Response of quantum spin networks to attacks

In section 2, we overview the wide variety of quantum complex networks in contrast to classical ones; describe our choice of imprinted networks in Hamiltonians; and explain the emergent MI network in the resulting quantum many-body ground states. In section 3, we describe the specific Hamiltonian we consider, and the numerical and analytical tools we use to study the system. In section 4, we apply complex network analysis to the emergent MI network in the ground state. In section 5, we overview attacks on quantum complex networks in general and describe in detail the attacks we consider here. In section 6, we determine the response of the MI networks to attack, and show how mean-field (MF) theory captures many of the results, independent of imprinted network choice. We summarize in section 7.

Our main results are summarised below:

• (a)  
  The emergent MI networks in the ground state do not satisfy the usual criteria for complexity, such as a power-law degree distribution or high clustering. Instead, for relatively dense imprinted networks, the average properties of the emergent networks are captured well by MF theory which predicts fully-connected emergent networks. MF theory does not capture the higher moments of these properties, and also fails to capture the emergent network properties for sparse imprinted networks.
• (b)  
  The lowest order moment of properties of the emergent MI network, for different dense imprinted networks, overlap with each other when plotted versus a dimensionless parameter in the problem.
• (c)  
  The above results hold both for the ground state and the attacked system.
• (d)  
  The average properties of the emergent MI network before and after the attack also collapse onto each other after an appropriate rescaling. Thus, the emergent MI network in the ground state of an imprinted complex network shows the same amount of robustness to projective measurement attacks as does that in the ground state of an imprinted random non-complex network.
• (e)  
  For sparse imprinted networks, the lowest order moment of emergent MI network's properties do not overlap with each other for different classes of networks. However, the average properties of the emergent MI network before and after the attack again collapse onto each other after an appropriate rescaling.

All these behaviors are unexpected, and completely different from the case of classical attacks on networks, which are known to depend strongly on the kind of network: random networks quickly fall apart, while complex networks maintain their properties and are therefore robust. We show that emergent quantum complex networks do not share these robustness properties: (1) the choice of network structure does not help with decoherence; and (2) targeted decoherence is no more effective than random decoherence. We explain these phenomena with MF theory.

A network is an abstract representation of connections (links) between agents (nodes). Networks are nearly everywhere around us, and have a variety of structures. Complex networks are a subset of networks with an emergent property, such as a non-trivial degree distribution or a short path length. Complex networks occur in many classical scenarios, ranging from telecommunication networks to computer networks, biological networks, social networks, and cognitive and semantic networks [4]. There are many models to generate networks, both complex and non-complex, that reproduce different properties of real-world networks such as those mentioned above. We will consider three of these models in section 2.2.

We now turn our attention to quantum networks. Quantum networks have been considered in at least three different settings. The first and most popular setting for quantum networks is the case where the links are physical, i.e., spins or qubits interact across either naturally occurring or engineered links [23–28]. In this work, we refer to this kind of network as the imprinted network. The second scenario where quantum networks have been studied is the case of networks whose links are represented by entangled states [6, 20, 21]. Loosely speaking, this network is defined on the quantum many-body state. In this work, we refer to this kind of network as the emergent network. A third setting is the case where the network is defined on only one part of the Hamiltonian, e.g. via perturbation theory. This third scenario has already been explored in experiments on electronic transport through quantum antidots [29]; and in an open quantum system the system can be simple with a complex environment, or vice-versa [30].

In this work, we study the complex network measures of the emergent network that exists in the quantum many-body ground state of a Hamiltonian defined on an imprinted network. We will describe the imprinted and emergent networks in detail in the following sections. We emphasize that the emergent network can have a very different structure from the imprinted network: complexity in the Hamiltonian may or may not give rise to complexity in the state, just as complexity can also arise out of non-complex simple systems. This is a non-trivial question that has to be addressed on a case-by-case basis, as we have separately considered for continuous-variable multimode quantum networks [31]. The general question of the relationship between complexity and emergence remains an open one [32].

