Quantum cellular automata part II: Robust complexity under analog and digital evolution in 1- and 2-dimensions

Quantum cellular automata are an extension of the classical cellular automata into quantum lattice systems. We consider symmetric, or totalistic, rules in 1- and 2-d quantum analog and digital evolution. We demonstrate the existence of a class of long-lived highly entangled states under noise models consistent with modern noisy intermediate-scale quantum computing fidelity levels. We showcase a robust emergent phenomenon in 1-d called the quantum entangled breather (QEB) which arises from a particular rule. Under certain rule admixtures, the QEB's lifetime exhibits power-law decay in the admixture ratio. The QEB's lifetime also exhibits a threshold-law in its Schmidt rank which indicates the QEB requires a minimum amount of entanglement to persist. We present 2- and 5-qubit gate decompositions of the aforementioned rules, and demonstrate the quantum digital evolution is resilient to depolarizing noise. In all cases, we compute network complexity dynamics based on mutual information and other adjacency matrices optimized for low gate count. We characterize trends in network structure using clustering and disparity fluctuations to quantify the complexity of the quantum states.