Discrete-Time Quantum Random Walk for Epidemiological Modeling

Abstract

We introduce a discrete-time quantum random walk (QRW) framework for spatial epidemic modelling on a two-dimensional square lattice and compare its dynamics to classical random-walk SIR models. In our model, each infected site spawns a quantum walker whose coherent evolution (controlled by an amplitude-splitting coin and conditional shifts) can infect visited susceptible sites with probability $p$ and persists for a lifetime of $τ$ steps. We perform extensive quantum simulations on finite lattices and compute the basic reproduction number $R_0$ across a broad grid of $(p,τ)$ values. Results show that QRW dynamics interpolate between diffusive and super-diffusive regimes: at low $p$ the QRW reproduces classical-like $R_0$, while at higher $p$ and $τ$ ballistic propagation and interference produce markedly larger $R_0$ and non-Gaussian spatial profiles. We compare the QRW $R_0$ range to empirical estimates from historical outbreaks and discuss parameter regimes where QRW offers a closer qualitative match than classical diffusion. We conclude that QRWs provide a flexible, conceptually novel toy model for exploring rapid or heavy-tailed epidemic spread.