  We construct noncommutative or `quantum' Riemannian geometry on the integers
$\Bbb Z$ as a lattice line $\cdots\bullet_{i-1}-\bullet_i-\bullet_{i+1}\cdots$
with its natural 2-dimensional differential structure and metric given by
arbitrary non-zero edge square-lengths $\bullet_i{\buildrel a_i\over
-}\bullet_{i+1}$. We find for general metrics a unique $*$-preserving quantum
Levi-Civita connection, which is flat if and only if $a_i$ are a geometric
progression where the ratios $\rho_i=a_{i+1}/a_i$ are constant. More generally,
we compute the Ricci tensor for the natural antisymmetric lift of the volume
2-form and find that the quantum Einstein-Hilbert action up to a total
divergence is $-{1\over 2}\sum \rho\Delta \rho$ where
$(\Delta\rho)_i=\rho_{i+1}+\rho_{i-1}-2\rho_i$ is the standard discrete
Laplacian. We take a first look at some issues for quantum gravity on the
lattice line. We also examine $1+0$ dimensional scalar quantum theory with mass
$m$ and the lattice line as discrete time. As an application, we compute
discrete time cosmological particle creation for a step function jump in the
metric by a factor $\rho$, finding that an initial vacuum state has at later
times an occupancy $<N>=(1-\sqrt{\rho})^2/(4\sqrt{\rho})$ in the continuum
limit, independently of the frequency. The continuum limit of the model is the
time-dependent harmonic oscillator, now viewed geometrically.