While previous studies have focused on using quantum networks to, for example, establish quantum communication over long distances, we focus on the complex networks properties of the emergent network in the quantum many-body ground state defined on imprinted networks. Our work thus opens a new avenue for using network science to study quantum systems.

Hamiltonians are usually defined on regular lattices with regular links in condensed matter and statistical physics, and have random links in random matrix theory, e.g. in nuclear physics or spin glasses in condensed matter physics. Our novelty is that we generalize these Hamiltonians to complex networks. Specifically, we define the Hamiltonian as $\hat{H}={\sum }_{\langle ij\rangle }{\hat{H}}_{ij}$, where the sum runs over the links in the network. We call the network with these links ${\hat{H}}_{ij}$ as the imprinted network. Hamiltonians defined on such non-regular imprinted networks have much more complex phases such as spin glasses already at the classical level, and also form the basis of Hopfield networks for artificial neural nets. The phase diagram of Hamiltonians defined on networks which are neither regular nor random is an unexplored frontier of research.

We consider imprinted networks generated from three different models—the Erdos–Renyi (ER) model, the Watts–Strogatz (WS) model, and the Barabasi–Albert (BA) model. The structure of the networks generated from these models differ significantly from the case of regular lattices. The ER model is random. The WS model is a small-world network, while the BA model shows power-law scaling and is thus called scale-free. The WS and BA models are therefore referred to as complex, in contrast to the ER model or regular lattices.

Networks in each of the above models are constructed and parameterized differently, as detailed below. In this paper, we will find that an important parameter describing the universal physics of the ground state of $\hat{H}$ is the average degree of each node in the imprinted network, $Z={\sum }_{i}{\tilde {k}}_{i}/n$, also called the coordination number, where ${\tilde {k}}_{i}$ is the degree of node i.

Networks generated by the ER model are parameterized by a number 0 ⩽ p ⩽ 1, which specifies the probability that any pair of nodes are connected by a link. The distribution of the degree ${\tilde {k}}_{i}$ of node i is binomial, and the average degree is peaked around Z = (n − 1)p.

Networks generated by the WS model are parameterized by two numbers, an even integer K which specifies the mean degree, and 0 ⩽ p ⩽ 1. Here, K is the mean degree of a node, and p is the probability that a link between two nodes (i, j) is removed and rewired to connect i to another node j'. When p = 0, WS networks are regular, and when p = 1, they are random and have a structure close to ER networks ⁶ .

Networks generated by the BA model are parameterized by an integer m. The BA model is a preferential attachment model in which every new node is attached to m existing nodes of the network, with probability of the new node attaching to an existing node i given by ${p}_{i}={\tilde {k}}_{i}/{\sum }_{j}{\tilde {k}}_{j}$. The BA model is scale-free, because upon adding nodes, it quickly generates a power-law degree distribution of P(k) ≈ k⁻³. The average degree of a BA network is fixed, Z = 2m(n − m)/n.

Figures 1(a)–(c) show examples of networks generated from the ER, WS, and BA models.

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Central to our work is the emergent network that we define in the ground state of $\hat{H}$. This emergent network is a fully connected network, whose nodes are the same as those of the imprinted network, and links are given by the quantum MI ${\mathcal{I}}_{ij}$ between spins i and j. Our main focus in this work is analyzing the emergent network, before and after network attacks, and quantifying the complexity of ${\mathcal{I}}_{ij}$ via complex network measures defined in section 2.4

The quantum MI ${\mathcal{I}}_{ij}$ between spins i and j is constructed from the one- and two-point von Neumann entropies S_i = − Tr(ρ_i  log  ρ_i ) and S_ij = − Tr(ρ_ij  log  ρ_ij ) as

$$\begin{equation}{\mathcal{I}}_{ij}=\frac{1}{2}\left({S}_{i}+{S}_{j}-{S}_{ij}\right),\end{equation} \tag{ 1 }$$

where ρ_i =  Tr_k≠i (ρ) and ρ_j =  Tr_k≠j (ρ) are the reduced density matrices for the spins labeled i and j in the quantum many-body ground state, and ρ_ij =  Tr_k≠i,j (ρ) is the reduced density matrix for the subsystem containing only the two spins i and j. We define ${\mathcal{I}}_{ii}=0\enspace \forall i$.

