  Recent work has shown that quantum computers can compute scattering
probabilities in massive quantum field theories, with a run time that is
polynomial in the number of particles, their energy, and the desired precision.
Here we study a closely related quantum field-theoretical problem: estimating
the vacuum-to-vacuum transition amplitude, in the presence of
spacetime-dependent classical sources, for a massive scalar field theory in
(1+1) dimensions. We show that this problem is BQP-hard; in other words, its
solution enables one to solve any problem that is solvable in polynomial time
by a quantum computer. Hence, the vacuum-to-vacuum amplitude cannot be
accurately estimated by any efficient classical algorithm, even if the field
theory is very weakly coupled, unless BQP=BPP. Furthermore, the corresponding
decision problem can be solved by a quantum computer in a time scaling
polynomially with the number of bits needed to specify the classical source
fields, and this problem is therefore BQP-complete. Our construction can be
regarded as an idealized architecture for a universal quantum computer in a
laboratory system described by massive phi^4 theory coupled to classical
spacetime-dependent sources.
