Quantum field theory provides the framework for the Standard Model of particle physics and plays a key role in many areas of physics. However, calculations are generally computationally complex and limited to weak interaction strengths. I shall describe a polynomial-time algorithm for computing, on a quantum computer, relativistic scattering amplitudes in massive scalar quantum field theories. The quantum algorithm applies at both weak and strong coupling, achieving exponential speedup over known classical methods at high precision or strong coupling. The study of such quantum algorithms may also help us learn more about the nature and foundations of quantum field theory itself.