While the quantum MI does not fully describe the system (a full description is given by the density matrix), there are still several motivations for using quantum MI. First, the pairwise MI is an upper bound to the squares of all two-body correlations along any direction [33]. This can be important since one is often only interested in knowing two-body correlations. Second, the MI network captures a lot of the richness of the system's quantum wave function, including long-range correlations and entanglement. Notably, previous works have shown that the variation of MI across the phase diagram mirrors the system's underlying phases [11–13, 22]. Third, the MI is a direct quantum analog of the most common complex network measures used in classical networks for EEG and fMRI [34], and is therefore an established choice from classical complex network theory and well understood. Finally, the MI taken as a complex network has proven useful in a number of quantum contexts, including complex dynamics [14].

Visualizing the full emergent MI network is cumbersome, and even more so when we consider an ensemble of imprinted networks. Therefore, we distill the MI in the emergent network by a few network measures. These are the degree k_i and clustering coefficient C_i on a node i, and shortest path length d_ij between i and j, generalized to analyze weighted networks as follows:

$$\begin{equation}\begin{aligned}{k}_{i}& =\sum\limits _{j}{\enspace \mathcal{I}}_{ij},\\ {C}_{i}& =\left(\sum\limits _{j\ne k}{\enspace \mathcal{I}}_{ij}{\mathcal{I}}_{jk}{\enspace \mathcal{I}}_{ki}\right)/\left(\sum\limits _{j\ne k}{\enspace \mathcal{I}}_{ij}{\mathcal{I}}_{ik}\right),\\ {d}_{ij}& ={\mathrm{min}}_{{i}_{1},{i}_{2},\dots {i}_{n}}P\left(i\to {i}_{1}\to {i}_{2}\to \dots \to j\right),\end{aligned}\end{equation} \tag{ 2 }$$

where $P\left(i\to {i}_{1}\to {i}_{2}\to \dots \to j\right)=1/{\mathcal{I}}_{i{i}_{1}}+1/{\mathcal{I}}_{{i}_{1}{i}_{2}}+\cdots +{\mathcal{I}}_{{i}_{n}j}$ is the length of the path from i to j via the nodes (i₁, i₂, ..., i_n ). Here, we have defined $1/{\mathcal{I}}_{ij}$ as the distance between two nodes i and j on the MI network via a direct link connecting them. Other network measures, such as disparity, geodesic distances, and various centrality measures can also yield useful information about the network [12]. Here we focus on k_i , C_i , and d_ij . We denote the corresponding measures for the imprinted networks as ${\tilde {k}}_{i}$, ${\tilde {C}}_{i}$, and ${\tilde {d}}_{ij}$.

The complex network measures also help us in classifying a network as complex or not. Complexity is not uniquely defined, and a working definition is often assumed to be a non-trivial topological feature such as a power-law degree distribution, or a higher clustering and shorter path length than random networks with the same average degree. In the following sections, we will use these working definitions to classify our emergent networks as complex or otherwise.

To illustrate the usefulness of these complex network measures, we briefly return to the imprinted networks. Figures 1(d)–(f) show the distributions of ${\tilde {k}}_{i}$, ${\tilde {C}}_{i}$, and ${\tilde {d}}_{ij}$ for an ensemble of imprinted ER, WS and BA networks. We clearly see that BA networks have a longer tail for ${\tilde {k}}_{i}$ than ER and WS networks, showing a hint of the power law decay already for n = 20. The BA networks also have a shorter path length on average than WS and ER networks.

We exactly diagonalize $\hat{H}$ (equation (3)) to find its ground state, and calculate ${\mathcal{I}}_{ij}$ and the measures defined on the MI network. For each imprinted network model, we consider an ensemble of 100 imprinted networks, and find moments of the network measures in this ensemble. We are limited to imprinted networks with 20 nodes, since the computations become exponentially harder for larger networks.

