Quantum field theory provides the framework for the Standard Model of particle physics and plays a key role in physics. However, calculations are generally computationally complex and limited to weak interaction strengths. I'll describe polynomial-time quantum algorithms for computing relativistic scattering amplitudes in both scalar and fermionic quantum field theories. The algorithms achieve exponential speedup over known classical methods. One of the motivations for this work comes from computational complexity theory. Ultimately, one wishes to know what is the computational power of our universe. Studying such quantum algorithms probes whether a universal quantum computer is powerful enough to represent quantum field theory; in other words, is quantum field theory in BQP? Conversely, one can ask whether quantum field theory can represent a universal quantum computer; is quantum field theory BQP-hard? We have shown that massive phi^4 theory can implement universal quantum computation and is thus BQP-complete.