We present and analyze an architecture for a European-scale quantum network using satellite links to connect Quantum Cities, which are metropolitan quantum networks with minimal hardware requirements for the end users. Using NetSquid, a quantum network simulation tool based on discrete events, we assess and benchmark the performance of such a network linking distant locations in Europe in terms of quantum key distribution rates, considering realistic parameters for currently available or near-term technology. Our results highlight the key parameters and the limits of current satellite quantum communication links and can be used to assist the design of future missions. We also discuss the possibility of using high-altitude balloons as an alternative to satellites.

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We numerically study the distribution of entanglement between the Dutch cities of Delft and Eindhoven realized with a processing-node quantum repeater and determine minimal hardware requirements for verifiable blind quantum computation using color centers and trapped ions. Our results are obtained considering restrictions imposed by a real-world fiber grid and using detailed hardware-specific models. By comparing our results to those we would obtain in idealized settings, we show that simplifications lead to a distorted picture of hardware demands, particularly on memory coherence and photon collection. We develop general machinery suitable for studying arbitrary processing-node repeater chains using NetSquid, a discrete-event simulator for quantum networks. This enables us to include time-dependent noise models and simulate repeater protocols with cut-offs, including the required classical control communication. We find minimal hardware requirements by solving an optimization problem using genetic algorithms on a high-performance-computing cluster. Our work provides guidance for further experimental progress, and showcases limitations of studying quantum-repeater requirements in idealized situations.

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The vision of a global network that enables quantum communications between any point on Earth is known as the quantum internet. One crucial element of this network is the use of quantum repeater chains, which have the potential to overcome transmission losses and implement entanglement or quantum key distribution protocols over extended distances. There are various proposals for quantum repeaters, but they can generally be evaluated based on two main figures of merit: the average time for end-to-end entanglement delivery and the associated average fidelity. However, characterizing these quantities can be difficult due to factors such as feedback loops, decoherence, entanglement generation being a probabilistic process, and the potential failure of subprotocols. In this talk, I will discuss algorithmic and analytical methods for computing these quantities for relevant families of protocols.

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The ability to distribute high-quality entanglement between remote parties is a necessary primitive for many quantum communication applications. A large range of schemes for realizing the long-distance delivery of remote entanglement has been proposed, for both bipartite and multipartite entanglement. For assessing the viability of these schemes, knowledge of the time at which entanglement is delivered is crucial. Specifically, if the communication task requires multiple remote-entangled quantum states and these states are generated at different times by the scheme, the earlier states will need to wait and thus their quality will decrease while being stored in an (imperfect) memory. For the remote-entanglement delivery schemes which are closest to experimental reach, this time assessment is challenging, as they consist of nondeterministic components such as probabilistic entanglement swaps. For many such protocols even the average time at which entanglement can be distributed is not known exactly, in particular when they consist of feedback loops and forced restarts. In this work, we provide improved analytical bounds on the average and on the quantiles of the completion time of entanglement distribution protocols in the case that all network components have success probabilities lower bounded by a constant. A canonical example of such a protocol is a nested quantum repeater scheme which consists of heralded entanglement generation and entanglement swaps. For this scheme specifically, our results imply that a common approximation to the mean entanglement distribution time, the 3-over-2 formula, is in essence an upper bound to the real time. Our results rely on a novel connection with reliability theory.

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Quantum communication enables the implementation of tasks that are unachievable with classical resources. However, losses on the communication channel preclude the direct long-distance transmission of quantum information in many relevant scenarios. In principle, quantum repeaters allow one to overcome losses. However, realistic hardware parameters make long-distance quantum communication a challenge in practice. For instance, in many protocols an entangled pair is generated that needs to wait in quantum memory until the generation of an additional pair. During this waiting time the first pair decoheres, impacting the quality of the final entanglement produced. At the cost of a lower rate, this effect can be mitigated by imposing a cutoff condition. For instance, a maximum storage time for entanglement after which it is discarded. In this article, we optimize the cutoffs for quantum repeater chains. First, we develop an algorithm for computing the probability distribution of the waiting time and fidelity of entanglement produced by repeater chain protocols which include a cutoff. Then, we use the algorithm to optimize cutoffs in order to maximize the secret-key rate between the end nodes of the repeater chain. In this article, we find that the use of the optimal cutoff extends the parameter regime for which secret key can be generated and, moreover, significantly increases the secret-key rate for a large range of parameters.@en

