Optimizing repeater schemes for the quantum internet

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, Dür, 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.

Figure 1

Elementary pair generation (EPG) between adjacent nodes ${\mathrm{QR}}_i$ and ${\mathrm{QR}}_{i+1}$. The schemes take a number of rounds $r=r^{*}$, even if entanglement is generated at an earlier round. See main text for further details.

Figure 2

Entanglement swapping between two entangled pairs between (multihop) links (${\mathrm{QR}}_i, {\mathrm{QR}}_j$) and (${\mathrm{QR}}_j, {\mathrm{QR}}_k$), indicated by a circle node. By performing a Bell state measurement on the two local states at ${\mathrm{QR}}_j$, the two entangled states turn into one entangled state between (${\mathrm{QR}}_i, {\mathrm{QR}}_k$). The schemes take a number of rounds $r=r^{*}$ even if the scheme succeeds at an earlier round. See main text for further details. Note that the distances over which the entanglement has been generated for the (multihop) links (${\mathrm{QR}}_i, {\mathrm{QR}}_j$) and (${\mathrm{QR}}_j, {\mathrm{QR}}_k$) need not be the same.

Figure 3

Example of a generic entanglement distillation protocol, transforming (possibly probabilistically) two entangled states to a single, more entangled state between nodes ${\mathrm{QR}}_i$ and ${\mathrm{QR}}_j$, using only local operations and classical communication. Distillation is indicated by a square node. The schemes take a number of rounds $r=r^{*}$ even if distillation succeeds at an earlier round. See main text for further details. Note that ${\mathrm{QR}}_i$ and ${\mathrm{QR}}_j$ do not have to be directly connected by a fiber.

Figure 4

Schematic description of how near-deterministic schemes are constructed from the protocols shown in Figs. 1, 2, 3. Here entanglement is generated between the nodes A and B, using an intermediate node labeled by QR. The overall structure is that of a binary tree (modulo the leaves indicating elementary pair generation, indicated by EPG), since swapping and distillation are always performed between exactly two schemes. Each subtree is required to succeed with probability at least $p_{\text{min}}$, which can be enforced by repeating the whole subtree for a number of attempts $r$. Here, the specific number of attempts is indicated by $r_b, b\in \{1,2,3,4,5\}$. The circular nodes indicate either elementary pair generation or swapping, and the square nodes indicate distillation.

Figure 5

An example of an elementary link implemented with an information processing platform. The two nodes are connected by a fiber with a beamsplitter in the middle and two detectors. For the case considered in this figure, the two nodes are nitrogen-vacancy centers in diamond. For both protocols, the two nodes both send one-half of an entangled state to the middle, which after interference and successful detection leads to a shared state between the two nodes. Figure taken with permission from [61].

Figure 6

Schematic description of an MP implementation. Top: The total distance $L$ is split into $N$ elementary links, each with a spectrally resolving BSM (indicated by $\nu \mathrm{RBSM}$) in the middle, and with two nodes (each indicated by REP) at the end point of the elementary links. Middle: Zoom in of a node. Each node contains two PDC sources of multiplexed bipartite entanglement, two quantum memories (indicated by QM) and a number-resolving Bell state measurement station (indicated by $N\text{RBSM}$). Bottom: Detailed view of QM and $N\text{RBSM}$. Each quantum memory not only stores (in the unit indicated by $\tau$), but can also perform a frequency shift (in a unit indicated by $\mathrm{\Delta }\nu$) and a frequency filter (indicated by the unit ${\nu }_0$), while each $N\text{RBSM}$ contains a beamsplitter and two single-photon detectors, which performs a Bell state measurement on the frequency-shifted photons. Illustration taken with permission from [38].

Figure 7

Results of the achieved fidelity and generation time for a total distance of 50 km using parameter set 1 (see Table 2) for information processing nodes, where we consider having 0 (green), 1 (purple), or 2 (yellow) of such intermediate nodes. The solid line corresponds to a heuristic optimization where we have excluded the double-click protocol, and the dotted line corresponds to a heuristic optimization with both the single- and double-click protocol. The double-click protocol does not provide a benefit for direct transmission, since the double-click protocol suffers more strongly from losses than the single-click protocol.

Figure 8

Maximum generation rates for several different values of the coherence times (1–10 s) and gate fidelities (0.98 to 1) and for several different target fidelities, for a distance of 50 km and a single information processing node. Down and to the right in the plot indicate better parameters. All the other parameters are fixed to those of set 2 (Table 2) or the base parameters (Table 1). The target fidelities are (a) $F=0.7$, (b) $F=0.8$, and (c) $F=0.9$, respectively. We also plot the gradient, indicating the direction and magnitude of steepest ascent. The blue ring and yellow diamond indicate the schemes we investigate in Fig. 9.

