  Integer arithmetic is the underpinning of many quantum algorithms, with
applications ranging from Shor's algorithm over HHL for matrix inversion to
Hamiltonian simulation algorithms. A basic objective is to keep the required
resources to implement arithmetic as low as possible. This applies in
particular to the number of qubits required in the implementation as for the
foreseeable future this number is expected to be small. We present a reversible
circuit for integer multiplication that is inspired by Karatsuba's recursive
method. The main improvement over circuits that have been previously reported
in the literature is an asymptotic reduction of the amount of space required
from $O(n^{1.585})$ to $O(n^{1.427})$. This improvement is obtained in exchange
for a small constant increase in the number of operations by a factor less than
$2$ and a small asymptotic increase in depth for the parallel version. The
asymptotic improvement are obtained from analyzing pebble games on complete
ternary trees.
