Structural vulnerability of quantum networks

Structural vulnerability of a network can be studied via two key notions in graph theory: articulation points (APs) and bridges, representing nodes and edges whose removal will disconnect the network, respectively. Fundamental properties of APs and bridges in classical random networks have been studied recently. Yet, it is unknown if those properties still hold in quantum networks. Quantum networks allow for the transmission of quantum information between physically separated quantum systems. They play a very important role in quantum computing, quantum communication, and quantum sensing. Here we offer an analytical framework to study the structural vulnerability of quantum networks in terms of APs and bridges. In particular, we analytically calculate the fraction of APs and bridges for quantum networks with arbitrary degree distribution and entangled qubits in pure states. We find that quantum networks with swap operations have lower fractions of APs and bridges than their classical counterparts. Moreover, we find that quantum networks under low-degree swap operations are substantially more robust against AP attacks than their classical counterparts. These results help us better understand the structural vulnerability of quantum networks and shed light on the design of more robust quantum networks.

Figure 1

Demonstration of a small quantum network and the 4-swap operation. (a) In this quantum network, edges (blue lines) represent entangled qubits (small green circles). Qubits are located in different spatial stations (gray circles), which represent nodes in the quantum network. The orange circle is a node of degree 4 that is going to be swapped. Inset: each edge (solid blue lines) contains two partially entangled pairs (dashed blue lines); blue circles are individual qubits contained in green circles; dashed lines of the same color are the same partially entangled states ($|\psi \rangle$). (b) The network after 4-swap operation. The yellow lines (dashed or solid) are new entangled pairs $|{\psi }^{\prime }\rangle$ with the same SCP as $|\psi \rangle$, generated by the swap operation (Bell measurement on the central orange node).

Figure 2

Topology changes of a quantum network under 3-swap operation. (a) Initial network generated by ER random graph. Here black and yellow nodes are normal nodes and APs, respectively. Blue and red edges are normal edges and bridges, respectively. (b) Network of (a) after 3-swap. Green edges are entangled states emerging after 3-swap. (c) Network of (b) when singlet conversion $p_s=0.5$ is applied to finalize edges (maximally entangled states). Connections in dash lines are entangled states deleted by singlet conversion. (d) Final topology of the network with less APs and bridges after singlet conversion.

Figure 3

New structural elements need to be considered in analyzing APs and bridges. Different dashed lines are different partial entangled states. Each dashed line is of the same SCP. Solid lines are edges after conversion. Red lines represent bridges. Points represent nodes, where yellow ones are APs and orange ones are isolated. Black lines and points are edges and nodes of uncertain state.

Figure 4

Articulation points and bridges in quantum networks generated from canonical random graphs. Points are simulation results in ER random graphs, which initially have one million nodes. $\mathrm{SCP}=p_s$ is the singlet conversion probability that a single entangled state will be converted to the maximally entangled state. $c$ is the mean degree of initial quantum networks. (a)–(c) Fraction of APs as a function of initial mean degree $c$ and $p_s=1.0,0.5,0.1$. (d)–(f) Fraction of bridges (normalized by initial total edges) as a function of initial mean degree $c$ and $p_s=1.0,0.5,0.1$.

Figure 5

Residual giant bicomponent (RGB) fraction as a function of initial mean degree $c$ in different networks. Dots are simulations in random graphs with one million nodes in different SCP $p_s=1.0,0.5$. Different colors represent various swap strategies. Vertical lines mark transition points. There is no RGB percolation at $\mathrm{SCP}=0.1$. (a) Fraction of RGB as a function of initial mean degree $c$ under SCP 1.0. The inset shows the distribution of RGB fraction at the transition point $c*=3.1$ over 1000 realizations. The two-peak distribution shows that this is the first order phase transition as the nonswap case [9]. (b) Fraction of RGB as a function of initial mean degree $c$ under SCP 0.5.

Figure 6

Phase diagram associated with the GAPR process for scale-free (SF) networks at $c=6$ with or without 3-swap. Colored symbols are critical points ($p_c$) at different $\gamma$ for SF networks. The two horizontal gray lines are critical results for ER networks ($\gamma \to \infty$ limit) under 3-swap. Here, GCC means percolation transition without AP removal. The results without swap are obtained from theoretical calculation [19], while 3-swap results are from simulations.

Figure 7

APs in different quantum networks generated from canonical random graphs. Points are simulation results in random graphs, which initially have one million nodes and mean degree $c=6.0$, and curves are analytical results. (a) ER random graph. (b) Scale-free networks with $\gamma =2.5$. (c) Scale-free networks with $\gamma =3.0$. (d) Scale-free networks with $\gamma =4.0$.

Figure 8

Bridges in different quantum networks generated from canonical random graphs. Points are simulation results in random graphs, which initially have one million nodes and mean degree $c=6.0$, and curves are analytical results. $p$ is the probability that a single entangled state will be converted to the maximally entangled state. Bridges are normalized by initial total edges when $c=6.0$. (a) ER random graph. (b) Scale-free networks with $\gamma =4.0$. (c) Scale-free networks with $\gamma =3.0$. (d) Scale-free networks with $\gamma =2.5$.

Figure 9

Comparison with null model after $q$-swap and singlet conversion. Points are simulation results as a function of the mean degree $c$ in initial ER random graphs of size $N={10}^6$. The solid curves are results of ER random graphs with mean degrees $c$ after $q$-swap and singlet conversion. (a)–(c) Fraction of APs and $p_s=1.0,0.5,0.1$. (d)–(f) Fraction of bridges and $p_s=1.0,0.5,0.1$.

Figure 10

RGB percolation transitions in different networks. Curves are simulation results in random graphs, which have one million nodes in two different SCP 1.0 and 0.5. Different colors represent various swap strategies. (a) Fraction of RGB as a function of initial mean degree $c$ under SCP 1.0. (b) Fraction of RGB as a function of initial mean degree $c$ under SCP 0.5.