  We study the computational complexity of certain integrable quantum theories
in 1+1 dimensions. We formalize a model of quantum computation based on these
theories. In this model, distinguishable particles start out with known momenta
and initial superposition of different configurations. Then the label of these
particles are measured at the end. We prove that additive approximation to
single amplitudes of these models can be obtained by the one-clean-qubit model,
if no initial superpositions are allowed. However, if arbitrary initial states
and non-adaptive intermediate measurements are allowed, we show that
conditioned on infinite polynomial hierarchy assumption it is hard to sample
from the output distribution of these models on a classical randomized
computer. A classical analogue of this model is also formalized and its
computational power is pinned down within the complexity classes below BPP and
NP.
