  Inspired by connections to two dimensional quantum theory, we define several
models of computation based on permuting distinguishable particles (which we
call balls), and characterize their computational complexity. In the quantum
setting, we find that the computational power of this model depends on the
initial input states. More precisely, with a standard basis input state, we
show how to approximate the amplitudes of this model within additive error
using the model DQC1 (the class of problems solvable with one clean qubit),
providing evidence that the model in this case is weaker than universal quantum
computing. However, for specific choices of input states, the model is shown to
be universal for BQP in an encoded sense. We use representation theory of the
symmetric group to partially classify the computational complexity of this
model for arbitrary input states. Interestingly, we find some input states
which yield a model intermediate between DQC1 and BQP. Furthermore, we consider
a restricted version of this model based on an integrable scattering problem in
1+1 dimensions. We show it is universal under postselection, if we allow
intermediate destructive measurements and specific input states. Therefore, the
existence of any classical procedure to sample from the output distribution of
this model within multiplicative error implies collapse of polynomial hierarchy
to its third level. Finally, we define a classical version of this model in
which one can probabilistically permute balls. We find this yields a complexity
class which is intermediate between L and BPP. Moreover, we find a
nondeterministic version of this model is NP-complete.
