2.1. Quantum State

A quantum mechanical system is described by its quantum state. A quantum state is an abstract object that provides a complete description of the system at that particular moment. When combined with the rules of the system's evolution in time, such as a quantum circuit, it also then provides a complete description of the system at all times. For the purposes of computing and networking, the classical equivalent of a quantum state would be a string or stream of logical bit values. These bits provide a complete description of what values we can read out from that string at that particular moment, and when combined with its rules for evolution in time, such as a logical circuit, we will also know its value at any other time.¶

Just like a single classical bit, a quantum mechanical system can be simple and consist of a single particle, e.g., an atom or a photon of light. In this case, the quantum state provides the complete description of that one particle. Similarly, just like a string of bits consists of multiple bits, a single quantum state can be used to also describe an ensemble of many particles. However, because quantum states are governed by the laws of quantum mechanics, their behaviour is significantly different to that of a string of bits. In this section, we will summarise the key concepts to understand these differences. We will then explain their consequences for networking in the rest of this document.¶

2.2. Qubit

The differences between quantum computation and classical computation begin at the bit level. A classical computer operates on the binary alphabet { 0, 1 }.  A quantum bit, called a qubit, exists over the same binary space, but unlike the classical bit, its state can exist in a superposition of the two possibilities:¶

|qubit⟩ = a |0⟩ + b |1⟩,¶

where |X⟩ is Dirac's ket notation for a quantum state (the value that a qubit holds) -- here, the binary 0 and 1 -- and the coefficients a and b are complex numbers called probability amplitudes. Physically, such a state can be realised using a variety of different technologies such as electron spin, photon polarisation, atomic energy levels, and so on.¶

Upon measurement, the qubit loses its superposition and irreversibly collapses into one of the two basis states, either |0⟩ or |1⟩. Which of the two states it ends up in may not be deterministic but can be determined from the readout of the measurement. The measurement result is a classical bit, 0 or 1, corresponding to |0⟩ and |1⟩, respectively. The probability of measuring the state in the |0⟩ state is |a|²; similarly, the probability of measuring the state in the |1⟩ state is |⁠b⁠|⁠², where |a|² + |b|² = 1. This randomness is not due to our ignorance of the underlying mechanisms but rather is a fundamental feature of a quantum mechanical system [Aspect81].¶

The superposition property plays an important role in fundamental gate operations on qubits. Since a qubit can exist in a superposition of its basis states, the elementary quantum gates are able to act on all states of the superposition at the same time. For example, consider the NOT gate:¶

NOT (a |0⟩ + b |1⟩) ➔ a |1⟩ + b |0⟩.¶

It is important to note that "qubit" can have two meanings. In the first meaning, "qubit" refers to a physical quantum system whose quantum state can be expressed as a superposition of two basis states, which we often label |0⟩ and |1⟩. Here, "qubit" refers to a physical implementation akin to what a flip-flop, switch, voltage, or current would be for a classical bit. In the second meaning, "qubit" refers to the abstract quantum state of a quantum system with such two basis states. In this case, the meaning of "qubit" is akin to the logical value of a bit, from classical computing, i.e., "logical 0" or "logical 1". The two concepts are related, because a physical "qubit" (first meaning) can be used to store the abstract "qubit" (second meaning). Both meanings are used interchangeably in literature, and the meaning is generally clear from the context.¶

2.3. Multiple Qubits

When multiple qubits are combined in a single quantum state, the space of possible states grows exponentially and all these states can coexist in a superposition. For example, the general form of a two-qubit register is¶

a |00⟩ + b |01⟩ + c |10⟩ + d |11⟩,¶

where the coefficients have the same probability amplitude interpretation as for the single-qubit state. Each state represents a possible outcome of a measurement of the two-qubit register. For example, |01⟩ denotes a state in which the first qubit is in the state |0⟩ and the second is in the state |1⟩.¶

Performing single-qubit gates affects the relevant qubit in each of the superposition states. Similarly, two-qubit gates also act on all the relevant superposition states, but their outcome is far more interesting.¶

Consider a two-qubit register where the first qubit is in the superposed state (|0⟩ + |1⟩)/sqrt(2) and the other is in the state |0⟩. This combined state can be written as¶

(|0⟩ + |1⟩)/sqrt(2) x |0⟩ = (|00⟩ + |10⟩)/sqrt(2),¶

where x denotes a tensor product (the mathematical mechanism for combining quantum states together).¶

