Quantum protocols commonly require a certain number of quantum resource states to be available simultaneously. An important class of examples is quantum network protocols that require a certain number of entangled pairs. Here, we consider a setting in which a process generates a quantum resource state with some probability p in each time step and stores it in a quantum memory that is subject to time-dependent noise. To maintain sufficient quality for an application, each resource state is discarded from the memory after w time steps. Let s be the number of desired resource states required by a protocol. We characterize the probability distribution X-{(w,s)} of the ages of the quantum resource states, once s states have been generated in a window w. Combined with a time-dependent noise model, knowledge of this distribution allows for the calculation of fidelity statistics of the s quantum resources. We also give exact solutions for the first and second moments of the waiting time \tau -{(w,s)} until s resources are produced within a window w, which provides information about the rate of the protocol. Since it is difficult to obtain general closed-form expressions for statistical quantities describing the expected waiting time \mathbb {E}(\tau -{(w,s)}) and the distribution X-{(w,s)}, we present two novel results that aid their computation in certain parameter regimes. The methods presented in this work can be used to analyze and optimize the execution of quantum protocols. Specifically, with an example of a blind quantum computing protocol, we illustrate how they may be used to infer w and p to optimize the rate of successful protocol execution.

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Small interconnected quantum processors can collaborate to tackle quantum computational problems that typically demand more capable devices. These linked processors, referred to as quantum nodes, can use shared entangled states to execute nonlocal operations. As a consequence, understanding how to distribute entangled states among nodes is essential for developing hardware and software. We analyze a protocol where entanglement is continuously distributed among nodes that are physically arranged in a regular pattern: a chain, a honeycomb lattice, a square grid, and a triangular lattice. These regular patterns allow for the modular expansion of networks for large-scale distributed quantum computing. Within the distribution protocol, we investigate how nodes can optimize the frequency of attempting entanglement swaps, trading off multiple entangled states shared with neighboring nodes for fewer states shared with non-neighboring nodes. We evaluate the protocol's performance using the virtual neighborhood size - a metric indicating the number of other nodes with which a given node shares entangled states. Employing numerical methods, we find that nodes must perform more swaps to maximize the virtual neighborhood size when coherence times are short. In a chain network, the virtual neighborhood size's dependence on swap attempt frequency differs for each node based on its distance from the end of the chain. Conversely, all nodes in the square grid exhibit a qualitatively similar dependence of the virtual neighborhood size on the swap frequency.

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We conduct a numerical investigation of fiber-based entanglement distribution over distances of up to 1600 km using a chain of processing-node quantum repeaters. We determine minimal hardware requirements while simultaneously optimizing over protocols for entanglement generation and entanglement purification, as well as over strategies for entanglement swapping. Notably, we discover that through an adequate choice of protocols the hardware improvement cost scales linearly with the distance covered. Our results highlight the crucial role of good protocol choices in significantly reducing hardware requirements, such as employing purification to meet high-fidelity targets and adopting a swap as soon as possible policy for faster rates. To carry out this analysis, we employ an extensive simulation framework implemented with NetSquid, a discrete-event-based quantum-network simulator, and a genetic-algorithm-based optimization methodology to determine minimal hardware requirements.@en

Quantum networks crucially rely on the availability of high-quality entangled pairs of qubits, known as entangled links, distributed across distant nodes. Maintaining the quality of these links is a challenging task due to the presence of time-dependent noise, also known as decoherence. Entanglement purification protocols offer a solution by converting multiple low-quality entangled states into a smaller number of higher-quality ones. In this work, we introduce a framework to analyse the performance of entanglement buffering setups that combine entanglement consumption, decoherence, and entanglement purification. We propose two key metrics: the availability, which is the steady-state probability that an entangled link is present, and the average consumed fidelity, which quantifies the steady-state quality of consumed links. We then investigate a two-node system, where each node possesses two quantum memories: one for long-term entanglement storage, and another for entanglement generation. We model this setup as a continuous-time stochastic process and derive analytical expressions for the performance metrics. Our findings unveil a trade-off between the availability and the average consumed fidelity. We also bound these performance metrics for a buffering system that employs the well-known bilocal Clifford purification protocols. Importantly, our analysis demonstrates that, in the presence of noise, consistently purifying the buffered entanglement increases the average consumed fidelity, even when some buffered entanglement is discarded due to purification failures.@en