To support and understand our exact diagonalization results, we also calculate the ground state of equation (3) within MF theory. This theory, widely applied in condensed matter and statistical physics, works by assuming that each particle feels its immediate neighborhood in the system as a mean field. In practice, this is done by approximating terms in the Hamiltonian by complex scalars, and self-consistently or variationally finding these scalars. It typically works well in systems with a large dimension D, i.e., particles arranged in a regular lattice with Z = 2D. We find that it is sufficient to have large Z for MF theory to be valid, and the regular lattice is not necessary. We show that MF theory accurately predicts the average properties of the MI network for dense imprinted random and complex networks, i.e. those with Z ≳ 4.

To derive the MI, we make the MF approximation that each spin lies in the x–z plane, and points at an angle θ to the z-axis. For simplicity, we first consider the case that all spins have the same Z. Then, the expectation value of equation (3) is

$$\begin{equation}\langle \hat{H}\rangle =-JZN\left(\frac{1}{2}\enspace {\mathrm{cos}}^{2}\enspace \theta +\lambda \enspace \mathrm{sin}\enspace \theta \right),\end{equation} \tag{ 4 }$$

where λ = h/(ZJ). We solve for θ by minimizing equation (4). We obtain two solutions for λ < 1, θ =  sin⁻¹  λ and θ = π −  sin⁻¹  λ, and one solution for λ > 1, θ = π/2. Then the MF ground state is

$$\begin{align}\left\vert \psi \right\rangle & =\frac{\left\vert mm\cdots \enspace \right\rangle +\left\vert - m- m\cdots \enspace \right\rangle }{\sqrt{2\left(1+{\mathrm{sin}}^{n}\enspace \theta \right)}}\quad \mathrm{f}\mathrm{o}\mathrm{r}\enspace \lambda {< }1,\\ \left\vert \psi \right\rangle & =\left\vert \to \to \dots \right\rangle \quad \mathrm{f}\mathrm{o}\mathrm{r}\enspace \lambda { >}1,\end{align} \tag{ 5 }$$

where $m=\sqrt{1-{\lambda }^{2}}{\Theta}\left(1-\lambda \right)$, $\left\vert m\right\rangle =\mathrm{cos}\enspace \frac{\theta }{2}\left\vert {\uparrow}\right\rangle +\mathrm{sin}\enspace \frac{\theta }{2}\left\vert {\downarrow}\right\rangle$, and $\left\vert - m\right\rangle =\mathrm{sin}\enspace \frac{\theta }{2}\left\vert {\uparrow}\right\rangle +\mathrm{cos}\enspace \frac{\theta }{2}\left\vert {\downarrow}\right\rangle$. Note that $\left\vert m\right\rangle$ and $\left\vert - m\right\rangle$ are not orthogonal, and ⟨m|− m⟩ =  sin θ, which explains the denominator on the first line of equation (5). For simplicity, we will assume that n is large, and therefore  sinⁿ   θ ≪ 1 for λ < 1.

The single-particle density matrix in the MF ground state is

$$\begin{equation}{\rho }_{i}=\frac{1}{2}\left(\begin{matrix}\hfill 1+\langle {\hat{\sigma }}^{z}\rangle \hfill & \hfill \langle {\hat{\sigma }}^{x}+i{\hat{\sigma }}^{y}\rangle \hfill \\ \hfill \langle {\hat{\sigma }}^{x}-i{\hat{\sigma }}^{y}\rangle \hfill & \hfill 1-\langle {\hat{\sigma }}^{z}\rangle \hfill \end{matrix}\right)=\frac{1}{2}\left(\begin{matrix}\hfill 1\hfill & \hfill \sqrt{1-{m}^{2}}\hfill \\ \hfill \sqrt{1-{m}^{2}}\hfill & \hfill 1\hfill \end{matrix}\right).\end{equation} \tag{ 6 }$$

