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The Power of Discrete Quantum Theories

We explore the implications of restricting the framework of quantum theory and quantum computation to finite fields. The simplest proposed theory is defined over arbitrary finite fields and loses the notion of unitaries. This makes such theories unnaturally strong, permitting the search of unstructured databases faster than asymptotically possible in conventional quantum computing. The next most general approach chooses finite fields with no solution to x^2+1=0, and thus permits an elegant complex-like representation of the extended field by adjoining i=sqrt(-1). Quantum theories over these fields retain the notion of unitaries and --- for particular problem sizes --- allow the same algorithms as conventional quantum theory. These theories, however, still support unnaturally strong computations for certain problem sizes, but the possibility of such phenomena decreases as the size of the field increases.