Communication between remote quantum computers enables tasks that are unachievable with their conventional counterparts, such as unconditionally-secure communication or quantum computing in the cloud. Bridging long distances, where the communication fundamentally suffers from loss, can be achieved by splitting up the distance into segments and positioning so-called quantum repeaters in between. In this thesis, we develop tools to analyse how a large class of quantum repeater schemes will perform when implemented on real hardware suffering from time-dependent noise, in particular imperfect quantum memories for storing quantum information.@en

In order to bring quantum networks into the real world, we would like to determine the requirements of quantum network protocols including the underlying quantum hardware. Because detailed architecture proposals are generally too complex for mathematical analysis, it is natural to employ numerical simulation. Here we introduce NetSquid, the NETwork Simulator for QUantum Information using Discrete events, a discrete-event based platform for simulating all aspects of quantum networks and modular quantum computing systems, ranging from the physical layer and its control plane up to the application level. We study several use cases to showcase NetSquid’s power, including detailed physical layer simulations of repeater chains based on nitrogen vacancy centres in diamond as well as atomic ensembles. We also study the control plane of a quantum switch beyond its analytically known regime, and showcase NetSquid’s ability to investigate large networks by simulating entanglement distribution over a chain of up to one thousand nodes.

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Long-distance quantum communication via entanglement distribution is of great importance for the quantum internet. However, scaling up to such long distances has proved challenging due to the loss of photons, which grows exponentially with the distance covered. Quantum repeaters could in theory be used to extend the distances over which entanglement can be distributed, but in practice hardware quality is still lacking. Furthermore, it is generally not clear how an improvement in a certain repeater parameter, such as memory quality or attempt rate, impacts the overall network performance, rendering the path toward scalable quantum repeaters unclear. In this work we propose a methodology based on genetic algorithms and simulations of quantum repeater chains for optimization of entanglement generation and distribution. By applying it to simulations of several different repeater chains, including real-world fiber topology, we demonstrate that it can be used to answer questions such as what are the minimum viable quantum repeaters satisfying given network performance benchmarks. This methodology constitutes an invaluable tool for the development of a blueprint for a pan-European quantum internet. We have made our code, in the form of NetSquid simulations and the smart-stopos optimization tool, freely available for use either locally or on high-performance computing centers.

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We provide two algorithms for computing the probability distribution of waiting time and fidelity in quantum repeater chains constructed from probabilistic components. Their polynomial runtimes improve upon existing algorithms’ exponential scaling.@en

Quantum communication enables a host of applications that cannot be achieved by classical communication means, with provably secure communication as one of the prime examples. The distance that quantum communication schemes can cover via direct communication is fundamentally limited by losses on the communication channel. By means of quantum repeaters, the reach of these schemes can be extended and chains of quantum repeaters could in principle cover arbitrarily long distances. In this work, we provide two efficient algorithms for determining the generation time and fidelity of the first generated entangled pair between the end nodes of a quantum repeater chain. The runtime of the algorithms increases polynomially with the number of segments of the chain, which improves upon the exponential runtime of existing algorithms. Our first algorithm is probabilistic and can analyze refined versions of repeater chain protocols which include intermediate entanglement distillation. Our second algorithm computes the waiting time distribution up to a pre-specified truncation time, has faster runtime than the first one and is moreover exact up to machine precision. Using our proof-of-principle implementation, we are able to analyze repeater chains of thousands of segments for some parameter regimes. The algorithms thus serve as useful tools for the analysis of large quantum repeater chain protocols and topologies of the future quantum internet.

