Noisy Cyclic Quantum Random Walk

Abstract: We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on the spectral and dynamical properties of the system. Using a three-parameter unitary coin, we control the spectral properties of the noiseless step operator on the unit circle in the complex plane. One parameter governs the probability amplitudes and induces two spectral bands, with a gap proportional to its value. The half-sum of the two phase parameters rotates the spectrum and induces twofold degeneracy under specific conditions. Degenerate spectra yield eigenstates with sinusoidal probability distributions, whereas non-degenerate spectra produce flat distributions. By using the eigenstate participation ratio, we predict the behavior of a walker under static phase noise in the coin and sites, showing a correlation between low participation ratios and localization, and high ratios with delocalization. Our results show that the average eigenstate participation ratio provides insights equivalent to computationally intensive mean squared displacement calculations. We observe a transition from super-diffusive to sub-diffusive behavior for uniformly distributed noise within the range $-\pi/3$ to $\pi/3$ and saturation of the mean square distance when the number of steps exceeds the graph size by an order of magnitude. Finally, we propose a quantum circuit implementation of our model.