I shall describe how a quantum computer can efficiently compute relativistic scattering amplitudes in quantum field theories. Conversely, a (1+1)-dimensional scalar quantum field theory can implement scalable, universal quantum computation.  Several ideas and techniques in this construction and its analysis have analogues in AMO physics.  A novel means of performing two-qubit unitary operations is required, because, as we demonstrate, two-qubit gates like those proposed for Josephson-junction qubits are not possible in the theory considered.  The implications of this research for computation and quantum field theory will be discussed, as well as AMO applications arising from it.