Non-local computation of quantum circuits with small light cones

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.