Restrictions imposed by existing infrastructure can make it hard to ensure an even spacing between the nodes of future fiber-based quantum networks. We investigate here the negative effects of asymmetric node placement by considering separately the placement of midpoint stations required for heralded entanglement generation, as well as of processing-node quantum repeaters in a chain. For midpoint stations, we describe the effect asymmetry has on the time required to perform one entangling attempt, the success probability of such an attempt, and the fidelity of the entangled states created. This includes accounting for the effects of chromatic dispersion on photon indistinguishability. For quantum-repeater chains, we numerically investigate how uneven spacing between repeater nodes leads to bottlenecks, thereby increasing both the waiting time and the time states are stored in noisy quantum memory. We find that while the time required to perform one entangling attempt may increase linearly with the midpoint's asymmetry, the success probability and fidelity of heralded entanglement generation and the distribution time and error rate for repeater chains all have vanishing first derivatives with respect to the amount of asymmetry. This suggests resilience of quantum-network performance against small amounts of asymmetry.@en

Distributing quantum states reliably among distant locations is a key challenge in the field of quantum networks. One-way quantum networks address this by using one-way communication and quantum error correction. Here, we analyze quantum circuit switching as a protocol to distribute quantum states in one-way quantum networks. In quantum circuit switching, pairs of users can request the delivery of multiple quantum states from one user to the other. After waiting for approval from the network, the states can be distributed either sequentially, forwarding one at a time along a path of quantum repeaters, or in parallel, sending batches of quantum states from repeater to repeater. Since repeaters can only forward a finite number of quantum states at a time, a pivotal question arises: is it advantageous to send them sequentially (allowing for multiple requests simultaneously) or in parallel (reducing processing time but handling only one request at a time)? We compare both approaches in a quantum network with a star topology. Using tools from queuing theory, we show that requests are met at a higher rate when packets are distributed in parallel, although sequential distribution can generally provide service to a larger number of users simultaneously. We also show that using a large number of quantum repeaters to combat channel losses limits the maximum distance between users, as each repeater introduces additional processing delays. These findings provide insight into the design of protocols for distributing quantum states in one-way quantum networks.@en

We consider the problem of multipath entanglement distribution to a pair of nodes in a quantum network consisting of devices with nondeterministic entanglement swapping capabilities. Multipath entanglement distribution enables a network to establish end-to-end entangled links across any number of available paths with preestablished link-level entanglement. Probabilistic entanglement swapping, on the other hand, limits the amount of entanglement that is shared between the nodes; this is especially the case when, due to practical constraints, swaps must be performed in temporal proximity to each other. Limiting our focus to the case where only bipartite entanglement is generated across the network, we cast the problem as an instance of generalized flow maximization between two quantum end nodes wishing to communicate. We propose a mixed-integer quadratically constrained program (MIQCP) to solve this flow problem for networks with arbitrary topology. We then compute the overall network capacity, defined as the maximum number of Einstein-Podolsky-Rosen (EPR) states distributed to users per time unit, by solving the flow problem for all possible network states generated by probabilistic entangled link presence and absence, and subsequently by averaging over all network state capacities. The MIQCP can also be applied to networks with multiplexed links. While our approach for computing the overall network capacity has the undesirable property that the total number of states grows exponentially with link multiplexing capability, it nevertheless yields an exact solution that serves as an upper bound comparison basis for the throughput performance of more easily implementable yet nonoptimal entanglement routing algorithms.

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In blind quantum computing (BQC), a user with a simple client device can perform a quantum computation on a remote quantum server such that the server cannot gain knowledge about the computation. Here, we numerically investigate hardware requirements for verifiable BQC using an ion trap as server and a distant measurement-only client. While the client has no direct access to quantum-computing resources, it can remotely execute quantum programs on the server by measuring photons emitted by the trapped ion. We introduce a numerical model for trapped-ion quantum devices in NetSquid, a discrete-event simulator for quantum networks. Using this, we determine the minimal hardware requirements on a per-parameter basis to perform the verifiable BQC protocol. We benchmark these for a five-qubit linear graph state, with which any single-qubit rotation can be performed, where client and server are separated by 50 km. Current state-of-the-art ion traps satisfy the minimal requirements on a per-parameter basis, but all current imperfections combined make it impossible to perform the blind computation securely over 50 km using existing technology. Using a genetic algorithm, we determine the set of hardware parameters that minimises the total improvements required, finding directions along which to improve hardware to reach our threshold error probability that would enable experimental demonstration. In this way, we lay a path for the near-term experimental progress required to realise the implementation of verifiable BQC over a 50 km distance.@en

