The main investigation in this thesis is the research on a future quantum internet. The focus is laid on the network structure of this quantum network. The network is rst examined by the investigation of the Kuramoto model for classical oscillators. The main property of the network is the ability to synchronise all nodes such that all oscillate with the same frequency. Several parameters of the oscillators are varied to verify why and when synchronisation takes place. After modelling the calculated equations of motion, an interesting conclusion arises. For a symmetric ring network, a stable conguration is not always found but by introducing asymmetric oscillators in the network, the synchronous state can be found. This conclusion leads to the idea that a quantum network requires certain dierences in its structure in order to guarantee transmission to take place. Furthermore, the existence of synchronisation heavily depends on the parameters used for the oscillators. The analysis is then continued in the quantum domain where optomechanical systems are introduced. At rst, two systems are connected to each other by a gaseous interaction and an electric interaction via a Dung circuit. The main investigation is again into the synchronisation, which implies that the operators belonging to both systems behave the same as a function of time. Then several synchronisation measures are introduced to measure the ability of the systems to synchronise. As predicted, both systems synchronise in terms of the operators for each system. Then a quantum network is introduced, where a complex yet ecient network is created. This quantum network is a small world network, where a transmitter node is able to connect to the receiver node by only a few links. Furthermore, multiple transmissions can take place at the same time and links which are not connected do not synchronise with the transmitter node. After modelling this network, the results are in compliance with the theory available. Challenges for a practical quantum network include the experimental basis of being able to utilise optomechanical systems outside laboratory circumstances. Also, the network should be tested for practical use and expanded in a much larger size both in theoretical analysis as well as in experiments.
The main investigation in this thesis is the research on a future quantum internet. The focus is laid on the network structure of this quantum network. The network is rst examined by the investigation of the Kuramoto model for classical oscillators. The main property of the network is the ability to synchronise all nodes such that all oscillate with the same frequency. Several parameters of the oscillators are varied to verify why and when synchronisation takes place. After modelling the calculated equations of motion, an interesting conclusion arises. For a symmetric ring network, a stable conguration is not always found but by introducing asymmetric oscillators in the network, the synchronous state can be found. This conclusion leads to the idea that a quantum network requires certain dierences in its structure in order to guarantee transmission to take place. Furthermore, the existence of synchronisation heavily depends on the parameters used for the oscillators. The analysis is then continued in the quantum domain where optomechanical systems are introduced. At rst, two systems are connected to each other by a gaseous interaction and an electric interaction via a Dung circuit. The main investigation is again into the synchronisation, which implies that the operators belonging to both systems behave the same as a function of time. Then several synchronisation measures are introduced to measure the ability of the systems to synchronise. As predicted, both systems synchronise in terms of the operators for each system. Then a quantum network is introduced, where a complex yet ecient network is created. This quantum network is a small world network, where a transmitter node is able to connect to the receiver node by only a few links. Furthermore, multiple transmissions can take place at the same time and links which are not connected do not synchronise with the transmitter node. After modelling this network, the results are in compliance with the theory available. Challenges for a practical quantum network include the experimental basis of being able to utilise optomechanical systems outside laboratory circumstances. Also, the network should be tested for practical use and expanded in a much larger size both in theoretical analysis as well as in experiments.
Quantum mechanics shows that if one is able to generate and manipulate entanglement over a distance, one is able to perform certain tasks which are impossible using only classical communication. Classical communication refers to what is used in the Internet of today. A quantum internet would therefore bring new capabilities to our highly connected world. These capabilities both involve (1) the ability to perform tasks with are provably impossible in the current Internet, such as unconditionally secure communication, and (2) the ability to perform certain tasks much more efficient, such as distributed (quantum) computing or extending the baseline of telescopes. To be able to build a quantum internet, two main components are needed: (i) hardware that can store, manipulate and entangle qubits and (ii) a software stack to control the hardware. The core task of both of these is to generate entanglement to be used by applications. In this thesis we focus on the latter, i.e. the development of software and protocols that enable entanglement generation using capable hardware. To enable a certain application, one can certainly, in theory, manually specify each operation the hardware should perform, involving micro-wave pulses, lasers etc. However, in practice this is not feasible, if not to say impossible, due to the complexity of the operations needed, especially in a distributed system such as a quantum network. What is needed is a software stack, which can help with abstracting complexity away in multiple layers. This allows for someone to program a protocol in one layer without knowing all the details of the lower layers. In particular, one can abstract away the hardware details, in order to make higher-layer protocols and applications hardware-agnostic. Therefore, to be able to build a universal, efficient and scalable quantum internet, a software stack is crucial. In chapter 2 we start discussing the networking part of a software stack. Namely, we introduce a network stack for a quantum internet, drawing parallels to the IP/TCP-suite of the classical Internet. We continue with proposing a service and interface of the lowest layer of the network stack: the link layer. The link layer is here responsible for generating entanglement between nodes in a quantum network which are directly connected by a quantum link, i.e. a fiber cable. When developing a protocol or application it is very useful to be able to run it. Both to see if the intended ideas make sense and also to check that the implementation is actually correct. Currently we do not have quantum hardware that exposes a full-fledge API that can be used to execute applications. For this reason, it is very useful to be able to instead simulate the hardware in a way that exposes the same API as the hardware being developed. In chapter 3 we introduce SimulaQron for this exact purpose. Any application of a quantum internet will need entanglement in one way or another. However, entanglement is generally hard to generate and is usually the bottleneck when executing an application. We would therefore like to make use of the generated entanglement in the most optimal way. To be able to do this we need to understand how entanglement can be transformed and distributed in a quantum network. We study the entanglement of a particular class of states called graph states in chapters 4 to 9 and how these states can be transformed in a quantum network.
