We explore finite-field frameworks for quantum theory and quantum
computation. The simplest theory, defined over unrestricted finite fields, is
unnaturally strong. A second framework employs only finite fields with no
solution to x^2+1=0, and thus permits an elegant complex representation of the
extended field by adjoining i=\sqrt{-1}. Quantum theories over these fields
recover much of the structure of conventional quantum theory except for the
condition that vanishing inner products arise only from null states;
unnaturally strong computational power may still occur. Finally, we are led to
consider one more framework, with further restrictions on the finite fields,
that recovers a local transitive order and a locally-consistent notion of inner
product with a new notion of cardinal probability. In this framework,
conventional quantum mechanics and quantum computation emerge locally (though
not globally) as the size of the underlying field increases. Interestingly, the
framework allows one to choose separate finite fields for system description
and for measurement: the size of the first field quantifies the resources
needed to describe the system and the size of the second quantifies the
resources used by the observer. This resource-based perspective potentially
provides insights into quantitative measures for actual computational power,
the complexity of quantum system definition and evolution, and the independent
question of the cost of the measurement process.
