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

%B Quantum Science and Technology %V 6 %P 045017 %8 9/29/2021 %G eng %U http://dx.doi.org/10.1088/2058-9565/ac1c41 %R 10.1088/2058-9565/ac1c41