Computational Complexity of Some Quantum Theories in $1+1$ Dimensions

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.