We perform a numerical study of the distribution of entanglement on a real-world fiber grid connecting the German cities of Bonn and Berlin. The connection is realized using a chain of processing-node quantum repeaters spanning roughly 900 kilometers. Their placement is constrained by the fiber grid we consider, resulting in asymmetric links. We investigate how minimal hardware requirements depend on the target application, as well as on the number of repeaters in the chain. We find that requirements for blind quantum computing are markedly different than those for quantum key distribution, with the required coherence time being around two and a half times larger for the former. Further, we observe a trade-off regarding how target secret-key rates are achieved when using different numbers of repeaters: comparatively low-quality entangled states generated at a high rate are preferred for higher numbers of repeaters, whereas comparatively high-quality states generated at a lower rate are favored for lower numbers of repeaters. To obtain our results we employ an extensive simulation framework implemented using NetSquid, a discrete-event simulator for quantum networks. These are combined with an optimization methodology based on genetic algorithms to determine minimal hardware requirements.@en

We report on the realization of a multi-node quantum network. Using the network, we have demonstrated three protocols; generation of a entangled state shared by all nodes, entanglement swapping and quantum teleportation between non-neighboring nodes.

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Network Utility Maximization (NUM) is a mathe-matical framework that has endowed researchers with powerful methods for designing and analyzing classical communication protocols. NUM has also enabled the development of distributed algorithms for solving the resource allocation problem, while at the same time providing certain guarantees, e.g., that of fair treatment, to the users of a network. We extend here the notion of NUM to quantum networks, and propose three quantum utility functions - each incorporating a different entanglement measure. We aim both to gain an understanding of some of the ways in which quantum users may perceive utility, as well as to explore structured and theoretically-motivated methods of simultaneously servicing multiple users in distributed quantum systems. Using our quantum NUM constructions, we develop an optimization framework for networks that use the single-photon scheme for entanglement generation, which enables us to solve the resource allocation problem while exploring rate-fidelity tradeoffs within the network topologies that we consider. We learn that two of our utility functions, which are based on distillable entanglement and secret key fraction, are in close agreement with each other and produce similar solutions to the optimization problems we study. While these two utilities place a higher emphasis on end-to-end fidelity, our third utility- based on entanglement negativity - has more favorable mathematical properties, and tends to place a higher value on the rate at which users receive entangled resources. These contrasting behaviors thus provide ideas regarding the suitability of quantum network utility definitions to different quantum applications.

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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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We study the limits of bipartite entanglement distribution using a chain of quantum repeaters that have quantum memories. To generate end-to-end entanglement, each node can attempt the generation of an entangled link with a neighbor, or perform an entanglement swapping measurement. A maximum storage time, known as cutoff, is enforced on the memories to ensure high-quality entanglement. Nodes follow a policy that determines when to perform each operation. Global-knowledge policies take into account all the information about the entanglement already produced. Here, we find global-knowledge policies that minimize the expected time to produce end-to-end entanglement. Our methods are based on Markov decision processes and value and policy iteration. We compare optimal policies to a policy in which nodes only use local information. We find that the advantage in expected delivery time provided by an optimal global-knowledge policy increases with increasing number of nodes and decreasing probability of successful swapping.

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Entanglement between quantum network nodes is often produced using intermediary devices - such as heralding stations - as a resource. When scaling quantum networks to many nodes, requiring a dedicated intermediary device for every pair of nodes introduces high costs. Here, we propose a cost-effective architecture to connect many quantum network nodes via a central quantum network hub called an entanglement generation switch (EGS). The EGS allows multiple quantum nodes to be connected at a fixed resource cost, by sharing the resources needed to make entanglement. We propose an algorithm called the rate control protocol, which moderates the level of competition for access to the hub's resources between sets of users. We proceed to prove a convergence theorem for rates yielded by the algorithm. To derive the algorithm we work in the framework of network utility maximization and make use of the theory of Lagrange multipliers and Lagrangian duality. Our EGS architecture lays the groundwork for developing control architectures compatible with other types of quantum network hubs as well as system models of greater complexity.

