  We numerically compute renormalized expectation values of quadratic operators
in a quantum field theory (QFT) of free Dirac fermions in curved
two-dimensional (Lorentzian) spacetime. First, we use a staggered-fermion
discretization to generate a sequence of lattice theories yielding the desired
QFT in the continuum limit. Numerically-computed lattice correlators are then
used to approximate, through extrapolation, those in the continuum. Finally, we
use so-called point-splitting regularization and Hadamard renormalization to
remove divergences, and thus obtain finite, renormalized expectation values of
quadratic operators in the continuum. As illustrative applications, we show how
to recover the Unruh effect in flat spacetime and how to compute renormalized
expectation values in the Hawking-Hartle vacuum of a Schwarzschild black hole
and in the Bunch-Davies vacuum of an expanding universe described by de Sitter
spacetime. Although here we address a non-interacting QFT using free fermion
techniques, the framework described in this paper lays the groundwork for a
series of subsequent studies involving simulation of interacting QFTs in curved
spacetime by tensor network techniques.
