Small-world complex network generation on a digital quantum processor

The universe exhibits complex emergent phenomena despite being governed by simple physical laws. Classical cellular automata (CA) demonstrate how complexity can arise from simple rules, inspiring investigation into quantum analogs. Quantum cellular automata (QCA) are computational models exhibiting emergent complexity from local unitary operators, offering potential for simulating strongly-correlated matter and exploring beyond-classical computation. However, classical computers are limited in simulating large quantum systems, hindering QCA research. This study leverages the power of digital quantum processors to experimentally explore QCA, specifically focusing on the Goldilocks rule, to investigate the emergence of complex networks.

Classical cellular automata have been extensively studied, demonstrating their ability to generate complex behavior from simple rules. Some CA are even Turing complete. The quantum mechanical nature of the universe motivates the study of quantum cellular automata (QCA) as a more fundamental model of complex systems. Previous work has shown that certain Goldilocks QCA generate mutual information networks with characteristics of small-world networks, including large clustering, short average path length, and broad node-strength distribution. Applications of QCA have been proposed in lattice discretization for the simulation of strongly-correlated matter and quantum field theories. However, the limitations of classical computers in simulating large quantum systems have hindered the exploration of QCA.

A one-dimensional Goldilocks QCA rule was simulated on a Sycamore-class superconducting processor (Weber). The experiment involved initializing a chain of qubits, applying QCA update cycles, and measuring the resulting state. The specific QCA rule used was a totalistic, three-site Goldilocks rule with a Hadamard activation unitary. The authors employed a suite of techniques to optimize circuit performance and mitigate errors, including moment alignment, spin-echo insertion, Floquet calibration, parasitic cphase compensation, and post-selection. Population dynamics were calculated and analyzed to understand the temporal evolution of the system. Classical Shannon mutual information between qubit pairs was calculated to characterize the network structure at each cycle. Network measures, including clustering coefficient, average path length, and node strength distribution, were computed and compared to both noise-free emulations and post-selected random states to determine the emergence of small-world network properties. Post-selection was applied based on the conservation of domain walls, a dynamical invariant of the Goldilocks rule.

The experimental results demonstrate the formation of small-world mutual information networks in the simulated Goldilocks QCA. Post-selection played a crucial role in revealing these characteristics by mitigating the effects of noise. The analysis of population dynamics showed coherent dynamics that persisted beyond a certain cycle depth due to the post-selection, while raw data showed rapid decoherence. The clustering coefficient remained significantly larger than that of a post-selected incoherent random state, indicating substantial network transitivity. Average path length in the post-selected data was one to two orders of magnitude smaller than in the raw data, demonstrating short global traversability. The node strength distribution of the post-selected data exhibited a flatter distribution compared to the random state, suggesting the presence of hubs in the network. These findings were consistent across various system sizes, with the largest system (23 qubits) involving 1056 √ISWAP gates.

The study successfully demonstrates the generation of small-world networks through the emergent dynamics of a QCA simulated on a noisy quantum processor. The results validate previous theoretical work suggesting the complexity generation potential of QCA and highlights the capability of near-term quantum processors for simulating such systems. The post-selection technique proved essential in unveiling the underlying complex network behavior by mitigating the influence of noise. The observation of small-world network characteristics, which are typically found in complex systems like social and biological networks, points to potential applications of QCA in modeling other complex systems and understanding emergent phenomena. These results also have implications for quantum computing architecture design and the development of robust quantum algorithms.

This research successfully demonstrates the generation of small-world mutual information networks within a quantum cellular automaton simulated on a noisy quantum processor. The use of post-selection was key in revealing this complex network structure. The findings validate the potential of QCA as models of complex systems and support the use of near-term quantum processors as powerful simulation tools. Future work could explore different QCA rules, investigate the impact of other noise mitigation techniques, and examine larger system sizes to further understand the scalability and limitations of this approach.

The study focused on a specific QCA rule and a particular quantum processor architecture. The post-selection technique, while effective, discards a significant portion of the data. Further research is needed to assess the generalizability of these findings to different QCA rules and quantum hardware platforms. The relatively small system sizes investigated limit the ability to make definitive conclusions about the scaling properties of the observed phenomena. The choice of classical Shannon mutual information as a metric, while suitable in this context, could be compared against quantum versions in future work.