Problem statement and notation

Consider a graph G(V, E) that denotes the topology of the repeater network. See Fig. 1b for an illustration. Each node v ∈ V is a repeater (blue circles), and each edge e ∈ E is a physical link connecting two repeater nodes. \(S(e) \in {\Bbb Z}^ +\) is an integer edge weight, which corresponds to the number of parallel (spatial, spectral, and/or polarization mode) channels across the edge e (shown using dashed lines). The number of memories at node v is \(\mathop {\sum}

olimits_{e \in {\cal N}(v)} {\kern 1pt} S(e)\) (see Fig. 1b), where the sum is over \({\cal N}\left( v \right)\), the set of nearest neighbor edges of v, with d(v) = \(\left| {{\cal N}\left( v \right)} \right|\) being the degree of node v.

To simplify our analysis, we assume that time is slotted and that each memory can hold a qubit perfectly for T ≥ 1 time slots, after which the stored qubit completely decoheres (T should be taken to be much smaller than the memory’s coherence time). Each time slot t, t = 1, 2, …, is divided into two phases: the “external” phase and the “internal” phase, which occur in that order. During the external phase, each of the S(e) pairs of memories across an edge e attempts to establish a shared entangled (EPR) pair. An entanglement attempt across any one of the S(e) parallel links across edge e succeeds with probability p 0 (e) ~ η(e),21,27 where η(e) ~ e−αL(e) is the transmissivity of a lossy optical channel of length L(e). Using two-way classical communication over edge e(u, v), neighboring repeater nodes u, v learn which of the S(e) parallel links (if any) succeeded in the external phase, in a given time slot.

Let us assume that neighboring repeaters pick upto one successfully created ebit (i.e., ignore multiple successes if any) as in refs. 29,31, in which case the probability that one ebit is established successfully across the edge e during the external phase is given by: p(e) = 1 − (1 − p 0 )S(e). Let us also assume S(e) = S, ∀e ∈ E, which in turn gives us p(e) = p, ∀e ∈ E. While our results in this paper can be adapted to any network topology, we will henceforth use the 2D regular square grid topology (Fig. 2) to illustrate the performance of our routing algorithms.

Fig. 2 Schematic of a square-grid topology. The blue circles represent repeater stations and the red circles represent quantum memories. Every cycle (time slot) of the protocol consists of two phases. a In the first (external) phase, entanglement is attempted between neighboring repeaters along all edges, each of which succeed with probability p (dashed lines). b In the second (internal) phase, entanglement swaps are attempted within each repeater node based on the successes and failures of the neighboring links in the first phase—with the objective of creating an unbroken end-to-end connection between Alice and Bob. Each of these internal connections succeed with probability q. Memories can hold qubits for T ≥ 1 time slots Full size image

One instance of the resulting external links created between repeater nodes after the external phase is shown in Fig. 2b using solid lines. In the internal phase of the time slot, entanglement swaps (BSMs) are attempted locally at each repeater node between pairs of qubit memories (red circles in Fig. 2). We associate these BSM attempts as internal links, i.e., links between memories internal to a repeater node, shown using dashed lines inside repeaters in Fig. 2b. If T > 1, a repeater node can attempt a BSM between qubits held in two memories that were entangled with their respective neighboring node’s qubits in two different time-slots. For minimizing the demands on memory coherence time,29,31 and to avoid requiring temporal switching, we will assume T = 1. So, BSMs will always be attempted between two qubits in distinct memories that were entangled with their respective counterparts at their respective neighboring nodes in the same time-slot. Each of these internal-link attempts succeed with probability q. Therefore, after the conclusion of one time-slot, along a path comprising k edges (and thus k − 1 repeater nodes), one ebit is successfully shared between the end points of the path with probability pkqk−1. The maximum number of ebits that can be shared between Alice (say, node a) and Bob (say, node b) after one time-slot is min{d(a), d(b)}, assuming S is the same over all edges. For the square-grid topology shown, the maximum number of ebits that can be generated between Alice and Bob in each time-slot is 4.

Alice and Bob only learn whether entanglement was generated after a communication lag proportional to the length of the end-to-end channel. For many applications including quantum key distribution,3,4 secure private-bid auctions,9 quantum digital fingerprinting43 and quantum-enhanced sensing,7 this does not affect the performance of the protocol, except for a latency. However, for some other applications like distributed quantum computing44 and quantum private queries45 that require knowledge of whether entanglement was generated in a particular time-slot, we will need an additional memory at Alice and Bob capable of holding entanglement during this latency period. A conventional linear repeater chain would also require these additional memories for these applications.

