  We introduce a linear optical technique that can implement ideal quantum
tele-amplification up to the $n^\mathrm{th}$ Fock state, where $n$ can be any
positive integer. Here tele-amplification consists of both quantum
teleportation and noiseless linear amplification (NLA). This simple protocol
consists of a beam-splitter and an $(n+1)$-splitter, with $n$ ancillary photons
and detection of $n$ photons. For a given target fidelity, our technique
improves success probability and physical resource costs by orders of magnitude
over current alternative teleportation and NLA schemes. We show how this
protocol can also be used as a loss-tolerant quantum relay for entanglement
distribution and distillation.