Similarly, the two-particle density matrix can be calculated to be

$$\begin{equation}{\rho }_{ij}=\frac{1}{4}\left(\begin{matrix}\hfill 1+{m}^{2}\hfill & \hfill \sqrt{1-{m}^{2}}\hfill & \hfill \sqrt{1-{m}^{2}}\hfill & \hfill 1-{m}^{2}\hfill \\ \hfill \sqrt{1-{m}^{2}}\hfill & \hfill 1-{m}^{2}\hfill & \hfill 1-{m}^{2}\hfill & \hfill \sqrt{1-{m}^{2}}\hfill \\ \hfill \sqrt{1-{m}^{2}}\hfill & \hfill 1-{m}^{2}\hfill & \hfill 1-{m}^{2}\hfill & \hfill \sqrt{1-{m}^{2}}\hfill \\ \hfill 1-{m}^{2}\hfill & \hfill \sqrt{1-{m}^{2}}\hfill & \hfill \sqrt{1-{m}^{2}}\hfill & \hfill 1+{m}^{2}\hfill \end{matrix}\right).\end{equation} \tag{ 7 }$$

Using equation (1), the MI between two nodes is

$$\begin{equation}{\mathcal{I}}_{\text{MF}}=\frac{1}{2}\left(1-\sqrt{1-{m}^{2}}\enspace {\mathrm{log}}_{2}\enspace \frac{1+\sqrt{1-{m}^{2}}}{1-\sqrt{1-{m}^{2}}}+\frac{2-{m}^{2}}{2}\enspace {\mathrm{log}}_{2}\enspace \frac{2-{m}^{2}}{{m}^{2}}\right).\end{equation} \tag{ 8 }$$

In this case, the MI network is a fully connected weighted network where all links have equal weights. The values of the network measures are ${k}_{i}/\left(n-1\right)={C}_{i}={\mathcal{I}}_{\text{MF}}$ and ${d}_{ij}=1/{\mathcal{I}}_{\text{MF}}$ for n ≫ 1.

The above analysis assumed for simplicity that all spins have the same Z. The systems we consider do not satisfy this condition. Then, our MF ansatz is modified to $\left\vert \psi \right\rangle =\left(\left\vert {m}_{1}{m}_{2}\dots \right\rangle +\left\vert -{m}_{1}-{m}_{2}\dots \right\rangle \right)/\sqrt{2}$ (up to a normalization constant). The self-consistent equation for m_i in this case is

$$\begin{equation}\frac{{m}_{i}}{\sqrt{1-{m}_{i}^{2}}}=\frac{J}{h}\sum\limits _{j\in \mathcal{N}\left(i\right)}{m}_{j},\end{equation} \tag{ 9 }$$

where $\mathcal{N}\left(i\right)$ is the set of neighbors of i. The solution to equation (9) is obtained from a set of coupled equations,

$$\begin{equation}{m}_{i}={\left(1+{\left(\frac{J}{h}\sum\limits _{j\in \mathcal{N}\left(i\right)}{m}_{j}\right)}^{-2}\right)}^{-1/2},\end{equation} \tag{ 10 }$$

which we solve by iterating equation (10) starting from an initial seed for {m_i }. The MF results in equations (6) and (7) are modified to

