  We propose a scheme to achieve quantum computation with neutral atoms whose
interactions are catalyzed by single photons. Conditional quantum gates,
including an $N$-atom Toffoli gate and nonlocal gates on remote atoms, are
obtained through cavity-assisted photon scattering in a manner that is robust
to random variation in the atom-photon coupling rate and which does not require
localization in the Lamb-Dicke regime. The dominant noise in our scheme is
automatically detected for each gate operation, leading to signalled errors
which do not preclude efficient quantum computation even if the error
probability is close to the unity.