The constant 1/sqrt(2) is called the normalisation factor and reflects the fact that the probabilities of measuring either a |0⟩ or a |1⟩ for the first qubit add up to one.¶

Let us now consider the two-qubit Controlled NOT, or CNOT, gate. The CNOT gate takes as input two qubits -- a control and a target -- and applies the NOT gate to the target if the control qubit is set. The truth table looks like¶

Now, consider performing a CNOT gate on the state with the first qubit being the control. We apply a two-qubit gate on all the superposition states:¶

CNOT (|00⟩ + |10⟩)/sqrt(2) ➔ (|00⟩ + |11⟩)/sqrt(2).¶

What is so interesting about this two-qubit gate operation? The final state is entangled.  There is no possible way of representing that quantum state as a product of two individual qubits; they are no longer independent. That is, it is not possible to describe the quantum state of either of the individual qubits in a way that is independent of the other qubit. Only the quantum state of the system that consists of both qubits provides a physically complete description of the two-qubit system. The states of the two individual qubits are now correlated beyond what is possible to achieve classically. Neither qubit is in a definite |0⟩ or |1⟩ state, but if we perform a measurement on either one, the outcome of the partner qubit will always yield the exact same outcome. The final state, whether it's |00⟩ or |11⟩, is fundamentally random as before, but the states of the two qubits following a measurement will always be identical. One can think of this as flipping two coins, but both coins always land on "heads" or both land on "tails" together -- something that we know is impossible classically.¶

Once a measurement is performed, the two qubits are once again independent. The final state is either |00⟩ or |11⟩, and both of these states can be trivially decomposed into a product of two individual qubits. The entanglement has been consumed, and the entangled state must be prepared again.¶

4.1. Challenges

A quantum network cannot be built by simply extrapolating all the classical models to their quantum analogues. Sending qubits over a wire like we send classical bits is simply not as easy to do. There are several technological as well as fundamental challenges that make classical approaches unsuitable in a quantum context.¶

4.2. Bell Pairs

Bell pair states are the entangled two-qubit states:¶

         |00⟩ + |11⟩,
         |00⟩ - |11⟩,
         |01⟩ + |10⟩,
         |01⟩ - |10⟩,

where the constant 1/sqrt(2) normalisation factor has been ignored for clarity. Any of the four Bell pair states above will do, as it is possible to transform any Bell pair into another Bell pair with local operations performed on only one of the qubits. When each qubit in a Bell pair is held by a separate node, either node can apply a series of single-qubit gates to their qubit alone in order to transform the state between the different variants.¶

Distributing a Bell pair between two nodes is much easier than transmitting an arbitrary quantum state over a network. Since the state is known, handling errors becomes easier, and small-scale error correction (such as entanglement distillation, as discussed in Section 4.4.3.1), combined with reattempts, becomes a valid strategy.¶

The reason for using Bell pairs specifically as opposed to any other two-qubit state is that they are the maximally entangled two-qubit set of basis states. Maximal entanglement means that these states have the strongest non-classical correlations of all possible two-qubit states. Furthermore, since single-qubit local operations can never increase entanglement, states that are less entangled would impose some constraints on distributed quantum algorithms. This makes Bell pairs particularly useful as a generic building block for distributed quantum applications.¶

4.4. The Life Cycle of Entanglement

Reducing the problem of quantum connectivity to one of generating a Bell pair has reduced the problem to a simpler, more fundamental case, but it has not solved it. In this section, we discuss how these entangled pairs are generated in the first place and how their two qubits are delivered to the end-points.¶

+---------+      +---------+      +---------+
|    A    |      |    B    |      |    C    |
|         |------|         |------|         |
|      X1~~~~~~~~~~X2   Y1~~~~~~~~~~Y2      |
+---------+      +---------+      +---------+

+---------+      +---------+      +---------+
|    A    |      |    B    |      |    C    |
|         |------|         |------|         |
|      X1~~~~~~~~~~~~~~~~~~~~~~~~~~~X2      |
+---------+      +---------+      +---------+

5.1. Challenges

This subsection covers the major fundamental challenges involved in building quantum networks. Here, we only describe the fundamental differences. Technological limitations are described in Section 5.4.¶

