We investigate the ground states of spin models defined on networks that we imprint (e.g., non-complex random networks like Erdos–Renyi, or complex networks like Watts–Strogatz, and Barabasi–Albert), and their response to decohering processes which we model with network attacks. We quantify the complexity of these ground states, and their response to the attacks, by calculating distributions of network measures of an emergent network whose link weights are the pairwise mutual information between spins. We focus on attacks which projectively measure spins. We find that the emergent networks in the ground state do not satisfy the usual criteria for complexity, and their average properties are captured well by a single dimensionless parameter in the Hamiltonian. While the response of classical networks to attacks is well-studied, where classical complex networks are known to be more robust to random attacks than random networks, we find counter-intuitive results for our quantum networks. We find that the ground states for Hamiltonians defined on different classes of imprinted networks respond similarly to all our attacks, and the attacks rescale the average properties of the emergent network by a constant factor. Mean field theory explains these results for relatively dense networks, but we also find the simple rescaling behavior away from the regime of validity of mean field theory. Our calculations indicate that complex spin networks are not more robust to projective measurement attacks, and presumably also other quantum attacks, than non-complex spin networks, in contrast to the classical case. Understanding the response of the spin networks to decoherence and attacks will have applications in understanding the physics of open quantum systems, and in designing robust complex quantum systems—possibly even a robust quantum internet in the long run—that is maximally resistant to decoherence.

我们研究了定义在我们印记的网络上的自旋模型的基态（例如，像 Erdos-Renyi 这样的非复杂随机网络，或像 Watts-Strogatz 和 Barabasi-Albert 这样的复杂网络），以及它们对我们建模的退相干过程的响应与网络攻击。我们通过计算紧急网络的网络度量分布来量化这些基态的复杂性及其对攻击的响应，该网络的链接权重是自旋之间的成对互信息。我们专注于投射测量旋转的攻击。我们发现基态的涌现网络不满足通常的复杂性标准，并且它们的平均特性可以被哈密顿量中的单个无量纲参数很好地捕获。虽然经典网络对攻击的反应得到了充分研究，在已知经典复杂网络对随机攻击比随机网络更稳健的情况下，我们发现我们的量子网络有反直觉的结果。我们发现在不同类型的印记网络上定义的哈密顿量的基态对我们所有的攻击都有相似的反应，并且这些攻击通过一个常数因子重新调整了涌现网络的平均属性。平均场理论解释了相对密集网络的这些结果，但我们也发现了远离平均场理论有效性的简单重新缩放行为。我们的计算表明，与经典情况相比，复杂自旋网络对投影测量攻击和其他量子攻击的鲁棒性并不比非复杂自旋网络强。