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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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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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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Noise and photon loss encountered on quantum channels pose a major challenge for reliable entanglement generation in quantum networks. In near-term networks, heralding is required to inform endpoints of successfully generated entanglement. If after heralding, entanglement fidelity is too low, entanglement purification may be utilized to probabilistically increase fidelity. Traditionally, purification protocols proceed as follows: generate heralded EPR pairs, execute a series of quantum operations on two or more pairs between two nodes, and classically communicate results to check for success. Purification may require several rounds while qubits are stored in memories, vulnerable to decoherence. In this work, we explore notions of optimistic purification, wherein classical communication required for heralding and purification is delayed, possibly to the end of the process. Optimism reduces the overall time EPR pairs are stored in memory, increasing fidelity while possibly decreasing EPR pair rate due to decreased heralding and purification failure. We apply optimism to the entanglement pumping scheme, ground- and satellite-based EPR generation sources, and current state-of-the-art purification circuits that include several measurement and purification checkpoints. We evaluate performance in view of a number of parameters, including link length, EPR source rate and fidelity; and memory coherence time. We show that while our optimistic protocol increases fidelity, the traditional approach may even decrease fidelity for longer distances. We study the trade-off between rate and fidelity under entanglement-based QKD, and find that optimistic schemes can yield higher rates compared to non-optimistic counterparts, with most advantages seen in scenarios with low initial fidelity and short coherence times.@en

Modular quantum computing architectures are a promising alternative to monolithic QPU (Quantum Processing Unit) designs for scaling up quantum devices. They refer to a set of interconnected QPUs or cores consisting of tightly coupled quantum bits that can communicate via quantum-coherent and classical links. In multi-core architectures, it is crucial to minimize the amount of communication between cores when executing an algorithm. Therefore, mapping a quantum circuit onto a modular architecture involves finding an optimal assignment of logical qubits (qubits in the quantum circuit) to different cores with the aim to minimize the number of expensive inter-core operations while adhering to given hardware constraints. In this paper, we propose for the first time a Quadratic Unconstrained Binary Optimization (QUBO) technique to encode the problem and the solution for both qubit allocation and inter-core communication costs in binary decision variables. To this end, the quantum circuit is split into slices, and qubit assignment is formulated as a graph partitioning problem for each circuit slice. The costly inter-core communication is reduced by penalizing inter-core qubit communications. The final solution is obtained by minimizing the overall cost across all circuit slices. To evaluate the effectiveness of our approach, we conduct a detailed analysis using a representative set of benchmarks having a high number of qubits on two different multi-core architectures. Our method showed promising results and performed exceptionally well with very dense and highly-parallelized circuits that require on average 0.78 inter-core communications per two-qubit gate.

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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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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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We study a quantum switch that distributes tripartite entangled states to sets of users. The entanglement switching process requires two steps: First, each user attempts to generate bipartite entanglement between itself and the switch, and second, the switch performs local operations and a measurement to create multipartite entanglement for a set of three users. In this work, we study a simple variant of this system, wherein the switch has infinite memory and the links that connect the users to the switch are identical. This problem formulation is of interest to several distributed quantum applications, while the technical aspects of this work result in new contributions within queueing theory. The state of the system is modeled as continuous-time Markov chain (CTMC), and performance metrics of interest (probability of an empty system, switch capacity, expectation, and variance of the number of qubit-pairs stored) are computed via the solution of a two-dimensional functional equation obtained by reducing it to a boundary value problem on a closed curve. This work is a follow-up of Nain et al. (Proc ACM Measure Anal Comput Syst(POMACS) 4, 2020) where a switch distributing entangled multipartite states to sets of users was studied, but only the switch capacity and the expected number of stored qubits were derived.

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Entanglement generation over large distances is a challenging task due to photon loss, which grows exponentially with link length (e.g., optical fiber). Moreover, EPR pair generation over noisy channels results in imperfect states. Thus, a link is associated with a success probability and an initial fidelity for successfully-generated entanglement. Over time, fidelity de-creases due to environmental noise. Entanglement purification is a way of probabilistically increasing state fidelity. Traditionally, purification protocols perform the following steps: generate two or more EPR pairs between two nodes, execute a series of quantum gates on both sides, and exchange results via classical messages. In this work, we propose a twist on the original entanglement pumping scheme, wherein a fixed number of purification steps are performed successively, before the classical information exchange. This modification introduces optimism to the overall procedure, such that classical communication is performed only at the end, to check the measurement results. The protocol is also optimistic about entanglement generation and does not wait for heralding signals before attempting purification. We study the effect of being optimistic on fidelity and the waiting time to generate an entangled pair as a function of the distance between network nodes. We observe an improvement in both fidelity and expected waiting time. We also study the effect of the number of purification steps on fidelity and latency and observe that typically no more than two or three purification steps are needed when limited by only two memories.

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Distributed quantum applications impose requirements on the quality of the quantum states that they consume. When analyzing architecture implementations of quantum hardware, characterizing this quality forms an important factor in understanding their performance. Fundamental characteristics of quantum hardware lead to inherent tradeoffs between the quality of states and traditional performance metrics such as throughput. Furthermore, any real-world implementation of quantum hardware exhibits time-dependent noise that degrades the quality of quantum states over time. Here, we study the performance of two possible architectures for interfacing a quantum processor with a quantum network. The first corresponds to the current experimental state of the art in which the same device functions both as a processor and a network device. The second corresponds to a future architecture that separates these two functions over two distinct devices. We model these architectures as continuous-time Markov chains and compare their quality of executing quantum operations and producing entangled quantum states as functions of their memory lifetimes, as well as the time that it takes to perform various operations within each architecture. As an illustrative example, we apply our analysis to architectures based on Nitrogen-Vacancy centers in diamond, where we find that for present-day device parameters one architecture is more suited to computation-heavy applications, and the other for network-heavy ones. We validate our analysis with the quantum network simulator NetSquid. Besides the detailed study of these architectures, a novel contribution of our work are several formulas that connect an understanding of waiting time distributions to the decay of quantum quality over time for the most common noise models employed in quantum technologies. This provides a valuable new tool for performance evaluation experts, and its applications extend beyond the two architectures studied in this work.

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