  Light states composed of multiple entangled photons - such as cluster states
- are essential for developing and scaling-up quantum computing networks.
Photonic cluster states with discrete variables can be obtained from
single-photon sources and entangling gates, but so far this has only been done
with probabilistic sources constrained to intrinsically-low efficiencies, and
an increasing hardware overhead. Here, we report the resource-efficient
generation of polarization-encoded, individually-addressable, photons in linear
cluster states occupying a single spatial mode. We employ a single
entangling-gate in a fiber loop configuration to sequentially entangle an
ever-growing stream of photons originating from the currently most efficient
single-photon source technology - a semiconductor quantum dot. With this
apparatus, we demonstrate the generation of linear cluster states up to four
photons in a single-mode fiber. The reported architecture can be programmed to
generate linear-cluster states of any number of photons with record scaling
ratios, potentially enabling practical implementation of photonic quantum
computing schemes.