1. Bell pairs are not equivalent to packets that carry payload.¶

   In most classical networks, including Ethernet, Internet Protocol (IP), and Multi-Protocol Label Switching (MPLS) networks, user data is grouped into packets. In addition to the user data, each packet also contains a series of headers that contain the control information that lets routers and switches forward it towards its destination. Packets are the fundamental unit in a classical network.¶

   In a quantum network, the entangled pairs of qubits are the basic unit of networking. These qubits themselves do not carry any headers. Therefore, quantum networks will have to send all control information via separate classical channels, which the repeaters will have to correlate with the qubits stored in their memory. Furthermore, unlike a classical packet, which is located at a single node, a Bell pair consists of two qubits distributed across two nodes. This has a fundamental impact on how quantum networks will be managed and how protocols need to be designed. To make long-distance Bell pairs, the nodes may have to keep their qubits in their quantum memories and wait until control information is exchanged before proceeding with the next operation. This signalling will result in additional latency, which will depend on the distance between the nodes holding the two ends of the Bell pair. Error management, such as entanglement distillation, is a typical example of such control information exchange [Nagayama21] (see also Section 4.4.3.3).¶

2. "Store and forward" and "store and swap" quantum networks require different state management techniques.¶

   As described in Section 4.4.1, quantum links provide Bell pairs that are undirected network resources, in contrast to directed frames of classical networks. This phenomenological distinction leads to architectural differences between quantum networks and classical networks. Quantum networks combine multiple elementary link Bell pairs together to create one end-to-end Bell pair, whereas classical networks deliver messages from one end to the other end hop by hop.¶

   Classical networks receive data on one interface, store it in local buffers, and then forward the data to another appropriate interface. Quantum networks store Bell pairs and then execute entanglement swapping instead of forwarding in the data plane. Such quantum networks are "store and swap" networks. In "store and swap" networks, we do not need to care about the order in which the Bell pairs were generated, since they are undirected. However, whilst the ordering does not matter, it is very important that the right entangled pairs get swapped, and that the intermediate measurement outcomes (see Section 4.4.2) are signalled to and correlated with the correct qubits at the other nodes. Otherwise, the final end-to-end entangled pair will not be created between the expected end-points or will be in a different quantum state than expected. For example, rather than Alice receiving a qubit that is entangled with Bob's qubit, her qubit is entangled with Charlie's qubit. This distinction makes control algorithms and optimisation of quantum networks different from those for classical networks, in the sense that swapping is stateful in contrast to stateless packet-by-packet forwarding. Note that, as described in Section 4.4.3.3, third-generation quantum networks will be able to support a "store and forward" architecture in addition to "store and swap".¶

3. An entangled pair is only useful if the locations of both qubits are known.¶

   A classical network packet logically exists only at one location at any point in time. If a packet is modified in some way, whether headers or payload, this information does not need to be conveyed to anybody else in the network. The packet can be simply forwarded as before.¶

   In contrast, entanglement is a phenomenon in which two or more qubits exist in a physically distributed state. Operations on one of the qubits change the mutual state of the pair. Since the owner of a particular qubit cannot just read out its state, it must coordinate all its actions with the owner of the pair's other qubit. Therefore, the owner of any qubit that is part of an entangled pair must know the location of its counterpart. Location, in this context, need not be the explicit spatial location. A relevant pair identifier, a means of communication between the pair owners, and an association between the pair ID and the individual qubits will be sufficient.¶

4. Generating entanglement requires temporary state.¶

   Packet forwarding in a classical network is largely a stateless operation. When a packet is received, the router does a lookup in its forwarding table and sends the packet out of the appropriate output. There is no need to keep any memory of the packet any more.¶

   A quantum node must be able to make decisions about qubits that it receives and is holding in its memory. Since qubits do not carry headers, the receipt of an entangled pair conveys no control information based on which the repeater can make a decision. The relevant control information will arrive separately over a classical channel. This implies that a repeater must store temporary state, as the control information and the qubit it pertains to will, in general, not arrive at the same time.¶

5.2. Classical Communication

In this document, we have already covered two different roles that classical communication must perform the following:¶

• Communicate classical bits of information as part of distributed protocols such as entanglement swapping and teleportation.¶
• Communicate control information within a network, including background protocols such as routing, as well as signalling protocols to set up end-to-end entanglement generation.¶

Classical communication is a crucial building block of any quantum network. All nodes in a quantum network are assumed to have classical connectivity with each other (within typical administrative domain limits). Therefore, quantum nodes will need to manage two data planes in parallel: a classical data plane and a quantum data plane. Additionally, a node must be able to correlate information between the two planes so that the control information received on a classical channel can be applied to the qubits managed by the quantum data plane.¶

