Faithfully Simulating Near-Term Quantum Repeaters

Quantum repeaters have long been established to be essential for distributing entanglement over long distances. Consequently, their experimental realization constitutes a core challenge of quantum communication. However, there are numerous open questions about implementation details for realistic near-term experimental setups. In order to assess the performance of realistic repeater protocols, here we present ReQuSim, a comprehensive Monte Carlo–based simulation platform for quantum repeaters that faithfully includes loss and models a wide range of imperfections such as memories with time-dependent noise. Our platform allows us to perform an analysis for quantum repeater setups and strategies that go far beyond known analytical results: This refers to being able to both capture more realistic noise models and analyze more complex repeater strategies. We present a number of findings centered around the combination of strategies for improving performance, such as entanglement purification and the use of multiple repeater stations, and demonstrate that there exist complex relationships between them. We stress that numerical tools such as ours are essential to model complex quantum communication protocols aimed at contributing to the quantum Internet.

Popular Summary

Quantum repeaters are a key technology for quantum networks, contributing to the vision of the quantum Internet because they make possible the reaching of long distances in quantum communication. While the basic mechanisms are established and well understood, there is a need to analyze advanced strategies that can cope with realistic noise and imperfections in the system. We have developed a comprehensive simulation tool for quantum repeaters that we use to analyze and enhance multiple methods aimed at considerably extending the reachable distances.

We not only demonstrate that our simulation can be applied to a variety of scenarios with a breadth of error models but also show how we can use this technique to give actionable guidance on the choice of strategy for a given setup. For one repeater station, we analyze two approaches that exchange a reduced rate for an improvement in fidelity and quantify in which parameter regimes employing a more complex protocol needing additional operations leads to a meaningful improvement in performance. Furthermore, by analyzing the behavior of a quantum repeater with multiple repeater stations, we show that that improving the quality of repeater stations can open up new options to modify the protocol, which result in a disproportionate improvement.

Our work improves the understanding of the intricacies of quantum repeater protocols and contributes to the effort of identifying good design principles for quantum repeaters, which would allow quantum communication over arbitrary distances.

Figure 1

An entanglement-swapping-based repeater protocol in the ReQuSim simulation framework. Running the protocol schedules first events on the event queue. The events on the queue are successively resolved and may schedule further events. An example of this is highlighted above for an entanglement purification event scheduling an entanglement swapping event.

Figure 2

The pseudocode for formulating a protocol and running the simulation. This example describes a protocol for a single repeater station that has access to multiple quantum memories to simultaneously attempt to establish an entangled connection for both repeater links and immediately performs entanglement swapping once successful qubits for both directions are in memory. The events are standardized ways to interact with the current state of the simulation. They have a method that specifies the associated quantum operation that is performed on the quantum states when they are resolved, e.g., a Bell-state measurement for the entanglement swapping event.

Figure 3

An illustration of a step-by-step update of the state of the simulation and the event queue. The source event (representing the successful establishment of an entangled pair between neighboring repeater stations) triggers the protocol to schedule two new entanglement swapping events, inserting them at the front of the event queue. Resolving the entanglement swapping events establishes a long-distance pair between the end stations. After each step, i.e., after every event that is resolved, the protocol again has the chance to react to the new situation.

Figure 4

The obtainable key rates for two repeater links with $\text{zero (blue)}$, $\text{one (red)}$, or $\text{two (green)}$ entanglement purification steps before entanglement swapping. (a) $F_{\mathrm{init}}=0.95$. Even though no entanglement purification protocol (EPP) is necessary for most distances, an EPP can help to extend the reachable distance. (b) $F_{\mathrm{init}}=0.935$. At worse initial fidelity, an EPP can improve the key rate for all distances. In both cases, it is also apparent that the number of purification steps needs to be managed carefully. For these parameters, using two purification steps is actually detrimental. The other parameters are as follows: $T_{\mathrm{dp}}=100\mathrm{ms}$, $P_{\mathrm{link}}=0.5$, and $p_d={10}^{-6}$, with four memories per repeater link.

Figure 5

The increase in the achievable distance (i.e., with a nonzero key rate) using a repeater protocol with one entanglement purification step compared to a repeater protocol without entanglement purification. The other parameters are as follows: $P_{\mathrm{link}}=0.5$ and $p_d={10}^{-6}$, with four memories per repeater link.

Figure 6

The achievable key rates for two repeater links with a combination of strategies, without entanglement purification and with cutoff times $t_{\mathrm{cut}}=\mathrm{\infty }\text{(red)}$ or $50\mathrm{ms}\text{(blue)}$. Alternatively, a protocol with one step of entanglement purification and cutoff times $t_{\mathrm{cut}}=\mathrm{\infty }\text{(green)}$ or $650\mathrm{ms}\text{(yellow)}$. Properly optimized cutoff times can clearly help in both cases. The other parameters are as follows: $T_{\mathrm{dp}}=1\mathrm{s}$, $P_{\mathrm{link}}=0.5$, and $F_{\mathrm{init}}=0.925$, with two memories per repeater link.

