We formulate the problem of finding the optimal entanglement swapping scheme in a quantum repeater chain as a Markov decision process and present its solution for different repeaters' sizes. Based on this, we are able to demonstrate that the commonly used "doubling" scheme for performing probabilistic entanglement swapping of probabilistically distributed entangled qubit pairs in quantum repeaters does not always produce the best possible raw rate. Focusing on this figure of merit, without considering additional probabilistic elements for error suppression such as entanglement distillation on higher "nesting levels," our approach reveals that a power-of-two number of segments has no privileged position in quantum repeater theory; the best scheme can be constructed for any number of segments. Moreover, classical communication can be included into our scheme, and we show how this influences the raw waiting time for different numbers of segments, confirming again the optimality of "nondoubling" in some relevant parameter regimes. Thus, our approach provides the minimal possible waiting time of quantum repeaters in a fairly general physical setting.
We formulate the problem of finding the optimal entanglement swapping scheme in a quantum repeater chain as a Markov decision process and present its solution for different repeaters' sizes. Based on this, we are able to demonstrate that the commonly used "doubling" scheme for performing probabilistic entanglement swapping of probabilistically distributed entangled qubit pairs in quantum repeaters does not always produce the best possible raw rate. Focusing on this figure of merit, without considering additional probabilistic elements for error suppression such as entanglement distillation on higher "nesting levels," our approach reveals that a power-of-two number of segments has no privileged position in quantum repeater theory; the best scheme can be constructed for any number of segments. Moreover, classical communication can be included into our scheme, and we show how this influences the raw waiting time for different numbers of segments, confirming again the optimality of "nondoubling" in some relevant parameter regimes. Thus, our approach provides the minimal possible waiting time of quantum repeaters in a fairly general physical setting.
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.
The rate at which quantum communication tasks can be performed using direct transmission is fundamentally hindered by the channel loss. Quantum repeaters allow one, in principle, to overcome these limitations, but their introduction necessarily adds an additional layer of complexity to the distribution of entanglement. This additional complexity---along with the stochastic nature of processes such as entanglement generation, Bell swaps, and entanglement distillation---makes finding good quantum repeater schemes nontrivial. We develop an algorithm that can efficiently perform a heuristic optimization over a subset of quantum repeater schemes for general repeater platforms. We find a strong improvement in the generation rate in comparison to an optimization over a simpler class of repeater schemes based on BDCZ (Briegel, D\"ur, Cirac, Zoller) repeater schemes. We use the algorithm to study three different experimental quantum repeater implementations on their ability to distribute entanglement, which we dub information processing implementations, multiplexed elementary pair generation implementations, and combinations of the two. We perform this heuristic optimization of repeater schemes for each of these implementations for a wide range of parameters and different experimental settings. This allows us to make estimates on what are the most critical parameters to improve for entanglement generation, how many repeaters to use, and which implementations perform best in their ability to generate entanglement.
Entanglement distribution via probabilistic entanglement swapping across a quantum repeater chain connecting two quantum nodes is a challenging problem. The difficulty lies in the exponential number of possible swapping structures within the repeater chain, necessitating efficient search algorithms, especially as the chain length increases. In this paper, we first explore the algorithmic design to facilitate the search for the optimal swapping structure along a repeater chain, aiming to maximize the entanglement distribution rate. Second, we examine the computational complexities of various algorithms and find that prior approaches exhibit excessively high complexities. Thus, we propose an efficient dynamic programming-based algorithm, FastHED, that leverages heuristics to expedite the search for the optimal swapping structure. Our theoretical analysis reveals that the upper bound of the proposed algorithm’s computational complexity isO(n(logn)3)(more precisely,O(n(logn)2loglogn)whenn≤229), a significant improvement over the existing algorithm with a complexity ofO(n2logn), where n denotes the repeater chain’s length. Additionally, we design a best-first framework to evaluate the performance of different algorithms. Numerical results show that our algorithm achieves a higher average entanglement distribution rate than existing algorithms.
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.
At the core of the quantum Internet lie quantum repeaters that enable remote end-to-end entanglement generation. Fundamentally, the entanglement generation rate and fidelity of quantum repeaters constitute the bottleneck for end-to-end performance. To achieve high rates, quantum repeaters employ quantum memory multiplexing. In a high-rate standard repeater, each memory sequentially generates an entanglement with its neighboring nodes and then applies entanglement swapping. This, however, results in low fidelity due to decoherence of the first-formed entanglement in the sequential generation process. By allocating different numbers of memories to simultaneously form entanglements with the left and right adjacent nodes, quantum repeaters reduce high waiting times and achieve high fidelity. In such a repeater, a mismatch problem arises due to the difference between the probabilistic number of generated entanglements on both sides. Consequently, some entanglements remain stored until opposite entanglements are available. The mismatch problem reduces the repeater rate and particularly the entanglement fidelity. In this paper, we consider the mismatch problem in an asymmetric repeater with different distances to its adjacent nodes. To mitigate the mismatch problem, we derive a dynamic optimal memory allocation. Under the optimal allocation, we derive statistical lower bounds on the achievable rate and fidelity. We demonstrate that the optimal allocation significantly improves the fidelity while maintaining a comparable rate to the standard repeater. In contrast, our results show that fixed memory allocation may be detrimental to the fidelity.
We develop entanglement swapping protocols and memory allocation methods for quantum repeater chains. Unlike most of the existing studies, the memory size of each quantum repeater in this work is a parameter that can be optimized. Based on Markov chain modeling of the entanglement distribution process, we determine the trade-off between the entanglement distribution rate and the memory size for temporal multiplexing techniques. We then propose three memory allocation methods that achieve entanglement distribution rates decaying polynomially with respect to the distance while using constant average memory slots per node. We also quantify the average number of memory slots required due to classical communication delay, as well as the delay of entanglement distribution. Our results show that a moderate memory size suffices to achieve a polynomial decay of entanglement distribution rate with respect to the distance, which is the scaling achieved by the optimal protocol even with infinite memory size at each node.