  The task of non-local quantum computation requires implementation of a
unitary on $n$ qubits between two parties with only one round of communication,
ideally with minimal pre-shared entanglement. We introduce a new protocol that
makes use of the fact that port-based teleportation costs much less
entanglement when done only on a small number of qubits at a time. Whereas
previous protocols have entanglement cost independent of the unitary or scaling
with its complexity, the cost of the new protocol scales with the non-locality
of the unitary. Specifically, it takes the form $\sim n^{4V}$ with $V$ the
maximum volume of a past light cone in a circuit implementing the unitary. Thus
we can implement unitary circuits with $V\sim O(1)$ using polynomial
entanglement, and those with $V\sim \mathrm{polylog}(n)$ using quasi-polynomial
entanglement. For a general unitary circuit with $d$ layers of $k$-qubit gates
$V$ is at most $k^d$, but if geometric locality is imposed it is at most
polynomial in $d$. We give an explicit class of unitaries for which our
protocol's entanglement cost scales better than any known protocol. We also
show that several extensions can be made without significantly affecting the
entanglement cost - arbitrary local pre- and post-processing; global Clifford
pre- and post-processing; and the addition of a polynomial number of auxiliary
systems.