Figure 9

Visualization of the two schemes indicated in the bottom of Fig. 8 by the blue ring (left) and the yellow diamond (right). The numbers indicate the number of nodes over which entanglement has been established, or elementary pair generation (EPG) has been performed. The “DC” indicates the double-click protocol, and the “$\theta ={\theta }^{*}$” indicates a single-click protocol with the $\theta$ parameter set to ${\theta }^{*}$. The “$r$” here indicates the number of rounds the corresponding subtree is attempted. Note the necessity of combining disparate schemes—in both cases the EPG protocols used are not the same, and the yellow diamond scheme requires a swap on a distilled and undistilled pair.

Figure 10

Maximum achieved generation rates for several different values of the success probabilities (i.e., we vary $p_{\text{det}}=p_{\text{em}}=p_{\text{PS}}$ simultaneously from 0.8 to 1) and gate fidelities (0.98 to 1), and for several different target fidelities, for a distance of 50 km and a single intermediate node for information processing platforms. Down and to the right in the plot indicate better parameters. All the other parameters are fixed to those of set 2 (Table 1) and the base parameters (Table 4). The target fidelities are (a) $F=0.7$, (b) $F=0.8$, and (c) $F=0.9$, respectively. We also plot the gradient, indicating the direction and magnitude of steepest ascent.

Figure 11

Secret-key generation using the six-state protocol, for several different values of (a) the coherence times (1–10 s) and gate fidelities (0.98 to 1) and (b) the success probabilities (i.e., we vary $p_{\text{det}}=p_{\text{em}}=p_{\text{PS}}$ simultaneously from 0.8 to 1) and gate fidelities (0.98 to 1) for a distance of 50 km and a single intermediate node for information processing platforms. Down and to the right in the plot indicate better parameters. All the other parameters are fixed to those of set 2 (Table 2) and the base parameters (Table 4). We also plot the gradient, indicating the direction and magnitude of steepest ascent.

Figure 12

Results of the achieved fidelity and generation time for total distances of 50 (a), 100 (b), 150 (c), and 200 (d) km using parameter sets 2 (solid), 3 (dashed-dotted), and 4 (dashed) (see Table 2) for information processing nodes, where we consider having 0 (green), 1 (purple), or 2 (yellow) of such intermediate nodes.

Figure 13

Visual representation of the schemes with the lowest nontrivial fidelity (a) and highest fidelity (b), for a distance of 200 km with information processing platforms using parameter set 4 (see Table 2) and two intermediate nodes or three hops. The numbers in the vertices indicate the number of nodes over which entanglement has been established. The “$\theta ={\theta }^{*}$” indicates a single-click protocol with the $\theta$ parameter set to ${\theta }^{*}$. The “$r$” indicates the number of rounds the corresponding subtree is attempted. We find that the second scheme performs distillation between two elementary pairs generated with a single- and double-click protocol, demonstrating the benefit of including such distillation protocols in our optimization.

Figure 14

Maximum generation rates for several different values of the success probabilities (i.e., we vary $p_{\text{det}}=p_{\text{em}}=p_{\text{PS}}$ simultaneously) and efficiency coherence times, and for several different target fidelities, for a distance of 15 km and a single node for MP platforms. Down and to the right in the plot indicate better parameters. All the other parameters are fixed to those of set 2 (Table 4) or the base parameters (Table 3). The target fidelities are (a) $F=0.7$, (b) $F=0.8$, and (c) $F=0.9$, respectively. We also plot the gradient, indicating the direction and magnitude of steepest ascent.

Figure 15

Results of the achieved fidelity and generation time for total distances of 50 (a), 100 (b), 150 (c), and 200 (d) km using parameter sets 2 (solid), 3 (dashed-dotted), and 4 (dashed) (see Table 4) for multiplexed elementary pair generation platforms, where we consider having 0 (green), 1 (purple), or 2 (yellow) of such intermediate nodes.

Figure 16

Secret-key generation using the six-state protocol, for several different values of the efficiency coherence times (${10}^{-2}$–${10}^0$ s), and number of modes (${10}^4$–${10}^7$), for a distance of 200 km and a single node for MP platforms. Down and to the right in the plot indicate better parameters. All the other parameters are fixed to those of set 2 (Table 4) and the base parameters (Table 3). We also plot the gradient, indicating the direction and magnitude of steepest ascent.

Figure 17

Results of the heuristic optimization for total distances of 200, 400, and 600 km, for MP platforms and ten intermediate nodes. We use parameter set 2 for MP platforms (see Table 3) as a baseline, where we set the success probability of the Bell state measurements to $\frac{3}{4},\frac{7}{8},\text{and}\frac{15}{16}$ in the first, second, and third column, respectively. We set the efficiency coherence time $T_{\text{coh}}$ to 1 and 10 in the first and second row, respectively.

Figure 18

Results of the achieved fidelity and generation time for total distances of 200 (a), 400 (b), 600 (c), and 800 (d) km using parameter set 4 (see Table 4) for MP platforms, where we consider having 1 (purple), 2 (yellow), 3 (blue), or 4 (orange) of such intermediate nodes.