The remainder of the paper is dedicated to finding the optimal strategy for each repeater node in order to decide which locally held qubits to attempt BSM(s) on during the internal phase of a time slot, based ideally only on knowledge of the outcomes (success or failure) of the nearest neighbor links, i.e., local link-state knowledge, during the respective preceding external phases. We will assume that each repeater node is aware of the overall network topology, as well as the locations of the K Alice-Bob pairs. The goal of the optimal repeater strategy will be to attain the maximum entanglement-generation rate (if there is a single Alice and Bob, i.e., K = 1) or the maximum rate-region for multiple flows (i.e., K > 1).

Multipath routing of a single entanglement flow

Entanglement routing with global link-state information

We begin with the assumption that global link-state knowledge is available at each repeater node, i.e., the state of every external link in the network after the external phase is known to every repeater in the network and can be used to determine the choice of which internal links to attempt within the nodes. Each memory can only be part of one entanglement swap, i.e., each red node can only be part of one internal edge. Consider the following greedy algorithm to choose the internal links: consider the subgraph induced by the successful external links and the repeater nodes (at the end of the external phase), and find in it the shortest path connecting Alice and Bob. If no connected path between Alice and Bob exists, no shared ebits are generated in that time slot. If a shortest path of length k 1 is found, all internal links along the nodes of that path are attempted, and the (conditional) probability of a shared ebit is generated by this path is the probability that all k 1 − 1 internal link attempts were successful, i.e., \(q^{k_1 - 1}\). We then remove all the (external and internal) links of the above path from the subgraph, and find a shortest path connecting Alice and Bob in the pruned subgraph. Note that instead of removing the links of the first path from the subgraph, we could simply search for a shortest path in the original subgraph but one that is edge disjoint from the previous path. If such a path exists, we again attempt all internal links at the nodes of this path, so the probability in the path contributes to the generation of an ebit (distinct from the ebit that may have been generated by the first path) is \(q^{k_2 - 1}\) where k 2 is the length of the second path; and so on.

The entanglement generation rate achieved using this greedy algorithm R g is the sum of expected rates (in ebits per time-slot) from these paths. Given the degree-4 nodes in a square grid topology, there can be a maximum of four edge disjoint paths between Alice and Bob. Figure 2b illustrates our greedy algorithm. Given the set of external links created, the shortest path has length k 1 = 4, the next path has length k 2 = 6, and no further paths can be found. The two edge-disjoint paths are highlighted in green. Hence, the internal links depicted with the dashed lines in Fig. 2b are attempted and the expected number of shared ebits generated in this time cycle is: \(q^{k_1 - 1} + q^{k_2 - 1}\). The net entanglement generation rate is the expectation of sums like the above (with up to four terms) over many random instantiations of the (p, 1 − p) external-link creations during the external phase of many time-slots. Evaluating this expected rate R g (p, q) achieved by the above routing strategy analytically as a function of the Alice-Bob distance (X 1 , X 2 ) is difficult, even for a square-grid topology.

The intuition behind this simple greedy algorithm is that the entanglement generation rate along a path of length k decays exponentially as qk−1, suggesting that attempting internal links to facilitate connections along the shortest path first would optimize the expected rate. However, it is possible to draw random instances of successes of external links, where either one of the two possible options—(1) picking the shortest path (which disrupts all other paths) and (2) picking two edge disjoint (but longer) paths—could yield either a larger or a smaller expected rate than the other, depending upon the value of q. If q is larger than a threshold, option (2) will have a larger expected rate and vice versa. Finding the global optimal rule remains an open problem. It is easy, however, to prove that the greedy algorithm achieves a rate within a factor of 4 of the optimum algorithm employing global link-state knowledge, R opt (p, q). Let us denote the length of the shortest path between Alice and Bob with Manhattan distance (X, Y) in the induced subgraph after the external phase, as n SP (p). This quantity is of interest in percolation theory, and is not known analytically, even for simple graph topologies. It undergoes a sharp transition (i.e., starts out large and suddenly jumps to a much smaller value) as p crosses the bond-percolation threshold p c of the graph G, from below to above. Clearly, \(R_{\mathrm{g}}(p,q) \ge {\Bbb E}[q^{n_{{\mathrm{SP}}}(p) - 1}]\) since using the shortest path is the first step of the greedy algorithm. Furthermore, since the optimal rule can create entanglement over a maximum of four edge-disjoint paths in each time-step, each of which must have a length no less than the length of the shortest path, \(R_{{\mathrm{opt}}}(p,q) \le {\Bbb E}[4q^{n_{{\mathrm{SP}}}(p) - 1}] \buildrel \Delta \over = R_{{\mathrm{opt}}}^{{\mathrm{(UB)}}}(p,q)\). Therefore,

