  This paper studies the problem of computing a linear approximation of
quadratic Wasserstein distance $W_2$. In particular, we compute an
approximation of the negative homogeneous weighted Sobolev norm whose
connection to Wasserstein distance follows from a classic linearization of a
general Monge-Amp\'ere equation. Our contribution is threefold. First, we
provide expository material on this classic linearization of Wasserstein
distance including a quantitative error estimate. Second, we reduce the
computational problem to solving an elliptic boundary value problem involving
the Witten Laplacian, which is a Schr\"odinger operator of the form $H =
-\Delta + V$, and describe an associated embedding. Third, for the case of
probability distributions on the unit square $[0,1]^2$ represented by $n \times
n$ arrays we present a fast code demonstrating our approach. Several numerical
examples are presented.
