Noisy Cyclic Quantum Random Walk

Abstract

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 ${\varphi }_s=\pi /3$, 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.

22 pages, 15 figures

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