  In recent work, Benjamin Schumacher and Michael D. Westmoreland investigate a
version of quantum mechanics which they call modal quantum theory. This theory
is obtained by instantiating the mathematical framework of Hilbert spaces with
a finite field instead of the field of complex numbers. This instantiation
collapses much the structure of actual quantum mechanics but retains several of
its distinguishing characteristics including the notions of superposition,
interference, and entanglement. Furthermore, modal quantum theory excludes
local hidden variable models, has a no-cloning theorem, and can express natural
counterparts of quantum information protocols such as superdense coding and
teleportation.
  We show that the problem of UNIQUE-SAT --- which decides whether a given
Boolean formula is unsatisfiable or has exactly one satisfying assignment ---
is deterministically solvable in any modal quantum theory in constant time. The
solution exploits the lack of orthogonality in modal quantum theories and is
not directly applicable to actual quantum theory.
