Semihierarchical quantum repeaters based on moderate lifetime quantum memories

The construction of large-scale quantum networks relies on the development of practical quantum repeaters. Many approaches have been proposed with the goal of outperforming the direct transmission of photons, but most of them are inefficient or difficult to implement with current technology. Here, we present a protocol that uses a semihierarchical structure to improve the entanglement distribution rate while reducing the requirement of memory time to a range of tens of milliseconds. This protocol can be implemented with a fixed distance of elementary links and fixed requirements on quantum memories, which are independent of the total distance. This configuration is especially suitable for scalable applications in large-scale quantum networks.

Figure 1

Schematic of the semihierarchical quantum repeater architecture. (a) Both ends of each elementary link contain multimode quantum memories and sources of entangled photon pairs, where each pair is emitted into a different optical mode. One member from each pair is stored in a multimode quantum memory while the other is sent to the center of the link over a quantum channel (orange lines) where they meet photons generated by the sources situated at the other end of the link. Photons occupying the same mode then undergo a mode-resolving Bell-state measurement (BSM) which is composed of a beam splitter (BS) and two single-photon detectors (assuming time-bin qubit encoding and temporal multiplexing is employed). A heralding signal, which originates from a successful BSM, is sent, via classical channel (green lines), to a central control unit located at the midpoint of the total distribution length. The central control unit waits until entanglement (of remote quantum memories) has been established over each elementary link (depicted by the dotted arrows), and then informs all quantum memories to implement the next step of the protocol. (b) The heralded photons are retrieved from adjacent memories such that they arrive indistinguishably at the BSM' to perform the ES. (c) The entanglement is distributed over the desired distance on the premise that ES operations at all nodes succeed. Otherwise, the whole process must start from scratch.

Figure 2

(a) Simulated performance of quantum repeater protocols based on different connection types of the elementary links. The quantity shown is the average time needed to distribute a single entangled pair for the given distance. A: the time required using direct transmission of 10-GHz single photons through fibers. B: the proposed protocol that uses semihierarchical structure. C: the protocol of Ref. [18]. D: the protocol of Ref. [20]. (b) The required average memory time of different quantum repeater protocols. The quantity shown is the average memory time needed to distribute a single entangled pair for the given distance. The letters refer to the same protocols as above. For all the curves we have assumed ${\eta }_M={\eta }_D=0.9$ and $m=100$. For curve B and curve D, the emission probability $\rho =0.9$. The shaded area of curve B represents a one standard deviation uncertainty of required memory time, the detailed derivation of which is contained in the Appendix. All curves are drawn by choosing the optimal link number.

Figure 3

Robustness of different protocols with respect to the reduction of mode number and emission probability. (a) The average time for the distribution of an entangled pair for a distance $L=1000$ km as a function of the mode number $m$. (b) The average time for the distribution of an entangled pair for a distance $L=1000$ km as a function of the emission probability $\rho$. Curve C is not included because the pair emission probability of the protocol proposed by Ref. [18] cannot be too high due to multiphoton errors. The letters refer to the same protocols as in Fig. 2. All curves are drawn by choosing the best link number to obtain the optimal distribution time.

Figure 4

(a) The average time needed to distribute a single entangled pair for a given distance in two scenarios. B: when choosing the best $n$ corresponding to total distance. E: when fixing the length of elementary links to 125 km. The curve E is drawn in two different styles–solid and dashed–and a change in styles means that the optimal link number increments by one. The bolded dots on the curve E represent nodes of the elementary links. Segments with the same style at both sides of one specific bolded dot means that the entanglement is transmitted via fibers from this node. (b) The memory time needed to distribute a single entangled pair for the given distance in two scenarios. B: when choosing the best $n$ corresponding to total distance. E: when fixing the length of elementary links to 125 km. The shaded areas represent the uncertainty of required memory time.