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 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 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.
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.
We study the capacity of a memory-based quantum repeater in entanglement swapping between two quantum links with either single or multiple memories, which we refer to as the end-to-end (E2E) entanglement throughput, subject to a constraint on the minimum fidelity. In order to approximate the E2E entanglement throughput, we adopt queueing models, where quantum links can have different characteristics: memory capacities, entanglement attempt rates and success probabilities, as well as classical communication latencies. We develop a model for estimating E2E entanglement fidelity, while taking into account the heterogeneous dephasing and depolarizing dynamics of quantum memories and Bell-state measurements in entanglement swapping as well as classical communication delays and noises. Finally, with the help of our models for approximating the E2E entanglement throughput and fidelity, we use the maximum waiting times of entanglements in quantum memories at the repeater as optimization variables to maximize the E2E entanglement throughput while ensuring required minimum E2E 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.
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 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.