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If two parties share sufficient entanglement, they are able to implement anychannel on a shared bipartite state via non-local quantum computation -- aprotocol consisting of local operations and a single simultaneous round ofquantum communication. Such a protocol can occur in the AdS/CFT correspondence,with the two parties represented by regions of the CFT, and the holographicstate serving as a resource to provide the necessary entanglement. Thisboundary non-local computation is dual to the local implementation of a channelin the bulk AdS theory. Previous work on this phenomenon was obstructed by thedivergent entanglement between adjacent CFT regions, and tried to circumventthis issue by assuming that certain regions are irrelevant. However, theabsence of these regions introduces violent phenomena that prevent the CFT fromimplementing the protocol. Instead, we resolve the issue of divergententanglement by using a finite-memory quantum simulation of the CFT. We showthat any finite-memory quantum system on a circular lattice yields a protocolfor non-local quantum computation. In the case of a quantum simulation of aholographic CFT, we carefully show that this protocol implements the channelperformed by the local bulk dynamics. Under plausible physical assumptionsabout quantum computation in the bulk, our results imply that non-local quantumcomputation can be performed for any polynomially complex unitary with apolynomial amount of entanglement. Finally, we provide a concrete example of aholographic code whose bulk dynamics correspond to a Clifford gate, and use ourresults to show that this corresponds to a non-local quantum computationprotocol for this gate.

Holography as a resource for non-local quantum computation