  It has recently been shown that vacuum expectation values and Feynman path
integrals can be regularized using Fourier integral operator $\zeta$-function,
yet the physical meaning of these $\zeta$-regularized objects was unknown.
  Here we show that $\zeta$-regularized vacuum expectations appear as continuum
limits using a certain discretization scheme. Furthermore, we study the rate of
convergence for the discretization scheme using the example of a
one-dimensional hydrogen atom in $(-\pi,\pi)$ which we evaluate classically,
using the Rigetti Quantum Virtual Machine, and on the Rigetti 8Q quantum chip
"Agave" device. We also provide the free radiation field as an example for the
computation of $\zeta$-regularized vacuum expectation values in a gauge theory.
