The effect of the thickness of the dielectric boundary layer that connects a material of refractive index n1 to another of index n2 is considered for the propagation of an electromagnetic pulse. A qubit lattice algorithm (QLA), which consists of a specially chosen non-commuting sequence of collision and streaming operators acting on a basis set of qubits, is theoretically determined that recovers the Maxwell equations to second-order in a small parameter ϵ. For very thin but continuous boundary layer the scattering properties of the pulse mimics that found from the Fresnel discontinuous jump conditions for a plane wave - except that the transmission to incident amplitudes are augmented by a factor of √n2/n1. As the boundary layer becomes thicker one finds deviations away from the discontinuous Fresnel conditions and eventually one approaches the expected WKB limit. However there is found a small but unusual dip in part of the transmitted pulse that persists in time. Computationally, the QLA simulations still recover the solutions to Maxwell equations even when this parameter ϵ→1. On examining the pulse propagation in medium n1,ϵ corresponds to the dimensionless speed of the pulse (in lattice units).