A Model for Quantum Autonomous Boolean Networks

Autonomous Boolean networks were first developed in the late 1960s as a tool for studying evolvability in complex systems. Notably Kauffman's NK model proved to be particularly useful for understanding genetic regulatory systems. It wasn't until the early 2000s that more extensive mathematical studies of these networks were undertaken and, while the NK model has been applied to a handful of quantum systems, a direct quantum analog has not been developed. In this work we develop just such a quantum model. In particular we focus on Boolean functions of two logical inputs. Due to requirements of unitarity, some functions require an additional ancilla qubit. As in Kauffman's model the network connections are chosen randomly and then set. While the quantum network evolves deterministically (as long as there is no measurement) and thus exhibits state cycles, unlike the classical state cycles, the length of the quantum state cycles appears to be independent of the number of nodes in the network and the lengths vary considerably. These networks also exhibit a cyclic entanglement structure. While these networks can reproduce the classical structures discovered by Kauffman when limited to purely classical inputs, they exhibit a much richer landscape of structure when extended to the quantum regime. We also explore the robustness of these networks to small perturbations and we show that, like their classical counterparts, they exhibit clustering phenomena.