$$R_{{\mathrm{opt}}}(p,q) \ge R_{\mathrm{g}}(p,q) \ge \frac{{R_{{\mathrm{opt}}}(p,q)}}{4},$$ (2)

i.e., the greedy rule achieves the same rate-vs.-distance scaling as the optimal algorithm that employs global link-state knowledge, and at worst is lower that the optimal rate only by a constant factor equaling the node degree.

In Fig. 3a we plot R g (p, q) as a function of the Alice-Bob X separation (measured in number of hops). We choose the X = Y direction here (45° with respect to the grid axes) but other directions show similar behavior, and a 3D plot with all directions is in Supplementary Fig. 1 in Supplementary Note 1. When q = 1 and p > p c (p c = 0.5 for the square lattice), a giant connected component is formed by the external links alone at the end of the first (external) phase of a time slot. Recall that the rate along a length k path is pkqk−1, where p ~ η is the transmissivity of each link. In the network case, when p > p c and q = 1, we find that the pk portion of the rate expression becomes immaterial for scaling with Alice-Bob distance. This behavior can be explained by percolation theory: in a large lattice in this regime, the probability of a connected path between Alice and Bob along successful external links in each time-slot approaches a non-zero constant as the Alice-Bob separation is increased. Furthermore, we numerically find that finite size effects do not have a significant impact on this behavior, even when the Alice-Bob separation is as small as 5 hops. So, if q = 1, R g (p, q) remains essentially distance invariant. When p < p c , the rate falls off exponentially with distance (even when q = 1). It is instructive to note here that the optimal rate (entanglement-generation capacity) achievable on a single length k path does not depend on k, and only on the transmissivity of the lossiest link in the path, i.e., C ~ η,20 but achieving this requires infinite-coherence-time quantum memories and ideal quantum operations at nodes. The multi-path gain in the p > p c regime lets us achieve a distance-independent rate, but with memories whose coherence times are no more than one time slot, and only using BSMs. The rates were calculated using Monte-Carlo simulations which resulted in some numerical noise as is apparent in Fig. 3.

Fig. 3 Entanglement generation rate vs. Alice-Bob X separation for different (p, q). We choose direction X = Y here but other directions show similar behavior, and a 3D plot with all directions is in Supplementary Fig. 1 in Supplementary Note 1; (a) R g (p, q) is the rate attained by the global-knowledge-based protocol. For q = 1, R g is distance independent when p is greater than the bond percolation threshold (0.5 for the square lattice). R(UB)(0.6) is the distance-independent rate upper bound for p = 0.6, achieving which requires perfect quantum processing at repeater nodes. R g (0.6, 1) is also distance independent, and within a factor 3.6 of R(UB)(0.6). With q < 1, e.g., R g (0.6, 0.9), the rate decays exponentially with distance. b R loc is attained by our local link-state knowledge protocol. The rate-distance scaling exponent of R loc is clearly worse than R g , but is superior to that of a linear repeater chain along the shortest path, R lin , demonstrating multi-path routing advantage even with local link-state knowledge Full size image

A general upper bound on the entanglement generation rate is given by the min-cut of the graph,20 and for a square lattice, is given by:

$$R^{{\mathrm{(UB)}}}(p) = - {\mathrm{log}}_2[(1 - p)^4].$$ (3)

R(UB)(0.6) is plotted in Fig. 3a. The known methods for achieving R(UB) require infinite coherence time memories and error-corrected quantum processors at each node. For our implementation (assuming global link state knowledge), R g (0.6, 1) is also plotted in Fig. 3a. Although our protocol only requires memories to hold entanglement for one time step, the multi path advantage gives us the same constant rate-distance scaling and within a factor of ~3.6 of R(UB)(0.6). The assumption of perfect BSMs is unrealistic and thus q < 1, in which case R g (p, q) falls off exponentially with distance; even when p > p c , as seen in the plot for R g (0.6, 0.9).