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Quantum communication enables the implementation of tasks that are unachievable with classical resources. However, losses on the communication channel preclude the direct long-distance transmission of quantum information in many relevant scenarios. In principle quantum repeaters allow one to overcome losses. However, realistic hardware parameters make long-distance quantum communication a challenge in practice. For instance, in many protocols an entangled pair is generated that needs to wait in quantum memory until the generation of an additional pair. During this waiting time the first pair decoheres, impacting the quality of the final entanglement produced. At the cost of a lower rate, this effect can be mitigated by imposing a cut-off condition. For instance, a maximum storage time for entanglement after which it is discarded. In this work, we optimize the cut-offs for quantum repeater chains. First, we develop an algorithm for computing the probability distribution of the waiting time and fidelity of entanglement produced by repeater chain protocols which include a cut-off. Then, we use the algorithm to optimize cut-offs in order to maximize secret-key rate between the end nodes of the repeater chain. We find that the use of the optimal cut-off extends the parameter regime for which secret key can be generated and moreover significantly increases the secret-key rate for a large range of parameters.

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It is well known that observing nonlocal correlations allows us to draw conclusions about the quantum systems under consideration. In some cases this yields a characterisation which is essentially complete, a phenomenon known as self-testing. Self-testing becomes particularly interesting if we can make the statement robust, so that it can be applied to a real experimental setup. For the simplest self-testing scenarios the most robust bounds come from the method based on operator inequalities. In this work we elaborate on this idea and apply it to the family of tilted Clauser-Horne-Shimony-Holt (CHSH) inequalities. These inequalities are maximally violated by partially entangled two-qubit states and our goal is to estimate the quality of the state based only on the observed violation. For these inequalities we have reached a candidate bound and while we have not been able to prove it analytically, we have gathered convincing numerical evidence that it holds. Our final contribution is a proof that in the usual formulation, the CHSH inequality only becomes a self-test when the violation exceeds a certain threshold. This shows that self-testing scenarios fall into two distinct classes depending on whether they exhibit such a threshold or not.

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Quantum communication brings radically new capabilities that are provably impossible to attain in any classical network. Here, we take the first step from a physics experiment to a quantum internet system. We propose a functional allocation of a quantum network stack, and construct the first physical and link layer protocols that turn ad-hoc physics experiments producing heralded entanglement between quantum processors into a well-defined and robust service. This lays the groundwork for designing and implementing scalable control and application protocols in platform-independent software. To design our protocol, we identify use cases, as well as fundamental and technological design considerations of quantum network hardware, illustrated by considering the state-of-the-art quantum processor platform available to us (Nitrogen-Vacancy (NV) centers in diamond). Using a purpose built discrete-event simulator for quantum networks, we examine the robustness and performance of our protocol using extensive simulations on a supercomputing cluster. We perform a full implementation of our protocol in our simulator, where we successfully validate the physical simulation model against data gathered from the NV hardware. We first observe that our protocol is robust even in a regime of exaggerated losses of classical control messages with only little impact on the performance of the system. We proceed to study the performance of our protocols for 169 distinct simulation scenarios, including trade-offs between traditional performance metrics such as throughput, and the quality of entanglement. Finally, we initiate the study of quantum network scheduling strategies to optimize protocol performance for different use cases.

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We present experimental and theoretical results on suspensions of silica rods in DMSO-water, subjected to an applied electric field. The experimental results indicate that, if the electrode used for generating the electric field is in direct contact with the suspension, a fraction of the rods close to the electrode surface does not stand parallel to the field but instead lies flat on the electrode when the field is switched on. To explain these results theoretically, we modify the coupled dipole method to include “image dipoles”, and find that a rod close to the electrode experiences not only the expected global potential energy minimum at an orientation parallel to the electric field, but also a local minimum several times the thermal energy in depth for orientations parallel to the electrode surface. Additionally, we indicate how the magnitude of the potential energy depends on the electric field strength and include results not only for negatively polarizable (which correspond to the aforementioned experimental system), but also for positively polarizable rods.

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