Robustness of periodicity in Grover walks under a magnetic vector potential

This paper investigates the stability of periodic Grover walks—a type of discrete-time quantum walk—when subjected to perturbations in the form of a magnetic vector potential. The authors model the magnetic vector potential within the framework of quantum graphs, where it acts as a phase factor on the edges of the graph. By treating this potential as a small perturbation (parameterized by strength β), the researchers aim to determine how the original periodic dynamics of the Grover walk deviate over time and whether these deviations can be captured by a simpler, continuous-time model.

The study establishes that the perturbed discrete-time dynamics converge to a continuous-time quantum walk as the perturbation strength β approaches zero. The authors derive a Hermitian matrix, H, that governs this continuous-time evolution. The spectral properties of this matrix, and consequently the robustness of the walk's periodicity, are shown to be fundamentally linked to the spectral structure of the graph's discriminant matrix. Specifically, the robustness is determined by the dimension of the kernel of H, which the authors decompose into subspaces related to simple eigenvectors, periodic components, and the graph's fundamental cycles.

Understanding the robustness of quantum walks is critical for quantum computation, where maintaining specific states or periodic behaviors is often necessary for algorithm performance. This work provides a rigorous mathematical bridge between discrete-time quantum walks and continuous-time approximations under external influences. By identifying that graphs with more fundamental cycles (higher first Betti number) tend to exhibit stronger robustness, the paper offers a structural guideline for designing quantum networks that are more resilient to environmental magnetic noise or similar perturbations.