The quantum version of the Hilhorst-van Leeuwen model i.e. an inhomogeneous Ising model in a transverse field with a marginal perturbation of the bulk interaction varying like l-1 with the distance l from the surface is studied numerically on a finite chain with symmetric interactions using fermion techniques and finite-size scaling at the bulk critical point. The surface exponents for the energy and the magnetisation are respectively xe=2 alpha +2, xm= alpha +1/2 when alpha >or= alpha c and xe=- alpha +1/2, xm=0 and x'm=- alpha -1/2 when alpha <or= alpha c where alpha c=-1/2 is the critical value of the perturbation amplitude below which the surface becomes ordered at the bulk critical point.