  We investigate the prospects for detecting a hidden sector at an $e^+ e^-$
collider. The hidden sector is assumed to be composed of invisible particles
that carry no charges under the Standard Model gauge interactions, and whose
primary interactions with ordinary matter are through the Higgs portal. We
consider both the cases when the decays of an on-shell Higgs into a pair of
hidden sector particles are kinematically allowed, and the case when such
decays are kinematically forbidden. We find that at collider energies below a
TeV, the most sensitive channel involves production of an on-shell or off-shell
Higgs in association with a Z boson, and the subsequent decay of the Higgs into
invisible hidden sector states. Focusing on this channel, we find that with
order a thousand inverse fb of data at 250 GeV, the decay branching fraction of
an on-shell Higgs to invisible hidden sector states can be constrained to lie
below half a percent. The corresponding limits on Higgs portal dark matter will
be stronger than the bounds from current and upcoming direct detection
experiments in much of parameter space. With the same amount of data at 500
GeV, assuming order one couplings, decays of an off-shell Higgs to hidden
sector states with a total mass up to about 200 GeV can also be probed. Both
the on-shell and off-shell cases represent a significant improvement in
sensitivity when compared to the Large Hadron Collider (LHC).