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We study the performance (rate and fidelity) of distributing multipartite entangled states in a quantum network through the use of a central node. Specifically, we consider the scenario where the multipartite entangled state is first prepared locally at a central node and then transmitted to the end nodes of the network through quantum teleportation. As our first result, we present leading-order analytical expressions and lower bounds for both the rate and fidelity at which a specific class of multipartite entangled states, namely, Greenberger-Horne-Zeilinger (GHZ) states, are distributed. Our analytical expressions for the fidelity accurately account for time-dependent depolarizing noise encountered by individual quantum bits while stored in quantum memory, as verified using Monte Carlo simulations. As our second result, we compare the performance to the case where the central node is an entanglement switch and the GHZ state is created by the end nodes in a distributed fashion. Apart from these two results, we outline how the teleportation-based scheme could be physically implemented using trapped ions or nitrogen-vacancy centers in diamond.

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We propose network benchmarking: a procedure to efficiently benchmark the quality of a quantum network link connecting quantum processors in a quantum network. This procedure is based on the standard randomized benchmarking protocol and provides an estimate for the fidelity of a quantum network link. We provide statistical analysis of the protocol as well as a simulated implementation inspired by nitrogen-vacancy center systems using Netsquid, a special purpose simulator for noisy quantum networks.

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Entangled states shared among distant nodes are frequently used in quantum network applications. When quantum resources are abundant, entangled states can be continuously distributed across the network, allowing nodes to consume them whenever necessary. This continuous distribution of entanglement enables quantum network applications to operate continuously while being regularly supplied with entangled states. Here, we focus on the steady-state performance analysis of protocols for continuous distribution of entanglement. We propose the virtual neighborhood size and the virtual node degree as performance metrics. We utilize the concept of Pareto optimality to formulate a multiobjective optimization problem to maximize the performance. As an example, we solve the problem for a quantum network with a tree topology. One of the main conclusions from our analysis is that the entanglement consumption rate has a greater impact on the protocol performance than the fidelity requirements. The metrics that we establish in this paper can be utilized to assess the feasibility of entanglement distribution protocols for large-scale quantum networks.

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Entanglement between quantum network nodes is often produced using intermediary devices - such as heralding stations - as a resource. When scaling quantum networks to many nodes, requiring a dedicated intermediary device for every pair of nodes introduces high costs. Here, we propose a cost-effective architecture to connect many quantum network nodes via a central quantum network hub called an Entanglement Generation Switch (EGS). The EGS allows multiple quantum nodes to be connected at a fixed resource cost, by sharing the resources needed to make entanglement. We propose an algorithm called the Rate Control Protocol (RCP) which moderates the level of competition for access to the hub's resources between sets of users. We proceed to prove a convergence theorem for rates yielded by the algorithm. To derive the algorithm we work in the framework of Network Utility Maximization (NUM) and make use of the theory of Lagrange multipliers and Lagrangian duality. Our EGS architecture lays the groundwork for developing control architectures compatible with other types of quantum network hubs as well as system models of greater complexity.

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The past decade has seen tremendous progress in experimentally realizing the building blocks of quantum repeaters. Repeater architectures with multiplexed quantum memories have been proposed to increase entanglement distribution rates, but an open challenge is to maintain entanglement fidelity over long-distance links. Here, we address this with a quantum router architecture comprising many quantum memories connected in a photonic switchboard to broker entanglement flows across quantum networks. We compute the rate and fidelity of entanglement distribution under this architecture using an event-based simulator, finding that the router improves the entanglement fidelity as multiplexing depth increases without a significant drop in the entanglement distribution rate. Specifically, the router permits channel-loss-invariant fidelity, i.e. the same fidelity achievable with lossless links. Furthermore, this scheme automatically prioritizes entanglement flows across the full network without requiring global network information. The proposed architecture uses present-day photonic technology, opening a path to near-term deployable multi-node quantum networks.

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