Robust quantum network architectures and topologies for entanglement distribution

Entanglement distribution is a prerequisite for several important quantum information processing and computing tasks, such as quantum teleportation, quantum key distribution, and distributed quantum computing. In this work, we focus on two-dimensional quantum networks based on optical quantum technologies using dual-rail photonic qubits for the building of a fail-safe quantum internet. We lay out a quantum network architecture for entanglement distribution between distant parties using a Bravais lattice topology, with the technological constraint that quantum repeaters equipped with quantum memories are not easily accessible. We provide a robust protocol for simultaneous entanglement distribution between two distant groups of parties on this network. We also discuss a memory-based quantum network architecture that can be implemented on networks with an arbitrary topology. We examine networks with bow-tie lattice and Archimedean lattice topologies and use percolation theory to quantify the robustness of the networks. In particular, we provide figures of merit on the loss parameter of the optical medium that depend only on the topology of the network and quantify the robustness of the network against intermittent photon loss and intermittent failure of nodes. These figures of merit can be used to compare the robustness of different network topologies in order to determine the best topology in a given real-world scenario, which is critical in the realization of the quantum internet.

Figure 1

All source stations in the network create two pairs of photons, with each pair in the Bell state ${\mathrm{\Psi }}^{+}$. The entanglement is created either between (a) diametrically opposite photons, (b) top and bottom photons, or (c) the left and right photons.

Figure 2

A two-dimensional network with the topology of a Bravais lattice containing $N=5$ branches of company $X$ and company $Y$ as well as $M=4$ columns of measurement terminals. Indicated in blue is the firing of a pair of entangled photons in the antidiagonal direction, while the green line indicates the simultaneous firing of photons in the diagonal direction, with the final source $S_4^5$ firing in the antidiagonal direction, in order to create entanglement between $X_1$ and $Y_4$.

Figure 3

Pairs of photons (indicated in blue) arriving from the source stations (small outer circles) at a measurement terminal (large central circle). Encircled photon pairs are measured. The measurement terminal measures either (a) pairs of diametrically opposite photons, (b) pairs of top and bottom photons, or (c) pairs of left and right photons.

Figure 4

Plots of ${\xi }_{N,M}^{\text{2D}}$ as a function of the total length $L$ and the number of branches $N$ for different values of the success probability $\gamma$ of each Bell measurement and $M$, the number of columns of measurement terminals in the network: (a) $\gamma =\frac{1}{2}, M=1$; (b) $\gamma =\frac{1}{2}, M=2$; (c) $\gamma =0.9, M=1$; and (d) $\gamma =1, M=2$.

Figure 5

An illustration of the robustness of the memory-free entanglement distribution protocol presented in Sec. 2a in two particular cases of nonfunctioning source stations and measurement terminals, which are enclosed in the shaded regions. Indicated are two possible paths from $X$ to $Y$. Despite the nonfunctioning nodes in 5, which do not allow shared entanglement between $X_2$ and $Y_3$ via the green path, the blue path is still available to share entanglement between $X_2$ and $Y_5$. Similarly, in 5, the blue path cannot be used to share entanglement but the green path can be used.

Figure 6

Given a graph (left), the corresponding quantum network architecture is illustrated explicitly for the circled portion of the graph (right) and consists of the following elements: members of the network placed at the nodes of the graph (gray circles), each possessing $d$ quantum memories (represented by red dots inside the nodes), where $d$ is the degree of the node, and source stations (white circles) placed at the midpoint of each edge of the graph that generate pairs of dual-rail single-photonic qubits (represented by blue dots) in the state ${\mathrm{\Psi }}^{+}$. Each qubit of the pair is fired from the source station in opposite directions along the edge of the graph. The quantum memories allow each member of the network to store the arriving qubits for later processing. Each member of the network has at least one measurement terminal, with a maximum of $\lceil \frac{d}{2}\rceil$, that can be used to perform Bell measurements for entanglement swapping on any two of the stored qubits.

Figure 7

Eleven Archimedean lattices in which all edges of the lattice have the same length. The lattices are named by listing the number of sides of the shapes surrounding each vertex, with repeated shapes indicated with an exponent. For example, $(3^4,6)$ means that every vertex is surrounded by four triangles and one hexagon. Members of the network, capable of storing photons with quantum memories, are represented by $•$. The source stations (not indicated) are placed at the midpoint of each edge between neighboring nodes; see Fig. 6.

Figure 8

Four different bow-tie lattices in which all edges of the lattice have the same length: (a) bow-tie I, (b) bow-tie II, (c) bow-tie III, and (d) bow-tie IV. Members of the network, capable of storing photons with quantum memories, are represented by $•$. The source stations (not indicated) are placed at the midpoint of each edge between neighboring nodes; see Fig. 6.

Figure 9

In this quantum network based on the $\left(3^6\right)$ Archimedean lattice there exists a path, i.e., a chain of successfully entangled nodes (indicated in magenta), between the two distant nodes $A$ and $B$ (indicated in blue). Through entanglement swapping at the intermediate nodes along the path, $A$ and $B$ can share entanglement. Due to the probabilistic nature of entanglement generation in our architecture, at any given time, sources firing throughout the network can lead to paths between multiple different pairs of nodes. Some of these other paths are indicated in green.

Figure 10

Unit-cell and inhomogeneous bond percolation connection probabilities for (a) the square $\left(4^4\right)$ Archimedean lattice, (b) the triangular $\left(3^6\right)$ Archimedean lattice, (c) the honeycomb $\left(6^3\right)$ Archimedean lattice, and (d) the bow-tie I lattice.