  We propose some protocols to implement various classes of bipartite unitary
operations on two remote parties with the help of repeater nodes in-between. We
also present a protocol to implement a single-qubit unitary with parameters
determined by a remote party with the help of up to three repeater nodes. It is
assumed that the neighboring nodes are connected by noisy photonic channels,
and the local gates can be performed quite accurately, while the decoherence of
memories is significant. A unitary is often a part of a larger computation or
communication task in a quantum network, and to reduce the amount of
decoherence in other systems of the network, we focus on the goal of saving the
total time for implementing a unitary including the time for entanglement
preparation. We review some previously studied protocols that implement
bipartite unitaries using local operations and classical communication and
prior shared entanglement, and apply them to the situation with repeater nodes
without prior entanglement. We find that the protocols using piecewise
entanglement between neighboring nodes often require less total time compared
to preparing entanglement between the two end nodes first and then performing
the previously known protocols. For a generic bipartite unitary, as the number
of repeater nodes increases, the total time could approach the time cost for
direct signal transfer from one end node to the other. We also prove some lower
bounds of the total time when there are a small number of repeater nodes. The
application to position-based cryptography is discussed.
