Noisy Cyclic Quantum Random Walk

We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on spectral and dynamical properties. Using a three-parameter unitary coin, we control the spectral structure of the noiseless step operator on the unit circle. One parameter induces two spectral bands separated by a gap proportional to its value, while the half-sum of the two phase parameters rotates the spectrum and enables twofold degeneracy under specific conditions. Degenerate spectra yield sinusoidal probability distributions; non-degenerate ones produce flat profiles. We introduce static phase noise on the sites and analyze its effects in two propagation regimes. In the walk-on-the-line regime, preceding a full graph traversal, we extract the spreading exponent $\beta$ from the step-resolved mean squared displacement. Low participation ratios correlate with sub-diffusive spread; high ratios indicate ballistic or super-diffusive evolution. Once the walker completes a cycle, finite-size effects dominate. In this walk-on-the-cycle regime, $\beta$ no longer characterizes the dynamics. Instead, we quantify localization using the coefficient of variation of the mean squared displacement. In both regimes, we observe a sharp crossover near static site noise $\phi_s = \pi/3$, marked by a drop in participation ratio, a transition from diffusive to sub-diffusive spread in the walk-on-the-line regime, and a reduced saturation level in the walk-on-the-cycle regime. Our results show that the eigenstate participation ratio is an efficient spectral diagnostic that anticipates localization across both regimes, offering an alternative to full dynamical simulations.