Quantum walks get trapped in random comb graphs, defying expectations

Scientists have uncovered unusual behaviour in quantum walks along random comb graphs. Unlike in regular structures, introducing randomness to the tooth lengths and positions causes the quantum particle to become trapped. This results in a 70% chance of the particle remaining confined within a limited area instead of spreading infinitely along the graph’s central 'spine'. The study focused on how quantum particles move across a comb graph—a structure with a straight central 'spine' and branching 'teeth'. When randomness was added to the tooth arrangement, interference effects emerged, disrupting the particle’s expected path. As a result, the wave function localised, preventing infinite travel along the spine.

Calculations revealed a minimal Lyapunov exponent, which defines the maximum distance a particle can travel before dispersing significantly. This value directly linked to the degree of localisation observed. The Hamiltonian’s spectrum further showed that eigenstates with energies above 4 exhibited a Lyapunov exponent, indicating how far the wave packet could spread. Analysis of the eigenstates confirmed that localised states clustered along the spine, while extended states spread across the entire graph. Despite the confinement along the spine, the walk could still extend infinitely along the teeth. The unitarity of the S-matrix ensured that probability remained conserved throughout the process. The findings stem from the interaction between the graph’s random topology and the quantum mechanical nature of the walk. This interplay creates a scenario where the particle has a measurable probability of staying trapped in a finite region.

The research demonstrates that random comb graphs can induce strong localisation in quantum walks. With a 0.7 probability of confinement, the results highlight how structural randomness alters quantum behaviour. These insights could influence future studies on quantum transport and localisation in complex networks.