  We improve the space complexity of Karatsuba multiplication on a quantum
computer from $O(n^{1.427})$ to $O(n)$ while maintaining $O(n^{\lg 3})$ gate
complexity. We achieve this by ensuring recursive calls can add their outputs
directly into subsections of the output register. This avoids the need to
store, and uncompute, intermediate results. This optimization, which is
analogous to classical tail-call optimization, should be applicable to a wide
range of recursive quantum algorithms.