5.3. Abstract Model of the Network

+-----+                                        +-----+
| App |- - - - - - - - - -CC- - - - - - - - - -| App |
+-----+                +------+                +-----+
| EN  |------ CL ------|  QR  |------ CL ------| EN  |
|     |------ QL ------|      |------ QL ------|     |
+-----+                +------+                +-----+

App - user-level application
EN - End Node
QL - Quantum Link
CL - Classical Link
CC - Classical Channel (traverses one or more CLs)
QR - Quantum Repeater

6.2. The Principles of a Quantum Internet

The principles support the goals but are not goals themselves. The goals define what we want to build, and the principles provide a guideline for how we might achieve this. The goals will also be the foundation for defining any metric of success for a network architecture, whereas the principles in themselves do not distinguish between success and failure. For more information about design considerations for quantum networks, see [VanMeter13.1] and [Dahlberg19].¶

1. Entanglement is the fundamental service.¶

   The key service that a quantum network provides is the distribution of entanglement between the nodes in a network. All distributed quantum applications are built on top of this key resource. Applications such as clustered quantum computing, distributed quantum computing, distributed quantum sensing networks, and certain kinds of quantum secure networks all consume quantum entanglement as a resource. Some applications (e.g., QKD) simply measure the entangled qubits to obtain a shared secret key [QKD]. Other applications (e.g., distributed quantum computing) build abstractions and operations that are more complex on the entangled qubits, e.g., distributed CNOT gates [DistCNOT] or teleportation of arbitrary qubit states [Teleportation].¶

   A quantum network may also distribute multipartite entangled states (entangled states of three or more qubits) [Meignant19], which are useful for applications such as conference key agreement [Murta20], distributed quantum computing [Cirac99], secret sharing [Qin17], and clock synchronisation [Komar14], though it is worth noting that multipartite entangled states can also be constructed from multiple entangled pairs distributed between the end nodes.¶

2. Bell pairs are indistinguishable.¶

   Any two Bell pairs between the same two nodes are indistinguishable for the purposes of an application, provided they both satisfy its required fidelity threshold. This observation is likely to be key in enabling a more optimal allocation of resources in a network, e.g., for the purposes of provisioning resources to meet application demand. However, the qubits that make up the pair themselves are not indistinguishable, and the two nodes operating on a pair must coordinate to make sure they are operating on qubits that belong to the same Bell pair.¶

3. Fidelity is part of the service.¶

   In addition to being able to deliver Bell pairs to the communication end-points, the Bell pairs must be of sufficient fidelity. Unlike in classical networks, where most errors are effectively eliminated before reaching the application, many quantum applications only need imperfect entanglement to function. However, quantum applications will generally have a threshold for Bell pair fidelity below which they are no longer able to operate. Different applications will have different requirements for what fidelity they can work with. It is the network's responsibility to balance the resource usage with respect to the applications' requirements. It may be that it is cheaper for the network to provide lower-fidelity pairs that are just above the threshold required by the application than it is to guarantee high-fidelity pairs to all applications regardless of their requirements.¶

4. Time is an expensive resource.¶

   Time is not the only resource that is in short supply (communication qubits and memory are as well), but ultimately it is the lifetime of quantum memories that imposes some of the most difficult conditions for operating an extended network of quantum nodes. Current hardware has low rates of Bell pair generation, short memory lifetimes, and access to a limited number of communication qubits. All these factors combined mean that even a short waiting queue at some node could be enough for a Bell pair to decohere or result in an end-to-end pair below an application's fidelity threshold. Therefore, managing the idle time of qubits holding live quantum states should be done carefully -- ideally by minimising the idle time, but potentially also by moving the quantum state for temporary storage to a quantum memory with a longer lifetime.¶

5. Be flexible with regards to capabilities and limitations.¶

   This goal encompasses two important points:¶

   • First, the architecture should be able to function under the physical constraints imposed by the current-generation hardware. Near-future hardware will have low entanglement generation rates, quantum memories able to hold a handful of qubits at best, and decoherence rates that will render many generated pairs unusable.¶
   • Second, the architecture should not make it difficult to run the network over any hardware that may come along in the future. The physical capabilities of repeaters will improve, and redeploying a technology is extremely challenging.¶