  We propose a framework to study models of computation of indeterministic
data, represented by abstract "distributions". In these distributions,
probabilities are replaced by "amplitudes" drawn from a fixed semi-ring $S$, of
which the non-negative reals, the complex numbers, finite fields $\mathbb
F_{p^r}$, and cyclic rings $\mathbb Z_k$ are examples. Varying $S$ yields
different models of computation, which we may investigate to better understand
the (likely) difference in power between randomised and quantum computation.
The "modal quantum states" of Schumacher and Westmoreland [arXiv:1010.2929] are
examples of such distributions, for $S$ a finite field. For $S = \mathbb F_2$,
Willcock and Sabry [arXiv:1102.3587] show that UNIQUE-SAT is solvable by
polynomial-time uniform circuit families consisting of invertible gates. We
characterize the decision problems solvable by polynomial uniform circuit
families, using either invertible or "unitary" transformations over cyclic
rings $S = \mathbb Z_k$, or (in the case that $k$ is a prime power) finite
fields $S = \mathbb F_k$. In particular, for $k$ a prime power, these are
precisely the problems in the class $\mathsf{Mod}_k\mathsf P$.

Computational Complexity;Quantum Physics