Spectral Criterion for Disorder-Free Localization of Quantum Walks on Hypercube: QBN Approach

We study disorder-free localization in a spin-coupled quantum walk on the hypercube within the quantum Bernoulli noises framework. We derive a general criterion for disorder-free localization: it occurs if and only if some spectral subspace of a reduced unitary evolution contains a vector whose position marginal probability is non-uniform. For the Grover walk with distance-dependent coupling ${\phi }_{\sigma }=\mathrm{\mid }\sigma \mathrm{\mid }\pi \mathrm{/}(n+1)$ in dimension $d=4$, we explicitly construct a non-uniform fixed point in the all $+1$ spin configuration. This implies that disorder-free localization occurs in our model.