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.
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.
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.
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.
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.
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.
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.