Entanglement routing with local link-state information

R g (p, q), the rate attained by the protocol described in the previous subsection that employs global link-state knowledge, is re-plotted in Fig. 3b. We also plot:

$$R_{{\mathrm{lin}}}(p,q) = p^{n_{{\mathrm{SP}}}(1)}q^{n_{{\mathrm{SP}}}(1) - 1},$$ (4)

the rate attained by a single linear repeater chain, where n SP (1) is the shortest-path length between Alice and Bob along the edges of the underlying square grid. The assumption of global link-state knowledge in large networks is unrealistic, as it requires memories whose coherence time increases with the network size due to the time required for the traversal of link-state information across the entire network. In this section, we describe a more realistic protocol in which knowledge of success and failure of an external link at each time slot is communicated only to the two repeater nodes connected by the link, as is the case in the analysis of many ‘second-generation’ linear repeater chains.26,27,29 Repeater nodes need to decide on which pair(s) of memories BSMs should be attempted (i.e., which internal links to attempt), based only on information about the states of external links adjacent to them. We assume that network topology and positions of Alice and Bob are known to each repeater station, and communicated classically beforehand.

Every repeater, except Alice and Bob which do not attempt any internal links, uses the same local rule, which is illustrated in Fig. 4 using the example of repeater u inside the dotted box. A repeater decides which internal edges to attempt based on the information of (1) which of the four neighboring external edges have been successfully created in the external phase and (2) The distance of it’s four neighbors to Alice and Bob. The distance of a repeater to Alice and Bob is denoted, respectively, by d A and d B . We use the \({\Bbb L}^2\) norm for both these distances (other distance metrics are discussed in Supplementary Note 3). The rules used to determine the internal links to be attempted at a repeater are:

If less than one of the neighboring external links is successful: no internal links are attempted, since this repeater node cannot be part of a path from Alice to Bob in that time slot.

If two or more neighboring external links are successful: of all the nearest neighbor nodes of u whose links to u were successful in that time slot, we label the one that has the minimum d A as v . Similarly, the neighbor with a successful external link with u and the minimum d B is labeled w . If two neighbors have the same values of d A and d B , an unbiased coin is tossed to determine the choice of v and w , to preserve symmetry in the protocol. If v and w are the same node, v (or w ) is replaced by node u ’s nearest-neighbor node with the next smallest value of d A (or d B ). The choice of whether to replace v or w is made in a manner that minimizes the sum of d A and d B from the eventually chosen two neighbors to connect. An internal link is attempted between the memories connected to v and w respectively, as shown in Fig. 4a.

If all four neighboring external links are successful: in addition to the internal link attempted in the previous point, an additional internal link is attempted between the remaining two memories as shown in Fig. 4b, since the addition of this internal link can only increase the entanglement generation rate.

Fig. 4 The entanglement swap rule used at the repeater C in the dotted box in the case of local link-state knowledge. A and B are the repeaters closest to Alice and Bob, respectively, with a direct edge to C. a If two or three links are up, the memories linked to A and B undergo an entanglement swap. b If four links are up, the remaining two memories also undergo an entanglement swap Full size image

The entanglement generation rate R loc (p, q) achieved by the above described local rule is plotted in Fig. 3b and compared to R g (p, q) and R lin (p, q). We use p = 0.6 and q = 0.9, the same values used for the global-information rate plots in Fig. 3a. As one expects, the rate-distance scaling of R loc is worse than that of R g . However, the rate-distance scaling exponent achieved by the local rule is superior to that of a linear chain, even though the physical elements employed to build the repeaters are identical. In other words,

$$R_g(p,q) > R_{{\mathrm{loc}}}(p,q) > R_{{\mathrm{lin}}}(p,q),$$ (5)

where each of the above three rates can be expressed as an exponential decay with the distance L, i.e., R ~ e−αL, where the exponents satisfy:

$$\alpha _g(p,q) < \alpha _{{\mathrm{loc}}}(p,q) < \alpha _{{\mathrm{lin}}}(p,q),$$ (6)

where it is known that α lin (p, q) < α fiber , the loss coefficient of a fiber, which is the rate scaling exponent when no repeaters are used.29 This is proven analytically in Supplementary Note 2A.