$$\begin{equation}\begin{aligned}{\rho }_{i}& =\frac{1}{2}\left(\begin{matrix}\hfill 1\hfill & \hfill \sqrt{1-{m}_{i}^{2}}\hfill \\ \hfill \sqrt{1-{m}_{i}^{2}}\hfill & \hfill 1\hfill \end{matrix}\right),\\ {\rho }_{ij}& =\frac{1}{4}\left(\begin{matrix}\hfill 1+{m}_{i}{m}_{j}\hfill & \hfill \sqrt{1-{m}_{i}^{2}}\hfill & \hfill \sqrt{1-{m}_{j}^{2}}\hfill & \hfill \sqrt{1-{m}_{i}^{2}}\sqrt{1-{m}_{j}^{2}}\hfill \\ \hfill \sqrt{1-{m}_{i}^{2}}\hfill & \hfill 1-{m}_{i}{m}_{j}\hfill & \hfill \sqrt{1-{m}_{i}^{2}}\sqrt{1-{m}_{j}^{2}}\hfill & \hfill \sqrt{1-{m}_{j}^{2}}\hfill \\ \hfill \sqrt{1-{m}_{j}^{2}}\hfill & \hfill \sqrt{1-{m}_{i}^{2}}\sqrt{1-{m}_{j}^{2}}\hfill & \hfill 1-{m}_{i}{m}_{j}\hfill & \hfill \sqrt{1-{m}_{i}^{2}}\hfill \\ \hfill \sqrt{1-{m}_{i}^{2}}\sqrt{1-{m}_{j}^{2}}\hfill & \hfill \sqrt{1-{m}_{j}^{2}}\hfill & \hfill \sqrt{1-{m}_{i}^{2}}\hfill & \hfill 1+{m}_{i}{m}_{j}\hfill \end{matrix}\right).\end{aligned}\end{equation} \tag{ 11 }$$

We will show that even though our assumption of uniform Z is invalid, the values of k_i , C_i , and d_ij are captured well by the solution in equation (8). We also capture the higher moments of k_i , C_i , and d_ij , such as the width of their distributions, using the generalized solution in equation (10).

We consider networks with n = 20 spins, and projectively measure 20% of the spins. We vary the measurement strength q, and the strategy used to choose the projected spins. We perform these attacks on the ground states of an ensemble of ER, WS, and BA networks, and calculate the complex network of the emergent MI network in all three cases. In addition to averaging over the ensemble of 100 networks for each network model, we also average over 50 realizations of the attack, i.e. 50 different combinations of the spins that we project, for the ground state on each imprinted network.

In figure 6, we consider the case that the networks are attacked by a projective measurement along $\hat{x}$, where the attacked spins are chosen randomly. Figure 6 plots the moments of the distributions of k_i /(n − 1), C_i , and d_ij after this attack. As in figure 4, panels (a)–(c) plot the average of k_i /(n − 1), C_i , and d_ij , panels (d)–(f) plot the width of their distributions, and panels (g)–(i) plot the skew. Panels (a)–(c) have eight curves each. The solid blue/red/green curves correspond to the case that the attacked nodes in each ER/WS/BA network are completely projected (q = 1) to the $\hat{x}$ direction. The dotted curves correspond to partially projecting the attacked spins to $\hat{x}$ with strength q = 0.5.

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Remarkably, we find that the three solid curves for exact results in figure 6(a)–(c) again overlap with each other when plotted versus λ, and the three dotted curves also overlap with each other. This occurs even though the imprinted networks have very different structures and different Z. Our calculations indicate that imprinted complex networks (such as WS and BA networks) do not show a more robust response to projective measurement along $\hat{x}$, as compared to non-complex (i.e. ER) networks. This is completely different from the case of classical network attacks, where the response of the system is highly dependent on whether the network is complex.

We also compare the exact results obtained from exact diagonalization with predictions from MF₀, shown as solid and dashed black curves in figures 6(a)–(c). When a node i is attacked with a projective measurement along $\hat{x}$, its magnetization $\langle {\hat{\sigma }}_{i}^{z}\rangle$ and all correlations $\langle {\hat{\sigma }}_{i}^{z}{\hat{\sigma }}_{j}^{z}\rangle$ get shrunk by (1 − q). Since $\langle {\hat{\sigma }}_{i}^{z}\rangle =0$ due to Z₂ symmetry, the node's single-particle density matrix after projection is left unchanged. The two-particle density matrix ρ_ij after an attack on node i becomes