Figure 7

The obtainable key rates when using protocols with $\text{two (blue)}$, $\text{four (red)}$, $\text{eight (green)}$, $\text{16 (yellow)}$, or $\text{32 (cyan)}$ repeater links. (a) With $F_{\mathrm{init}}=1.0$, the utilization of more repeater links is generally favorable. (b) Even with small constant overheads per repeater link ($F_{\mathrm{init}}=0.998$), trade-offs become apparent. The other parameters are as follows: $T_{\mathrm{dp}}=10\mathrm{ms}$, $P_{\mathrm{link}}=0.5$, and $p_d={10}^{-6}$.

Figure 8

Key rates for $\text{eight (green)}$ or $\text{16 (yellow)}$ repeater links, with varying initial fidelities: $F_{\mathrm{init}}=0.996\text{ (crosses), }0.997\text{ (plus signs)},\text{or }0.998\text{ (circles)}$. Improving some experimental parameters can enable alternative strategies. For example, when connecting end stations 125 km apart, improving $F_{\mathrm{init}}$ from $0.996$ to $0.998$ allows changing from an eight-link protocol to a 16-link protocol, which improves the obtainable key rate more than the increase in $F_{\mathrm{init}}$ alone. The other parameters are as follows: $T_{\mathrm{dp}}=10\mathrm{ms}$, $P_{\mathrm{link}}=0.5$, and $p_d={10}^{-6}$.

Figure 9

Entanglement purification can extend achievable ranges for quantum repeaters with multiple links. Key rates without entanglement purification (circles) and with one step of entanglement purification at the lowest level (crosses) when using $\text{two (blue)}$, $\text{four (red)}$, $\text{eight (green)}$, $\text{16 (yellow)}$, or $\text{32 (cyan)}$ repeater links. The parameters are as follows: $T_{\mathrm{dp}}=100\text{ ms}$, $F_{\mathrm{init}}=0.99$, $P_{\mathrm{link}}=0.5$, and $p_d={10}^{-6}$, with two quantum memories per repeater link at each station.

Figure 10

The impact of improving certain hardware parameters for a custom error model, using a repeater protocol with four repeater links and one entanglement purification step at the lowest repeater level. Two quantum memories per connected link are available at each station. The parameters are as follows: $P_{\mathrm{link}}=0.01$, $p_d={10}^{-6}$, and $F_{\mathrm{init}}=0.99$. For a total distance of $\text{20 (blue)}$, $\text{35 (red)}$, $\text{50 (green)}$, $\text{75 (yellow)}$, or $\text{100 (cyan)}$ km: (a) varying memory quality with a fixed $p_{\mathrm{gate}}=0.99$; (b) varying quality of two-qubit gates with fixed $T_{\mathrm{damp}}=1\mathrm{s}$ (no positive key rate for $75$ or $100$ km).

Figure 11

The scaling of the run times of the simulation with the number of repeater links. This is for a setup with single memories and no entanglement purification, as in Sec. 4b. Two methods of scheduling events are compared: a general adjustable method indicative of the protocols used in the rest of this work $\text{(blue)}$ and a specialized one for this scenario optimized for scaling $\text{(red)}$.

Figure 12

The obtainable key rates for the two-link repeater protocol and parameters described in Ref. [9]. A comparison of the analytical formulas (lines) and our simulation (dots).

Figure 13

Key rates for node-sends-photons protocols with (a) currently available or (b) future parameters. The experimental platforms are NV centers $\text{(dark blue)}$, silicon-vacancy ($\mathrm{Si}$V) centers $\text{(red)}$, Quantum dots $\text{(green)}$, $\mathrm{Ca}$ or $\mathrm{Yb}$ ions $\text{(yellow)}$, and $\mathrm{Rb}$ atoms $\text{(light blue)}$, with parameters taken from Ref. [21]. A comparison of the analytical formulas (lines) and our simulation (dots). The solid black line is the PLOB bound and the solid gray line represents $\sqrt{\eta }$ (the asymptotic behavior and upper bound [61] of the ideal one-repeater setup). The two dashed lines are $P_{\mathrm{link}}\eta /2$ for $P_{\mathrm{link}}=0.7$ and $0.1$.

Figure 14

The obtainable key rates for the multimemory protocol with two repeater links in Ref. [23]. A comparison of the analytical formulas (lines) and our simulation (dots). Results for using one $\text{(dark blue)}$, $\text{ten (red)}$, $\text{100 (green)}$, $\text{400 (yellow)}$, or $\text{1000 (light blue)}$ memories per repeater link. The orange dashed line is the repeaterless PLOB bound and the gray dashed line is the upper bound for the one-repeater rate [61].