How to embed Pearson distance into Euclidean space

Question

I have a lot of numerical vectors, each of dimension 1000. I would like to compare them according to their Pearson distance. This works fine but comparing all vectors to each other is quadratic time and too slow. Ideally I would like to be able to perform efficient approximate nearest neighbour searches instead.

If I could embed the vectors into Euclidean space then I could use standard tools to do this.

Is there a way to embed vectors from a space using the Pearson distance into Euclidean space?