Entanglement distribution rate (EDR) is a fundamental performance metric for quantum networks. Existing EDR models often focus on expected throughput, but expectation alone is insufficient for Quality-of-Service (QoS)-aware provisioning, where a network must satisfy user demands with a specified probability rather than only on average. This paper studies the distribution of end-to-end EDR over linear quantum repeater chains. Existing works for exact distribution calculation becomes computationally expensive as chain width and length increase. To overcome this, we develop a moment-based analytical approximation that represents each intermediate segment throughput as a Gaussian distribution with matched mean and variance. The algorithm runs in O(L) time for a chain of length L while retaining the accuracy of cumulative distribution information needed for QoS probability queries. Evaluations against exact recursive computation show that the approximation preserves the main throughput trends, achieves speedups up to 6,000×, and maintains modest distributional error in the tested regimes. These results demonstrate that the proposed approximation provides an efficient analytical foundation for QoS-aware EDR analysis and future optimization of quantum repeater chains.
Entanglement distribution rate (EDR) is a fundamental performance metric for quantum networks. Existing EDR models often focus on expected throughput, but expectation alone is insufficient for Quality-of-Service (QoS)-aware provisioning, where a network must satisfy user demands with a specified probability rather than only on average. This paper studies the distribution of end-to-end EDR over linear quantum repeater chains. Existing works for exact distribution calculation becomes computationally expensive as chain width and length increase. To overcome this, we develop a moment-based analytical approximation that represents each intermediate segment throughput as a Gaussian distribution with matched mean and variance. The algorithm runs in O(L) time for a chain of length L while retaining the accuracy of cumulative distribution information needed for QoS probability queries. Evaluations against exact recursive computation show that the approximation preserves the main throughput trends, achieves speedups up to 6,000×, and maintains modest distributional error in the tested regimes. These results demonstrate that the proposed approximation provides an efficient analytical foundation for QoS-aware EDR analysis and future optimization of quantum repeater chains.
A quantum network distributes quantum entanglements between remote nodes, and is key to many applications in secure communication, quantum sensing and distributed quantum computing. This paper explores the fundamental trade-off between the throughput and the quality of entanglement distribution in a multi-hop quantum repeater network. Compared to existing work which aims to heuristically maximize the entanglement distribution rate (EDR) and/or entanglement fidelity, our goal is to characterize the maximum achievable worst-case fidelity, while satisfying a bound on the maximum achievable expected EDR between an arbitrary pair of quantum nodes. This characterization will provide fundamental bounds on the achievable performance region of a quantum network, which can assist with the design of quantum network topology, protocols and applications. However, the task is highly non-trivial and is NP-hard as we shall prove. Our main contribution is a fully polynomial-time approximation scheme to approximate the achievable worst-case fidelity subject to a strict expected EDR bound, combining an optimal fidelity-agnostic EDR-maximizing formulation and a worst-case isotropic noise model. The EDR and fidelity guarantees can be implemented by a post-selection-and-storage protocol with quantum memories. By developing a discrete-time quantum network simulator, we conduct simulations to show the characterized performance region (the approximate Pareto frontier) of a network, and demonstrate that the designed protocol can achieve the performance region while existing protocols exhibit a substantial gap.
Losses are one of the main bottlenecks for the distribution of entanglement in quantum networks, which can be overcome by the implementation of quantum repeaters. The most basic form of a quantum repeater chain is the swap ASAP repeater chain. In such a repeater chain, elementary links are probabilistically generated and deterministically swapped as soon as two adjacent links have been generated. As each entangled state is waiting to be swapped, decoherence is experienced, turning the fidelity of the entangled state between the end nodes of the chain into a random variable. Fully characterizing the (average) fidelity as the repeater chain grows is still an open problem. Here, we analytically investigate the case of equally-spaced repeaters, where we find exact analytic formulae for all moments of the fidelity up to 25 segments. We obtain these formulae by providing a general solution in terms of a generating function ; a function whose n'th term in its Maclaurin series yields the moments of the fidelity for n segments. We generalize this approach as well to a global cut-off policy – a method for increasing fidelity at the cost of longer entanglement delivery times – allowing for fast optimization of the cut-off parameter by eliminating the need for Monte Carlo simulation. We furthermore find simple approximations of the average fidelity that are exponentially tight, and, for up to 10 segments, the full distribution of the delivered fidelity. We use this to analytically calculate the secret-key rate, both with and without binning methods.
