  We use the worldline numerics technique to study a cylindrically symmetric
model of magnetic flux tubes in a dense lattice and the non-local Casimir
forces acting between regions of magnetic flux. Within a superconductor the
magnetic field is constrained within magnetic flux tubes and if the background
magnetic field is on the order the quantum critical field strength, $B_k =
\frac{m^2}{e} = 4.4 \times 10^{13}$ Gauss, the magnetic field is likely to vary
rapidly on the scales where \acs{QED} effects are important. In this paper, we
construct a cylindrically symmetric toy model of a flux tube lattice in which
the non-local influence of \acs{QED} on neighbouring flux tubes is taken into
account. We compute the effective action densities using the worldline numerics
technique. The numerics predict a greater effective energy density in the
region of the flux tube, but a smaller energy density in the regions between
the flux tubes compared to a locally-constant-field approximation. We also
compute the interaction energy between a flux tube and its neighbours as the
lattice spacing is reduced from infinity. Because our flux tubes exhibit
compact support, this energy is entirely non-local and predicted to be zero in
local approximations such as the derivative expansion. This Casimir-Polder
energy can take positive or negative values depending on the distance between
the flux tubes, and it may cause the flux tubes in neutron stars to form
bunches.
  In addition to the above results we also discuss two important subtleties of
determining the statistical uncertainties within the worldline numerics
technique and recommend a form of jackknife analysis.