$$\begin{equation}{\rho }_{ij}=\frac{1}{4}\left(\begin{matrix}\hfill 1+{m}^{2}\left(1-q\right)\hfill & \hfill \sqrt{1-{m}^{2}}\hfill & \hfill \sqrt{1-{m}^{2}}\hfill & \hfill \left(1-q\right)\left(1-{m}^{2}\right)\hfill \\ \hfill \sqrt{1-{m}^{2}}\hfill & \hfill 1-{m}^{2}\left(1-q\right)\hfill & \hfill \left(1-{m}^{2}\right)\hfill & \hfill \sqrt{1-{m}^{2}}\hfill \\ \hfill \sqrt{1-{m}^{2}}\hfill & \hfill \left(1-{m}^{2}\right)\hfill & \hfill 1-{m}^{2}\left(1-q\right)\hfill & \hfill \sqrt{1-{m}^{2}}\hfill \\ \hfill \left(1-q\right)\left(1-{m}^{2}\right)\hfill & \hfill \sqrt{1-{m}^{2}}\hfill & \hfill \sqrt{1-{m}^{2}}\hfill & \hfill 1+{m}^{2}\left(1-q\right)\hfill \end{matrix}\right).\end{equation} \tag{ 12 }$$

If both nodes i and j are attacked, then ρ_ij has a similar form to equation (12), with (1 − q) replaced by (1 − q)². The MI can then be calculated using equations (1) and (12).

The black curves in figures 6(a)–(c) plot the MF₀ prediction for the mean value of k_i /(n − 1), C_i , and d_ij . These quantitatively differ from the exact results, but are qualitatively similar. MF₀ does not correctly predict any higher moments of the distributions.

In figure 7, we pick the nodes to be attacked preferentially, with the probability of a node i to be attacked given by k_i /∑_j k_j . The attacks are again projective measurements along $\hat{x}$ with q = 0.5 (dotted) or q = 1 (solid). We find almost no difference to the case of random attacks. This is a surprising finding, since the response of complex networks to classical network attacks is known to depend on whether the attacked nodes are picked randomly or preferentially.

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Our calculations highlight two striking aspects. First, they indicate that (imprinted) complex networks respond similarly to projective measurement attacks as non-complex networks, and respond similarly when the attacks are random or preferential. This is evident from the overlapping curves for the three imprinted networks in figures 6 and 7, and is completely different from the case of classical network attacks—network attacks with a classical counterpart, such as deleting nodes—where the response to the attacks depend heavily on the structure of the imprinted networks. The main reason for this is that the attacks are made on the emergent networks, and these networks do not seem to be complex. Moreover, MF₀ as well as the general MF theory qualitatively capture the mean values of k_i /(n − 1), C_i , and d_ij , although MF₀ does not capture any higher moments, and the general MF theory has quantitative disagreements for all higher moments of the distributions. The accuracy of MF and MF₀ is likely due to the large degree of nodes in the imprinted networks, consistent with its accuracy in regular lattices with a large Z in condensed matter systems. As we show later, the results might deviate from MF theory when the degree is made smaller.

The second striking aspect of our calculation, illustrated in figure 8, is that we also observe a collapse of the curves for average k_i when it is normalized by the average k_i (h = 0), for the cases before attacks and after attacks (for all the four kinds of attacks considered in figures 6–B3 and B2–7). This indicates that projective measurement attacks only rescale the average value of k_i before the attack by a constant factor, irrespective of the Hamiltonian parameters or the imprinted network's structure.

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Finally, we consider an ensemble of imprinted networks that are expected to lie outside the validity of MF theory. To this end, we consider ER networks with p = 0.1, WS networks with K = 2 and p = 0.5, and BA networks with m = 1, all of which have a smaller Z than before. We only consider connected networks, and discard disconnected networks. Figures 9(a), (d), and (g) plots the moments of the density k_i /(n − 1) for these networks. We clearly see that the mean density is not captured by MF₀. Unlike the case of dense imprinted networks, we also see here a difference between the mean value of k_i in the three classes of networks.

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Performing projective attacks on these networks, we find that MF₀ theory again does not capture the mean density. Figures 9(b), (e) and (h) plot the moments of the density k_i /(n − 1) after 20% of the nodes are randomly picked and projected along $\hat{x}$, and figures 9(c), (f) and (i) plot the moments of k_i /(n − 1) after 20% of the nodes are preferentially picked and projected along $\hat{x}$.

