Exact diffusion coefficients for quantum--classical hybrid walks

One-dimensional quantum--classical hybrid walks are introduced by combining quantum and classical steps under periodic and random protocols. Exact diffusion coefficients are derived for both protocols throughout the diffusive regime. The exact results show how diffusion depends on the average frequency and temporal arrangement of classical steps. As the average frequency approaches zero, both diffusion coefficients diverge inversely with the frequency. At matched average frequencies, the ratio of the diffusion coefficients for the random and periodic protocols approaches two. A finite-mode approximation is also developed and tested against the exact diffusion coefficient for the random protocol. The exact results for the one-dimensional model provide a foundation for future studies of more general quantum--classical hybrid walks.