Quantum mechanics over sets

In models of QM over finite fields (e.g., Schumacher's ``modal quantum theory'' MQT), one finite field stands out, ${\rm Z}_{2}$, since ${\rm Z}_{2}$ vectors represent sets. QM (finite-dimensional) mathematics can be transported to sets resulting in quantum mechanics over sets or QM/sets. This gives a full probability calculus (unlike MQT with only zero-one modalities) that leads to a fulsome theory of QM/sets including ``logical'' models of the double-slit experiment, Bell's Theorem, QIT, and QC. In QC over ${\rm Z}_{2}$ (where gates are non-singular matrices as in MQT), a simple quantum algorithm (one gate plus one function evaluation) solves the Parity SAT problem (finding the parity of the sum of all values of an n-ary Boolean function). Classically, the Parity SAT problem requires 2$^{\mathrm{n}}$ function evaluations in contrast to the one function evaluation required in the quantum algorithm. This is quantum speedup but with all the calculations over ${\rm Z}_{2}$ \textit{just like classical computing}. This shows definitively that the source of quantum speedup is \textit{not} in the greater power of computing over the complex numbers, and confirms the idea that the source is in superposition.