Figure 19

Visual representation of the schemes with the lowest nontrivial fidelity (a) and highest fidelity (b), respectively, for a distance of 800 km with MP platforms using parameter set 4 (see Table 4) and four intermediate nodes or five hops. The “${\mathrm{N}}_{\text{s}}$= ${\mathrm{N}}_{\text{s}}^{*}$” indicates the elementary pair generation (EPG) protocol with mean photon number ${\mathrm{N}}_{\text{s}}^{*}$ used for MP platforms discussed in the main text. The “$r$” here indicates the number of rounds the corresponding subtree is attempted. Note that the second scheme requires a swap between two schemes over multihop links of length 5 and 2 at the end.

Figure 20

Optimization results for a total distance of 4000 km, using a combination of multiplexed and information processing platforms. We use parameter sets 4 from both the multiplexed and information processing part of the platform (see Tables 2 and 4). The solid lines are the optimization without the bisection heuristic discussed in Sec. 2d, while the dotted lines are with the bisection heuristic.

Figure 21

Results of the heuristic optimization for a total distance of 800 km, where we compare the three implementations considered in this paper, using five (solid) or ten (dashed) intermediate nodes. We use parameter sets 4 from information processing (IP) platforms, multiplexed elementary pair generation (MP) platforms, and the combination of the two ($\mathrm{IP}+\mathrm{MP}$). The two crosses in the plot indicate the schemes depicted in Figs. 27 and 28, respectively.

Figure 22

Optimized schemes for a distance of 15 km using a single node with the IP parameter set 2 (see Table 2) for several different pairs of ${\varepsilon }_F$ and ${\varepsilon }_p$. Note that as ${\varepsilon }_F$ and ${\varepsilon }_p$ approach zero, the curves converge.

Figure 23

The runtime of the algorithm for the optimizations performed in Fig. 22. Notice the logarithmic scale, indicating the strong growth rate as ${\varepsilon }_F$ and ${\varepsilon }_p$ become smaller.

Figure 24

Optimized schemes for a distance of 300 km using four intermediate nodes with the IP parameter set 4 (see Table 2) for several different pairs of ${\varepsilon }_{\text{swap}}$ and ${\varepsilon }_{\text{distill}}$. (a) Optimization results with banded swapping heuristic. The difference between ${\varepsilon }_{\text{swap}}$ = ${\varepsilon }_{\text{distill}}=0.1$ and 0.2 is minimal, differing only slightly for very low fidelities. (b) Optimization results with naive swapping heuristic. Note that the curves converge in a significantly poorer fashion than in part level (a).

Figure 25

Algorithm runtimes for several different values of ${\varepsilon }_{\text{swap}}$ and ${\varepsilon }_{\text{distill}}$. The right, solid bars are for the optimization with the heuristic for swapping found in Eq. (6), while the left, hatched bars are for the optimization where we only swap between states that are ${\varepsilon }_{\text{swap}}$ close in fidelity. We observe that both heuristics lead to approximately the same runtime behavior, while the results with the banded swapping heuristic are closer to optimal.

Figure 26

Comparison between the optimization results of the full optimization with the optimization over BDCZ schemes. We consider a repeater chain over 200 km with three intermediate nodes. The parameters used are parameter set 4 for the intermediate nodes and parameter set 2 for Alice and Bob. BDCZ schemes are those schemes that only perform swapping and distillation between two schemes that have used the same sequences of protocols. Contrary to the comparison with BDCZ schemes in [20] we allow for an optimization over the different ways of generating elementary pairs, i.e., we vary the number of attempts $r$ and the $\theta$ parameter. We observe that the schemes found with the full optimization outperform BDCZ schemes, achieving a faster generation time by a factor of $\approx 10$, and extending the maximal achievable fidelity by a small margin.

Figure 27

Visual representation of the scheme with the lowest nontrivial achieved fidelity for a distance of 800 km with a combination of IP and MP platforms using parameter sets 4 (see Tables 2 and 4) and ten intermediate nodes or eleven hops. Elementary pair generation is indicated by EPG, and the mean photon number used is indicated by the $N_s$.

Figure 28

Visual representation of the scheme that achieves a fidelity of $F=0.9605$ in time $T=17.7$ ms (indicated by the cross in Fig. 21), for a distance of 800 km with a combination of IP and MP platforms using parameter sets 4 (see Tables 2 and 4) and ten intermediate nodes or eleven hops. Elementary pair generation is indicated by EPG, and the mean photon number used is indicated by the $N_s$. The structure of the scheme is nonhierarchical, which can most clearly be seen in the final swap operation, which happens between two multihop links of lengths 4 and 9.

Figure 29

Optimization results for total distances of 200, 400, 600, and 800 km, using ten intermediate nodes. We use IP parameter set 2 as a baseline, where we set the gate fidelities to be 0.99, 0.995, and 0.999 in the first, second, and third column, respectively. We set the coherence times $T_{\text{deph}}\text{and}{T}_{\text{depol}}$ to 10, 50, and 100 s in the first, second, and third row, respectively.