Is there a common mathematical foundation for quantum and classical computation?

Question

Is there a (natural) formalism that neatly and insightfully includes classical and quantum computation?

I have a mostly sound basis in quantum mechanics (as a physics grad), and a much less sound but still non-zero understanding of the foundations of computation - Turing machines and transition functions, and something something lambda calculus. While the maths of each doesn't seem too bad, they're pretty different from each other, with the smooth vector spaces, unitary operators, and projections of the quantum world seeming to have little in common with the chunky discrete bits and gates of classical computers beyond analogy.

So is there some common ground between them, a maximal mathematical core $x$ so $x + y = $ classical computation and $x + z = $ quantum computation?

Answer 1

Classical computation is a subset of quantum computation. To see it, just consider circuits in which all gates and operations are permutation matrices, the inputs are product states of the type $|q_1,...,q_n\rangle$ with $q_i\in\{0,1\}$, and the measurements are in the computational basis. You then get back a deterministic device that can only perform classical reversible computation. Just stop looking at some of the outputs and you can perform arbitrary logic gates.

You just don't hear about it very often in the context of quantum computation because this is exactly the kind of thing you want to avoid: quantum computers are useless in the regime in which classical devices can perform the same things.