Nevertheless, as shown in figure 10, we again find that when the average k_i is rescaled by the average k_i (h = 0), the curves before and after projective attacks collapse onto each other. This indicates that even when MF does not capture the distributions of k_i (and possibly other network measures too), the dominant effect of projective attacks on the average measure is only a simple rescaling.

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The small network size that we consider is a potential concern in comparing the robustness of complex and random networks to random versus targeted attacks. We are limited to n = 20 for calculating the ground state with exact diagonalization. Larger networks can be studied using more sophisticated numerical techniques, or realized in experiment. Network size is, however, much less of a concern for calculating measures of classical networks. Exploiting this, we investigate the robustness of complex and random networks to classical attacks at n = 20 and n = 54 in appendix A, as well as consider a larger ensemble of classical networks. For the classical attack, we delete nodes, since this resembles projectively measuring a node with q = 1, which sets the MI between the measured node and all other nodes as 0. We find that BA networks with n = 54 are more robust to random deletions than targeted deletions, as expected, and this robustness is less prominent for networks with n = 20. An experimental realization of the quantum networks considered in this paper will conclusively demonstrate whether the robustness to random classical attack, i.e. node deletions, is also extended to projective measurement attacks.

Quantum networks are subject to many kinds of attacks. In the following, we briefly sketch some forms of attack for future study.

Decoherence effects can be modeled by a quantum trajectories approach or by a master equation, for example the Lindblad master equation. There are many sources of possible decoherence. For the master equation approach, each form of decoherence is identified by Lindblad operators. We list a couple below.

Single-qubit dephasing along $\hat{x}$ is modeled by the master equation

$$\begin{equation}{\partial }_{t}\rho =-i\left[\hat{H},\rho \right]+\gamma \sum\limits _{i}{\hat{\sigma }}_{i}^{x}\rho {\hat{\sigma }}_{i}^{x}-\rho .\end{equation} \tag{ 13 }$$

Collective dephasing is modeled by the master equation

$$\begin{equation}{\partial }_{t}\rho =-i\left[\hat{H},\rho \right]+\gamma \left({\hat{\sigma }}_{\text{tot}}^{x}\rho {\hat{\sigma }}_{\text{tot}}^{x}-\frac{1}{2}{\left({\hat{\sigma }}_{\text{tot}}^{x}\right)}^{2}\rho -\frac{1}{2}\rho {\left({\hat{\sigma }}_{\text{tot}}^{x}\right)}^{2}\right).\end{equation} \tag{ 14 }$$

Both these forms of dephasing suppress density matrix elements that are off-diagonal in the $\hat{x}$ basis. We expect these to have the same effect as the projective measurements we consider, which also suppress off-diagonal elements.

Depolarizing is modeled by the master equation

$$\begin{equation}{\partial }_{t}\rho =-i\left[\hat{H},\rho \right]+\gamma \left({2}^{-n}-\rho \right).\end{equation} \tag{ 15 }$$

The solution to this equation is $\rho \left(t\right)={\text{e}}^{-\gamma t}\enspace {\text{e}}^{-\text{i}\hat{H}t}\rho \enspace {\text{e}}^{\text{i}\hat{H}t}+\left(1-{\text{e}}^{-\gamma t}\right)/{2}^{n}$. The two-particle density matrices, which are used to define ${\mathcal{I}}_{ij}$, also have the same form. Insofar as MF theory qualitatively captures the mean value of k_i /(n − 1), C_i , and d_ij before the attack, we expect it to capture them after depolarizing attack as well.

Recently, it has been shown for the TIM that a multichannel Lindblad approach is needed to correctly predict equilibration times, which provide a bound on decoherence times [42]. Thus attacks due to a reservoir may address global eigenstates, i.e., the entire wave function—not just local qubits. We expect the same issues to arise for quantum complex networks based on spins with an Ising coupling.

Other attack scenarios include the usual removal of nodes one finds in classical complex network robustness studies, probes of reduced Hilbert spaces on a subset of qubits, and using a single immersed qubit as a probe of complex network structure [30].