Universal Gates in Other Universes

Abstract

I describe a new formalization for computation which is similar to traditional circuit models but which depends upon the choice of a family of [semi]groups – essentially, a choice of the structure group of the universe of the computation. Choosing the symmetric groups results in the reversible version of classical computation; the unitary groups give quantum computation. Other groups can result in models which are stronger or weaker than the traditional models, or are hybrids of classical and quantum computation.

One particular example, built out of the semigroup of doubly stochastic matrices, yields classical but probabilistic computation, helping explain why probabilistic computation can be so fast. Another example is a smaller and entirely ℝeal version of the quantum one which uses a (real) rotation matrix in place of the (complex, unitary) Hadamard gate to create algorithms which are exponentially faster than classical ones.

I also articulate a conjecture which would help explain the different powers of these different types of computation, and point to many new avenues of investigation permitted by this model.