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.
Quantum repeater chains will form the backbone of future quantum networks that distribute entanglement between network nodes. Therefore, it is important to understand the entanglement distribution performance of quantum repeater chains, especially their throughput and latency. By using Markov chains to model the stochastic dynamics in quantum repeater chains, we offer analytical estimations for long-run throughput and on-demand latency of continuous entanglement distribution. We first study single-link entanglement generation using general multiheralded protocols. We then model entanglement distribution with entanglement swapping over two links, using either a single- or a double-heralded entanglement generation protocol. We also demonstrate how the two-link results offer insights into the performance of general2k-link nested repeater chains. Our results enrich the quantitative understanding of quantum repeater network performance, especially the dependence on system parameters. The analytical formulae themselves are valuable reference resources for the quantum networking community. They can serve as benchmarks for quantum network simulation validation or as examples of quantum network dynamics modeling using the Markov chain formalism.
The standard approach to realize a quantum repeater relies upon probabilistic but heralded entangled state manipulations and the storage of quantum states while waiting for successful events. In the literature on this class of repeaters, calculating repeater rates has typically depended on approximations assuming sufficiently small probabilities. Here we propose an exact and systematic approach including an algorithm based on Markov chain theory to compute the average waiting time (and hence the transmission rates) of quantum repeaters with arbitrary numbers of links. For up to four repeater segments, we explicitly give the exact rate formulas for arbitrary entanglement swapping probabilities. Starting with three segments, we explore schemes with arbitrary (not only doubling) and dynamical (not only predetermined) connections. The effect of finite memory times is also considered and the relative influence of the classical communication (of heralded signals) is shown to grow significantly for larger probabilities. Conversely, we demonstrate that for small swapping probabilities the statistical behavior of the waiting time in a quantum repeater cannot be characterized by its average value alone and additional statistical quantifiers are needed. For large repeater systems, we propose a recursive approach based on exactly but still efficiently computable waiting times of sufficiently small subrepeaters. This approach leads to better lower bounds on repeater rates compared to existing schemes.
We analyze an entanglement-based quantum key distribution (QKD) architecture that uses a linear chain of quantum repeaters employing photon-pair sources, spectral-multiplexing, linear-optic Bell-state measurements, multimode quantum memories, and classical-only error correction. Assuming perfect sources, we find an exact expression for the secret-key rate, and an analytical description of how errors propagate through the repeater chain, as a function of various loss-and-noise parameters of the devices. We show via an explicit analytical calculation, which separately addresses the effects of the principle nonidealities, that this scheme achieves a secret-key rate that surpasses the Takeoka-Guha-Wilde bound---a recently found fundamental limit to the rate-vs-loss scaling achievable by any QKD protocol over a direct optical link---thereby providing one of the first rigorous proofs of the efficacy of a repeater protocol. We explicitly calculate the end-to-end shared noisy quantum state generated by the repeater chain, which could be useful for analyzing the performance of other non-QKD quantum protocols that require establishing long-distance entanglement. We evaluate that shared state's fidelity and the achievable entanglement-distillation rate, as a function of the number of repeater nodes, total range, and various loss-and-noise parameters of the system. We extend our theoretical analysis to encompass sources with nonzero two-pair-emission probability, using an efficient exact numerical evaluation of the quantum state propagation and measurements. We expect our results to spur formal rate-loss analysis of other repeater protocols and also to provide useful abstractions to seed analyses of quantum networks of complex topologies.