64 has been improved upon.89,90 Notably,90 uses a single quantum scissor as both the entanglement swapping and purification operation, with the reduction in operations yielding a favorable impact on the distributed secret key rate. While there are very practical reasons to consider CV systems in the implementation of future quantum networks, the state of the quantum repeater field for CV remains elementary compared to DV schemes. Further development in quantum memories and error correcting codes for CV quantum states would be a promising improvement in this direction.
APPENDIX C: REPEATERS OR NOT?
Now, when would quantum repeater systems be superior to other approaches? Many say it is when they beat the repeaterless performance bounds91 but there is an apparent fallacy here due to the incorrect interpretation of what repeaters are. Logically, repeaterless must mean systems without repeaters rather than a scheme that beats direct transmission of a signal between two nodes (without processing in between). Repeaterless approaches must allow relays/amplifiers which will allow them to naturally beat the direct transmission schemes—hence the reason care must be taken. With quantum relays incorporating loss-based error correction and low local gate errors, quantum messages may be able to be sent over long distances (far greater distances than direct transmission would allow). However, there will come a point where the noise inherent in those local gates limits the communications distance and repeaters become a necessity. Thus, for global networks, quantum repeaters are going to be a necessity.