This Thesis contributes to the theory of entanglement distribution in quantum networks, analyzing the generation of long-distance entanglement in particular.We consider that neighboring stations share one partially entangled pair of qubits, which emphasizes the difficulty of creating remote entanglement in realistic settings.The task is then to design local quantum operations at the stations, such that the entanglement present in the links of the whole network gets concentrated between few parties only, regardless of their spatial arrangement.First, we study quantum networks with a two-dimensional lattice structure, where quantum connections between the stations (nodes) are described by non-maximally entangled pure states (links).We show that the generation of a perfectly entangled pair of qubits over an arbitrarily long distance is possible if the initial entanglement of the links is larger than a threshold.This critical value highly depends on the geometry of the lattice, in particular on the connectivity of the nodes, and is related to a classical percolation problem.We then develop a genuine quantum strategy based on multipartite entanglement, improving both the threshold and the success probability of the generation of long-distance entanglement.Second, we consider a mixed-state definition of the connections of the quantum networks.This formalism is well-adapted for a more realistic description of systems in which noise (random errors) inevitably occurs.New techniques are required to create remote entanglement in this setting, and we show how to locally extract and globally process some error syndromes in order to create useful long-distance quantum correlations.Finally, we turn to networks that have a complex topology, which is the case for most real-world communication networks such as the Internet for instance.Besides many other characteristics, these systems have in common the small-world feature, stating that any two nodes are separated by a few links only.Based on the theory of random graphs, we propose a model of quantum complex networks, which exhibit some totally unexpected properties compared to their classical counterparts.5.1 Propagation of a large GHZ state through the lattice . . .5.2 Bit-flip error correction . . . . . . . . . . . . .
Synchronisation has been explored extensively for classical systems from the mid-1600s, and is a phenomenon widely understood in many diverse systems. In the age of technology miniaturisation, there are great incentives to develop an understanding of quantum synchronisation, where quantum fluctuations become the dominant source of noise and non-classical features become important. This thesis considers different types of synchronisation in systems that can display highly non-linear, quantum behaviours. Phase synchronisation for a micromaser coupled to an external drive is initially considered, and it is found that for low photon numbers a semiclassical approximation underestimates phase synchronisation, but a perturbation theory can be accurate for sufficiently weak coupling in either a quantum or semiclassical regime. When a small detuning between the drive and micromaser is introduced, frequency entrainment is also found to exist. Then two micromasers are coupled with each other, in two different configurations, and synchronisation through a relative phase preference is found for each. A semiclassical approximation once again underestimates the phase synchronisation. Entanglement and mutual information are found to behave differently to the relative phase preference depending on each system's parameters. Finally, a mean-field theory is considered for many coupled van der Pol oscillators and micromasers respectively. As the coupling between the oscillators is increased, a critical point is reached and the systems undergo a transition to a phase synchronised state. This bistability appears via a saddle-node bifurcation, and in addition to examining a variety of properties before and after the transition the critical scaling behaviour was investigated close to the transition.
Quantum Networking (QN) has the potential to revolutionize many areas of science and technology with the groundbreaking capabilities in communication and computation that it introduces. The quantum Internet is different from the classical Internet in that it is based on quantum bits (qubits), the fundamental units, rather than classical bits. This opens up the possibility of qubits being able to perform complex computation and communication tasks in ways that go beyond what is possible with classical systems because they can exist in multiple states at once through the principle of superposition. This is important for the future of quantum applications, in particular for satellite-based systems, to enable secure and efficient communication over long distances. However, there are still significant challenges to achieving scalable and reliable time synchronization and integration of quantum systems with existing classical networking infrastructures. This project focuses on these challenges. It will design, model, and experimentally evaluate a hardware clock system for integrating digital control boards in Quantum Networks, with an emphasis on scalable and reliable time synchronization architecture. It will model and benchmark time synchronization architectures using high-precision time measurement instruments and versatile microcontroller platforms. These microcontrollers mimic QN nodes and allow the experimentation of quantum communication protocols to evaluate how well the proposed solution can be scaled and performed. This work provides a foundation for addressing the technical issues of Quantum Networks. It is a step towards scalable quantum communication technologies and the development of future aerospace systems for secure global communication.
Abstract We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary N > 2 identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.
We consider a network of quantum oscillators in which quantum states are distributed among connected nodes by means of unitary transformations. The distributed states interact with each local state according to a time-dependent interaction Hamiltonian, which is modeled by a Hermitian operator constructed in terms of the states themselves, thereby introducing nonlinear network interactions. For qubit nodes of differing natural frequencies, we show numerically that for a sufficiently large coupling constant, synchronization of quantum nodes occurs in which the spins of all qubits are mutually aligned with a common frequency of oscillation, following initial transient configurations. We discuss the significance of quantum synchronization as a means to create copies of unknown quantum states.