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Cellular automata are interacting classical bits that display diverseemergent behaviors, from fractals to random-number generators toTuring-complete computation. We discover that quantum cellular automata (QCA)can exhibit complexity in the sense of the complexity science that describesbiology, sociology, and economics. QCA exhibit complexity when evolving under"Goldilocks rules" that we define by balancing activity and stasis. OurGoldilocks rules generate robust dynamical features (entangled breathers),network structure and dynamics consistent with complexity, and persistententropy fluctuations. Present-day experimental platforms -- Rydberg arrays,trapped ions, and superconducting qubits -- can implement our Goldilocksprotocols, making testable the link between complexity science and quantumcomputation exposed by our QCA.

Entangled quantum cellular automata, physical complexity, and Goldilocks rules