  If two parties share sufficient entanglement, they are able to implement any
channel on a shared bipartite state via non-local quantum computation -- a
protocol consisting of local operations and a single simultaneous round of
quantum communication. Such a protocol can occur in the AdS/CFT correspondence,
with the two parties represented by regions of the CFT, and the holographic
state serving as a resource to provide the necessary entanglement. This
boundary non-local computation is dual to the local implementation of a channel
in the bulk AdS theory. Previous work on this phenomenon was obstructed by the
divergent entanglement between adjacent CFT regions, and tried to circumvent
this issue by assuming that certain regions are irrelevant. However, the
absence of these regions introduces violent phenomena that prevent the CFT from
implementing the protocol. Instead, we resolve the issue of divergent
entanglement by using a finite-memory quantum simulation of the CFT. We show
that any finite-memory quantum system on a circular lattice yields a protocol
for non-local quantum computation. In the case of a quantum simulation of a
holographic CFT, we carefully show that this protocol implements the channel
performed by the local bulk dynamics. Under plausible physical assumptions
about quantum computation in the bulk, our results imply that non-local quantum
computation can be performed for any polynomially complex unitary with a
polynomial amount of entanglement. Finally, we provide a concrete example of a
holographic code whose bulk dynamics correspond to a Clifford gate, and use our
results to show that this corresponds to a non-local quantum computation
protocol for this gate.