The scaling advantage of R loc over R lin arises because the local rule allows the entanglement-generation flow between Alice and Bob to find different (and potentially simultaneously multiple) paths in different time slots, and does not have to rely on all links along a linear chain to be successful. This is analogous to multi-path routing in a classical computer network. The contour plot in Supplementary Fig. 1d in Supplementary Note 1 further illustrates this point: there is a noticeable enhancement of R loc along the X = Y line because the diagonal direction contains the largest spatial density of possible paths between Alice and Bob. The scaling advantage over R lin persists in any direction, i.e., along Y = 0 as well.

Sweeping over different values of p and q, we find that the multi-path advantage relative to a linear repeater chain increases as p decreases from unity, but there is little relative improvement as q is varied (see Supplementary Note 2B).

Clearly, other distance metrics (e.g., \({\Bbb L}^p\) norm for p ≥ 1) can be used in lieu of the \({\Bbb L}^2\) norm in the algorithm described above. In Supplementary Note 3, we present a recursive numerical evaluation technique to find the rate-optimal distance metric, which can be applied to any network topology. For planar network topologies, we find that the \({\Bbb L}^2\) norm is near-optimal for our local routing algorithm.

An analytical enumeration of the expected number of edge-disjoint paths as a function of p between Alice and Bob separated by a given distance (X, Y) in a bond-percolation instance (i.e., with p > p c ) of a network is an open question, the solution of which will enable a firmer quantitative understanding of the multi-path advantage in entanglement generation in a repeater network.

Simultaneous entanglement flows

In this section, we consider simultaneous entanglement-generation flows between two Alice-Bob pairs, using local link state knowledge at all repeater nodes. Consider two pairs Alice 1 - Bob 1 (red nodes) and Alice 2 - Bob 2 (green nodes) as shown in the two scenarios in Fig. 5. In Fig. 5a, the shortest paths connecting the two Alice-Bob pairs do not cross, but they do in Fig. 5b. In both cases, they are placed at the four corners of a 6 × 6 square grid, embedded within a large square grid network. Denote by R 1 and R 2 the entanglement generation rates achieved by the two Alice-Bob pairs respectively. We first consider the case of non-intersecting flows shown in Fig. 5a. A simple strategy is for every single repeater node (including the nodes labeled as the two Alices and Bobs) to use the local rule described in the previous section tailored to support the Alice 1-Bob 1 flow for a fraction, λ, of the time slots and to support the Alice 2-Bob 2 flow for the remaining 1 − λ fraction. For p = q = 0.9, the rate region attained by varying λ ∈ [0, 1] is depicted with the blue line in Fig. 5b, which we refer to as single-flow time-share. However, if every repeater with the exception of the Alices and Bobs carry out the above time-sharing strategy, even when all repeater nodes support flow 1, there is still some ‘left-over’ non-zero R 2 that is attained. This multi-flow time-share rate region is shown using the red line in Fig. 5b.

Fig. 5 a Multi-flow routing for two Alice-Bob pairs that lie along the sides of a 6 × 6 square, embedded in a 100 × 100 grid; (b) rate region (R 1 , R 2 ) with different rules at repeater nodes, each employing local link-state knowledge, for p = q = 0.9. c Multi flow routing when the Alice-Bob paths cross (d) multiflow rate region for two local-knowledge rules Full size image

In Fig. 6, for the case that a single Alice and a single Bob are separated by 6 hops on the square grid, we plot a color map of p usage , the probability a given repeater node is involved in a successful creation of a shared ebit generated between Alice and Bob when our local rule (for multi-path entanglement routing described in the previous section) is employed. We observe that only the repeaters lying in a small spatial region surrounding the straight line joining Alice and Bob are used significantly.

Fig. 6 A heat map plotting p usage , the probability that a given repeater node is involved in a successful creation of a shared ebit generated between Alice and Bob, separated by 6 hops in an underlying square grid topology, when our local rule is employed. We assume p = 0.9 and q = 0.9 Full size image

This observation motivates a multi-flow spatial-division rule for routing multi-flow entanglement, in which we divide the network between two spatial regions corresponding to the two flows, as shown in Fig. 5a. Any repeater in the red shaded region follows the local rule tied to the Alice 1 - Bob 1 flow while repeaters in the green region operate with the local rule tied to the Alice 2 - Bob 2 flow. The placement of the boundary determines the rates R 1 and R 2 . The rate region attained is plotted with the yellow line in Fig. 5b. This significantly outperforms time sharing. The two flows can co-exist and operate with a very small reduction from their individual best rates, because the repeaters they respectively benefit the most from, form (almost) disjoint sets.