Emergence of drifted diffusion in quantum walks with subspace restart

Abstract

Restart of a quantum process is typically modeled as a global reinitialization that erases the system's entire history. Here we introduce subspace restart, a protocol that periodically resets only the internal degrees of freedom while preserving the spatial distribution, as a tunable knob for the quantum-to-classical crossover. Using the discrete-time quantum walk as an example, we show that this selective reset drives the walker into an engineered drifted-diffusion regime. This phenomenon can be understood by a Huygens-Fresnel mechanism, where each restart fragments the wave function into a set of independent secondary sources to screen long-range correlations and isolate a robust classical backbone, whose drift and diffusivity are set by the geometric orientation of the initial coin and the restart period. Residual quantum interference, confined to effective light cones, survives only as a short-range correction that renormalizes these coefficients and imprints periodic modulations on the cumulants. Our results establish subspace restart as a route to controlling the quantum-to-classical crossover in synthetic lattices.